1 Cosmological Mass-Defect A New Effect of General Relativity Dmitri Rabounski Abstract: This study targets the change of mass of a mass-bearing parti...

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The Abraham Zelmanov Journal — Vol. 4, 2011

§1. Problem statement. In 2008, I presented my theory of the cosmological Hubble redshift [1]. According to the theory, the Hubble redshift was explained as the energy loss of photons with distance due to the work done against the field of global non-holonomity (rotation) of the isotropic space, which is the home of photons∗. I arrived at this conclusion after solving the scalar geodesic equation (equation of energy) of a photon travelling in a static (non-deforming) universe. The calculation matched the observed Hubble law, including its non-linearity. My idea now is that, in analogy to photons, we could as well consider mass-bearing particles. Let’s compare the isotropic and non-isotropic geodesic equations, which are the equations of motion of particles. According to the chronometrically invariant formalism, which was introduced in 1944 by Abraham Zelmanov [3–5], any four-dimensional quantity is observed as its projections onto the time line and three-dimensional spatial section of the observer†. The projected (chronometrically invariant) equations for non-isotropic geodesics have the form [3–5] m dm m − 2 Fi vi + 2 Dik vi vk = 0 , dτ c c k d(mvi ) i n k − mF i + 2 m Dki + A·i k· v + m△nk v v = 0 , dτ

(1.1) (1.2)

while the projected equations for isotropic geodesics are

dω ω ω − 2 Fi ci + 2 Dik ci ck = 0 , dτ c c k d(ωci ) i n k − ωF i + 2 ω Dki + A·i k· c + ω△nk c c = 0 . dτ

(1.3) (1.4)

Thus, according to the chronometrically invariant equations of motion, the factors affecting the particles are: the gravitational inertial force Fi , the angular velocity Aik of the rotation of space due to its non-holonomity, the deformation Dik of space, and the non-uniformity of space (expressed by the Christoffel symbols ∆ijk ). ∗ The four-dimensional pseudo-Riemannian space (space-time) consists of two segregate regions: the non-isotropic space, which is the home of mass-bearing particles, and the isotropic space inhabited by massless light-like particles (photons). The isotropic space rotates with the velocity of light under the conditions of both Special Relativity and General Relativity, due to the sign-alternating property of the space-time metric. See [2] for details. † Chronometric invariance means that the projected (chronometrically invariant) quantities and equations are invariant along the spatial section of the observer.

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As is seen, the non-isotropic geodesic equations have the same form as the isotropic ones. Only the sublight velocity vi and the relativistic mass m are used instead of the light velocity ci and the frequency ω of a photon. Therefore, the factors of gravitation, non-holonomity, and deformation, presented in the scalar geodesic equation, should change the mass of a moving mass-bearing particle with distance just as they change the frequency of a photon. Relativistic mass change due to the field of gravitation of a massive body (the space of Schwarzschild’s mass-point metric) is a textbook effect of General Relativity, well verified by experiments. It is regularly deduced from the conservation of energy of a mass-bearing particle in the stationary field of gravitation [6, §88]. However, this method of deduction can only be used in stationary fields [6, §88], wherein gravitation is the sole factor affecting the particle. In contrast, the new method of deduction of the relativistic mass change with distance I propose herein — through integrating the scalar geodesic equation, based on the chronometically invariant formalism, — is universal. This is because the scalar geodesic equation contains all three factors changing the mass of a moving mass-bearing particle with distance (these are gravitation, non-holonomity, and deformation), and these factors are presented in their general form, without any limitations. Therefore the suggested method of deduction can equally be applied to calculating the relativistic mass change with distance travelled by the particle in any particular space metric known due to the General Theory of Relativity. In the next paragraphs of this paper, we will apply the suggested method of deduction to the main (principal) cosmological metrics. As a result, we will see how a mass-bearing particle changes its mass with the distance travelled in most of these spaces, including “cosmologically large” distances where the relativistic mass change thus becomes cosmological mass-defect. §2. The chronometrically invariant formalism in brief. Before we solve the geodesic equations in chronometrically invariant form, we need to have a necessary amount of definitions of those quantities specifying the equations. According to the chronometrically invariant formalism [3–5], these are: the chr.inv.-vector of the gravitational inertial force Fi , the chr.inv.-tensor of the angular velocity of the rotation of space Aik due to its non-holonomity (non-orthogonality of the time lines to the three-dimensional spatial section), the chr.inv.-tensor of the deformation of space Dik , and the chr.inv.-Christoffel symbols ∆ijk (they

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manifest the non-uniformity of space) ∂vi ∂w 1 , − Fi = √ g00 ∂xi ∂t Aik

Dik

1 = 2

∂vk ∂vi − k ∂xi ∂x

+

1 (Fi vk − Fk vi ) , 2c2

(2.1)

(2.2)

√ ∗ 1 ∗∂hik ∂ ln h ik D =− , D = h Dik = , (2.3) 2 ∂t ∂t ∗ ∗ ∗ ∂hjm ∂hkm ∂hjk 1 . (2.4) + − = him ∆jk,m = him 2 ∂xk ∂xj ∂xm

1 ∗∂hik = , 2 ∂t

∆ijk

√ w g00 = 1 − 2 , c

ik

They are expressed through the chr.inv.-differential operators ∗

∂ 1 ∂ =√ , ∂t g00 ∂t

∗ ∂ ∂ 1 ∂ = + 2 vi , i i ∂x ∂x c ∂t ∗

(2.5)

as well as the gravitational potential w, the linear velocity vi of space rotation due to the respective non-holonomity, and also the chr.inv.metric tensor hik , which are determined as cg0i √ vi = − √ , (2.6) w = c2 (1 − g00 ) , g00 hik = − gik +

