The Hamiltonian formulation of General Relativity: myths and reality

The Hamiltonian formulation of General Relativity: myths and reality N. Kiriushcheva∗ and S.V. Kuzmin† Faculty of Arts and Social Science, Huron University College, N6G 1H3 and

arXiv:0809.0097v2 [gr-qc] 12 Jul 2011

Department of Applied Mathematics, University of Western Ontario, N6A 5B7, London, Canada

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Abstract A conventional wisdom often perpetuated in the literature states that: (i) a 3+1 decomposition of space-time into space and time is synonymous with the canonical treatment and this decomposition is essential for any Hamiltonian formulation of General Relativity (GR); (ii) the canonical treatment unavoidably breaks the symmetry between space and time in GR and the resulting algebra of constraints is not the algebra of four-dimensional diffeomorphism; (iii) according to some authors this algebra allows one to derive only spatial diffeomorphism or, according to others, a specific fielddependent and non-covariant four-dimensional diffeomorphism; (iv) the analyses of Dirac [Proc. Roy. Soc. A 246 (1958) 333 ] and of ADM [Arnowitt, Deser and Misner, in “Gravitation: An Introduction to Current Research” (1962) 227 ] of the canonical structure of GR are equivalent. We provide some general reasons why these statements should be questioned. Points (i-iii) have been shown to be incorrect in [Kiriushcheva et al., Phys. Lett. A 372 (2008) 5101 ] and now we thoroughly re-examine all steps of the Dirac Hamiltonian formulation of GR. By direct calculation we show that Dirac’s references to space-like surfaces are inessential and that such surfaces do not enter his calculations. In addition, we show that his assumption g0k = 0, used to simplify his calculation of different contributions to the secondary constraints, is unwarranted; yet, remarkably his total Hamiltonian is equivalent to the one computed without the assumption g0k = 0. The secondary constraints resulting from the conservation of the primary constraints of Dirac are in fact different from the original constraints that Dirac called secondary (also known as the “Hamiltonian” and “diffeomorphism” constraints). The Dirac constraints are instead particular combinations of the constraints which follow directly from the primary constraints. Taking this difference into account we found, using two standard methods, that the generator of the gauge transformation gives diffeomorphism invariance in four-dimensional space-time; and this shows that points (i-iii) above cannot be attributed to the Dirac Hamiltonian formulation of GR. We also demonstrate that ADM and Dirac formulations are related by a transformation of phase-space variables from the metric gµν to lapse and shift functions and the three-metric gkm , which is not canonical. This proves that point (iv) is incorrect. Points (i-iii) are mere consequences of using a non-canonical change of variables and are not an intrinsic property of either the Hamilton-Dirac approach to constrained systems or Einstein’s theory itself.

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∗

Electronic address: [email protected]

†

Electronic address: [email protected]

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“On ne trouvera point de Figures dans cet Ouvrage. Les m´ethodes que j’y expose ne demandent ni constructions, ni raisonnemens g´eom´etriques ou m´ecaniques, mais seulement des op´erations alg´ebriques, assuj´eties a` une marche r´eguli`ere et uniforme. Ceux qui aiment ˇl Analyse, verront avec plaisir la M´ecanique en devenir une nouvelle branche, et me sauront gr´e dˇ en avoir ´etendu ainsi le domaine.” J. L. Lagrange, “M´ecanique Analytique” (1788)

The reader will find no figures in this work.

The

methods which I set forth do not require either constructions or geometrical or mechanical reasonings, but merely algebraic operations subjected to a regular and uniform rule of procedure. Those who are fond of Mathematical Analysis will observe with pleasure Mechanics becoming one of its new branches and they will be grateful to me for having thus extended its domain. I.

INTRODUCTION

We begin our paper with words written more than two centuries ago by Lagrange in the preface to the first edition of the “M´ecanique Analytique” [1] (the first English translation appeared in 1997 [2]) because they express our standpoint in analyzing of the Hamiltonian formulation of General Relativity (GR). The results previously obtained by others are reconsidered and classified as either “myth” or “reality” depending on whether they were obtained by what Lagrange called a regular and uniform rule of procedure, or by geometrical or some other reasonings. The results and conclusions constructed using such reasonings must be checked by explicit calculation without which they are meaningless and could be misleading, contradicting the rules of procedure and the essential properties of GR. Originating more than half a century ago, the Hamiltonian formulation of GR is not a 4

new subject. It began with advances in the Hamiltonian formulation of singular Lagrangians due to the pioneering works of Dirac [3] and Bergmann with coauthors [4, 5] on generalized (constrained) Hamiltonian dynamics1 . We restrict our discussion to the original Einstein metric formulation of GR. The firstorder, affine-metric, form [17, 18] (an English translation of this and a few other Einstein’s papers can be found in [19]) will be just briefly touched; but the analysis presented here can and must be extended to a affine-metric form and to other formulations (e.g. Einstein-Cartan formulation of GR). In chronological order (which is also ranked inversely in popularity) the Hamiltonian formulation of metric GR was considered by Pirani, Schild, and Skinner (PSS) [20], Dirac [21], and Arnowitt, Deser, and Misner (ADM) [22] and references therein. The relationship among these formulations has not been analyzed; and some authors have adopted to using the name “Dirac-ADM” or they refer to Dirac when actually working with the ADM Hamiltonian. This presumes equivalence of the Dirac and ADM formulations. These two, as we will demonstrate, are not equivalent. The Dirac conjecture [11], that knowing all the first-class constraints is sufficient to derive the gauge transformations, was made only after the appearance of [20–22] and became a well defined procedure only later [23, 24]. In the most general form (with chains of constraints of any length) and with application to field theories this procedure was considered for the first time by Castellani [25] (for alternative approaches see [26–28]; problems or, at least, limitations of these approaches are discussed in [29]). Deriving the gauge invariance of GR from the complete set of the first-class constraints should also be viewed as a crucial consistency condition that must be met by any Hamiltonian formulation of the theory; yet, this requirement did not attract much attention and it is not discussed in textbooks on GR, where a Hamiltonian formulation is presented (e.g. [30, 31]). In books on con1

There are references on the Rosenfeld work [6] but only in a few old papers [5, 7–9] and neither in the first review on this subject [10] nor in monographs [11–14] including the newest one [15]. We always wondered what is the Rosenfeld contribution. Recently, due to an effort of Salisbury [16], the paper by Rosenfeld [6] became available for non-German speaking readers and now anyone can evaluate his contribution and learn about his views on a Hamiltonian formulation of gauge invariant Lagrangians. In [16] the original article [6] and its English translation are accompanied by quite extensive and in many cases helpful comments, but we want to warn the reader that some of them reflect the opinion of the commentator, in particular, about the “crucial error” of Rosenfeld. We will discuss this later.

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straint dynamics [12–15], even if such a procedure is discussed [14, 15], it is not applied to the Hamiltonian formulation of GR. Recently this question was again brought to light by Mukherjee and Saha [32] who applied the method of [27] to the ADM Hamiltonian with the sole emphasis on presenting the method of deriving the gauge invariance, not on the results themselves. In [32] there appears a first complete derivation of the gauge transformations from the constraint structure of the ADM Hamiltonian. The expected transformation of the metric tensor is [33]

δgµν = −ξµ;ν − ξν;µ ,

(1)

where ξµ is the gauge parameter and the semicolon “;” signifies the covariant derivative. In the literature on the Hamiltonian formulation of GR, the word “diffeomorphism” is often used as equivalent to the transformation (1), which is similar to gauge transformations in ordinary field theories. This meaning is employed in our article2 . The expected transformation (1) does not follow from the constraint structure of ADM Hamiltonian [22] and a field-dependent and non-covariant redefinition of gauge parameters is needed3 to present the transformations of [32] in the form of (1), i.e. 0

ξ = −g


00 1/2

ε⊥ ADM ,

k

ξ =

εkADM


1/2 ⊥ g 0k εADM . + 00 −g 00 g

(2)

The field-dependent redefinition of gauge parameters (2) goes back to work of Bergmann and Komar [34] where it was presented for the first time. The same redefinition of gauge parameters (2), but in a less transparent form, was obtained for the ADM Hamiltonian by Castellani [25] for the transformation of the g0µ components of the metric tensor to illustrate his procedure for the construction of the gauge generators. This redefinition of gauge parameters was also discussed from different points of view in [35–38] in an attempt to justify the necessity of such changes or even make them compulsory. We would like to return to footnote 1 about the Rosenfeld contribution at the dawn of the Hamiltonian analysis. From the beginning he restricts his interest to field-independent gauge parameters. This approach differs from what is advocated by Salisbury with coauthors [39]. As an 2

3

In mathematical literature the term diffeomorphism refers to a mapping from one manifold to another which is differentiable, one-to-one, onto, with a differentiable inverse. More detail on the derivation of (2) is given in the last Section where application of Castellani’s procedure k to the ADM Hamiltonian is re-examined (ε⊥ ADM and εADM are gauge parameters of ADM formulation).

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application of a general but ‘not-yet’ procedure, Rosenfeld considered a tetrad formulation of gravity and, according to Salisbury, made a “crucial error” that was corrected by Pons, Salisbury, and Shepley [39] “by introducing a coordinate symmetry transformation group whose elements depend on metric” [16]. But the same “crucial error” emerges from Dirac’s treatment of metric gravity; as we will demonstrate, the Dirac formulation allows one to restore gauge invariance without the need for field-dependent gauge parameters or the use of the field-dependent redefinition, i.e. it supports the equivalence of the Lagrangian and Hamiltonian methods. Such an equivalence is leitmotif and cornerstone of the Rosenfeld treatment of gauge invariant systems. Exactly the disappearance of this equivalence in the ADM formulation leads the authors of [39] to introduce the“diffeomorphism-induced gauge symmetry” to support ADM variables. So, in such a case the “crucial error” of Rosenfeld is simply that he did not use ADM variables4 . A common feature of different approaches [32, 34–39] is that they consider only the ADM Hamiltonian [22] because this is a formulation that does not lead to (1). According to the conclusion of [34], the transformation (1) and the one with parameters that depend on the fields (2) are distinct. In [36] this transformation is called the “specific metric-dependent diffeomorphism”. The authors of [32] have a brief and ambiguous conclusion about (2): “[it will] lead to the equivalence 5 between the diffeomorphism and gauge transformations” and, at the same time, “demonstrate the unity of the different symmetries involved”; these are contradictory statements. Soon after appearance of [32], Samanta [41] posed the question “whether it is possible to describe the diffeomorphism symmetries without recourse to the ADM decomposition”. To answer this question, he derived the transformation (1) starting from the Einstein-Hilbert (EH) Lagrangian (not the ADM Lagrangian) and applying the Lagrangian method for recovering gauge symmetries based on the use of certain gauge identities that appear in [13]. It is important that (1) follows exactly from this procedure without the need of a fielddependent and non-covariant redefinition of the gauge parameters, which would be necessary in [25, 32, 34–39] where the ADM Hamiltonian is used. The question of the equivalence of 4

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This point is explicitly stated in the historical papers of Salisbury and comments, for example, in [40]: “Had he [Rosenfeld] expressed this orthonormal tetrad in terms of the lapse and shift functions introduced by Arnowitt, Deser and Misner he would have obtained the triad form of their ADM Hamiltonian. [11]” (The Salisbury’s reference [11] is our [22] where neither the tetrad nor the triad form of the ADM Hamiltonian is discussed.) Here and everywhere in this article the Italic in quotations is ours.

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(1) and (2) does not even arise in the approach of [41]. In [41] the diffeomorphism transformations were also derived by applying the same method to the first-order, affine-metric, formulation [17] of GR. The conclusion of [41] that “the ADM splitting, which is essential for discussing diffeomorphism symmetries, is bypassed ” contradicts the obtained result. Firstly, any feature that is “essential ” cannot be “bypassed ”. Secondly, the transformations derived from the ADM Hamiltonian in [32] are not those of [41]. It is not a “bypass” because the “destination”, of having the invariance of (1), is changed. Comparison of [32] and [41] allows us to conclude that ADM decomposition is not only inessential but incorrect because the EH Lagrangian leads directly to diffeomorphism invariance without the need of any field-dependent and non-covariant redefinition of gauge parameters; and a correct Hamiltonian formulation should give the same result. This discrepancy between these two recent results vindicates Hawking’s old statement [42] “the split into three spatial dimensions and one time dimension seems to be contrary to the whole spirit of relativity”, the more recent statements of Pons [37]: “Being non-intrinsic, the 3+1 decomposition is somewhat at odds with a generally covariant formalism, and difficulties arise for this reason”, and Rovelli [43]: “The very foundation of general covariant physics is the idea that the notion of a simultaneity surface all over the universe is devoid of physical meaning”. There is another statement in [41] that can also be found in many places “it is well known that this decomposition plays a central role in all Hamiltonian formulations of general relativity”. This sentence combined with Hawking’s “spiritual” statement forces one to conclude that the Hamiltonian formulation by itself contradicts the spirit of GR. This resonates with Pullin’s conclusion [44] that “Unfortunately, the canonical treatment breaks the symmetry between space and time in general relativity and the resulting algebra of constraints is not the algebra of four diffeomorphism”. We will show in this paper that the canonical formalism is in fact consistent with the diffeomorphism (1) when the Dirac constraint formalism is applied consistently and that the discrepancies between the ADM formalism and (1) can be explained. The difference of the results [32] and [41] which were obtained by different methods also implies the non-equivalence of the Lagrangian and Hamiltonian formulations. In all field theories (e.g., Maxwell or Yang-Mills) the Hamiltonian and Lagrangian formulations produce the same result for gauge invariance, so for GR to differ seems unnatural. Could 8

this be a peculiar property of GR? Is GR a theory in which the Hamiltonian and Lagrangian formulations lead to different results or was a “rule of procedure” broken somewhere? Recently, in collaboration with Racknor and Valluri [45], we demonstrated that, by following the most natural first attempt of PSS [20] and by applying the rules of procedure [3, 11–14, 25], the Hamiltonian formulation of GR (without any modifications of the action or change of variables) leads to consistent results. The gauge transformation of the metric tensor was derived using the method of [25] and, without any field-dependent redefinitions of gauge parameters, it gives exactly the same result as the Lagrangian approach of [41], as it should. In the Hamiltonian formulation of GR given in [45] the algebra of constraints is the algebra of “four diffeomorphism”, in contradiction to the general conclusion of [44], which was based on a particular, ADM, formulation. The procedure of passing to a Hamiltonian formulation in field theories based on the separation of the space and time components of the fields and their derivatives (defined on the whole space-time, not on some hypersurface) is not equivalent to the separation of space-time into space and time. For example, by rewriting the Einstein equations in components (as was done before Einstein introduced his condensed notation), we do not abandon covariance even if it is not manifest. In addition, such explicit separation of the space and time components and the derivatives of the fields does not affect space-time itself and is not to be associated with any 3+1 decomposition, slicing, splitting, foliation, etc. of space-time. The final result for the gauge transformation of the fields can be presented in covariant form when using the Hamiltonian formulation of ordinary field theories (e.g., YangMills, Maxwell), as well as in GR [45]. In any field theory, after rewriting its Lagrangian in components, the Hamiltonian formulation for singular Lagrangians follows a well defined procedure. Such a procedure is based on consequent calculations of the Poisson brackets (PB) of constraints with the Hamiltonian using the fundamental PBs of independent fields. In the case of field theories they are 

q x0 , x , p y 0 , y x0 =y0 = δ (x − y) .

(3)

This is a local relation that does not presume any extended objects or surfaces. Again, as with separation into components, this locally defined canonical PB does not affect space-time and is not related to a space-like surface or any other hypersurface because (3) is zero for x 6= y in the whole space-time; and there is no information in (3) that, using mathematical 9

language, can allow one to classify two separate points as points on a particular space-like surface or on any surface. The canonical procedure does not itself lead to the appearance of any hypersurfaces; in [45] there are no references to such surfaces and the result is consistent with the Lagrangian formulation of [41]. Such surfaces are either a phantom of interpretation or canonical procedure was abandoned by their introduction. The discussion of an interpretational approach is not on the main road of our analysis of the Hamiltonian formulations of GR. However, the routes of such an approach6 are quite interesting: one starting from the basic equations of the ADM formulation, according to [46], “would like to understand intuitively their geometrical and physical meaning and derive them from some first principles rather than by a formal rearrangement of Einstein’s law”. By taking this approach, a formal rearrangement (which is a “rule of procedure”) is replaced by some sort of intuitive understanding. As a result, a new language is created which “is much closer to the language of quantum dynamics than the original language of Einstein’s law ever was” [46]. This language allows one “to recover the old comforts of a Hamiltonianlike scheme: a system of hypersurfaces stacked in a well defined way in space-time, with the system of dynamic variables distributed over these hypersurfaces and developing uniquely from one hypersurface to another” [47]. Such an interpretation, although ‘reasonable’ from the point of view of classical Laplacian determinism, is hard to justify from the standpoint of GR [48]. In GR, an entire spatial slice can only be seen by an observer in the infinite future [49] and an observer at any point on a space-like surface does not have access to information about the rest of the surface (this is reflected in the local nature of (3) in field theories). It would be non-physical to build any formalism by basing it on the development in time of data that can be available only in the infinite future and trying to fit GR into a scheme of classical determinism and nonrelativistic Quantum Mechanics with its notion of a wave function defined on a space-like slice. The condition that a space-like surface remains space-like obviously imposes restrictions on possible coordinate transformations, thereby destroying four-dimensional symmetry, and, according to Hawking, “it restricts the topology of space-time to be the product of the real line with some three-dimensional manifold, whereas one would expect that quantum gravity would allow all possible topologies 6

We have to confess that we found it hard to understand approaches which are not analytical and, to avoid any misinterpretations, we will merely quote their advocates. A reader interested in such approaches can find more details in the articles we cite.