1 vi vk , c2

hik = − g ik ,

hik = δki ,

(2.7)

while the derivation parameter of the equations is the physical observable time 1 √ (2.8) dτ = g00 dt − 2 vi dxi . c This is enough. We now have all the necessary equipment to solve the geodesic equations in chronometrically invariant form. §3. Local mass-defect in the space of a mass-point (Schwarzschild’s mass-point metric). This is an empty space∗, wherein a spherical massive island of matter is located, thus producing a spherically symmetric field of gravitation (curvature). The massive island is ∗ In the General Theory of Relativity, we say that a space is empty if it is free of distributed matter — substance or fields, described by the right-hand side of Einstein’s equations, — except for the field of gravitation, which is the same as the field of the space curvature described by the left-hand side of the equations.

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approximated as a mass-point at distances much larger than its radius. The metric of such a space was introduced in 1916 by Karl Schwarzschild [7]. In the spherical three-dimensional coordinates x1 = r, x2 = ϕ, x3 = θ, the metric has the form dr2 rg 2 2 c dt − − r2 dθ2 + sin2 θ dϕ2 , ds2 = 1 − r r 1 − rg

(3.1)

where r is the distance from the mass-island of the mass M , rg = 2GM is c2 the corresponding gravitational radius of the mass, and G is the worldconstant of gravitation. As is seen from the metric, such a space is free of rotation and deformation. Only the field of gravitation affects massbearing particles therein. √ Differentiating the gravitational potential w = c2 (1 − g00 ) with respect to xi , we obtain c2 ∂g00 1 ∂w = − , (3.2) Fi = √ g00 ∂xi 2g00 ∂xi wherein, according to the metric (3.1), we should readily substitute g00 = 1 −

rg . r

(3.3)

Thus the gravitational inertial force (2.1) in the space of Schwarzschild’s mass-point metric has the following nonzero components F1 = −

c2 rg 1 , 2r2 1 − rg r

F1 = −

c2 rg 2r2

(3.4)

which, if the mass-island is not a collapsar (r ≫ rg ), are F1 = F 1 = −

GM . r2

(3.5)

As a result, the scalar geodesic equation for a mass-bearing particle (1.1) takes the form dm m (3.6) − 2 F1 v1 = 0 , dτ c 1 dr . This equation transforms into dm where v1 = dτ m = c2 F1 dr, thus we dr obtain the equation d ln m = − GM . It solves, obviously, as c2 r 2 GM

m = m0 e

c2 r

≃ m0

GM 1+ 2 c r

.

(3.7)

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According to the solution, a spacecraft’s mass measured on the surface of the Earth (M = 6.0 ×1027 gram, r = 6.4 ×108 cm) will be greater than its mass measured at the distance of the Moon (r = 3.0 ×1010 cm) by a value of 1.5 ×10−11 m0 due to the greater magnitude of the gravitational potential near the Earth. This mass-defect is a local phenomenon: it decreases with distance from the source of the field, thus becoming negligible at “cosmologically large” distances even in the case of such massive sources of gravitation as the galaxies. This is not a cosmological effect, in other words. It is known as the gravitational mass-defect in the Schwarzschild mass-point field, which is just one of the basic effects of the General Theory of Relativity. The reason why I speak of this well-known effect herein is that this method of deduction — through integrating the scalar geodesic equation, based on the chronometically invariant formalism, — differs from the regular deduction [6, §88], derived from the conservation of energy of a particle travelling in a stationary field of gravitation. §4. Local mass-defect in the space of an electrically charged mass-point (Reissner-Nordstr¨ om’s metric). Due to the suggested new method of deduction — through integrating the scalar geodesic equation, based on the chronometically invariant formalism, — we can now calculate mass-defect in the space of Reissner-Nordstr¨ om’s metric. This is a space analogous to the space of the mass-point metric with the only difference being that the spherical massive island of matter is electrically charged: in this case, the massive island is the source of both the gravitational field (the field of the space curvature) and the electromagnetic field. Therefore such a space is not empty but filled with a spherically symmetric electromagnetic field (distributed matter). Such a space has a metric which appears as an actual extension of Schwarzschild’s mass-point metric (3.1). The metric was first introduced in 1916 by Hans Reissner [8] then, independently, in 1918 by Gunnar Nordstr¨ om [9]. It has the form dr2 rg rq2 2 2 2 − r2 dθ2 + sin2 θ dϕ2 , (4.1) ds = 1 − + 2 c dt − rq2 rg r r 1 − r + r2

where r is the distance from the charged mass-island, rg = 2GM is the c2 corresponding gravitational radius, M is its mass, G is the constant of Gq2 gravitation, rq2 = 4πε 4 , where q is the corresponding electric charge, 0c 1 and 4πε0 is Coulomb’s force constant. As is seen from the metric, such a space is free of rotation and deformation. The gravitational inertial

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force is, in this case, determined by both Newton’s force and Coulomb’s force according the component g00 of the metric (4.1) which is g00 = 1 −

rq2 rg + 2, r r

(4.2)

thus we obtain F1 = − 2 1− F1 = − −

c2 2

c2

rg r

+

rq2 r2

2 rq2 rg − r2 r3

2 rq2 rg − r2 r3

.