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of space-time including those which are not product” [42]. This restriction, imposed by the slicing of space-time, must be lifted at the quantum level [50]; but, from our point of view, avoiding it at the outset seems to be the most natural cure for this problem. The usual interpretation of the ADM variables, constraints, and Hamiltonian obviously contradicts the spirit of relativity. With restrictions on coordinate transformations, which are imposed by such an interpretation, it is quite natural to expect something different from a diffeomorphism transformation, as was found in [25, 32]. Any interpretation, whether or not it contradicts the spirit of GR, cannot provide a sufficiently strong argument to prove or disprove some particular result or theory, because an arbitrary interpretation cannot change or affect the result of formal rearrangements. The transformation, which is different from the diffeomorphism that follows from the ADM Hamiltonian is the result of a definite procedure [25, 32] and is based on calculations performed with their variables and their algebra of constraints. From the beginning we will not use the language of 3+1 dimensions, so as to avoid the necessity of getting ourselves “out of space and back into space-time” [51] at the end of the calculations. In any case, it would likely be impossible to do so after we have gone beyond the point of no return on such a road. We must re-examine the derivation of ADM Hamiltonian right from the start. It is difficult to compare the results of [45] directly with those of ADM because some additional modifications of the original GR Lagrangian were performed by ADM and it is not easy to trace them according to the “rules of procedure”. We will start with the work of Dirac [21], where all modifications and assumptions are explicitly stated making it possible for them to be checked and analyzed. In addition, Dirac’s canonical variables are components of the metric tensor which are the same as those used in [45], where diffeomorphism invariance was derived directly from the Hamiltonian and constraints. Moreover, in [52] two Hamiltonian formulations, based on the linearized Lagrangians of [20, 45] and [21], were considered. Despite there being different expressions for the primary and secondary constraints, these two formulations have the same algebra of PBs among the constraints, and with the Hamiltonian; therefore, they have the same gauge invariance. This is exactly what one can expect in the case of full GR, provided one makes no deviation from canonical procedure. In analyzing the ADM formulation we will follow a different path. We will not start from the GR Lagrangian, but instead compare the final results of Dirac and ADM and try to determine what deviation from the canonical procedure leads to the transformations 11

found in [25, 32], which are distinct from those of (1). In the next Section we shall thoroughly re-examine the Dirac derivation of the GR Hamiltonian [21] with emphasis on the effect of his modifications of the action and of the other simplifying assumptions he made. In particular, we will investigate whether space-like surfaces actually play any role in his derivation, or if they just serve as an illustration, which can be completely disregarded from the standpoint of the canonical procedure, as in [45]. In Section III, using Castellani’s procedure and the results of Section II (Dirac’s constraints), we derive the gauge transformations of the metric tensor, diffeomorphism (1), without the need of a field-dependent redefinition of gauge parameters. This result is identical to that found in [41] for metric GR using the Lagrangian method and in [45] for the Hamiltonian formulation of PSS. The same result is obtained by application of the method [27] used for the ADM Hamiltonian in [32] when it is applied to the Dirac Hamiltonian. Some peculiarities of such methods that cannot be seen in ordinary field theories are briefly discussed and related to the peculiarities of diffeomorphism invariance as it compares to the gauge invariance in ordinary theories. Finally, we consider the ADM Hamiltonian formulation of GR. In the last Section IV we demonstrate that the ADM formulation follows from Dirac’s by a change of variables. The canonicity of this change of variables (the ADM lapse and shift functions) is analyzed. Based on this analysis, the general and more restrictive criteria for a canonical transformation in the case of singular gauge invariant theories are discussed.

II.

ANALYSIS OF DIRAC DERIVATION

In [45] the GR Hamiltonian, constraints, closure of the Dirac procedure, and the diffeomorphism transformation of the metric tensor were derived without any reference to space-like surfaces, the use of any 3+1 decomposition of space-time, or slicing, splitting, foliation, etc., as well as without modifications of the Lagrangian or the introduction of any new variables. (The canonical variables of [45] are components of the metric tensor.) Dirac, when considering the Hamiltonian formulation of GR in [21], also used the metric tensor as a canonical variable; but he made frequent references to space-like surfaces. If such surfaces, which according to Hawking [42] contradict the whole spirit of General Relativity, are the part of Dirac’s calculations, then one has to expect transformations different from diffeomorphism and similar to the one found in [32] from the ADM Hamiltonian. Our main 12

interest is to find out, by following all the steps of Dirac’s derivation of the Hamiltonian, the place where (if anywhere) space-like surfaces enter his derivation or where (if anywhere) his approach deviates from a regular and uniform rule of canonical procedure. If there is no deviation, one should then obtain the diffeomorphism invariance (1), the same as found in [45]. This would resemble what happens in linearized GR, as discussed in [52].

A.

Dirac’s modification of Lagrangian and primary constraints

In [21], Dirac started the Hamiltonian formulation from the “gamma-gamma” part of the Einstein-Hilbert (EH) Lagrangian (Eq. (D8))7 (e.g., see [33, 53]) LG = where

√


1√ −gg µν Γρµν Γσρσ − Γσµρ Γρνσ = −ggµν,ρ gαβ,σ B µνραβσ 4



B µνραβσ = g µα g νβ − g µν g αβ g ρσ + 2 g µρ g αβ − g µα g βρ g νσ .

(4)

(5)

The same Lagrangian was used in [20] and [45]. This is a covariant Lagrangian of a local field theory in four(or any)-dimensional space-time, and space-like surfaces or any other hypersurfaces are not intrinsic to such a formulation. The primary constraints (the φ-equations of [21]) that follow from (4) are φµ0 = pµ0 −

δLG ≈ 0, δgµ0,0

(6)

where pµν are momenta conjugate to gµν . The exact form of φµ0 can be found in [20, 45] (Greek subscripts run from 0 to d−1 and Latin ones from 1 to d−1 where d is the dimension of space-time). In addition to eliminating the second order derivatives of the metric tensor present in the Ricci scalar in passing from the EH Lagrangian to its gamma-gamma part (4) so that [53]

LEH = 7

√

−gR = LG + ∂µ V µ ,

(7)

We will refer on Dirac equations quite often and use the convention, Eq. (D#), to mean equation # from [21].

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Dirac made an additional change to the Lagrangian in order to eliminate the second term in (6). The modified Lagrangian is obtained by adding two total derivatives which are non-covariant (Eq. (D15)) ∗

L = LG +



√


g v0 −gg ,v 00 g 00



,0

−



√


g v0 −gg ,0 00 g 00



.

(8)

,v

This change does not affect the equations of motion, but leads to simple primary constraints (Eq. (D14)) φµ0 = pµ0 ≈ 0.

(9)

It was shown in [52], that the linearized version of the modified (8) and unmodified Lagrangians (4), despite leading to different expressions for the constraints and the Hamiltonian, result in the same constraint structure, the same number of first-class constraints, and the same gauge invariance, which is the linearized version of diffeomorphism (1). This is what one can also expect in the case of full GR. According to [21], the simplification (9) “can be achieved only at the expense of abandoning four-dimensional symmetry” which is obviously correct for this modification of the Lagrangian (8); yet Dirac’s further conclusion that “four-dimensional symmetry is not a fundamental property of the physical world” is too strong and has to be clarified. Of course, four-dimensional symmetry of the Lagrangian is destroyed by the modification (8); but this change does not affect the equations of motion, which are the same as the Einstein equations. Consequently, for the equations of motion, not only four-dimensional symmetry is preserved, but also general covariance8 . If fourdimensional symmetry is preserved in the equations of motion, which are invariant under general coordinate transformations, then diffeomorphism should be recovered in the course of the Hamiltonian analysis, as in [45]. The new Lagrangian L∗ differs from the original one (4) only for terms linear in the time derivatives of a metric (i.e. ‘velocities’), which are the parts responsible for the simplification of the primary constraints. We then have L∗ = LG (2) + L∗ (1) + LG (0) , 8

(10)

The term “four-dimensional” symmetry used by Dirac probably reflects the fact that the gamma-gamma part of the Lagrangian, quadratic in first order derivatives, is not generally covariant after the elimination of terms with second order derivatives in the full EH Lagrangian (7).

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where the numbers in brackets indicate the order in velocities (for the Hamiltonian and constraints it will indicate the order in momenta). The exact form of L∗ (1) is given by Eq. (D18). This Lagrangian is used to pass to the Hamiltonian H = gαβ,0 pαβ − L∗ .

(11)

With the modification of (8) the part of the Lagrangian LG (2) + L∗ (1), as was shown by Dirac, can be written as LG (2) + L∗ (1) = LX (0) −

√

−g

1 rsab 0 0 E Γrs Γab g 00

(12)

where Γµαβ is the Christoffel symbol 1 Γµαβ = g µν (gαν,β + gβν,α − gαβ,ν ) 2

(13)

E rsab = ers eab − era esb

(14)

and

with eαβ = g αβ −

g 0α g 0β . g 00

(15)

Note, that in the second order formulation, Γµαβ , E αβµν , and eαβ are just short notations and none of them denote a new and/or independent variable. Some comments about (12) are in order. The careful reader will definitely wonder how the parts of the Lagrangian which are quadratic and linear in velocities can have contributions without velocities, LX (0); the direct calculation of LG (2) + L∗ (1) does not have such contributions (see Dirac’s unnumbered equation preceding (D19))
1√ −gE rasb grs,0gab,0 g 00 + 2grs,0gab,v g v0 − 4grs,0gaβ,b g β0 . 4

(16)

Dirac completed this expression to the square in velocities, leading to the compact form of (12). Working with (16) instead of (12), will of course not change the results and actually

15

has no calculational advantage. However, we keep (12) so as to compare our calculations with those of Dirac. The LX (0) in (12) (explicitly given by (24)) is independent of the velocities. The only part of (11) that has dependence on grs,0 is √

1 rsab 0 0 E Γrs Γab . g 00 ∗ Performing the variation δgδLrs,0 , we obtain (see (D18-D21)) grs,0prs +

prs =

√

−gE rsab Γ0ab =

−g

(17)

  1√ −gE rsab g 00 (ga0,b + gb0,a − gab,0 ) + g 0k (gak,b + gbk,a − gab,k ) . 2 (18)

Equation (18) is easy to solve for gab,0 due to the invertability of E rsab r s E rsab Iabmn = δm δn ,

(19)

where the inverse to E rsab in any space-time dimension d (except d = 2) is

Iabmn = This result gives

1 gab gmn − gam gbn . d−2

(20)

2 g 0k rs p I + g + g + (gmk,n + gnk,m − gmn,k ) . (21) rsmn m0,n n0,m −gg 00 g 00 After substitution of (21) into (17) (note that (18) can be solved for Γ0ab thus making the gmn,0 = − √

calculations shorter) we obtain the total Hamiltonian HT = g00,0 p00 + 2g0k,0p0k + HG ,

(22)

where HG (the canonical part of the Hamiltonian) is given by Eqs. (D33, D34)) as, i h 1 rs ab uv rs rs HG = − 00 √ Irsab p p + gu0 e p grs,v − 2 (p grv ),s − LX (0) − LG (0) , g −g

(23)

√   1 −g rsab  u0 α0 v0 β0 LX (0) = E g g − (g + g ) g g g − (g + g ) g , rs,u rα,s sα,r ab,v aβ,b bβ,a 4 g 00

(24)

with LX (0) (Eq. (D19)) and LG (0) (Eq. (D8)):

16

LG (0) =

1√ −ggµν,k gαβ,t B µνkαβt . 4

(25)

Note that the second term of (23), the part linear in the momenta, arises only after some rearrangement, and B µνkαβt in (25) was defined in (5). The direct substitution of grs,0 into grs,0prs (the only part of (11) that leads to terms linear in the momenta) gives 2pmn gm0,n +

g 0k mn p (2gmk,n − gmn,k ) , g 00

which after integration by parts and using

− 2gm0



pmn ,n

g 0k g 00

= −g0m emk leads to

  1 grk,n − grn,k p + 2 (pmn gm0 ),n . 2

(27)

  1 = gm0 emv Hv , grk,n − grn,k p 2

(28)

mk rn

+e

(26)

The first term of (27) can be written in the form given by Dirac (D41)

− 2gm0 with



pmn ,n

mk rn

+e

Hv = prs grs,v − 2 (prs grv ),s .

(29)

We note that in obtaining the expression for the Hamiltonian (23), all direct calculations with the initially modified Lagrangian (12) were performed by Dirac without any reference to space-like surfaces or any additional restrictions or assumptions.

B.

Dirac’s secondary constraints

The next step in the canonical procedure is to find the time development of the primary constraints and see if there are any secondary constraints (or χ-equations in Dirac’s terminology). Poisson brackets (PBs) among the primary constraints are obviously zero,  0α 0β p ,p = 0. The PBs of the primary constraints (9) with the total Hamiltonian (22) are  0σ p , HT =

δ HG = χ0σ , δg0σ

where we keep Dirac’s convention for the fundamental PB (Eq. (D11)),

17

(30)


 αβ 1 α β δµ δν + δµβ δνα δ3 (x − x′ ) . p (x) , gµν (x′ ) = 2

(31)

According to Dirac, “the second term of (D33) [HG (0) = −LX (0) − LG (0) in our (23)] is very complicated and a great deal of labour would be needed to calculate it directly” and instead of performing the variation

δ H δg0σ G

(0), he uses some arguments (see (D23-

D27)) related to the displacements of surfaces of constant time, and thus he infers that the Hamiltonian “must be of the form” (see (D28)) H = g 00


− 12

HL + gr0 ers Hs .

The explicit form of HL and Hs was not given at this stage but a statement about their independence of the g0µ was made based on some arguments which are very general and independent of the particular form of the Lagrangian, i.e. they have no connection with his initial modifications of LG leading to L∗ . And, even in the linearized case [52], without these modifications, the secondary constraints have a dependence on g0µ ; this dependence can be also observed in full GR [45]. In any case, the explicit form of the constraints cannot be found using such arguments and explicit calculations are needed; one has to use a well defined rule of procedure to find them, i.e. we must calculate

δ H δg0σ G

(0) . Dirac performed

these calculations using an additional simplifying assumption (see below) and this result has to be analyzed and compared to what follows from direct calculations. According to Dirac, there are no contributions from HG (0) to a vector constraint (Hr = ers Hs in Dirac’s notation) which presumably comes from the time development of

the corresponding primary constraint φr0 = pr0 (30). Furthermore, HL , which comes from the time development of the primary constraint, φ00 , can be calculated with the additional

simplifying assumption gr0 = 0, which gives (Eq. (D36)): g r0 = 0, g rs = ers , g 00 =

1 . g00

(32)

As a result, all of LX (0), along with the biggest part of LG (0), is dropped from his calculations. According to Dirac [21], the equation for HL “must hold also when gr0 does not vanish”. It is important to check this assumption by direct calculation because if the result of

δ H δg0σ G

(0) is the same as that of Dirac’s, then the simplifying assumption of (32), along

with any references to surfaces of constant time, has nothing to do with his final result. 18

In such a case, Hawking’s criticism of formulations based on the introduction of space-like surfaces, which is in contradiction with the whole spirit of General Relativity and restricts topology of space-time [42], cannot be applied to the Dirac analysis of GR. This also means that the transformations (1) should be derivable in the Dirac Hamiltonian formulation, as was done in the Lagrangian formulation [41] or for the Hamiltonian formulation obtained in [45]. If the results following from the assumption of (32) are different from those where the assumption is not made, then we cannot use (32) as an extra condition in the midst of the calculations and we have to go back to the original Lagrangian to introduce this condition from the outset. This is the rule followed in ordinary constraint dynamics; all imposed constraints must be solved at the Lagrangian level, or added to the Lagrangian using Lagrange multipliers, before performing a variation and/or considering the Hamiltonian formulation. For example, when Chandrasekhar considers the Hamiltonian for Schwarzschild spacetime he, first of all, writes the Lagrangian using this metric and only then passes to the Hamiltonian formulation [54]. Similarly, the condition (32) corresponds to a particular coordinate system, one which is static [33, 53]; and, of course, the momenta p0k , which are conjugate to the eliminated variables g0k cannot appear in such a formulation. Note that the initial modification of the Lagrangian (8) is irrelevant in a static coordinate system as the last two terms in (8) are zero when g 0k is zero. For field theories, especially generally covariant ones, there is an additional restriction: the unambiguous canonical formulation must be performed without explicit reference to ambient space-time by making an a priori choice of a particular coordinate system or subclass of coordinate systems [55], i.e. without destroying the main feature of a theory from the beginning. To find out whether or not Dirac’s formulation is correct or any reference to surfaces of constant time and the simplifications of (32) (or (D36) of [21]} are relevant to his actual results, we perform a “great deal of labour” to find the functional derivatives

δ δg0σ

separately

for each contribution of HG (0) = −LX (0) − LG (0) and to compare the results with those obtained by Dirac. For LG (0) in (25), we find that
 0σ 1√ 0σ αβkt 0t αβkσ 0β tkσα g E − g E − g E χ0σ (0) = p , −L (0) = − −gg αβ,kt G G 2 19

+

  1√ −ggµν,k gαβ,t C σµνkαβt (eee) + C σµνkαβt (ee) , 4

(33)

where the C’s are combinations of terms of different order in eαβ (note that the terms of first and zero orders in eαβ cancel)

C

and

σµνkαβt

(eee) = g

0σ

  1 µναβ kt ktαν µβ αβνt µk − E e +E e + 2E e 2



+ g 0k eβµ E σνtα + eσν E αtβµ + g 0α eµν E σβtk + 2eνβ E σtµk − 2eνt E σβµk

C σµνkαβt (ee) = g 0σ + 00 g



(34)


g 0α g β0 E tkµσ g ν0 − 2E ktµσ g ν0 − E µνσt g k0 00 g

 1 µναβ k0 t0 ktµβ α0 ν0 µtkα β0 ν0 αβkµ ν0 t0 . − E g g +E g g +E g g + 2E g g 2

(35)

When σ = 0, the expression (33) is considerably simplified (this is because eαβ or E µναβ equal to zero if at least one index is zero):

χ00 G

   √ 1 µναβ kt 1√ 00 αβkt 1 00 ktαν µβ αβνt µk − E −ggαβ,kt g E + −ggµν,k gαβ,t g (0) = − e +E e + 2E e 2 4 2  1 µναβ k0 t0 ktµβ α0 ν0 µtkα β0 ν0 αβkµ ν0 t0 g g +E g g +E g g + 2E g g . − E 2

(36)

According to Dirac, this LG (0) is the only source of contributions to the scalar constraint and he constructed it using the simplifying assumption of (32) and later concluded that it “must hold also when g0r does not vanish”. Let us check this assertion by explicitly separating all space and time indices in (36)

χ00 G

+

  √ 1 mnpq kt 1√ ktpn mq pqnt mk 00 mnkt 1 00 e +E e + 2E e − E (0) = − −ggmn,kt g E + −ggmn,k gpq,tg 2 4 2


1√ 
 1√ −ggm0,k g0q,t g 00 g 00 E ktmq + E mtkq + −ggm0,k gpq,t g 00 E pqkmg t0 + g p0 E ktmq + E mtkq 4 2 20

  1√ 1 mnpq k0 t0 ktmq p0 n0 mtkp q0 n0 pqkm n0 t0 . (37) g g +E g g +E g g + 2E g g + −ggmn,k gpq,t − E 2 2 Some terms in (37) have explicit dependence on the space-time components of the metric tensor and these components will disappear only if condition (32) is imposed. For χ0k G (0) there are even more such contributions. Even with condition (32), the result is not zero and this part of the Hamiltonian, LG (0), contributes to the vector constraint. Now let us find contributions coming from the second part, LX (0). After a rearrangement of the terms given in (24) into a form which is more suitable for calculation, we obtain  √
1 −g 1 rskb E grs,ugkb,v g u0 g v0 − gkβ,bgrs,u E rskb + E rsbk g u0 g β0 LX (0) = 00 2 g 2

For the

δLX (0) δg0σ


 +grα,s gkβ,b E rskb + E rsbk g β0 g α0 .

(38)

part we calculate

 0σ χ0σ X (0) = p , −LX (0) =
1√ σ −gδkσ −grs,ubE rskb g u0 + grα,sb E rskbg α0 + C σ (eee) + CIσ (ee) + CII (ee) . 2

The variation

δLX (0) δg0σ

(39)

obviously produces contributions which are only third and second

order in eαβ as in (33). For terms of third order we find C σ (eee) = −

1√ −ggµν,k gαβ,t 4



 × g 0k eβµ E σνtα + eσν E αtβµ + g 0α eµν E σβtk + 2eνβ E σtµk − 2eνt E σβµk

(40)

and in second order we have two contributions: the first proportional to g 0σ

CIσ

 

1 √ g 0σ 1 rsab u0 v0 β0 rsab rsba u0 β0 −g 00 (ee) = grs,u gab,v E g g − gaβ,b g E +E grs,u g − grβ,s g 4 g 2 (41)

and the second with an index σ on E rsσk

21

σ CII

 
t0
1√ g 0ν g 0µ 1 rsσk rskσ β0 σkrt σktr (ee) = −ggµν,k 00 grs,t E . (42) +E g − grβ,t g E +E 4 g 2

Note, that we cannot present the part quadratic in eαβ (41), (42) in a compact form, where terms with derivatives are a common factor, because of the mixture of four and three indices, which is the result of the original noncovariant modification (8) of the Lagrangian. When performing these calculations we have to consider all possible combinations separately. It is not difficult to confirm that χ0σ X (0) is not zero; even with assumption (32), there 0k are contributions to both constraints χ00 X (0) and χX (0). Consequently, Dirac’s conjecture,

if made separately for LX (0) and LG (0), is not correct; but, when both parts are combined, the contribution of zeroth order to the secondary constraint is greatly simplified 0σ χ0σ (0) = χ0σ G (0) + χX (0) =

 
1 1√ mnkt 0σ mnpq kt ktpn mq pqnt mk −gmn,kt E + gmn,k gpq,t −E −gg e + 2E e + 4E e . 2 4

(43)

The χ00 (0)-part is the same with or without condition (32) and χ0k (0) is given by (43) with σ = k (it is zero when (32) is imposed). Frequently (43) is written in a different form which is based on the following observation: if in the expression for the four-dimensional Ricci scalar R
1 R = g αβ g µν Rαµβν = gαβ,µν g αµ g βν − g αβ g µν − gαβ,γ gµν,ρ 4
× g αβ g µν g γρ − 3g αµ g βν g γρ + 2g αρ g βν g γµ + 4g αγ g µρ g βν − 4g αγ g βρg µν ,

(44)

we keep only the spatial indices and change the covariant component of g km to ekm or, equivalently, impose the conditions (32), we obtain the expression shown in square brackets of (43), which is often called the three-dimensional Ricci scalar R(3) . Equation (43) gives contributions to the secondary constraints of zeroth order in the momenta pkm . There are obviously such contributions in χ0k . Dirac’s vector constraint, Hr , does not have such contributions, so it is not directly related to the time development of the corresponding primary constraint p0k (we will discuss this later). 22

For χ00 (0), the equation (43) has to be compared to the corresponding expression of Dirac’s (D39):

XL (0) = −B + where B (D38)

B=

n√

−gg

001/2

grs,u E

rusv

o

,v

 1√ 1/2 −gg 00 grs,u gab,v E rasb euv + 2E ruab esv 4

(45)

(46)

is a part of full expression (5) where after passing to “e−form” (substituting eαβ from (15) instead of g αβ in (5)) only the terms cubic in eαβ are present. Terms quadratic and linear in eαβ are neglected, which results from the simplifying assumption because all non-cubic terms have either g 0k or the derivatives g0k,m. In his final expression (45) Dirac keeps ekm , not g km, which is consistent with his statement that this has to be true without the simplifying assumption which removes the difference between ekn and g kn. In addition, we keep g = det (gµν ) in all equations. Dirac used J 2 = − det (gµν ) and K 2 = − det (gkm ) (or

now more familiar notation 4 g for det (gµν ) and g (or g(3) ) for det (gkm ) ) which are related p √ by g 00 J 2 = K 2 or −g = − det (gkm ) /g 00. By differentiating the second term of (45), it is not difficult to derive the relation 1 1/2 χ00 (0) = g 00 XL (0) . 2

(47)

Dirac’s scalar constraint HL (HL (0) = XL (0)) is not the result of a direct calculation of

{p00 , HG }. This difference is not important for the proof of closure of the Dirac procedure

and one can always consider linear combinations of non-primary constraints. However, the difference between primary and non-primary constraints becomes important when their linear combinations are used to construct a gauge generator; this was discussed in great detail in [29]. For Castellani’s procedure [25] (or any other procedure) for finding gauge transformations we have to be careful with the redefinition of constraints, as we will demonstrate in the next Section (see also [29]). Until now, we have been concerned with the most complicated contributions to the secondary constraints which are zeroth order in the momenta. Let us now consider the contributions to all orders. In the other two orders we obtain (using (23))

23

1 1 g 0σ δ HG (2) = √ χ (2) = δg0σ 2 −g g 00 0σ



 1 gra gsb − grs gab prs pab , 2

  1 uv rs δ σ us uv rs HG (1) = −δu p,s − e p grs,v + e p grv,s . χ (1) = δg0σ 2 0σ

(48)

(49)

Note, as χ00 (1) = 0 there are no contributions linear in the momenta to the scalar constraint, but χ0k (2) 6= 0, unless we impose (32).

χ0σ (0) was already calculated in (43). For the full scalar constraint, χ00 , the relation

(47) is preserved in all orders 1 1/2 χ00 = g 00 HL . 2

(50)

The vector constraint χ0k has non-zero contributions in all orders of the momenta, if (32) is not imposed. Before we continue to compare our direct calculations with those of Dirac, let us try to present the canonical Hamiltonian as a linear combination of the secondary constraints we calculated above. We approach this problem by considering different orders in the momenta. The highest order is the second and the result is easily obtained from the first terms of (23) and (48) 1 HG (2) = 00 √ g −g

(using g0σ g 0σ = δ00 = 1).

  1 gra gsb − grs gab prs pab = 2g0σ χ0σ (2) 2

(51)

By considering (27), which is equivalent to the second terms of (23) and (49), we have in first order HG (1) = −2gu0



pus ,s

1 − euv prs grs,v + euv prs grv,s 2



= 2g0σ χ0σ (1) .

(52)

HG (2) and HG (1) are of the same form and we anticipate HG (0) is also in this form. Unfortunately, this is not obvious and we have to perform some calculations to show it. To preserve the structure found in (51), (52), we will demonstrate that HG (0) = −LG (0) − LX (0) = 2g0σ χ0σ (0) + (...),k .

(53)

Note, that for HG (1) given by (27) we also obtain (52) only up to a total spatial derivative. Using (25), (38), and (43) we have 24

−LG (0) − LX (0) − 2g0σ χ0σ (0) =  √

−gE

mnki

gmn,i −

√

  0k 0µ 0i 0µ νi g g νk g g −e . −ggµν,i e g 00 g 00 ,k

(54)

This equation demonstrates that the relations found for HG (2) and HG (1) are also valid for HG (0) and the canonical Hamiltonian can be written in terms of χ0σ as HG = 2g0σ χ0σ .

(55)

Of course, this is correct up to total temporal (see (8)) and spatial (see (8), (27), and (54)) derivatives. The modification of the initial Lagrangian (8) was proposed by Dirac while (54) is obtained in the course of preserving relations found among contributions of higher order in the momenta to the constraints and the Hamiltonian. It would be very difficult to guess (54) without knowing the final result. Such an additional integration appearing in (54), is very often performed at the Lagrangian level. For example, in the book by Gitman and Tyutin [13], in addition to Dirac’s (8) (which are B and first term of C i of Eq. (4.4.12) in [13]), the integrations of (54) were performed at the Lagrangian level (the second and third terms of C i ). The integrations of (54) can be derived only in the course of the Hamiltonian procedure, but such integrations (if they are known) are also correct when applied to the Lagrangian because (going back to Dirac’s derivation) it is clear that LX (0) was constructed before the elimination of the velocities (i.e., at the Lagrangian level). How is this “covariant”9 form of HG (55) (which is equivalent to what was found in [45]) related to Dirac’s expression for the Hamiltonian? Are they equivalent? The relationship between scalar constraints χ00 and HL was found in (50); we now consider the relation between the vector constraints. Let us inspect the form of our constraints calculated to different orders appearing in (48), (49), and (43). There are simple relations between the contributions of different orders to χ00 and χ0k : 9

We use the term “covariant” to distinguish constraints χ0σ that follow directly from time development of the primary constraints poσ and lead to a compact and symmetric form (55) for the canonical part of the Hamiltonian. This is “covariance” in the sense of [56], i.e. constraints are contracted with components of a true tensor (components goσ ), contrary to a similar form used in the ADM formulation (see, for example, [39], Eq. (4.12): H = N µ Hµ ) where N µ is neither a true vector nor components of any true tensor.

25

χ0k (2) =

g 0k 00 g 0k 00 00 0k 0k 0k χ (2) , χ (1) = 0, χ (1) = ψ , χ (0) = χ (0) g 00 g 00

that allow one to write (to all orders) 0k

χ

=ψ

0k

g 0k 00 + 00 χ g

(56)

with

ψ

0k

=

−pks ,s

kv rs

−e p



 1 grs,v − grv,s . 2

(57)

Solving (56) for ψ 0k gives a combination of the constraints χ00 and χ0k which were originally calculated from the time development of the corresponding primary constraints. In terms of this combination of constraints ψ 0k and χ00 , we obtain a different form of the canonical Hamiltonian

HG = 2

1 00 χ + 2g0k ψ 0k . g 00

(58)

This form of HG is easy to compare with Dirac’s, because his vector constraint is simply related to ψ 0k 2ψ 0k = eks Hs .

(59)

1 1 1/2 χ0k = eks Hs − g0s esk g 00 HL . 2 2

(60)

For χ0k we find

Equation (59), together with (47), demonstrates the equivalence of the two different forms of HG given in (55) and (58) to Dirac’s canonical Hamiltonian HG = 2g0σ χ0σ = 2


1 00 0k 00 −1/2 χ + 2g ψ = g HL + gr0 ers Hs . 0k 00 g

(61)

We would like to emphasize that Dirac’s constraints are not a direct result of the time development of the primary constraints φ0σ which produce χ0σ ((50) and (60)). The only place known to us where this is stated is in the book by Gitman and Tyutin (Eq. (4.4.19) of [13]); but Dirac’s particular combinations of constraints and the corresponding form of the Hamiltonian are usually used. 26

The linear approximation of χ0σ gives exactly the constraints of linearized GR [52]. In the linearized case there is no difference between χ0k and ψ 0k ; therefore linearized gravity can provide little “guidance” to full GR, in contrast to what was emphasized by ADM in [57]. Any such guidance has to be taken cautiously.

C.

Closure of Dirac procedure for the GR Hamiltonian

To demonstrate closure of the Dirac procedure, any form of the canonical Hamiltonian (61) is suitable as they are all equivalent; and any linear combination of secondary (non1/2

primary) constraints can be used for this purpose (e.g., χ00 = 21 g 00

HL and ψ 0k = 21 ers Hs ).

When using Castellani’s procedure to derive the gauge transformations generated by firstclass constraints, we will consider those secondary constraints that directly follow from the corresponding primary ones and the PBs of secondary constraints with the total Hamiltonian, not just with its canonical part (this is also discussed in the next Section). All of Dirac’s secondary constraints have zero PBs with the primary constraints. In constraint dynamics this means that Lagrange multipliers cannot be found at this stage. As the PB of the secondary constraints with the canonical part of the Hamiltonian is zero or proportional to constraints, the procedure is closed. This is exactly the case here when we are taking into account the algebra10 of PBs among Dirac’s combinations of the secondary constraints:

 ′ {HL (x) , HL (x′ )} = ers (x) Hs (x) δ,r(x) (x − x′ ) − ers x Hs (x′ ) δ,r(x′ ) (x − x′ ) , {Hs (x) , HL (x′ )} = HL (x) δ,s(x) (x − x′ ) ,

(62)

{Hr (x) , Hs (x′ )} = Hs (x) δ,r(x) (x − x′ ) − Hr (x′ ) δ,s(x′ ) (x − x′ ) . When dealing with the “covariant” secondary constraints χ0σ , the multipliers are also not determined, but now we have 10

This algebra is called “hypersurface deformation algebra” or “Dirac algebra” and can be found in slightly different forms in many places, e.g. [25, 32, 47, 58–60], but it was not calculated by Dirac for the GR Hamiltonian.

27

 0σ 0γ 1 σγ 00 (63) χ ,p = g χ . 2 The closure of Dirac’s procedure is obviously preserved when using the “covariant” constraints because χ0σ and Dirac’s constraints are simply related by (50) and (60). This can also be shown by direct calculation of {χ0σ , HG } without any reference to Dirac’s combinations of constraints and their algebra. We choose the second path because to prove closure we need the PB with the total Hamiltonian, and in such a calculation we avoid working with derivatives of delta functions. This is exactly the PB that we need in implementing Castellani’s procedure (see next Section). In addition, the second equation of (62), as it is usually presented in the literature, is not symmetric in form, contrary to the rest of the equations. The symmetric form of this equation was presented in [61, 62]. Applying Dirac’s procedure to the first-order formulation of GR, the same algebra emerges, but for tertiary constraints where all equations are symmetric [29, 63, 64] (see also the discussion on this algebra in [29]). The calculations of PBs between the secondary constraints and the Hamiltonian are very involved and to perform them we found it more convenient to work in the intermediate stages with χ00 and ψ 0k . This allows us to sort out terms uniquely, and at the final stage we can express the result, using (56), in terms of the “covariant” constraint χ0σ . The details of such calculations are given in [45, 65]. We arrive to the following PBs of χ0σ with the canonical part of the Hamiltonian

and



g 0α g 0β 1 2 + gαβ,k χ0k − g 0b g00,b χ00 χ00 , H = − √ Ikmrb pkm g0a eab χ0r + χ0k ,k 00 −g g 2



(64)


g 0k 2 1 1 ak 0r ab 0k χ ,H = √ 2gra p χ − gab p χ − 00 √ Itmrb ptm g0a eab χ0r 00 −g g g −g 0k



+ g 0k g00,t χ0t + 2g0p,t g pk χ0t +

g 0p kq 1 g (gpq,r + grp,q − grq,p ) χ0r − g km g00,m χ00 . 00 g 2

(65)

Of course, we can present (64) and (65) as one “covariant” equation


 0σ 1 1 g 0σ 2 σ am 0r ab 0m √ 2g p χ − g p χ + δ0σ χ0k χ , H = − 00 √ Itmrb ptm g0a eab χ0r + δm ra ab ,k g −g −g g 00 28

g 0p 1 − g σb g00,b χ00 + g 0σ g00,t χ0t + 2g0p,t g pσ χ0t + 00 g σq (gpq,r + grp,q − grq,p ) χ0r . 2 g

(66)

These equations, (64), (65) or (66), along with (63), provide proof of the closure of the Dirac procedure: higher order (tertiary) constraints do not appear and multipliers cannot be found because  0σ χ , HT ∼ χ0σ . D.

Summary

The Dirac Hamiltonian for GR, which is based on the modified Lagrangian of (10) and the simplifying assumption (32), is equivalent to the result of direct calculations given in (55) which are performed without any reference to surfaces of constant time. All the equivalent forms of the Hamiltonian of (61) are only the consequence of an initial modification that does not affect the equations of motion and preserves the four-dimensional symmetry. It is natural to expect that Dirac’s Hamiltonian formulation, which is obtained without any a priori assumptions and restrictions (e.g. surfaces of constant time), has to preserve another manifestation of four-dimensional symmetry: invariance under the diffeomorphism transformation (1). Such a demonstration, given in the next Section, is an important consistency check of our results. All constraints are first-class and thus to find the generators of the gauge transformation, we have to consider “chains” of constraints. This means that one has to work, not with some combination of the constraints χ0σ , but with the exact results for {φ0σ , HT } and {χ0σ , HT }. These expressions are complicated, especially (66), but their correctness can be verified if they lead to diffeomorphism invariance. A simple preliminary check of (66) is that the linearized version of this equation gives  0k  00 χ , H = 0. χ , H = −p0k ,k ,

0k This is equivalent to the results of [52] (note, that in the linearized case χ0k lin = ψlin ) and the

constraint structure leads to a linearized version of diffeomorphism invariance. To summarize, the reality of Dirac’s formulation, that is based on a modification of the initial Lagrangian which does not affect the equations of motion, is as follows: notwithstand29

ing Dirac’s references to space-like surfaces, all of his calculations were performed without use of any such surfaces. Consequently, Hawking’s statement [42] about the contradiction of the Hamiltonian formulation, based on splitting space-time into three spatial dimensions and one time dimension, is not applicable to Dirac’s Hamiltonian formulation of GR, which does preserve the spirit of GR. Our own criticism of Dirac’s formulation in [66] was not correct as we based it only on the ‘interpretational’ aspects of his work. This faux pas is also an illustration of how interpretations or some geometrical (or any other) reasonings can be dangerous if the “rule of procedure” referred by Lagrange is neglected. Dirac’s simplifying assumption, g0k = 0 and (32), for constructing zeroth order in momenta contributions to the secondary constraints is not correct with respect to the individual parts given in (33), (39); but remarkably when these parts are combined together in (43), they are equivalent to his final expression. His secondary constraints do not follow directly from the time development of the primary constraints but rather they are particular combinations of the “covariant” secondary constraints χ0σ which are the result of direct calculation of PBs of primary constraints with the canonical Hamiltonian. His secondary constraints cannot be directly used to find gauge transformations (Dirac himself did not consider this question). In the next Section we will show that the generator built from the “covariant” constraints gives the four-dimensional diffeomorphism (1), and we can say that the “covariant” constraints of the Dirac formulation and their algebra is “the algebra of four diffeomorphisms” [44].