,

(4.3)

(4.4)

If the massive island is not a collapsar (r ≫ rg ), and it bears a weak electric charge (r ≫ rq ), we have 2 rq2 GM c2 rg Gq 2 1 =− 2 + F1 = F 1 = − − . (4.5) 2 3 2 r r r 4πε0 c2 r3 Thus, the scalar geodesic equation for a mass-bearing particle (1.1) takes the form dm m − 2 F1 v1 = 0 , (4.6) dτ c Gq2 1 dr where v1 = dτ . It transforms into d ln m = − cGM dr, which 2 r 2 + 4πε c4 r 3 0 solves, obviously, as ! Gq2 GM − 1 2 2 GM 1 Gq 2 c r 2r 4πε0 c4 . (4.7) ≃ m0 1 + 2 − 2 m = m0 e c r 2r 4πε0 c4 As is seen from the solution, we should expect a mass-defect to be observed in the space of Reissner-Nordstr¨ om’s metric. Its magnitude is that of the mass-defect of the mass-point metric (the second term in the solution) with a second-order correction — the mass-defect due to the electromagnetic field of the massive island (the third term). The magnitude of the correction decreases with distance from the source of the field (a charged spherical massive island) even faster than the massdefect due to the field of gravitation of the massive island. Therefore, the mass-defect in the space of Reissner-Nordstr¨ om’s metric we have obtained here is a local phenomenon, not a cosmological effect. Note that this is the first case, where a mass-defect is predicted due to the presence of the electromagnetic field. Such an effect was not considered in the General Theory of Relativity prior to the present study.

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A note concerning two other primary extensions of Schwarzschild’s mass-point metric. Kerr’s metric describes the space of a rotating masspoint. It was introduced in 1963 by Roy P. Kerr [10] then transformed into suitable coordinates by Robert H. Boyer and Richard W. Lindquist [11]. The Kerr-Newman metric was introduced in 1965 by Ezra T. Newman [12,13]. It describes the space of a rotating, electrically charged mass-point. These metrics are deduced in the vicinity of the point-like source of the field: they do not contain the distribution function of the rotational velocity with distance from the source. As a result, when taking into account the geodesic equations to be integrated in the space of any one of the rotating mass-point metrics, we should introduce the functions on our own behalf. This is not good at all: our choice of the functions, based on our understanding of the space rotation, can be true or false. We therefore omit calculation of mass-defect in the space of a rotating mass-point (Kerr’s metric), and in the space of a rotating, electrically charged mass-point (the Kerr-Newman metric). §5. No mass-defect present in the rotating space with selfclosed time-like geodesics (G¨ odel’s metric). This space metric was introduced in 1949 by Kurt G¨odel [14], in order to find a possibility of time travel (realized through self-closed time-like geodesics). G¨ odel’s metric, as was shown by himself [14], satisfies Einstein’s equations where the right-hand side contains the energy-momentum tensor of dust and also the λ-term. This means that such a space is not empty, but filled with dust and physical vacuum (λ-field). Also, it rotates so that time-like geodesics are self-closed therein. G¨odel’s metric in its original notation, given in his primary publication [14], is " # 1 e2˜x 2 2 0 2 x ˜1 0 2 1 2 2 2 3 2 ds = a (d˜ x ) + 2 e d˜ x d˜ x − (d˜ x ) + (d˜ x ) − (d˜ x ) , (5.1) 2 where a = const > 0 [cm] is a constant of the space, determined through c2 1 so that a2 = 8π Gρ = − 2λ , Einstein’s equations as λ = − 2a1 2 = − 4πcGρ 2 and ρ is the dust density. G¨odel’s metric in its original notation (5.1) is expressed through the dimensionless Cartesian coordinates d˜ x0 = a1 dx0 , 1 1 1 1 1 2 2 3 3 d˜ x = a dx , d˜ x = a dx , d˜ x = a dx , which emphasize the meaning of the world-constant a of such a space. Also, this is a constant-curvature space wherein the curvature radius is R = a12 = const > 0. We now move to the regular Cartesian coordinates ad˜ x0 = dx0 = cdt, 1 1 2 2 3 3 ad˜ x = dx , ad˜ x = dx , ad˜ x = dx , which are more suitable for the calculation of the components of the fundamental metric tensor, thus

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manifesting the forces acting in the space better. As a result, we obtain G¨ odel’s metric in the form x1 a

e

2x1 a

(dx2 )2 − (dx3 )2 . 2 As is seen from this form of G¨odel’s metric, ds2 = c2 dt2 + 2 e

cdtdx2 −(dx1 )2 +

(5.2)

x1

g00 = 1 ,

g02 = e a ,

g01 = g03 = 0 ,

(5.3)

thus implying that such a space is free of gravitation, but rotates with a three-dimensional linear velocity vi (determined by g0i ) whose only nonzero component is v2 . The velocity v2 (actually, the component g02 ) manifests the cosine of the angle of inclination of the line of time x0 = ct to the spatial axis x2 = y. Therefore the lines of time are non-orthogonal to the spatial axis at each single point of a G¨odel space, owing to which local time-like geodesics are the elements of big circles (self-closing timelike geodesics) therein. The nonzero v2 also means that the shift of the whole three-dimensional space along the axis draws a big circle. This velocity, according to the definition of vi (2.6) provided by the chronometrically invariant formalism, is x1

v2 = − ce a ,

(5.4)

which, obviously, does not depend on time. Therefore, in the space of G¨ odel’s metric, the second (inertial) term of the gravitational inertial force Fi (2.1) is zero as well as the first (gravitational) term. The metric is also free of deformation: the spatial components gik of the fundamental metric tensor do not depend on time therein. As a result, we see that no one of the factors changing the mass of a mass-bearing particle according to the scalar geodesic equation (whose factors are gravitation, non-holonomity, and deformation of space) is present in the space of G¨odel’s metric. We therefore conclude that massbearing particles do not achieve mass-defect with the distance travelled in a G¨ odel universe. §6. Cosmological mass-defect in the space of Schwarzschild’s metric of a sphere of incompressible liquid. This is the internal space of a sphere filled, homogeneously, with an incompressible liquid. The preliminary form metric of such a space was introduced in 1916 by Karl Schwarzschild [15]. He however limited himself to the assumption that the three-dimensional components of the fundamental metric tensor should not possess breaking (discontinuity). The general form of this metric, which is free of this geometric limitation, was deduced in 2009

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by Larissa Borissova: see formula (3.55) in [16], or (1.1) in [17]. It is !2 r r κρ0 a2 κρ0 r2 1 2 3 1− ds = c2 dt2 − − 1− 4 3 3 −

dr2 1−

κ ρ0 r 2 3

− r2 dθ2 + sin2 θ dϕ2 ,

(6.1)

3M where κ = 8πc2G is Einstein’s gravitational constant, ρ0 = M V = 4π a3 is the density of the liquid, a is the sphere’s radius, and r is the radial coordinate from the central point of the sphere. The metric manifests that such a space is free of rotation and deformation. Only gravitation affects mass-bearing particles therein. It is determined by !2 r r κρ0 a2 κρ0 r2 1 3 1− . (6.2) − 1− g00 = 4 3 3

Respectively, the gravitational inertial force (2.1) in the space of the generalized Schwarzschild metric of a sphere of incompressible liquid has the following nonzero components F1 = − q 3 1−

κ ρ0 r 2 3

c2 κρ0 r q q 2 3 1 − κ ρ30 a − 1 −

κ ρ0 r 2 3

,

q 2 c2 κρ0 r 1 − κ ρ30 r 1 , F =− q q 2 2 3 3 1 − κ ρ30 a − 1 − κ ρ30 r

(6.3)

(6.4)

while the remaining components of the force are zero, because, as is seen from the metric (6.1), the component g00 , which determines the force, is only dependent on the radial coordinate x1 = r. Thus the scalar geodesic equation for a mass-bearing particle (1.1) takes the form dm m (6.5) − 2 F1 v1 = 0 , dτ c dr , while F1 is determined by (6.3). This equation transwhere v1 = dτ forms, obviously, into d ln m = c12 F1 dr, thus

κρ0 r q d ln m = − q κ ρ0 r 2 3 1− 3 3 1−

dr κ ρ0 3

a2

−

q

1−

κ ρ0 r 2 3

.

(6.6)

Dmitri Rabounski

Meanwhile, r d 3

κρ0 a2 − 1− 3

r

κρ0 r2 1− 3

!

=

147

κρ0 r dr q , 2 3 1 − κ ρ0 r

(6.7)

3

therefore the initial equation transforms into ! r r κρ0 a2 κρ0 r2 , d ln m = − d ln 3 1 − − 1− 3 3

(6.8)

which solves as 3

q 1−

m = m0 q 3 1−

κ ρ0 a2 3

κ ρ0 a2 3

−1 q . 2 − 1 − κ ρ30 r

(6.9)

Because the world-density is quite small, ρ0 ≈ 10−29 gram/cm3 or even less than it, and Einstein’s gravitational constant is very small as well, κ = 8πc2G = 1.862 ×10−27 cm/gram, the obtained solution (6.9) at distances much smaller than the radius of such a universe (r ≪ a), takes the simplified form κρ0 r2 . (6.10) m = m0 1 − 12 As such, mass-defect in a spherical universe filled with incompressible liquid is negative. The magnitude of the negative mass-defect increases with distance from the observer, eventually taking the ultimately high numerical value at the event horizon. Hence, this is definitely a true instance of cosmological effects. We will therefore further refer to this effect as the cosmological mass-defect. In other words, the more distant an object we observe in such a universe is, the less is its observed mass in comparison to its real restmass measured near this object. If our Universe would be a sphere of incompressible liquid, the massdefect would be negligible within our Galaxy “Milky Way” (because ρ0 and κ are very small). However, it would become essential at distances of even the closest galaxies: an object located as distant as the Andromeda Galaxy (r ≃ 780 ×103 pc ≃ 2.4 ×1024 cm) would have a negative cosmological mass-defect equal, according to the linearized solution (6.10), 2 to κ ρ120 r ≈ 10−8 of its true rest-mass m0 . At the ultimate large distance in such a universe, which is the event horizon r = a, the obtained solution (6.9) manifests the ultimately high

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mass-defect m = m0

3

q 2 1 − κ ρ30 a − 1 q . 2 2 1 − κ ρ30 a

(6.11)

§7. Cosmological mass-defect in the space of a sphere filled with physical vacuum (de Sitter’s metric). Such a space was first considered in 1917 by Willem de Sitter [18,19]. It contains no substance, but is filled with a spherically symmetric distribution of physical vacuum (λ-field). Its curvature is constant at each point: this is a constantcurvature space. Its metric, introduced by de Sitter, is dr2 λr2 c2 dt2 − ds2 = 1 − − r2 dθ2 + sin2 θ dϕ2 , (7.1) 2 3 1 − λr3 which contains the λ-term of Einstein’s equations. Such a space is as well free of rotation and deformation, while gravitation is only determined by the λ-term λr2 . (7.2) g00 = 1 − 3 Respectively, the sole nonzero components of the gravitational inertial force (2.1) in such a space are F1 =