III.

THE GAUGE GENERATOR AND TRANSFORMATION OF THE METRIC

TENSOR

The knowledge of the complete set of first-class constraints (primary, p0σ ,(9), and secondary, χ0σ = χ0σ (2) + χ0σ (1) + χ0σ (0), where contributions of different order in momenta are given by (48), (49) and (43)), as well as the PBs between the primary and secondary constraints (63), and the exact form of the closure (66) are sufficient to find the generators of the gauge transformations. This possibility is Dirac’s old conjecture [11] which became a well developed algorithm and exists in a few variations [25–27]11 . We follow the work of 11

Some methods have to be used with caution because they are sensitive to the choice of linear combinations of non-primary constraints (see discussion in [29]).

30

Castellani [25] where the first application of such a method to Yang-Mills theory and ADM gravity12 was considered.

A.

Castellani’s procedure

Castellani’s procedure is based on a derivation of the generator of gauge transformations which is defined by chains of first-class constraints. One starts with primary first-class (n)

(n)

constraint(s), i = 1, 2, ..., and construct the chain(s) ξi Gi(n) where ξi

is the nth order time

derivative of the gauge parameter ξi (n = 0, 1, ...). The maximum value of n corresponds to the length of the chain (e.g., n = 0, 1, 2 for the system with tertiary constraints). The number of gauge parameters ξi is equal to the number of primary first-class constraints. Note, that these chains are an unambiguous construction once the primary constraints are defined; the remaining members of the chain are uniquely determined. Of course, one can use different linear combinations of non-primary constraints, the generator will have a different form but the gauge transformations will be the same. More details about the problems arising when linear combinations of primary constraints are used in constructing of a gauge generator can be found in [29]. From this point, we specialize to the Dirac Hamiltonian formulation of GR with n = 0, 1 and i = 0, 1, ..., (d − 1); if d = 4 there are four primary and four secondary constraints13 . The functions Gi(n) are calculated as follows

Gσ(1) (x) = p0σ (x) ,

Gσ(0)

 (x) = + p0σ (x) , HT +

Z

αγσ (x, y) p0γ (y) d3 y

(67)

(68)

where the functions αγσ (x, y) have to be chosen in such a way that the chain beginning with Gσ(1) in (67) ends on the primary constraint surface  σ G(0) , HT = primary. 12

13

(69)

In [25] the author referred to the Dirac formulation of GR but in fact considered the ADM formulation. The non-equivalence of these two formulations will be discussed in the next Section. The following calculations, as well as the results of the previous Section, are valid in all dimensions, except d = 2.

31

The generator G (ξσ ) is given by G (ξσ ) = ξσ Gσ(0) + ξσ,0 Gσ(1) .

(70)

There are some peculiarities that arise when applying this algorithm to GR that cannot be seen in simpler cases like Maxwell, Yang-Mills or linearized GR theories. We will consider them in detail below. Firstly, we comment on the use of different linear combinations of secondary (nonprimary) constraints. The term {p0σ (x) , HT } in (68) is uniquely defined by primary constraints. After Dirac’s modification (8) of the original Lagrangian, these remain just the momenta p0σ conjugate to the g0σ components of the metric tensor. Straightforward calculation of the PBs of these primary constraints with the Hamiltonian gives  00 1 1/2 p (x) , HG = χ00 = g 00 HL , 2  0k 1 1 00
−1/2 0k g 0k g g HL , p (x) , HG = χ0k = ψ 0k + 00 χ00 = eks Hs + g 2 2

and these expressions, χ00 and χ0k , will be used when the gauge generators are derived. Of course, one can use Dirac’s combinations, HL and Hs , but only with appropriate coeffi-

cients or in appropriate combinations because of the following inequalities: {p00 , HG } = 6 HL ,  0k p , HG 6= eks Hs . We are not aware of any other situation where one must consider

combinations of secondary constraints in ordinary field theories, the exception is only the first-order, affine-metric, GR [29] where such combinations lead to some computational advantages but can also be avoided. Such a situation does not appear, for example, when the gauge generator for Yang-Mills theory is constructed [25]. In this case, the first term of (68) is just a secondary first-class constraint which is the result of direct calculation of the PB of primary constraints with the Hamiltonian. In the Hamiltonian formulation of GR it is quite common (if not exclusive) to use the Dirac combinations of constraints, but one has to be careful when gauge generators are constructed using HL and Hs . Secondly, in both the Maxwell and Yang-Mills theories it is possible to choose the functions αγσ (x, y) so that chains truly end with zero in (69) [25]. It does not happen in GR, and chains end only on the surface of the primary constraints. The effect of such a difference will be seen in our calculation of the gauge generators and the associated transformations. 32

Thirdly, the total Hamiltonian should be used in Castellani’s procedure, not just its canonical part (see (68), (69)). Again, in linearized GR, Yang-Mills and Maxwell theories this difference is irrelevant because in these theories the PBs of secondary constraints with primary ones are zero. This is not the case for full GR as can be seen from (63) and similarly from equation (16) of [45]. Finally, a purely technical comment. There is a change of sign in front of the first term of (68) relative to that used in [25]. This is the result of Dirac’s convention for the fundamental brackets in (31) (it is the negative of the fundamental brackets used in [25]).

B.

Castellani’s generator

To construct the generator (70) we have to find functions αγσ (x, y) using the condition (69)  σ G(0) , HT =  0σ χ (x) , HT +

Z



0σ

χ (x) +

Z

αγσ

0γ

3

(x, y) p (y) d y, HT

 σ αγ (x, y) , HT p0γ (y) d3 y +

Z



=

 αγσ (x, y) p0γ (y) , HT d3 y.

(71)

Part of the first term has already been calculated and {χ0σ (x) , HG } is given by (66). For

the part involving the primary constraints, p0γ , we use (63) which gives  −

1 χ0σ (x) , g00,0 p00 + 2g0m,0 p0m + HG = g00,0 g 0σ χ00 + g0m,0 g σm χ00 2

σ
1 δm g 0σ 2 tm ab 0r √ √ I p g e χ + 2gra pam χ0r − gab pab χ0m + δ0σ χ0k tmrb 0a ,k 00 00 g −g −g g

1 g 0p − g σb g00,b χ00 + g 0σ g00,t χ0t + 2g0p,t g pσ χ0t + 00 g σq (gpq,r + grp,q − grq,p ) χ0r . 2 g

(72)

The second term of (71) is irrelevant because it is automatically zero on the surface of primary constraints, as required by (69). For the last term of (71), taking into account the zero value of the PB among primary constraints, we find that 33

 0γ  p , HT = p0γ , HG = χ0γ .

(73)

This illustrates the advantage of using the “covariant” constraints χ0γ for deriving the generator. With (72)-(73), we can now read off the functions αγσ (x, y) from (71) that compensates (72) 1 −αγσ (x, y) = g00,0 (x) g 0σ (x) δγ0 (x, y) + g0m,0 g σm δγ0 2 −

σ
g 0σ 2 1 δm tm ab r k √ √ I p g e δ + 2gra pam δγr − gab pab δγm + δ0σ δγ,k tmrb 0a γ 00 00 g −g −g g

1 g 0p − g σb g00,b δγ0 + g 0σ g00,t δγt + 2g0p,t g pσ δγt + 00 g σq (gpq,r + grp,q − grq,p ) δγr , 2 g

(74)

where δγσ (x, y) ≡ δγσ δ (x − y) and the arguments x and y are explicitly written only in the

first term. Contracting αγσ (x, y) with the primary constraints and performing integration in the second term of (68) we obtain 1 Gσ(0) = χ0σ − g00,0 g 0σ p00 − g0m,0 g σm p00 2 +

σ
1 δm g 0σ 2 tm ab 0r √ √ I p g e p − 2gra pam p0r − gab pab p0m − δ0σ p0k tmrb 0a ,k 00 00 g −g −g g

1 g 0p + g σb g00,b p00 − g 0σ g00,t p0t − 2g0p,tg pσ p0t − 00 g σq (gpq,r + grp,q − grq,p) p0r . 2 g

(75)

Equation (75) completes the calculation of the generator (70). Now the transformation of fields can be found by calculating their PB with the generator14

δ (f ield) = {f ield, G} .

(76)

For the time-time component of the metric tensor, g00 , we obtain (using Dirac’s convention of (31) and keeping only part of the generator (70) with terms proportional to p00 ) 14

Some authors defined transformations as δ (f ield) = {G, f ield} which seems to be more natural. However, we keep the convention of Castellani that, of course, affects only an overall sign in the final result, which can always be incorporated into the gauge parameters (this is not a field-dependent redefinition, as is used in [32]).

34

δg00 = {g00 , G} = −

δ G= δp00

    δ 1 0σ 00 1 σb 00 σm 00 00 − 00 −ξσ g00,0 g p + g0m,0 g p − g g00,b p = + ξ0,0 p δp 2 2

− ξ0,0 +



   1 1 1 1 00 0m k0 km ξ0 + ξk − g 0b g00,b ξ0 − g km g00,m ξk . g00,0 g + g0m,0 g g00,0 g + g0m,0 g 2 2 2 2 (77)

Let us compare this result with diffeomorphism invariance (1) which can be written in an equivalent form, that is more convenient for comparison with our calculations: δ(dif f ) gµν = −ξµ,ν − ξν,µ + g αβ (gµβ,ν + gνβ,µ − gµν,β ) ξα .

(78)

Taking µ = ν = 0 and explicitly separating space and time indices, we have



δ(dif f ) g00 = −2ξ0,0 + g 00 g00,0 + 2g 0k g0k,0 − g 0k g00,k ξ0 + g k0g00,0 + 2g kmg0m,0 − g kmg00,m ξk .

(79)

We see that (79) is equivalent to (77) up to a numerical factor 2,

2 {g00 , G} = δ(dif f ) g00 ,

(80)

that can be incorporated into the gauge parameter by a rescaling ξσ → 2ξσ . Similarly, for the space-time components, g0k , we have δg0k = {g0k , G} =   0σ σ
δ g 1 δm 2 am 0r ab 0m − 0k −ξσ − 00 √ Itmrb ptm g0a eab p0r + √ 2g p p − g p p + δ0σ p0m ra ab ,m δp g −g −g g 00 g 0p +g g00,t p + 2g0p,tg p + 00 g σq (gpq,r + grp,q − grq,p) p0r g 0σ

0t

pσ 0t

1 1 = − ξ0,k − ξk,0 2 2 35



+ ξm,0 p

0m



=

 0σ
g 1 1 2 1 σ 2gka pam δm − gab pab δkσ + ξσ − 00 √ Itmkb ptm g0a eab + √ 00 2 g −g −g g 0σ

+g g00,k + 2g0p,k g

pσ

 g 0p σq + 00 g (gpq,k + gkp,q − gkq,p) . g

(81)

There is a difference between the transformation (81) and the transformation of the time-time component (77) as the momenta pab are present in (81). Using the definition of momenta (18) and re-expressing ekm in terms of g km by (15), we obtain 1 1 δg0k = − ξ0,k − ξk,0 2 2 +

+

 1  00 g g00,k + g 0m (g0m,k + gkm,0 − g0k,m ) ξ0 2

 1  m0 g (g00,k + gk0,0 − g0k,0 ) + g mn (g0n,k + gkn,0 − g0k,n ) ξm , 2

(82)

which again equals δ(dif f ) gµν as given in (78) with µν = 0k; which is true provided we again rescale ξσ by a factor of 2. The last transformation to be checked is the transformation of the space-space components

δgkm = {gkm , G} = −

δ G (ppq ) . δpkm

(83)

The relevant part of the generator (70) which has an explicit dependence on ppq is

pg



0σ

G (p ) = ξσ χ


g 0σ 2 1 1 tm ab 0r σ am 0r ab 0m + 00 √ Itmrb p g0a e p − δm √ 2gra p p − gab p p g −g −g g 00 (84)

where parts of the secondary constraints (χ00 (2), χ0k (2) and χ0k (1)) will also contribute to the final result. The variation of the last two terms in (84) gives contributions proportional to the primary constraints (which equal zero on the surface of primary constraints). The only relevant parts of the generator for δgkm are given by
G = ξ0 χ00 (2) + ξk χ0k (2) + χ0k (1) . 36

(85)

Performing variation of (85) with respect to pkm , using the expression for momenta given in (18), and reverting from ekm to g km using (15), we obtain 1 δgkm = − (ξk,m + ξm,k ) 2 1 1 + g 00 (gk0,m + gm0,k − gkm,0 ) ξ0 + g p0 (gk0,m + gm0,k − gkm,0 ) ξp 2 2 1 1 + g 0p (gkp,m + gmp,k − gkm,p ) ξ0 + g pq (gkq,m + gmq,k − gkm,q ) ξp . 2 2

(86)

It is not difficult to check that up to the same numerical factor 2 (as occurred in (80)) this is equivalent to δ(dif f ) gµν (78) with µν = km. We see that transformations of the time-time and space-time components of the metric tensor are exactly equivalent to a diffeomorphism and the space-space components give a diffeomorphism only on the surface of primary constraints. Such a deviation from ordinary field theories like Yang-Mills can be expected because the derivation of generators is performed (i.e., the functions αγσ (x, y) are found) only on a surface of primary constraints. This is a consequence of the peculiarities of diffeomorphism transformations that will be discussed at the end of this Section. Returning to Dirac’s statement [21] about abandoning four–dimensional symmetry in his approach; we can see that it is restrictive and only related to his initial modification of the Lagrangian (8). This abandoning of four-dimensional symmetry does not appear, neither in linearized [52] nor in full GR [45]. The Dirac Hamiltonian formulation of GR, as we demonstrated in this Section, allows one to derive the gauge transformation of the metric tensor that can be put in covariant form and four-dimensional symmetry is preserved. The exact meaning of common statements, such as the one that is found in [44], “unfortunately, the canonical treatment breaks the symmetry between space and time in general relativity”, must also be clarified in light of our results. Of course, and this is a property of the Hamiltonian approach itself, the four-dimensional symmetry is not broken, it just is not manifest. For any generally covariant theory with first-class constraints we can only make a conclusion about abandoning such a symmetry if the gauge invariance that is derived from the first-class constraints cannot be presented in covariant form. In next Section we will show that it happens in the case of the ADM formulation. If the symmetry possessed by the 37

original Lagrangian disappears in the Hamiltonian formulation, then it should be regarded as a very strong indication that there is a mistake in the formulation; this is not a problem with the initial Lagrangian or with the Hamilton-Dirac method. From this point of view, if the “canonical treatment breaks the symmetry”, then such a treatment is not canonical.

C.

Alternative derivation of a gauge generator

Let us return to the derivation of the gauge transformations. Our derivation of the transformation was based on an application of Castellani’s method [25]. There are at least two variations of it: one of them is based on the extended Hamiltonian [14, 26], where all firstclass constraints are included, and the other, [27], is based only on the total Hamiltonian (i.e. only primary first-class constraints are included) as in Castellani’s case. The equivalence of the algorithms [26] and [27] was discussed in [27] and a comparison of methods [26] and [25] was made in [26]. Primary constraints play a special role in all of these methods. The need to include multipliers associated with the primary constraints was also emphasized in [14, 26] and their importance in gauge transformations was demonstrated by some simple examples. In [27], the multipliers are also important elements of this method (see below). Recently, the method of [27] was applied to the ADM Hamiltonian in [32], where the transformation of the metric tensor was derived and was shown to differ in form from (1). For completeness, we apply the method used in [32] to the Dirac Hamiltonian formulation of GR to demonstrate that the derivation of the diffeomorphism transformation in the Hamiltonian approach is not an artifact of a particular procedure for finding a gauge generator. At the same time this demonstration will illustrate the equivalence between the two different methods described in [25] and [27]15 , as well as the equivalence of both of them to the Lagrangian treatment of this problem in [41]. The total Hamiltonian is the starting point of the method outlined in [27] H T = H c + λµ φ µ .

(87)

For a system with only irreducible secondary first-class constraints and no tertiary con15

We would like to note that the method [27] is sensitive to the choice of linear combination of nonprimary constraints and gives equivalent to [25] result only when using “covariant” constraints of Dirac’s formulation (see more detail in [29]).

38

straints (so that {φµ , Hc } = χµ ) the generator of gauge transformations is simply G = ηµ φµ + ξµ χµ

(88)

with two sets of parameters, ηµ and ξµ (twice the number of primary constraints), which are related by (see Eq. (17) of [27]):     νγ νγ ν ν . + λγ C(s)µ − ην W(s)µ + λγ B(s)µ 0 = ξµ,0 − ξν V(s)µ

(89)

Here Wµν , Vµν , Cµνγ , and Bµνγ are the structure functions of the involutive algebra (see Eqs. (2) and (3) of [27]) ν ν {Hc , φν } = W(p)µ φµ + W(s)µ χµ ,

(90)

ν ν {Hc , χν } = V(p)µ φµ + V(s)µ χµ ,

(91)

νγ νγ {φν , φγ } = C(p)µ φµ + C(s)µ χµ ,

(92)

νγ νγ {φν , χγ } = B(p)µ φµ + B(s)µ χµ .

(93)

The indices (p) and (s) indicate structure functions associated with the primary φµ and secondary constraints χµ , respectively. Note, that (89) involves only functions with the subscript (s), i.e. structure functions related to the primary constraints are not present, so that this equation is valid on the primary constraint surface. This is similar to what happens in Castellani’s procedure. To find the generator (88), one has to solve (89) for ην in terms of ξν ; and as in Castellani’s procedure the number of independent parameters becomes equal to the number of primary constraints. For the Dirac Hamiltonian of GR, which is obtained after modification of the gamma-gamma part (with no effect on the equations of motion, canonical variables gµν , and conjugate momenta pµν ) we have the simple primary constraints (9) φ0µ = p0µ and the secondary constraints (for explicit expressions see (48), (49) and (43))  0µ φ , Hc = χ0µ . 39

(94)

This allows us to write the Hamiltonian in a compact and symmetric form Hc = 2g0µ χ0µ . The possibility of solving (89) for ην , which is an ordinary algebraic equation, depends on νγ ν the structure functions W(s)µ and C(s)µ . In the case of the Dirac Hamiltonian formulation

of GR (see (94)) they are νγ ν W(s)µ = δµν , C(s)µ =0

(95)

that give for solution of (89) for ηµ :   νγ ν ηµ = ξµ,0 − ξν V(s)µ + λγ B(s)µ .