λc2 r , 3 1 − λr2 3

F1 =

λc2 r, 3

(7.3)

while the remaining ones are zero: the component g00 , which determines gravitation, in de Sitter’s metric (7.1) is dependent only on the radial coordinate x1 = r. This is a non-Newtonian gravitational force which is due to the λ-field (physical vacuum). Its magnitude increases with distance: if λ < 0, this is a force of attraction, if λ > 0 this is a force of repulsion. Thus the scalar geodesic equation for a mass-bearing particle (1.1) in this case has the form dm m − 2 F1 v1 = 0 , dτ c

(7.4)

dr where v1 = dτ , with Fi determined by (7.3). It transforms, obviously, into d ln m = c12 F1 dr, which is

d ln m =

λr dr . 3 1 − λr2 3

(7.5)

Dmitri Rabounski

Because

149

λr2 2λr dr d ln 1 − =− , 3 3 1 − λr2

(7.6)

λr2 1 , d ln m = − d ln 1 − 2 3

(7.7)

3

the initial equation takes the form

which solves as

m0 m= q . 2 1 − λr3

(7.8)

Because, according to astronomical estimates, the λ-term is quite small as λ 6 10−56 cm−2 , at small distances this solution becomes λr2 . (7.9) m = m0 1 + 6 As is seen from the obtained solution, a positive mass-defect should be observed in a de Sitter universe: the more distant the observed object therein is, the greater is its observed mass in comparison to its real restmass measured near the object. The magnitude of this effect increases with distance with respect to the object under observation. In other words, this is another cosmological mass-defect. For instance, suppose our Universe to be a de Sitter world. Consider an object, which is located at the distance of the Andromeda Galaxy (r ≃ 780 ×103 pc ≃ 2.4 ×1024 cm). In this case, with λ 6 10−56 cm−2 and according to the linearized solution (7.9), the mass of this object registered in our observation should be greater than its true rest-mass m0 for 2 a value of λr6 6 10−8 . However, at the event horizon r ≈ 1028 cm, which is the ultimately large distance observed in our Universe according to the newest data of observational astronomy, the magnitude of the massdefect, according to the obtained exact solution (7.8), is expected to be very high, even approaching infinity. Therefore, the one of experimenta crucis answering the question “is our Universe a de Sitter world or not?” would be a substantially high positive mass-defect of distant galaxies and quasars. §8. No mass-defect present in the space of a sphere filled with ideal liquid and physical vacuum (Einstein’s metric). This cosmological solution was introduced by Albert Einstein in his famous presentation [20], held on February 8, 1917, wherein he introduced relativistic cosmology. This solution implies a closed spherical space, which is

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filled with homogeneous and isotropic distribution of ideal (non-viscous) liquid and physical vacuum (λ-field). It was not the first of the exact solutions of Einstein’s equations, found by the relativists, but the first cosmological model — this metric was suggested (by Einstein) as the most suitable model of the Universe as a whole, answering the data of observational astronomy known in those years. The metric of such a space, known also as Einstein’s metric, has the form ds2 = c2 dt2 −

dr2 − r2 dθ2 + sin2 θ dϕ2 , 2 1 − λr

(8.1)

which is similar to de Sitter’s metric (7.1), with only the difference being that Einstein’s metric has g00 = 1 and there is no numerical coefficient 4π Gρ 1 3 in the denominator of g11 . Herein λ = c2 , i.e. the cosmological λ-term has the opposite sign compared to that of G¨odel’s metric. As is seen, in Einstein’s metric, g00 = 1 ,

g01 = g02 = g03 = 0 ,

(8.2)

thus implying that such a space is free of gravitation and rotation. It is also not deforming: the three-dimensional components gik do not depend on time therein. So, the metric contains no one of the factors changing the mass of a mass-bearing particle according to the scalar geodesic equation. This means that mass-bearing particles do not achieve mass-defect with the distance travelled in the space of Einstein’s metric. §9. Cosmological mass-defect in the deforming spaces of Friedmann’s metric. This space metric was introduced in 1922 by Alexander Friedmann as a class of non-stationary solutions to Einstein’s equations aimed at generalizing the static homogeneous, and isotropic cosmological model suggested in 1917 by Einstein. Spaces of Friedmann’s metric can be empty, or filled with a homogeneous and isotropic distribution of ideal (non-viscous) liquid in common with physical vacuum (λ-field), or filled with one of the media. In a particular case, it can be dust. This is because the energy-momentum tensor of ideal liquid transforms into the energy-momentum tensor of dust by removing the term containing pressure (in this sense, dust behaves as pressureless ideal liquid). Friedmann’s metric in the spherical three-dimensional coordinates has the form dr2 2 2 2 2 2 2 2 2 ds = c dt − R , (9.1) + r dθ + sin θ dϕ 1 − κr2

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where R = R(t) is the curvature radius of the space, while κ = 0, ± 1 is the curvature factor∗. In the case of κ =− 1, the four-dimensional space curvature is negative: this manifests an open three-dimensional space of the hyperbolic type. The case of κ = 0 yields zero curvature (flat three-dimensional space). If κ = + 1, the four-dimensional curvature is positive, giving a closed three-dimensional space of the elliptic type. The non-static cosmological models with κ = +1 and κ = −1 were considered in 1922 by Friedmann in his primary publication [21] wherein he pioneered non-stationary solutions of Einstein’s equations, then in 1924, in his second (last) paper [22]. However, the most popular among the cosmologists is the generalized formulation of Friedmann’s metric, which contains all three cases κ = 0, ± 1 of the space curvature as in (9.1). It was first considered in 1925 by Georges Lemaˆıtre [23,24], who did not specify κ, then in 1929 by Howard Percy Robertson [25], and in 1937 by Arthur Geoffrey Walker [26]. Friedmann’s metric in its generalized form (9.1) containing κ = 0, ± 1 is also conventionally known as the Friedmann-Lemaˆıtre-Robertson-Walker metric. A short note about the dimensionless radial coordinate r used in Friedmann’s metric (9.1). In a deforming (expanding or compressing) space, the regular coordinates change their scales with time. In particular, if the space deforms as any expanding or compressing spherical space, the regular radial coordinate will change its scale. To remove this problem, Friedmann’s metric is regularly expressed through a “homogeneous” radial coordinate r as in (9.1)†. It comes as the regular radial coordinate (circumference measured on the hypersphere), which is then divided by the curvature radius whose scale changes with time accordingly. As a result, the homogeneous radial coordinate r (“reduced” circumference) does not change its scale with time during expansion or compression of the space. Let’s have a look at Friedmann’s metric (9.1). We see that g00 = 1 ,

g0i = 0 ,

gik = gik (t) ,

(9.2)

hence, such a space is free of gravitation and rotation, while its threedimensional subspace deforms. Therefore, the scalar geodesic equation ∗ This form of Friedmann’s metric, containing the curvature factor κ, was introduced due to the independent studies conducted by Lemaˆıtre [23, 24] and Robertson [25], following Friedmann’s death in 1925. † Sometimes, Cartesian coordinates are more reasonable for the purpose of calculation. In this case, Friedmann’s metric is expressed through the “homogeneous” Cartesian coordinates, which are derived in the same way from the regular Cartesian coordinates, and which are also dimensionless. See Zelmanov’s book on cosmology [4] and his paper [5], for instance.

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(1.1) for a mass-bearing particle which travels in the space of Friedmann’s metric (we assume that it travels along the radial coordinate r with respect to the observer) takes the form dm m + 2 D11 v1 v1 = 0 , dτ c

(9.3)

dr [sec−1 ], while only the space deformation along the radial where v1 = dτ coordinate, which is D11 , affects the mass of the particle during its motion. According to Friedmann’s metric, dτ = dt due to g00 = 1 and g0i = 0. Thus the scalar geodesic equation (9.3) transforms into

d ln m = −

1 D11 r˙ 2 dt . c2

(9.4)

Unfortunately, this equation, (9.4), cannot be solved alone, as well as the scalar geodesic equation in any deforming space: the deformation term of the equation contains the velocity of the particle which is unknown and is determined by the space metric. We find the velocity from the vectorial geodesic equation (1.2), which for a mass-bearing particle travelling in the radial direction r in the space of Friedmann’s metric (9.1) takes the form dv1 1 dm 1 + v + 2 D11 v1 + △111 v1 v1 = 0 . dτ m dτ

(9.5)

To remove m from the vectorial geodesic equation (9.5), we make a substitution of the scalar equation (9.3). We obtain a second-order differential equation with respect to r, which has the form r¨ + 2 D11 r˙ + ∆111 r˙ 2 −

1 D11 r˙ 3 = 0 . c2

(9.6)

According to the definitions of Dik (2.3) and ∆iik (2.4), we calculate D11 , D11 , and ∆111 in the space of Friedmann’s metric. To do it, we use the components of the chr.inv.-metric tensor hik (2.7) calculated according to Friedmann’s metric (9.1). After some algebra, we obtain h11 =

R2 , 1 − κr2

h22 = R2 r2 ,

h33 = R2 r2 sin2 θ , R 6 r4 sin2 θ , 1 − κr2 1 h33 = 2 2 2 . R r sin θ

h = det khik k = h11 h22 h33 = h11 =

1 − κr2 , R2

h22 =

1 , R2 r 2

(9.7) (9.8) (9.9)

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As a result, we obtain, in the general case of an arbitrary space of Friedmann’s metric, D11 =

∂R R R˙ R , = 1 − κr2 ∂t 1 − κr2 ∆111 =

D11 =

R˙ , R

D=

3R˙ , R

κr , 1 − κr2

(9.10) (9.11)

thus our equation (9.6) takes the form r¨ +

2 R˙ RR˙ κr r˙ 2 − 2 r˙ + r˙ 3 = 0 . 2 R 1 − κr c (1 − κr2 )

(9.12)

This equation is non-solvable being considered in the general form as here. To solve this equation, we should simplify it by assuming particular forms of the functions κ and R = R(t). The curvature factor κ can be chosen very easily: with κ = 0 we have a deforming flat universe, κ = +1 describes a deforming closed universe, while κ =− 1 means a deforming open universe. The curvature radius as a function of time, R = R(t), appears due to that fact that the space deforms. This function can be found through the tensor of the space deformation Dik , whose trace √ √ ∗ 1 ∗∂ h 1 ∗∂V ∂ ln h =√ = (9.13) D = hik Dik = ∂t V ∂t h ∂t yields the speed of relative deformation (expansion or compression) of the volume of the space element [4, 5]. The volume of a space element, which plays the key rˆole in the formula, is calculated as follows. A parali i i lelepiped built on the vectors r(1) , r(2) , . . . , r(n) in an n-dimensional Eui i clidean space has its volume calculated as V = ± det kr(n) k = ±|r(n) |. We 2 i i k i k thus have an invariant V = |r(n) ||r(m) i | = |r(n) ||hik r(m) | = |hik r(n) r(m) |, where hik ≡− gik according to Euclidean geometry. Thus, we obtain (dV )2 = |hik dxi(n) dxk(m) | = |hik ||dxi(n) ||dxk(m) | = h |dxi(n) ||dxk(m) |. Finally, we see that the volume of a differentially small element of an Euclidean √ space is calculated as dV = h |dxi(n) |. Extending this method into a Riemannian space such as the basic space (space-time) of the General √ Theory of Relativity, we obtain dV = −g |dxα (ν) |. In particular, the volume of a√three-dimensional (spatial) differentially small element therein is dV = h |dxi(n) |, or, if the parallelepiped’s edges meet the (curved) √ spatial coordinate axes, dV = h dx1 dx2 dx3 . The total volume of an extended space element is a result of integration of dV along all three spatial coordinates. Thus, in an arbitrary three-dimensional space, which