(96)

νγ ν The structure functions V(s)µ for GR are complicated (see (66)), for B(s)µ we have (see (63))

1 νγ B(s)µ = g νγ δµ0 , 2 and (96) becomes 1 ν ηµ = ξµ,0 − ξν V(s)µ − ξν g0γ,0 g νγ δµ0 . 2

(97)

νγ Note that the structure functions B(s)µ do not equal zero and the Lagrange multipliers,

which are the velocities g0σ,0 that cannot be expressed in term of p0σ enter (97) explicitly (λ0 = g00,0 and λk = 2g0k,0).16 After the substitution of (97) into (88) we obtain a oneparameter ξν (the number of components equals to the number of primary constraints) generator 1 ν G (ξν ) = −ξν g00,0 g ν0 φ0 − ξν g0m,0 g νm φ0 + ξν,0 φν − ξν V(s)µ φµ + ξν χν . 2

(98)

This expression has to be compared with the generator found using Castellani’s procedure G = ξν,0Gν(1) + ξν Gν(0) .

(99)

The third term of (98) is exactly the same as the first term of (99); the first, second, and last terms of (98) are also the same as in Castellani’s approach (see the first line of (75)). The fourth term of (98), to be compared with our result obtained using Castellani’s approach, is 16

Expressions for the multipliers came from the Legendre transformation and from the fact that the modified Lagrangian is independent of g0µ.0 , so that H = gνµ,0 pνµ − L = g00,0 p00 + 2g0µ,0 p0µ + ...

40

ν ν −ξν V(s)µ φµ . The structure function, V(s)µ , originates from the calculation of the PB of (91),

which in the case of a field theory is ν

{Hc , χ (x)} =

Z

ν V(s)µ (x, y) χµ (y) d3 y.

(100)

The direct calculation of {Hc , χν } gives terms proportional to χµ and its derivatives (for its explicit form see (66)) {Hc , χν } = Kµν χµ + Mµνk χµ,k .

(101)

This equation is usually presented in the following form in order to find the structure functions  Z  ∂ ν νk {Hc , χ } = Kµ (x) δ (x − y) − Mµ (x) δ (x − y) χµ (y) d3 y. ∂yk ν

(102)

This is the standard form for an intermediate result in such calculations (e.g. see [25]). To find the generator (98), it is not necessary to rewrite (101) in the form of (102); and ν the direct substitution χµ → φµ into (101) gives the corresponding part, −ξν V(s)µ φµ , of the

generator (98). This is equivalent to the second and third lines of (75). If the novel method of [27] is applied to the Dirac Hamiltonian of GR and not to the ADM Hamiltonian, as was done in [32], then the generator (98) is equivalent to the one obtained using the old procedure of [25]; and so they generate the same gauge transformations when these two methods are applied to the same Hamiltonian. The peculiarities of Castellani’s procedure, when it is applied to the Hamiltonian of GR, were discussed at the beginning of this Section and illustrated by a derivation of (1) using the methods [25] and [27]. They are originated from the algebra of constraints either in Dirac’s formulation, given in this Section, or in the ‘covariant’ formulation of [45]. This algebra is different from that of ordinary gauge theories. It reflects the peculiarities of diffeomorphism invariance if compared to the gauge invariance of ordinary gauge theories. We now briefly consider this topic.

41

D.

General coordinate transformations and diffeomorphism

In the Introduction, we restrict our discussion to a particular meaning of diffeomorphism given in (1) that is generally accepted in literature on GR and which is similar to the usual gauge transformations. It is in exactly this sense that diffeomorphism invariance can be derived from the Hamiltonian approach. Now we would like to describe this transformation without recourse to the Hamiltonian formulation as it is usually presented in textbooks (e.g. see [33]); and we wish to reveal how a difference between these two views of diffeomorphism invariance manifests itself. This exercise will also demonstrate the connection between (1) and general coordinate transformations. The principle of general covariance, the cornerstone of GR, puts severe restrictions on the possible forms of the Lagrangian. The simplest is the Einstein-Hilbert Lagrangian [33, 53]17

LEH =

√

−gR =

√

−gg µν Rµν =

√

  −gg µν Γαµν,α − Γαµα,ν + Γαµν Γβαβ − Γαµβ Γβαν .

The EH action and the Einstein equations are invariant under a general coordinate transformation x′µ = f µ (xν )

(103)

and the corresponding transformation of the metric tensor

g

′µν

∂x′µ ∂x′ν αβ g (x) . (x ) = ∂xα ∂xβ ′

(104)

For infinitesimal transformations xµ → x′µ = xµ + ξ µ (x)

(105)


ν µα µ αν g ′µν (x′ ) = g µν (x) + ξ,α g (x) + ξ,α g (x) + O ξ 2 .

(106)

(104) can be written as

17

Of course, it is not unique and there are many posibilities: such as Lovelock gravity [67] or f (R).

42

Note that the components ξ µ form a true vector [53, 68], in contrast to the parameters ε⊥ and εk which appear in the ADM formulation of [37] (see (2) and its derivation in the next Section). If we consider ξ µ (x) as being a small parameter and restrict our interest to the first-order contributions in ξ µ , then the exact invariance18 of the EH action and the Einstein equations of motion is lost as only the inclusion of all terms in the expansion of (106) will preserve it. In addition, if we want to present (106) in a form similar to a gauge transformation, in which the invariance with respect to replacement of field variables is written in the same coordinate frame of reference [69], we should write both sides of (106) in the same coordinate system. This can be done by an additional approximation, using the Taylor expansion of ′ gµν (x′ ):


µν α g ′µν (x′ ) = g ′µν (xα + ξ α (x)) = g ′µν (x) + g,α ξ + O ξ2 ∂g ′µν (x′ ) α where in the second term (which is already linear in ξ ) we used ∂x′α ′

µν = g,α + O (ξ).

ν µα µ αν µν α δg µν = g ′µν (x) − g µν (x) = ξ,α g (x) + ξ,α g (x) − g,α ξ

(108)

x =x

(107)

Combining (106) and (107) and keeping only terms linear in ξ α , we obtain

which is equivalent to δg µν = ξ µ;ν + ξ µ;ν .

(109)

By repeating similar calculations, or by using δ (gµα g αν ) = 0, one can find transformations for the covariant components of a metric tensor δgµν = −ξµ;ν − ξν;µ = −ξµ,ν − ξν,µ + 2Γαµν ξα .

(110)

This equation is just (1) (or its equivalent form (78) that was more convenient for comparison with our calculations). With ξ µ , being a true vector, the transformations (109) and (110) are generally covariant (these are covariant derivatives of a true vector), so they are independent 18

We use the phrase “exact invariance” when a variation of the Lagrangian and equations of motion under the gauge transformation gives zero exactly. We will see that for GR the invariance is not “exact” as the variation of the Lagrangian equals to the total derivative, while the variation of the Einstein equations is proportional to the equations themselves.

43

of the choice of coordinate system; these are the transformations we derived from the Dirac Hamiltonian of GR and in [45]. Similarly, we can obtain the transformation of the Christoffel symbols. Using the relation between Γαµν and gµν in (13) and the transformation δgµν in (110), we obtain β α α − Γαµβ ξ,νβ − Γανβ ξ,µ − ξ,µν . δΓαµν = −ξ β Γαµν,β + Γβµν ξ,β

(111)

α = ∂µ ∂ν ξ α ) imNote that the presence of second-order derivatives of the parameters (ξ,µν

mediately allows one to come to a general conclusion about the constraint structure of the Hamiltonian formulation of GR in the first-order form, the Einstein affine-metric formulation of [17], where g µν and Γαµν are treated as independent fields.19 The presence of the second-order derivatives of the parameters in the transformation δΓαµν implies that the generators must have the same order of derivatives, i.e. the tertiary constraints must appear in such a formulation. Of course, direct calculations confirmed this simple observation [29, 63, 64, 66, 73–75]. In the first discussion of the Hamiltonian formulation of the firstorder form of GR which was presented in [57], the tertiary constraints did not appear because some first-class constraints were solved before closure of Dirac’s procedure was reached; this procedure is not a consistent implementation of Dirac’s procedure for treating constrained systems. This fact was clearly demonstrated in the Appendix of [61] and was discussed in [66]. The loss of tertiary constraints is also due to a misleading analogy between the first-order formulations of Electrodynamics and linearized GR appearing in [57, 76]. In the first-order formulation of Electrodynamics, where the field strength is treated as an independent variable, there is no increase in the order of the derivatives of the gauge parameters in the generator of gauge transformations because the variation of the field strength is zero. In contrast, the variation of Γαµν under diffeomorphism transformations, (111), is not zero and cannot even be written in covariant form as δgµν was in (110). This characteristic is consistent with Γαµν not being a true tensor [33, 77]20 . Also there is no linear combination of the first-order derivatives of the metric tensor that is exactly invariant under diffeomorphism 19

20

This formulation was originally introduced by Einstein [17] (for English translation see [19]), but mistakenly attributed to Palatini [70] (see also Palatini’s original paper [71] and its English translation [72]). The Γα µν behaves like a tensor only with respect to linear coordinate trasformations [33]. Probably, this was the origin of the analogy between Electrodynamics and linearized GR and of the conjecture that this analogy should be extended to full GR [57].

44

transformations, as well as under general coordinate transformations21 . This invariance is also related to the fact that a generally covariant action for GR cannot be built from terms only quadratic in the first-order derivatives of the metric tensor; and the simplest generally covariant, EH Lagrangian, is proportional to a Ricci scalar and this involves second-order derivatives of the metric tensor. This is why the affine-metric formulation of GR leads unavoidably to tertiary constraints and consequently increases the length of the chain of constraints and the order of the derivatives in the gauge generator. This general arguments were recently confirmed by direct calculations in [29, 63, 64]. Let us compare (111) with the gauge invariance of Yang-Mills theory. We have the field a strength Fµν whose variation does not vanish, in contrast to Electrodynamics,

a c δFµν = f abc θb Fµν ,

with f abc a totally antisymmetric structure constant and θb a gauge parameter. We do a a not have exact invariance for Fµν (i.e. δFµν 6= 0), but the gauge parameters enter only

linearly (without derivatives). Thus in the first-order formulation of Yang-Mills theory, if we consider the field strength as an independent variable, there is no increase in the length of the chains of constraints needed to accommodate these transformations. In GR it is not possible to build any combination of first-order derivatives which is exactly invariant under diffeomorphism; and it is also impossible to find a combination whose variation is proportional to the gauge parameter or its first-order derivatives. The transformation (1) (or other equivalent forms) is written in the same coordinate system and, because this combination is a true tensor, it is independent of the coordinate transformations. In this sense it is “analogous to the gauge transformation” [78, 79]; but the absence of combinations of derivatives that do not lead to an increase of the order of the derivatives of gauge parameters in the generator is a distinct property of GR. In addition, the Lagrangian of GR is not exactly invariant (δL 6= 0) under a diffeomorphism transformation, in contrast to Maxwell and Yang-Mills theories. From (108) and (111) and by using 21

Actually, such a combination exists which is a true tensor: this is the covariant derivative of the metric tensor, gµν;γ , but it identically equals zero in GR.

45

Rµν = Γαµν,α − Γαµα,ν + Γαµν Γβαβ − Γαµβ Γβαν we can find the transformations of Rµν and R = g αβ Rαβ : ρ δRµν = −ξ ρ Rµν,ρ − ξ,µ Rνρ − ξ,νρ Rµρ , δR = −ξ ρ R,ρ .

(112)

From the above relations it is not difficult to confirm that the transformation of the EH Lagrangian under a diffeomorphism gives the total divergence

δ

√



√ −gR = −ξ µ −gR ,µ .

(113)

This lack of exact invariance is distinct from what occurs in the Maxwell and YangMills theories, but other models exist with Lagrangians which are also invariant up to a total divergence, e.g. Topologically Massive Electrodynamics (TME) of [80]. (See [81– 83] for a discussion of its first-order formulation.) Despite differences in the invariance property of Lagrangians, which can be either exact as in ordinary Electrodynamics or up to a total divergence as in TME [80], the equations of motion are exactly invariant under gauge transformations. In GR the transformation of the equations of motion is proportional to the equations themselves [69]. Using (112) we can easily find transformations of the Einstein field equations ρ δGµν = −ξ ρ Gµν,ρ − ξ,µ Gνρ − ξ,νρ Gµρ

(114)

1 Gµν = Rµν − gµν R 2

(115)

where

is the Einstein tensor. So, in GR under diffeomorphism transformations (1), the equations of motion are invariant only ‘on-shell’ which does not contradict the principle of gauge invariance: a solution to the equations of motion maps into a solution. This last result, the ‘on-shell’ invariance of the equations of motion might cause some confusion because (1) was obtained from infinitesimal coordinate transformations (105), which are a particular case of general coordinate transformations (103). The EH action and the Einstein equations are exactly invariant under (103) and therefore, under (105) (in fact, the Einstein equations were originally postulated so as to satisfy (103)). After writing (105) in the same coordinate 46

system and using the approximations (106) and (107), exact invariance is lost. Such a mapping, (114), by itself, is not peculiar to GR because a similar property is present in the Yang-Mills theory where the transformation of the equations of motion is δ (Dµ Faµν ) = fabc θb Dµ Fcµν .

(116)

As in GR, the Yang-Mills field equation is only invariant ‘on-shell’; but this transformation is proportional to the gauge parameter, whereas the transformation of δGµν in (114) also contains derivatives of the gauge parameter. The peculiar algebra of constraints in GR is related to this increase in the order of the derivatives of the gauge parameters in the transformations of the equations of motion, as well as the impossibility of finding a combination of derivatives of the metric tensor which is either exactly invariant or whose variation does not require derivatives of the gauge parameters.

E.

Noether’s differential identities and diffeomorphism

In ordinary field theories gauge invariance is derivable from the corresponding actions without any reference to coordinate transformations; and GR being a field theory is no exception. All properties of a system are encoded in its action and the presence of gauge symmetries reflects the fact that some relations exist between equations of motion. This is stated in Noether’s theorems [84] (see English translation of [84] in [85]) that allows us to find differential identities if a symmetry of an action is known. The converse of both theorems also holds; that is why we can try to build such identities without a priori knowledge of the gauge invariance(s) of a particular action. Such a construction for covariant theories is a very simple iterative procedure: starting from Euler derivatives (variation of the Lagrangian with respect to basic field(s)) of an action one can try to build a basic differential identity by differentiating the Euler derivative (ED) and then finding combinations of EDs that identically give zero. This procedure works for all field theories and we demonstrated it in detail by deriving the gauge invariance of the first-order Einstein-Cartan action in [86], where the basic differential identities lead to translational and rotational invariances in the tangent space [86]. We can also apply this method to GR. Using the variational principle (without any references to a coordinate system) we find a variation of (7) with respect to only the independent 47

field variable gαβ . That gives us the Euler derivative: E

αβ

√ δLEH = = −g δgαβ



1 αβ g R − Rαβ 2



√ = − −gGαβ ,

(117)

where Gαβ is the contravariant Einstein tensor. Note that the equations of motion of GR, √ which are in fact the Euler derivative, E αβ , differ from Einstein’s equation Gαβ = 0 by −g. This difference is not important if we want to obtain solutions of Einstein’s equations (or equations of motion) as we always assume that det (gµν ) 6= 0; but in deriving the identities and the corresponding transformations of fields, the Euler derivative must be used. The √ combination −gGαβ in (117) appears naturally, in contrast to the method employed in [41], where the gauge identity has to be imposed from the outset. We start to build the basic differential identity from the derivative of (117)
√ √ 1√ −gg µν gµν,α Gαβ − −g∂α Gαβ . ∂α E αβ = ∂α − −gGαβ = − 2 Compare (118) with (117) as in [86] (where details can be found), one obtains √ ∂α E αβ = − −gΓβαλ Gαλ = −Γβαλ E αλ .

(118)

(119)

Equally well (and in a much shorter way) the identity (119) can be obtained using the Bianci αβ identity: Gαβ + Γααλ Gβλ + Γβαλ Gαλ = 0. Note that the equations of motion were ;α = ∂α G

not imposed in deriving the Bianci identity [33]. This is important as a differential identity should be satisfied ‘off-shell’, without imposing equations of motion. Finally, the differential identity for the EH action is: I β = ∂α E αβ + Γβαλ E αλ = 0.

(120)

Now, forming a scalar density by contracting I β (which is a contravariant vector density) with the gauge parameter of the corresponding tensorial dimension (a covariant vector in this case) Z

I β εβ d4 x = 0

(121)

we find (using integration by parts and eliminating a surface term)

Z

β

4

I εβ d x =

Z 

∂α E

αβ

+

Γβαλ E αλ



4

εβ d x =

48

Z 

 1 γ − (∂α εβ + ∂β εα ) + Γαβ εγ E αβ d4 x 2 (122)

(as E αβ is symmetric in α and β). Z

E αβ δgαβ d4 x = 0 and using

1 δgαβ = − (∂α εβ + ∂β εα ) + Γγαβ εγ 2

(123)

Comparing (121) with the variation of the action δS =

(122) we can read off the transformation of the metric tensor:

where Γγαβ is just a short-hand notation for (13). The transformation (123) coincides with the diffeomorphism (110) if we redefine εα = 2ξα . This result follows from the action without reference to any properties of the coordinate transformations or the transformations of fields. Note that in (123) the numerical factor 2 appears, which is the same as we obtained in the gauge transformations when applying Castellani’s procedure. This consideration, based on Noether’s second theorem, is an illustration of how diffeomorphism invariance appears in GR in exactly the same sense (in terms of differential identities) as in other field theories.

F.