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is a subspace of the entire space-time, we obtain √ √ ∗ 1 ∗∂ h 1 ∗∂V 1 ∗∂a u ∂ ln h =√ = =γ =γ , D= ∂t ∂t V ∂t a ∂t a h

(9.14)

where a is the radius of the extended volume (V ∼ a3 ), u is the linear velocity of its deformation (positive in the case of expansion, and negative in the case of compression), and γ = const is the shape factor of the space (γ = 3 in the homogeneous isotropic models [4, 5]). Taking this formula into account, I would like to introduce two main types of the corresponding space deformation, and two respective types of the function R = R(t). They are as follows. A constant-deformation (homotachydioncotic) universe. Each single volume V of such a universe, including its total volume and differential volumes, undergoes equal relative changes with time∗ u 1 ∗∂V = γ = const . (9.15) V ∂t a If such a universe expands, the linear velocity of the expansion increases with time. This is an accelerated expanding universe. In contrast, if such a universe compresses, the linear velocity of its compression decreases with time: this is a decelerated compressing universe. ˙ ˙ In spaces of Friedmann’s metric, D = 3RR (9.10). Once R R = A = const that means D = const, we have R1 dR = Adt that means d ln R = Adt. As a result, denoting R0 = a0 , we obtain that D=

R = a0 e At ,

R˙ = a0 Ae At

(9.16)

in this case. Substituting the solutions into the general formulae (9.10), we obtain that, in a constant deformation Friedmann universe, D=

3R˙ = 3A = const , R

R R˙ a20 Ae 2At = , 1 − κr2 1 − κr2 R˙ D11 = = A = const . R

D11 =

(9.17) (9.18) (9.19)

∗ I refer to this kind of universe as homotachydioncotic (oµoταχυδιoγκωτικ´ o). This terms originates from homotachydioncosis — oµoταχυδι´ oγκωσης — volume expansion with a constant speed, from ´ oµo which is the first part of ´ oµoιoς (omeos) — the same, ταχ´ υ τητα — speed, δι´ oγκωση — volume expansion, while compression can be considered as negative expansion.

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A constant speed deforming (homotachydiastolic) universe. ∗ ∂a = const. Such a universe deforms with a constant linear velocity∗ u = ∂t As a result, the radius of any volume element changes linearly with time a = a0 + u t (the sign of u is positive in an expanding universe, and negative in the case of compression). Thus, relative change of such a volume is expressed, according to the general formula (9.14), as u ut u D=γ 1− . (9.20) ≃γ a0 + ut a0 a0 We see that deformation of such a universe decreases with time in the case of expansion, and increases with time if it compresses. ˙ u (9.20), because D = 3RR in spaces of Friedmann’s With D = a0γ+ut u metric, we arrive at the simplest equation R3 dR = a0γ+ut dt. It obviously solves, in the Friedmann case (γ = 3), as R = a0 + ut. Thus we obtain R = a0 + ut ,

R˙ = u .

(9.21)

As a result, substituting the solutions into the general formulae (9.10), we obtain, in a constant-speed deforming Friedmann universe, D=

3u 3R˙ = , R a0 + ut

R R˙ (a0 + ut) u = , 2 1 − κr 1 − κr2 u R˙ = . D11 = R a0 + ut

D11 =

(9.22) (9.23) (9.24)

In reality, space expands or compresses as a whole so that its volume undergoes equal relative changes with time. Therefore, if our Universe really deforms — expands or compresses — it is a space of the homotachydioncotic (constant deformation) kind. Therefore, we will further consider a constant-deformation Friedmann universe as follows. Consider the vectorial geodesic equation (9.12) in the simplest case of Friedmann universe, wherein κ = 0. This is a flat three-dimensional space which expands or compresses due to the four-dimensional curvature which, having a radius R, is nonzero. In such a Friedmann universe ∗ I refer to this kind of universe as homotachydiastolic (oµoταχυδιαστoλικ´ oς). Its origin is homotachydiastoli — oµoταχυδιαστoλ´ η — linear expansion with a constant speed, from ´ oµo which is the first part of ´ oµoιoς — the same, ταχ´ υ τητα — speed, and διαστoλ´ η — linear expansion (compression is the same as negative expansion).

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(κ = 0, D = 3A = const), while taking into account that under the condition of constant deformation we have R = a0 e At and R˙ = a0 Ae At (9.16), the vectorial geodesic equation (9.12) takes the most simplified form r¨ −

a20 Ae 2At 3 r˙ + 2A r˙ = 0 . c2

(9.25)

dp ′ Let’s introduce a new variable r˙ ≡ p. Thus r¨ = dr dt dr = pp , where dp p = dr . Thus re-write the initially equation (9.25) with the new variable. We obtain a2 Ae 2At 3 pp′ − 0 2 p + 2Ap = 0 . (9.26) c Assuming that p 6= 0, we reduce this equation by p. We obtain ′

p′ −

a20 Ae 2At 2 p + 2A = 0 . c2

By introducing the denotations a =− form this equation into the form

a20 Ae2At c2

(9.27)

and b = − 2A we trans-

p′ + ap2 = b .