Summary

To summarize: firstly, the diffeomorphism transformation (1) is related to coordinate transformations and as a consequence to general covariance; secondly, the Hamiltonian formulation of GR is performed using the same “rule of procedure” as in ordinary gauge theories, allowing one to obtain the same transformation (1); thirdly, the same transformation can be derived using purely Lagrangian methods, which are valid for all field theories, without any reference to a coordinate transformations or relying to the Hamiltonian methods. The resulting invariance makes these transformations distinct from ordinary gauge theories because of the presence of the derivatives of the gauge parameters (compare (116) and (114) for transformations of the equations of motion). Such distinct transformations should manifest themselves in all steps of the procedure and this is exactly what we have found. For example, the non-zero PB among primary and secondary constraints in (63) and the ‘on-shell’ of primary constraints result of (84) are properties that are not observed in ordinary gauge theories. These peculiarities are present in Dirac’s formulation as considered in this article, as well as in [45]; and both of these formulations allow one to derive the diffeomorphism invariance (1). At least two of the above mentioned properties, which are related to (63) and (84), are absent in the ADM formulation; and the transformations derived from the ADM Hamiltonian are different from a diffeomorphism. Diffeomorphism, as a gauge 49

transformation according to Noether’s theorems, reflects a very special connection between the equations of motion; and such a relationship specific to Einstein’s GR cannot disappear in the Hamiltonian analysis. A comparison between the Dirac and ADM formulations will be made in the next Section.

IV.

THE DIRAC HAMILTONIAN OF GR VERSUS THE ADM HAMILTONIAN

OF GEOMETRODYNAMICS A.

Prelude

In the previous Section we demonstrated that, following the “rule of procedure” as applied to Dirac’s Hamiltonian of GR (61) (with the covariant metric gµν and corresponding conjugate momentum, pµν , as the fundamental canonical variables) the diffeomorphism invariance (1) can be derived using Castellani’s procedure [25] or the method of [27]. The same result was recently obtained in the Hamiltonian formulation of GR [45] without using Dirac’s modifications (8) and in the Lagrangian approach of [13] by Samanta [41]. This result is exactly what one would expect for the invariance of GR, as well as the equivalence of the results in the Hamiltonian and Lagrangian approaches. In addition, because Dirac’s non-covariant modification of the Lagrangian does not change the equations of motion, four-dimensional symmetry is preserved and it is reflected in the covariant form of the transformations (1). We have also demonstrated that Dirac’s references to space-like surfaces are not part of his actual calculations. As a result, Hawking’s statement [42] that “the split into three spatial dimensions and one time dimension seems to be contrary to the whole spirit of relativity” is not related to Dirac’s formulation, where only manifest invariance (but not the invariance itself) is broken by explicitly considering different components of the metric tensor. We have seen that working with the original Einstein variable, the metric tensor, we have the Hamiltonian formulation of GR that is consistent with diffeomorphism symmetry, and the spirit of GR is “alive”. There exists a more popular Hamiltonian than Dirac’s Hamiltonian which is based on a different set of variables: the lapse and shift functions and the space-space components of the metric tensor. It is frequently, but mistakenly called the Dirac Hamiltonian (e.g. [25])

50

and even its variables are called “Dirac’s lapse and shift” [35]22 . This formulation (with the lapse and shift functions) is due to Arnowitt, Deser and Misner (see [22] and references therein). The name “Dirac-ADM” is also not correct; moreover, despite the apparent similarities between the Dirac and ADM Hamiltonians, they are different (see below). The appropriate and best known names for the ADM formulation are “ADM gravity” and geometrodynamics, as opposed to Einstein gravity. The gauge transformation derived from the ADM Hamiltonian by the methods of [25, 27], using ADM variables and constraints, is not diffeomorphism invariance (1). This fact was recently demonstrated very clearly using the method of [27] in [32] and was discussed in our Introduction. The field-dependent redefinition of the parameters (2) for the ADM Hamiltonian distinct from the exact result that was obtained in Section III from the Dirac formulation, where no such a redefinition was used, and in [41] and [45] where the question of equivalence with diffeomorphism does not arise. We feel that it is insufficient to say that the formulations of [20, 45] and that of Dirac [21], both of which use the metric tensor as the canonical variable, correctly describe the Hamiltonian of GR and one is obliged to work with the original Einstein variables. One should not try to recast GR into a description of the motion of surfaces; more precisely, one should not change variables to make such an interpretation plausible. It is necessary to understand why two such closely related Hamiltonians, those of Dirac and ADM, which are mistakenly said to be equivalent, produce different results. This is a general question related to the Hamiltonian formulation of singular Lagrangians. The understanding of the peculiarities of the Hamiltonian method for constrained systems can prevent from repeating some mistakes that have been made when considering formulations of such theories. The analysis of the ADM formulation is also interesting from another point of view, as it provides a very instructive example of what might be called an “interpretational” approach to physics. In the original work of ADM (e.g., [22]), an interpretation of the variables they introduced was proposed and in later work, this interpretation became the cornerstone of that formulation. Attempts were made to “derive” results starting from that interpretation, i.e. by elevating the interpretation to the level of first principles [47]. Such 22

The names “lapse”and “shift” were introduced neither by ADM nor by Dirac and appeared only later, for example, in [30, 47]. To the best of our knowledge, it was Wheeler who coined the names of these variables, “lapse”and “shift” in [60].

51

an approach exhibits lack of rigour and relies on some geometrical reasoning. The essence of this approach is perfectly expressed in the following quotation from [87]: “I capture as much of the classical theory as I can by pictorial visualization”23 and “The reader is encouraged to follow the broad outlines and not worry about technical details.” The “advantage” of such an approach is that it cannot be disproved; yet it prevents one from obtaining any reliable prediction or result. For example, if we accept Dirac’s references to space-like surfaces as a part of his formalism, then Hawking’s statement that introduction of a family of spacelike surfaces “seems to be contrary to the whole spirit of relativity” [42] forces us to reject this formulation as an inappropriate formulation of GR. And yet, as we have demonstrated, Dirac’s formulation leads to a direct restoration of diffeomorphism invariance and, because of this, it is consistent with the spirit of GR and is the correct Hamiltonian formulation of GR. This demonstration shows how a purely interpretational consideration can lead to a wrong conclusion and that the interpretational approach without having to “worry about technical details” is meaningless. Dirac’s derivation follows the “rule of procedure” and allows us to check any interpretation by explicit calculations. If something is constructed only using pure interpretation, then the final result cannot be checked by calculations and can only be analyzed by comparing its consistency with general principles. An “interpretation” cannot serve as a ground to disprove a result and in fact could be wrong, as in the case of Dirac’s formulation in which he makes references to space-like surfaces that were not used in his calculations. A general understanding of the limitations of the interpretational approach probably provides an explanation of why Hawking’s words [42], spoken almost thirty years ago on the occasion of the centenary of Einstein’s birth, were not enough to cause people to immediately abandon the ADM formulation. Yet more disturbing, is that this “interpretational” language has completely prevailed in the Hamiltonian formulation of GR. As an example, consider the work of Samanta [41] in which he used the Lagrangian formulation, and where there are no surfaces of constant time, space-like, slicing, etc.; then his position is abruptly altered when he refers to the Hamiltonian formulation of the same theory and states that “slicing is essential for Hamiltonian formulation”. This assertion is obviously wrong as slicing is not essential and the 23

This becomes a well-known pedagogical approach in teaching conceptional (not mathematical) physics for non-science students.

52

Hamiltonian formulations of GR obtained without any reference to slicing gives a consistent result, as we have demonstrated in the previous Section and in [45]. The general trend in Physics and the main goal of many physicists is the unification of theories and methods on all possible levels; but even now, when we are a few years away from the 100th jubilee of the discovery of GR [88] there remains an inconsistency when discussing different formulations (Lagrange and Hamilton-Dirac) of the same theory, Einstein’s GR! This leads to erroneous observations, such as “It is worth noting that generalized Hamiltonian dynamics is not completely equivalent to Lagrangian formulation of the original theory. In the Hamiltonian formalism the constraints generate transformations of phase space variables; however, the group of these transformations does not have to be equivalent to the group of gauge transformations of Lagrangian theory” [90]. We consider it important to understand and find explicitly where and why the ADM Hamiltonian formulation contradicts the spirit of GR, and why it cannot be associated with Einstein’s theory (i.e. it is not the Hamiltonian formulation of GR, but rather the Hamiltonian formulation of distinct formulation: “geometrodynamics”). One can talk about abandoning the spirit of GR if one is discussing a different formulation built on different principles (such as [91]) but the ADM formulation has been given the appearance of having a formal basis on GR (according to the ADM papers and the presentation in many textbooks); indeed the summary of ADM’s work in [22] has the title “Dynamics of General Relativity”. The mathematical manifestation of the spirit of Einstein’s GR is the general covariance of Einstein’s equations of motion for the metric tensor. Einstein’s GR is a field theory and the methods used in ordinary field theories, those of Lagrange and Hamilton, if applied correctly, must not destroy its spirit. This is exactly what we want to investigate: where was “a regular and uniform rule of procedure” broken in the ADM approach? In Hamiltonian language, we want to see where the canonical procedure was destroyed by passing from the Dirac Hamiltonian to the ADM Hamiltonian and why the ADM formulation with their variables is not equivalent to GR or, in other words, is not a canonical formulation of GR. It does not seem possible to start from the Lagrangian of GR, where surfaces are not present, and then after introducing new variables have such surfaces. We will follow a path distinct from the interpretational approach and pay attention to technical details by using the “uniform rule of procedure” in analyzing the ADM formulation. In the previous Section we demonstrated that, when we are using the rule of procedure, surfaces do not appear in 53

either Dirac’s calculations or in [45]. We will not repeat the calculations of the previous Section or of [45] for the ADM formulation, but instead we will analyze how the ADM formulation is related to that of Dirac.

B.

Relationship between Dirac and ADM Hamiltonians

Let us compare the two Hamiltonians of Dirac and ADM. Castellani himself considered the GR and Yang-Mills theories as illustrative examples of his algorithm for finding gauge transformations. Referring to Dirac’s book [11], Castellani [25] started with the statement “from the Hilbert action one derives the Hamiltonian”24 ADM H ADM = NH⊥ + N i HiADM + N,0 Π + N,0i Πi ,

(124)

where Π and Πi are momenta conjugate to N and N i , respectively. This Hamiltonian never appears in Dirac’s book or in any of his papers. Dirac’s derivation of the Hamiltonian of GR is in the article [21] that we discussed in previous Sections. It is different from (124) and given by (61) which is the canonical part of the Hamiltonian, and his primary constraints are the momenta p0µ conjugate to g0µ (9). Equation (124) is, in fact, the ADM result [22] and we have indicated so by using the superscript ‘ADM’. In order to compare the results of the Dirac and ADM formulations we use a slightly different convention for the Dirac Hamiltonian which appeared in a subsequent article of Dirac (see Eq. (7) of [92] for the canonical part of Hamiltonian) HD = −g 00


−1/2

HL + gr0 ers Hs + g00,0 p00 + 2g0k,0p0k .

(125)

In this convention, g00 is negative [92]. Let us compare the secondary constraints of the two formulations. HiADM , the “diffeomorphism” constraint, is given in many sources (e.g. see Eq. (3.14b) of [22]) as k HiADM = gik HADM = −2gik Πkj |j 24

Such a Hamiltonian can be “derived” without recourse to the EH Lagrangian. We refer the reader interested in “visualization” or in the geometrical reasoning behind a derivation or geometrical meaning of this equation to the numerous figures in [47]. We disregard such approaches as inadequate for any proofs.

54

where Πkj is a momentum conjugate to the spatial metric gkj . The symbol “|” seems to indicate the covariant derivative with respect to gkj [22]; but the definition of the particular covariant derivative used in ADM is non-standard and is not easily found (see Eq. (5.1) of [93] and Eq. (2.3b) of [94] which are consequently papers six and twelve in their series) kj lm k Πkj |j ≡ Π,j + Π Γlm

which gives kj kj HiADM = −2gik Πkj ,j − 2Π gik,j + Π gjk,i .

This is exactly the Dirac constraint (see Eq. (D41) of [21]) or our (29)). We note that the ADM definition of a covariant derivative of a contravariant second rank tensor mimics a covariant derivative of a contravariant vector (the first rank tensor) or a covariant derivative of the tensor density [53]. If we use the standard definition of a covariant derivative of the second rank tensor [33, 53, 77] we will obtain a different result. This result can only be presented as a standard covariant derivative (but with respect to the spatial metric gik only) if we write it as HiDirac

=

HiADM

= −2gis

(see Eq. (37) of [95] or Eq. (E.2.34) of [31]).

p

det gkm



Πls √ det gkm



;l

ADM The scalar, “Hamiltonian”, constraint H⊥ is given by Eq. (3.14b) of [22] ADM H⊥

−

√

√ =− g



3

1 g R+ √ g 3

R+g



−1



1 2 Π − Πij Πij 2



=

 1 gik gjm − gij gkm Πij Πkm . 2

(126)

Using (50) (employing the convention of [92] where g 00 is negative), from (48) we obtain the second term of (126). Also from (43), and taking into account (44), the first term of (126) follows. Thus, this constraint is also equivalent to Dirac’s HL .

Dirac’s combinations HL and Hs of the “covariant” secondary constraints χ0σ (given in

ADM (50), (60)) are exactly the same as the ADM secondary25 constraints H⊥ and HiADM 25

Actually, in the ADM formulation they appear as primary [22, 97]. We will return to this later.

55

ADM H⊥ = HL , HiADM = Hi .

(127)

The only difference between the first two terms of (124) and (125) is that the field-dependent coefficients in front of Dirac combinations of constraints are called new variables by ADM [22]. These are the lapse and shift functions N ≡ −g 00


−1/2

N i ≡ gj0eji = −

,

(128)

g 0i . g 00

(129)

In addition, Dirac’s eji (15) (the reciprocal of gji ) is called the “three-dimensional” metric g ji [22]. (In fact, g ji is the space-space component of the full four-dimensional metric in fourdimensional space-time.) To distinguish eji from the space-space components of Einstein’s four-dimensional contravariant metric tensor, the latter is defined to be 4 g ji [22]. Dirac’s notation is more transparent as it arises in his derivation of the Hamiltonian that was analyzed in Section II.26 The confused notion (in the literature) that these two formulations are equivalent, is understandable, especially in light of relation (127). Another reason is that in many presentations of the ADM Hamiltonian (124), such as in [44], the primary constraints are ignored by imposing the idea that the lapse and shift variables are merely the Lagrange multipliers for the constraints H⊥ and Hi and that they can be treated as ‘primary’ rather than ‘secondary’ constraints [39]. Firstly, in such an approach even derivation of the gauge transformations of all components of the metric tensor becomes impossible as we are no longer dealing with the full phase space of the Hamiltonian (e.g., see the remark on p. 3288 of [37]). The methods of derivation of Castellani [25] and of [27] (which is used in [32]) employ the complete phase space and so all fields and their conjugate momenta must be included. Secondly, dropping primary constraints contradicts the methods of constraint dynamics: primary constraints are first-class and must not be solved as this does not allow to restore the gauge symmetry present in the Lagrangian. Only second-class constraints can be solved and then some of the phase-space variables can be eliminated provided PBs are replaced 26

The ADM renaming was probably introduced to underline the geometrical interpretation of their variables (see [60]).

56

by Dirac brackets. All this means that if we derive generators of gauge transformations, ADM following any procedure using H⊥ and HiADM as the first-class primary constraints, we

will have generators independent of the momenta conjugate to the “multipliers”, so that the gauge transformation of, for example N, would be zero

δN = {N, G} = 0.

(130)

This result means that δg 00 = 0, which is different from a diffeomorphism transformation at all. Without the primary constraints the time derivatives of the lapse function, for example, would vanish according to the Hamiltonian formulation, {N, H} = 0. Thus, N is constant in time, yet if we use the total Hamiltonian (125), we obtain {N, HT } = N,0 . Finally, in the Hamiltonian approach, the primary constraints come from varying the ADM Lagrangian with respect to velocities, and if we follow this rule, then H⊥ and HiADM are

not primary constraints. We will consequently work with the total Hamiltonian. Let us continue to compare these two formulations. The relation between their primary constraints is as yet unclear and we shall return to this later. Form (128) and (129) the metric and its inverse are (e.g., see [25, 30]) 

gµν = 

i

j

2

gij N N − N gij N gij N

j

gij

j





 , g µν = 

− N12 Ni N2

Ni N2 3 ij

g −

N iN j N2



.

(131)

The fundamental PBs of the canonical variables, the components of the covariant metric tensor and their corresponding conjugate momenta, for Dirac’s Hamiltonian are [21]  αβ 1 α β
p (x) , gµν (x′ ) = δµ δν + δνα δµβ δ3 (x − x′ ) , 2

(132)

whereas for the ADM approach the fundamental PBs are (e.g., see [25, 32, 99])
 1 k l ′ δi δj + δil δjk δ3 (x − x′ ) = ∆kl gij (x) , Πkl (x′ ) = ij δ3 (x − x ) , 2 57

(133)

 i N (x) , Πj (x′ ) = δji δ3 (x − x′ ) ,

(134)

{N (x) , Π (x′ )} = δ3 (x − x′ ) . (135)  Other PBs presumably equal zero (i.e. N (x) , Πkl (x′ ) = 0, etc.) if these variables are to

be canonical.

We now investigate why the Dirac and ADM approaches to the canonical treatment of GR lead to diffeomorphism transformations in the former case and to transformations that correspond to a diffeomorphism only after a non-covariant field-dependent redefinition of gauge parameters in the latter case.

C.

Complete treatment of the ADM formulation using Castellani algorithm

In Section III we have demonstrated that the gauge transformations that follow from Dirac’s Hamiltonian can be derived using both the methods of [25] and [27]. The method [27] was applied to ADM gravity in [32]. The result of [32] is not new, but it is probably the first complete consideration of how one can derive the gauge transformations from the constraint structure of the ADM formulation. The application of Castellani’s method to the ADM Hamiltonian given in Appendix of [25] is opaque and incomplete as the relation between the diffeomorphism and the ADM parameters is not explicitly given and only the transformations of g0µ are found. The calculations themselves were performed in an unnatural way - in order to find the transformations of the metric tensor, the ADM variables were expressed in terms of the metric in the generator. For completeness, we shall use Castellani’s approach to find the gauge generator with the ADM Hamiltonian and compare it with the result of [32]. In addition, we will show some of the peculiarities in this calculation which are related to the somewhat confusing notation used by ADM. According to Castellani’s procedure, the generators of a gauge transformation can be constructed for the Hamiltonian using the so-called algebra of the Dirac constraints (62) (unnumbered equation preceding Eq. (29) of [25])27 27

Starting from here we are considering the ADM formulation and do not use the superscript ADM .

58

{H⊥ , H} = N,r ers Hs + (Ners Hs ),r + (N r H⊥ ),r , {Hi , H} = N,i H⊥ + N,ij Hj + N j Hi These lead to the generator (see Eq. (29) of [25])




,j

(136)

.

(137)

Z nh  i

 ⊥ ij ij j ⊥ G=− ε H⊥ + N,i e Πj + NΠi e ,j + ΠN ,j + ε,0 Π h   io
j j i i + ε Hi + N,i Πj + N Πi ,j + N,i Π + ε,0 Πi d3 x.

(138)

There are some differences between these equations and the corresponding ones of [25] as we use Dirac’s notation, ers , which in ADM is called the three-dimensional metric. Actually, this renaming is sloppy and confusing because (see (131)) 4 g km = g 0k g 0m g 00

3

g km −

N kN m N2

=

3

g km +

which, when solved for 3 g km, is equivalent to Dirac’s ekm (15). We keep ers to avoid

the temptation to raise spatial indices with this tensor. We can do this for the spatial metric (as they are inverses) but we cannot do this for derivatives, as was done in [25] in the second term of (138) where N ,j Πj is correct only if we consider N ,j as short for N,i eij . This is because, according to the standard rules of raising indices in GR, ∂ j = g jµ∂µ = g ji ∂i + g j0∂0 = eji ∂i +

g 0i g 0j ∂i + g j0∂0 , 00 g

(139)

and ∂ j = eji ∂i

(140)

only if g j0 = 0 which is Dirac’s simplifying assumption (32). If we use (140) instead of (139) we would obtain a different result. The generator (138) allows one to find the transformations of the ADM fields (N, N i and gkm ) and then by using the definition of these variables (128), (129), we can formally find the transformations of gµν . We do this in natural order - we first find the transformations of the ADM variables using the generator in terms of the ADM variables and then revert to the metric tensor. Transformations of the ADM fields are calculated using δ (f ield) = {f ield, G}. 59

Starting with the simplest variable, N, we obtain j ⊥ i δADM N = {N, G} = ε⊥ ,j N − ε,0 − ε N,i −1/2

which using N = (−g 00 )

gives

δADM N = and so we find

δADM g

00

= 2 −g


00 +3/2

(141)

δN = 2 −g


−3/2 1 −g 00 δADM g 00 2


00 +3/2

   

⊥ 0j 00 ⊥ i 00 −1/2 ε,j −g /g − ε,0 − ε −g . ,i

(142)

This differs from the diffeomorphism transformation (109): 00 00 δ(dif f ) g 00 = 2ξ 0,0 − ξ 0g,0 − ξ k g,k .

In ADM variables we cannot restore diffeomorphism invariance; the most that can be done is to present (142) in a form similar to a diffeomorphism: δADM g 00 = 2 −g 00


+1/2  0j ⊥  i 00 g ε,j + g 00 ε⊥ ,0 − ε g,i

and using ∂ 0 = g 0µ ∂µ = g 00 ∂0 + g 0k ∂k this becomes δADM g 00 = 2 −g 00


+1/2

ε⊥,0 − εi g,i00 =

  i,0 h h i


g 0k ⊥ 00 00 +1/2 ⊥ 00 +1/2 00 k ⊥ 00 +1/2 g,k . 2 ε −g g,0 − ε + 00 ε −g − ε −g g

The combinations in square brackets “correspond” to the diffeomorphism parameters ξ 0 = ε⊥ −g 00 ξ k = εk + or


+1/2

=

1 ⊥ ε , N


gk ⊥ Nk ⊥ 00 +1/2 k ε −g = ε − ε g 00 N

60

(143)

(144)

ε⊥ = Nξ 0 = −g 00


−1/2

εk = ξ k + N k ξ 0 = ξ k −

ξ 0,

(145)

g 0k 0 ξ . g 00

(146)

Equations (143)-(146) are equivalent to the result of [32] (where the methods of [27] were used) and also to what is presented in [34, 37]. The relations (143), (145) can be found in the Appendix of Castellani’s article [25], but (144), (146) were not given there explicitly. This field-dependent redefinition of gauge parameters provides a “correspondence” [32], but not an equivalence with the diffeomorphism, that follows directly from consideration of the Dirac Hamiltonian. For the next variable, N k , we obtain h i  kj j k k j k δADM N k = N k , G = − ε⊥ N,j ejk − ε⊥ Ne + ε N − ε N + ε ,j ,j ,j ,0

(147)

0k

and using N k = − gg00 we find δADM g 0k =

  g 0k jk j k k j k δADM g 00 + g 00 ε⊥ N,j ejk − ε⊥ ,j Ne + ε N,j − ε,j N + ε,0 . 00 g

After expressing N and N k in terms of the metric

δADM g

"

+g 00 ε⊥

h

−g

i
00 −1/2

,j

ejk − ε⊥ ,j −g

0k

g 0k = 00 δADM g 00 g


00 −1/2

#  0k   0j  g g ekj + εj − 00 − εk,j − 00 + εk,0 , g g ,j

we again have an invariance that is not a diffeomorphism (109) 0k 0k δ(dif f ) g 0k = ξ 0,k + ξ k,0 − ξ 0 g,0 − ξ m g,m .

If we perform the field-dependent change of parameters (145), (146) we again can present δADM g 0k in the form of a diffeomorphism transformation. The transformation of the space-space components gkm was not considered in [25] because, according to the author, it is well-known that (138) generates a diffeomorphism transformation of gkm. Let us check this statement; 61

δgkm = {gkm, G} = −

 δ  ⊥ ε H⊥ + εi Hi km δΠ

(148)

which, keeping only the Πpq -dependent part of secondary constraints, gives

 

1 1 pq 00 −1/2 pq i i −g gip gjq − gij gpq 2∆ij Π − 2 ε g ∆pq −ε √ ip km km + ε (2gpi,q − gpq,i ) ∆km . ,q −g 2 (149) ⊥

We must express Πpq in terms of gij and its derivatives; using Eq. (7-3.9b) of [22] Πij =

p −4 g

4 0 Γpq


− gpq 4 Γ0rs g rs g ip g jq

which is (taking into account that the “three-dimensional quantity” g ip in ADM is Dirac’s eip ) Πij =

√


√ −g Γ0pq eip ejq − Γ0rs ers eij = − −gE ijab Γ0ab .

(150)

This expression is equivalent to Dirac’s pij (18). Substituting (150) into (149) and using (14), (19) and (20), we obtain δADM gkm = −ε⊥ −g 00


−1/2

2Γ0km − 2 εi gip




,q

pq i ∆pq km + ε (2gpi,q − gpq,i ) ∆km ,

or in the explicit form, using (18), for the ADM formulation δADM gkm = ε⊥

 1  Nk,m + Nm,k − gkm,0 − N i (gki,m + gmi,k − gkm,i ) N − εi,m gik − εi,k gim − εi gkm,i

(151)

and for Dirac’s variables

δADM gkm = −ε⊥ −g 00


−1/2 

g 00 (gk0,m + gm0,k − gkm,0) + g 0i (gki,m + gmi,k − gkm,i )

− εi,m gik − εi,k gim − εi gkm,i .

 (152)

This is again different from the transformation of the spatial components of the metric tensor under a diffeomorphism (110) 62

0 0 i i δ(dif f ) gkm = −gkm,0 ξ 0 − gk0 ξ,m − gm0 ξ,k − gkm,i ξ i − gki ξ,m − gmi ξ,k .

(153)

Again, only after the field-dependent redefinition of parameters (145), (146) we can obtain a “correspondence” between δADM gkm and δ(dif f ) gkm . (Note that both parameters have to be redefined despite the apparent equivalence of the last three terms in both equations (152) and (153).) So, as in the case of Dirac’s Hamiltonian, both methods [25, 27] produce the same result (2) for the ADM Hamiltonian. We would like to note that even a spatial diffeomorphism does not follow directly from the ADM formulation despite what is often stated in the literature (e.g. [39, 44]). If the lapse and shift functions are treated as ‘multipliers’28 and the secondary constraints are considered as being primary (the contradictions that result from such manipulations have already been discussed), this would not correct this problem because a spatial diffeomorphism does not follow. In this case, according to Castellani’s procedure, the generator is simply G = ε ⊥ H⊥ + ε i Hi

(154)

which is equivalent to (148). Even in such a “formulation” the gauge parameters have to be redefined and, in addition, the transformations of lapse and shift functions equal zero (see, e.g. (130)). There is only one way to “derive” spatial diffeomorphism invariance and it explains the origin of the term “diffeomorphism constraint”. It behooves us to warn the reader that such a “derivation” has nothing to do with any procedure. If, in addition to eliminating primary constraints and promoting secondary to being primary (which leads to the generator (154)), we also consider only the second term of this generator, then

28

 δgkm = gkm, εi Hi

(155)

The authors of [30] in the “Historical remark” on p. 486 stated that ”The great payoff of this work [ADM] was recognition of the lapse and shift functions of equation (21.40) [the same as in (21.42) or our (131)] as Lagrange multipliers, the coefficients of which gave directly and simply Dirac’s constraints.” As we have shown, “the great payoff” of this recognition is that the ADM formulation lost the connection with GR and the gauge transformations derived from it are different from diffeomorphism. We would also like to mention that in the same “Historical remark” the authors wrote: “Dirac paid no particular attention to any variational principle”. The interested reader is encouraged to look at Dirac’s papers, especially at [21], to recognize that this is not correct.

63

will give the spatial diffeomorphism (see the second line of (152)). The only possible explanation of why such manipulations were accepted is that it seems to follow the “guidance” which comes from linearized gravity. From the derivation of the gauge transformations for linearized gravity in [52], it is clear that only one part of the generator which is proportional to χ0n contributes to the transformation of the space-space components of the metric tensor, but this is not the case for full GR. Ironically, the result (155), which is just the consequence of a series of manipulations that contradict any consistent procedure, is often presented as being the “problem” of the Hamiltonian formulation of GR: “Hamiltonian” and “diffeomorphism” constraints are treated in a different manner [44, 98]. Furthermore, the conclusion is drawn, based on (155), that “the diffeomorphism constraint can be shown to be associated with the invariance of general relativity under spatial diffeomorphism” [44] (see also [99]). Finally, because (155) leads to a spatial diffeomorphism, this result is interpreted as “disappearance of DiffM” and considered as “the problem that has worried many people working in geometrodynamics for so long” [100]. Statements similar to some of the quotations above can be found in many articles. They are based on questionable manipulations; but, at the same time, they clearly demonstrate that some authors correctly consider that the restoration of diffeomorphism invariance for all components of the metric tensor is to be expected (however, “this expectation has never been fully realized...” [100])29 and its absence is taken to be a deficiency or a contradiction arising in the Hamiltonian formulation. To have the possibility of restoring transformations (whatever it may be) of all variables one must work in the full phase space and use all of the first-class constraints. We disregard “approaches” leading to (154) and (155) and return to the analysis of the total Hamiltonians (124) and (125).

D.

ADM change of variables and canonicity

We now ask why, in the ADM case, we must redefine the gauge parameters (143)-(146) to have a “correspondence” with diffeomorphism, whereas from the Dirac Hamiltonian (and also from the formulation without Dirac’s modifications [45]) diffeomorphism arises directly. 29

This expectation is fully realized in [45] and in the previous Section.

64

Dirac’s Hamiltonian was obtained from the Lagrangian of GR, after some integrations that do not affect the equations of motion; and the ADM Hamiltonian apparently follows from the same Lagrangian. The two Hamiltonian formulations of the same Lagrangian ought to be equivalent and should give the same gauge transformation, which does not happen in the case of the Dirac and ADM approaches. It is well-known that different sets of phase-space variables can be used to describe Hamiltonian systems. In the ordinary Classical Mechanics of non-singular systems (e.g., [101, 102]), for a given Hamiltonian H (qi , pi ) and the Hamilton equations, qi = {qi , H} =

∂H ∂H , pi = {pi , H} = − , ∂pi ∂qi

(156)

we can pass to another set of phase-space variables (Qi , Pi ):

qi = qi (Qk , Pk ) , pi = pi (Qk , Pk ) , K (Qk , Pk ) = H (qi (Qk , Pk ) , pi (Qk , Pk )) ,

(157)

such that Qi = {Qi , K} =

∂K ∂K , Pi = {Pi , K} = − . ∂Pi ∂Qi

(158)

If this system of Hamilton equations can be solved, then one can return to the original variables by using the inverse of (157)

Qi = Qi (qk , pk ) , Pi = Pi (qk , pk ) .

(159)

In Classical Mechanics, even for non-singular systems, it is well-known, that [101] “...at first sight we might think that arbitrary point transformations of the phase space are now at our disposal. This would mean that the 2n coordinates qi and pi can be transformed into some new Qi and Pi by any functional relations we please. This, however, is not the case.” For the non-singular systems, the necessary and sufficient condition that the transformation (157) to the new set of variables (Qi , Pi ) is a canonical transformation (and keeps the two formulation equivalent), is

{Qi , Qk }Q,P = {Qi (p, q) , Qk (p, q)}p,q = 0,

65

{Pi , Pk }Q,P = {Pi (p, q) , Pk (p, q)}p,q = 0,

(160)

{Qi , Pk }Q,P = {Qi (p, q) , Pk (p, q)}p,q = δik . The change of phase-space variables (canonical transformations) for unconstrained Hamiltonians is an old and well established topic that can be found in many textbooks on Classical Mechanics. For constrained Hamiltonians the situation is different and even the number of papers (e.g. see [103, 104]) that discuss the general questions or particular examples (which are mainly final dimensional and artificial) of such changes is minuscule compared to the number of articles in which such changes are used without any analysis of their consequences. Such changes are especially common in Hamiltonian formulations of GR (in both from Einstein to ADM [22] and Einstein-Cartan to ADM-inspired formulations [105, 106], with a rare exception like [107]). For the Hamiltonian and Lagrangian treatments of the Einstein-Cartan action without any change of variables see [86, 108, 109]. The Dirac’s generalization of the Hamiltonian formulation to constrained systems leads to the system of equations which is similar to (156):

qi = {qi , HT } , pi = {pi , HT } , where HT also includes the primary constraints. We restrict our discussion to gauge invariant systems with only first-class constraints. For such systems, the conditions that should be imposed on possible changes of variables in phase space seems to be more restrictive. The reason for this is that gauge invariance depends on all of the first-class constraints and their PB algebra [25–27]. In addition, it is related to the total Hamiltonian of a particular model, in contrast to unconstrained systems where the conditions (160) are, in fact, independent of the Hamiltonian. For gauge invariant systems the change of variables (159) must preserve gauge invariance, i.e. gauge invariance derived in the (Qi , Pi ) variables, after using the inverse transformations, must produce the same result as with the original (qi , pi ) variables. The set of phase-space transformations (157) that preserves this property is the equivalent set or, as in non-singular case, we can call such changes canonical transformations. In [29] the special role of primary constraints is emphasized and discussed in detail in both derivation of a gauge generator and in canonical transformations as, for example, in first-order formulation 66

of GR the primary constraints are momenta which are phase-space variables and unjustified manipulations with them are not allowed. In linearized versions of the Dirac formulation (see the previous Sections) and that of [20] which were considered in [52], both formulations, despite having different constraints and Hamiltonians, lead to the same gauge invariance. The relation between the two formulations was discussed and it was shown that they are related by canonical transformations (160) [65], which is exactly the condition known to be needed for non-singular Hamiltonians [101, 102]. Despite there being far more complicated expressions for constraints and transformations, a similar relation (160) exists between the non-linearized Hamiltonian formulation of Dirac considered in the present paper and [45]; and these, as we have shown, also have the same gauge invariance. These examples demonstrate that the ordinary condition for the transformation to be canonical, which is known for unconstrained Hamiltonians is also valid for the general (constrained) case; i.e. it is a necessary condition as before. Yet, it is not sufficient and we shall demonstrate this fact by way of example. Surprisingly, we have been unable to find in the literature, either a discussion of the relations between the ADM and Dirac phase-space variables or the transformations between them, which is strange as many authors presume their equivalence by calling this formulation “Dirac-ADM”. In [110] the authors called the variables N, N i and gkm “an equivalent set” which “is analytically convenient and geometrically more significant”. In footnote 5 of [110] (appeared in 1959) it is stated that “The properties of these variables are discussed in detail in a forthcoming paper by C. Misner” that we have not been able to find. The convenience and significance are not our present concern, but the question of equivalence of the ADM and Dirac sets of variables is important. We are interested in the equivalence of gauge transformations in the two approaches, i.e. we must work in the full phase space. The complete relation between these two sets of variables is not known, but there is one PB,  N (x) , Πkl (x′ ) , that can be easily checked. The space-space components of the metric
tensor and corresponding momenta are the same in both formulations, i.e. Πkl = Πkl pkl = pkl . The Πkl of ADM given by (150) is equivalent to pkl of Dirac given by (18). It is sufficient  to check using the Dirac variables gµν , pµν the PB N (x) , Πkl (x′ ) which, if ADM variables

are canonical, must give zero [101, 102]. Using the corresponding fundamental PBs (133)(135) we obtain:

67

 n
−1/2 kl o N (x) , Πkl (x′ ) = −g 00 ,p

g,p

=−


−1/2 1
−3/2 0k 0l δ −g 00 −g 00 = g g 6= 0. δgkl 2 (161)

Once again, we have the result that depends on Dirac’s simplifying assumption (32): if we impose g 0k = 0, then the PB of (161) gives zero. In general this PB is not zero
and the transformation from (gµν , pµν ) to (N, Π), (N i , Πi ), and gkm , Πkm therefore is not canonical. One PB is enough to show that the transformation from gµν to ADM variables

is not canonical, irrespective of the results we might obtain for the PBs among the other phase-space variables. We note that equation (161) gives zero in the static coordinate system, but in this case the corresponding components, g0k , and their conjugate momenta have to be dropped out of the formalism from the beginning, as in the case of the Hamiltonian formulation in the Schwarzschild metric [54] where only four components of the metric tensor are left in the Lagrangian before passing to the Hamiltonian. This simple calculation, (161), allows us to conclude that the two Hamiltonians of Dirac and ADM are not related by a canonical transformation and the respective failure of the ADM variables and the Hamiltonian to produce a diffeomorphism transformation is a manifestation of this non-equivalence. Moreover, (161) shows that the ADM variables are not the canonical variables of GR. The converse statement is also true and the metric tensor is not a canonical variable of the ADM formulation. The ADM Hamiltonian formulation might be considered as a model (geometrodynamics or ADM gravity) without any reference to Einstein GR, but in this case a “correspondence” between the two transformations (143)-(146) is, in fact, meaningless. The transformations that follow from ADM are given by (141), (147), and (151) and in the absence of canonicity we cannot return to the transformations of the metric tensor. There is another characteristic that supports the loss of connection with the original variables: it is impossible to find any redefinition of Π and Πk in terms of Dirac’s phase space variables (whether they satisfy (160) or not) that can transform his total Hamiltonian (125) into the ADM total Hamiltonian (124). In general, no algorithm exists for finding a canonical transformation but the canonicity of a given transformation can be checked. There are no canonical transformations for Hamiltonians that involve only a change of generalized coordinates, as this change must be

68

accompanied by transformations of the momenta that can be found by using the following procedure [101]. If the transformations of the generalized coordinates (fields) are given, one can find the corresponding transformations of the momenta that will guarantee that the new coordinates and momenta are canonical and satisfy (160) using the relation

pi δqi = Pi δQi .

(162)

Let us see what we can obtain from this relation for the ADM change of variables pαβ δgαβ = ΠδN + Πk δN k + Πkm δgkm. By performing the variations δQ =

δQ δgαβ δgαβ

pαβ = Π

using (128)-(129), we find that

δN δN k δgkm + Πk + Πkm δgαβ δgαβ δgαβ

which gives p00 = −Π p0m = Π and p

pq


1/2 1 −g 00 , 2


−1/2 0m 1 1 −g 00 g + Πk ekm , 2 2


−3/2 0p 0q 1 1 g g + Πk = −Π −g 00 2 2

Now solving for the Π’s:

Π = −2 −g 00



 g 0p kq g 0q kp + Πpq . e + 00 e 00 g g


−1/2

p00 ,

Πn = 2gmn p0m + 2g0n p00 ,

Πpq = ppq +

g 0q g 0p 00 g 0p 0q g 0q 0p p − 00 p − 00 p . g 00 g 00 g g

(163)

(164)

(165)

(166)

(167)

(168)

Note that, to have Πpq = ppq , as in the case of the Dirac and ADM Hamiltonians, we again must impose the condition g 0k = 0. 69

Only if equations (166)-(168) are taken together with the relations for the generalized coordinates (128)-(129), are the transformations canonical. It is not difficult to check that the canonical properties of PBs are preserved, and as an example, the PB that we considered in (161) gives

pq

′

{N (x) , Π (x )} =



−g


00 −1/2

g 0q g 0p g 0p g 0q , p + 00 00 p00 − 00 p0q − 00 p0p g g g g pq



= 0, g,p

as it should for variables that are connected by canonical transformations. For non-singular Lagrangians and their corresponding Hamiltonians, the ADM change of variables accompanied by the change of momenta (163)-(165) would be sufficient to obtain the new set of canonical variables. If GR were a non-singular theory, these transformations would guarantee equivalence between the two formulations; but for the constrained Hamiltonian this is not the case. If the canonical transformation of (131), (163)-(165) are performed in the Dirac Hamiltonian, we will not obtain the ADM Hamiltonian, and we will not obtain a consistent result. In particular, for a description of a constrained system, the total Hamiltonian, HT , is important as it includes all the primary constraints. In the Dirac formulation, these are (22): g00,0 p00 + 2g0k,0p0k .

(169)

Substitution of (131), (163)-(165) into this equation gives  1 N kN j j mk N,0 Π + + gkj,0 Π + N Πm e 2 N which is nonsensical. The first two terms are equivalent to (169) but an extra term appears, N,0m Πm



which is zero only if N k = 0. In the course of the Hamiltonian procedure the space-space velocities have already been eliminated in favour of their corresponding momenta but now they reappear and it is not clear what to do with them at this stage. If we treat them on the same footing as the rest of the terms with time derivatives, we must specify their coefficients as primary constraints, which would then give a total of ten primary constraints. The same change of variables in the canonical part of the Hamiltonian will produce contributions that are quadratic in all momenta. Without further analysis we see that this will probably lead to some contradictions, as it is clear that the constraint structure of the Hamiltonian is changed and one would expect second-class constraints, etc. 70

The foregoing example, (128)-(129) and (166)-(168), clearly demonstrates that the condition (160), which is necessary and sufficient for the transformation to be canonical in the case of non-singular Lagrangians, is not sufficient for singular Lagrangians. Actually, for the ADM change of variables, the non-canonical nature of the transformations is immediately clear even from (161) and this is not related to a singular structure of the Lagrangian of GR. The possible existence of additional restrictions beyond (160) is understandable, and for the Hamiltonian with first-class constraints it can be expected. Gauge transformations are derived from the first-class constraints and the whole PB algebra of constraints plays a key role in this derivation. It is not enough to have the same number of constraints in the two formulations. The ADM change of variables keeps the same number of constraints as the Dirac formulation; but the PB algebra of constraints is affected. The simplest example is a PB among primary and secondary constraints, which are zero in the ADM case and proportional to the constraint χ00 in the case of the Dirac Hamiltonian (63). From a mathematical point of view the ADM formulation is just the result of a non-canonical change of variables in Dirac’s Hamiltonian of GR and if Dirac’s formulation allows one to derive the diffeomorphism transformations, then the ADM formulation, because of this noncanonical change of variables, does not allow one to restore either the full diffeomorphism invariance for all components of gµν or even for its spatial part (as it is usually claimed) without a non-covariant and field-dependent redefinition of gauge parameters. In addition, we observed that some equations in the ADM formulation are true only if g0k = 0. So the ADM change of variables is somehow also related to a static coordinate system, but in a mysterious way: g0k is not zero at the outset, but later should be set equal to zero so that some subsequent relations are made valid. We conclude that the Dirac Hamiltonian of GR was obtained by following the “rule of procedure” and, because of this, it is automatically canonical and the use of the adjective “canonical” is a tautology. It is not equivalent to the ADM Hamiltonian, which, as we have demonstrated, is the result of a non-canonical change of variables. The ADM formulation is obtained by abandoning the “rule of procedure” and, consequently, only by a canonization can it be ironically called the “canonical formulation of GR”.

71

E.

Dirac and ADM Lagrangians and equations of motion

One can argue that the ADM Hamiltonian is not obtained from the Dirac Hamiltonian. There are no references in [22] to Dirac’s article [21], the only reference where a derivation of Dirac’s Hamiltonian has been considered. In their culminating paper [22] and in 13 preceding articles it is mentioned only once in [97] and in a different context. However, according to both Dirac and ADM, their respective Hamiltonians are derived from the same theory - GR. In both cases some modifications of the EH action were performed. In Dirac’s case these modifications are explicitly stated, in the ADM case it is more difficult to trace what has been done. There is a statement in [22] that ADM formulation is started from “the Palatini Lagrangian” (see footnote 19 about “Palatini” formulation). But in the first-order, affinemetric, formulation of GR tertiary constraints appear and diffeomorphism invariance follows from all the first-class constraints [29, 63, 64]; such constraints, as well as diffeomorphism invariance, are absent in the ADM Hamiltonian formulation. Let us try to obtain the ADM Lagrangian from the Dirac Lagrangian. We have discussed above how, in the course of a Hamiltonian analysis that follows Dirac, the possibility of additional integrations by parts appears when the Hamiltonian is to be expressed as a linear combination of secondary constraints (see our discussion after (55)). The modified Lagrangian can be written in the following form (up to total derivatives as in (54))

LDirac


1/2 rsab 1p = det gkm −g 00 E 4 ×

+

p

"

g 0q −grq 00 g

det gkm −g

# "    g 0p g 0k g 0p + −gbp 00 − gab,0 + 00 (gak,b + gbk,a − gab,k ) −gap 00 g g g ,b ,a

#   g 0q g 0m + −gsq 00 − grs,0 + 00 (grm,s + gsm,r − grs,m) g g ,s ,r




00 −1/2



gmn,kt E

mnkt


1 mnpq kt ktnp mq pqnt mk . e − 2E e − 4E e + gmn,k gpq,t E 4 (170)

Such a Lagrangian can be also constructed if, in addition to Dirac’s modification (8), the integrations of (54) were performed at the Lagrangian level [13]. This Lagrangian can be easily obtained from Dirac’s Hamiltonian by performing the inverse Legendre transformation and eliminating momenta in terms of ‘velocities’. 72

By performing the change of variables of (131) in (170), we obtain the ADM Lagrangian

LADM =

h i 1p 1 det gkm E rsab (gap N p ),b + (gbp N p ),a − gab,0 − N k (gak,b + gbk,a − gab,k ) 4 N h i × (grq N q ),s + (gsq N q ),r − grs,0 − N m (grm,s + gsm,r − grs,m)

+

p



det gkm N gmn,ktE

mnkt


1 mnpq kt ktnp mq pqnt mk . e − 2E e − 4E e + gmn,k gpq,t E 4

(171)

Or, by using the intrinsic and extrinsic curvatures, 3 Rrs and Krs , respectively, it can be written in more familiar form [31, 99]

LADM = with

p
det gkm N E rsab Krs Kab +3 R

Krs =

(172)


1 Nr|s + Ns|r − grs,0 2N

where, as before, “|” means covariant derivative with respect to three dimensional metric (Nr|s = Nr,s − Γkrs Nk ). Now using (171) or (172) we can easily obtain the ADM Hamiltonian, which is the same as (124). However, following such a detour we cannot avoid the question of whether the ADM variables are canonical. The transformations of the metric tensor derived from the Hamiltonian formulation of the Dirac Lagrangian (170) and from the Hamiltonian formulation of the ADM Lagrangian (171) are different, so the destination is changed. This detour is a wrong turn or perhaps a dead end road. We combine the results of these two formulations into a compact visual form30 : 30

This ‘pictorial visualization’ is based on results of calculations, not the other way around.

73

LDirac (q)

q=q(Q)

=⇒

LADM (Q)

⇓ HDirac (p, q)

⇓ 6=

HADM (P, Q)

⇓

(173)

⇓ δADM Q ⇓ Q (q)

δdif f q

6=

δADM q q=q(Q)

It is reasonable to expect that if by a change of variables =⇒ we can obtain a new equivalent Lagrangian and find the corresponding Hamiltonian, then the two Hamiltonians (in both the new and the original variables) should also be equivalent, be related to each other by a canonical transformation, and necessarily lead to the same gauge invariance. If the Hamiltonians are not related by a canonical transformation, then the two Lagrangians are not equivalent. This is the natural conclusion that one can make. In particular, application of the Lagrangian methods [13] used by Samanta [41] to derive the diffeomorphism invariance of GR, when applied to the ADM Lagrangian, does not give the diffeomorphism transformations [111] but rather the same transformations as obtained in its Hamiltonian treatment in [32]. This is a consequence of the well-known equivalence of the Lagrangian and Hamiltonian formulations for any system, either non-singular or singular [13, 112]: the “vertical” equivalence of (173). If a “horizontal” equivalence is broken, either for the two Hamiltonians or for the gauge transformations, then it is broken everywhere, including at the Lagrangian level. The Hamiltonian formulations of the linearized versions of the Dirac and gamma-gamma (PSS) Lagrangians lead to the same algebra of constraints and gauge transformations (although the Hamiltonians themselves and the constraints are different). But the two Hamiltonians are related by a canonical transformation [52]. Such transformations also exist in the case of the corresponding full Dirac and the gamma-gamma formulations of GR [65]. The equivalence of the PB algebra of constraints can be easily seen by comparing [45] to the results of Section II. The explicit canonical transformation and its effect on constraints, their algebra and structure functions is given in [65]. The main subject of the present paper is the Hamiltonian formulation of GR in second74

order form and, in particular, a comparison of the formulations related by a change of variables. However, we think that some comments on similar changes made to its Lagrangian should be given. It is often stated that in a Lagrangian formulation of a model any field redefinition is legitimate provided it is invertible (i.e. it has a non-zero Jacobian). For singular Lagrangians, and especially gauge invariant ones, this is obviously not a sufficient condition. One additional restriction in gauge invariant cases is the preservation of the rank of Hessian as this gives us the number of gauge parameters (if all constraints are first-class). We cannot have two equivalent formulations if they have a different number of gauge parameters, and we cannot, by a change of variables, eliminate some gauge invariance or create a new gauge invariance. If we were to make such a change, we can of course, treat the new Lagrangian as some different model, but we cannot relate it to the original one as any connection with the original theory is lost. Obviously these two conditions, non-zero Jacobian and the preservation of the rank of Hessian, are necessary, but not sufficient. The ADM change of variables satisfies them both, but leads to different gauge transformations (compare [45] and Section III versus [32] and (143)-(146)). A change of variables in singular (in particular, gauge invariant) Lagrangians is a much more restrictive procedure if one intends to preserve its equivalence with the initial formulation. One way is to rely on the Hamiltonian method and check if the new variables are canonical and the two total Hamiltonians are equivalent, including the primary constraints. We must also check whether the entire algebra of constraints is equivalent, as this algebra is responsible for the gauge transformations. This can be considered as a confirmation of Dirac’s statement [11] “I feel that there will always be something missing from them [non-Hamiltonian methods] which we can only get by working from a Hamiltonian, or maybe from some generalization of the concept of a Hamiltonian”. However, we think that some criteria for the equivalence between two sets of variables for singular Lagrangians can be formulated at the pure Lagrangian level. At the Lagrangian level, a gauge invariance is related to the existence of gauge identities [13, 84, 85] and an inappropriate change of fields can modify or even destroy them. This echoes the conclusion of Isham and Kuchar [100] that “... space-time diffeomorphism has somehow got lost in making the transition from the Hilbert action to the Dirac-ADM action”31 . The Dirac Lagrangian 31

We have demonstrated the inequivalence of the Dirac and ADM formulations. In [100] the authors

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gives the same equations of motion as the EH Lagrangian and consequently the same differential identity (120) and transformation of the metric tensor (123) will be obtained. It is interesting to investigate what differential identities will follow from the ADM Lagrangian, but for non-covariant variables such calculation are extremely cumbersome. The alternative path is to find what differential identities are needed to describe diffeomorphism invariance of the ADM variables and after that to check whether these identities are satisfied if the Euler derivatives of the ADM Lagrangian are substituted (this work is in progress and will be reported elsewhere). Regge and Teitelboim in [113] stated: “...the usual Hamiltonian H0 [the ADM Hamiltonian] not only does not give the correct equations of motion (Einstein’s equations) but, worse than that, it (H0 ) gives no well-defined set of equations of motion at all.” One confirmation of their words comes from Numerical Relativity.

F.

ADM formulation and Numerical Relativity

There is another indication of the incorrectness of the ADM change of variables at the Lagrangian level that comes from Numerical Relativity. In almost all methods of numerical integration of the Einstein equations the starting point is the ADM 3+1 decomposition. So at the outset the Einstein equations are replaced by the ADM equations [22]. It was shown that, in contrast to the Einstein equations which are strongly (strictly) hyperbolic (SH) [114], the ADM equations are weakly hyperbolic (WH) (e.g., see [115–117]). This change in the type of equations is related to the different constraint structure and different transformations derived from a non-equivalent Hamiltonian. There is a fundamental difference between SH and WH systems of PDEs: the former are well-posed and convergent, whereas the latter are not well-posed and are divergent [118]. There is “evidence” that ADM-based algorithms are numerically unstable. As is indicated in [119]: “The common lore these days is, however, that the standard Arnowitt-Deser-Misner (ADM) formulation is the one which most easily suffers instabilities”. Or in [118]: “It took several years to realize that such instabilities were not associated with the numerical algorithms but rather with the mathematical structure of the ADMY [ADM [22] and York [120]] system itself”. This is not just an additional indication of the incorrectness of the ADM change of variables, but also a demonstration discussed geometrodynamics (not the Dirac formulation) which uses ADM variables and their statements should be applied to the ADM action only.

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that even as a model, geometrodynamics is an ill defined formulation. In general, the change in the type of equations, from SH to WH, or a change of “level of hyperbolicity” [121], is an indication that in the process of transforming from the Einstein to the ADM equations, some “damage” was done (as there is no longer a complete set of eigenvectors associated with the characteristic matrix [122]). The proposed “cure” [119] of such a “damage” in most approaches lies in a modification (or “adjusting”) of the ADM equations by adding terms involving constraints [123] (trying to restore what has been lost) or by using different choices of the lapse and shift functions [119] to give the ADM system of equations well-posedness (or “quasi well-posedness” [119]). From our point of view, the best “cure” of this “illness” is to return to the original Einstein equations and constraints (or differential identities) that preserve diffeomorphism invariance.

G.

Summary

The main point of this relatively long Section is not to prove that the ADM variables are not canonical variables for GR (as is shown by the one simple PB (161)), but to demonstrate and discuss the restrictive conditions that must be made on change of variables in any Hamiltonian formulation of a singular Lagrangian, using GR as an example. A blind change of variables in singular systems without performing a thorough analysis and without developing mathematical criteria for such changes can lead to a wrong result. All new variables that are introduced in such cases, regardless of their physical or geometrical meaning, regardless of what new names were given to reflect their interpretation, or after whom new variables were named, must be carefully analyzed if one wants to keep all the properties of the original theory intact or, in other words, if one intends to study the original theory and not a substitute, which can be ill defined, even as unrelated to the original theory model. One should under no circumstances attribute to the original theory the contradictions or problems that arise after such inappropriate changes are made; and never project any novel result or discovery obtained by abandoning a “regular and uniform rule of procedure” into the original theory or to Nature Herself.

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V.

ACKNOWLEDGEMENT

The authors are grateful to A.M. Frolov, T. Hyland, M. Kidd, S.R. Valluri and, especially, to D.G.C. McKeon and A.V. Zvelindovsky for numerous discussions and suggestions during the preparation of our paper. We are thankful to P.G. Komorowski for pointing out, guiding in and providing references on results of numerical relativity. The partial support of the Huron University College Faculty of Arts and Social Science Research Grant Fund is greatly acknowledged. We would also like to express our gratefulness to our Teacher, the late Prof. A.V. Zatovsky, who long ago taught us the first course in Theoretical Physics, Analytical Mechanics, in the spirit inherited from Lagrange.

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