(9.28)

This is Riccati’s equation: see Kamke [27], Part III, Chapter I, §1.23. 2a2 A2 e2At > 0. The solution We assume a natural condition that ab = 0 c2 of Riccati’s equation under ab > 0, and with the initially conditions ξ ≡ r(t0 ) and η ≡ r˙0 = r(t ˙ 0 ), is √ √ r˙0 ab + b tanh ab (r − r0 ) √ r˙ = p = √ , (9.29) ab + a r˙0 tanh ab (r − r0 ) where we immediately assume r(t0 ) = 0 and r˙0 = r(t ˙ 0 ) = 0, then extend the variables a and b according to our denotations. We obtain √ √ √ br tanh ab 2 cr 2 a0 AeAt √ r˙ = tanh . (9.30) = a0 eAt c ab Let’s now substitute this solution into the initial scalar geodesic equation (9.4). We obtain √ 2 a0 AeAt 2 2 d ln m = − 2Ar tanh dt, (9.31) c thus we arrive at an integral which has the form √ Z 2 a0 AeAt 2 2 ln m = − 2A r tanh dt + B , c

B = const. (9.32)

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This integral in non-solvable. We can only qualitatively study it. So. . . the solution should have the following form: √ R 2 a A eAt m = m0 e

−2A

r 2 tanh2

0

c

dt

.

(9.33)

We see that, in an expanding Friedmann universe (A > 0), the particle’s mass m decreases, exponentially, with the distance travelled by it. In a compressing Friedmann universe (A < 0), the mass increases, exponentially, according to the travelled distance. In any case, the magnitude of the mass-defect increases with distance from the object under observation. So, this is another instance of cosmological mass-effect. So, we have obtained that cosmological mass-defect should clearly manifest in the space of even the simplest Friedmann metric. Experimental verification of this theoretical conclusion should manifest whether, after all, we live in a Friedmann universe or not. The vectorial geodesic equation (9.12) with κ = +1 or κ =−1 is much more complicated than the most simplified equation (9.25) we have considered in the case of κ = 0. It leads to integrals which are not only non-solvable by exact methods, but also hard-to-analyze in the general form (without simplification). Therefore, I see two practical ways of considering cosmological mass-defect in the closed and open Friedmann universes (κ =±1, respectively). First, the consideration of a very particular case of such a universe, with many simplifications and artificially determined functions. Second, the application of computer-aided numerical methods. Anyhow, these allusions are beyond the scope of this principal study. §10. Conclusions. As is well-known, mass-defect due to the field of gravitation is regularly attributed to the generally covariant formalism, which gives a deduction of it through the conservation of the energy of a particle moving in a stationary field of gravitation [6, §88]. In other words, this well-known effect is regularly considered per se. In contrast, the chronometrically invariant formalism manifests the gravitational mass-defect as one instance in the row of similar effects, which can be deduced as a result of integrating the scalar geodesic equation (equation of energy) of a mass-bearing particle. This new method of deduction has been suggested herein. It is not limited to the very particular case of the Schwarzschild mass-point field as is the case of the aforementioned old method. The new method can be applied to

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a particle travelling in the space of any metric theoretically conceivable due to the General Theory of Relativity. Herein, we have successfully applied this new method of deduction to the main (principal) cosmological metrics. In the space of Schwarzschild’s mass-point metric, the obtained solution coincides with the known gravitational mass-defect [6, §88] whose magnitude increases toward the gravitating body. A similar effect has been found in the space of an electrically charged mass-point (ReissnerNordstr¨ om’s metric), with the difference being that there is a massdefect due to both the gravitational and electromagnetic fields. The presence of an electromagnetic field in the mass of a particle was never considered in this fashion prior to the present study. No mass-defect has been found in the rotating space of G¨odel’s metric, and in the space filled with a homogeneous distribution of ideal liquid and physical vacuum (Einstein’s metric). This means that a massbearing particle does not achieve an add-on to its mass with the distance travelled in a G¨ odel universe or in an Einstein universe. The other obtained solutions manifest a mass-defect of another sort than that in the case of the mass-point metric. Its magnitude increases with the distance travelled by the particle. Thus this mass-defect manifests itself at cosmologically large distances travelled by the particle. We therefore refer to it as the cosmological mass-defect. According to the calculations presented in this study, cosmological mass-defect has been found in the space of Schwarzschild’s metric of a sphere of incompressible liquid, in the space of a sphere filled with physical vacuum (de Sitter’s metric), and in the deforming spaces of Friedmann’s metric (empty or filled with ideal liquid and physical vacuum). In other words, a mass-bearing particle travelling in each of these spaces changes its mass according to the travelled distance. The origin of this effect is the presence of gravitation, non-holonomity, and deformation of the space wherein the particle travels (if at least one of the factors is presented in the space): these are only three factors affecting the mass of a mass-bearing particle according to the scalar geodesic equation. In other words, a particle which travels in the field gains an additional mass due to the field’s work accelerating the particle, or it loses its own mass due to the work against the field (depending on the condition in the particular space). All these results have been obtained only due to the chronometrically invariant formalism, which has led us to the new method of deduction through integrating the scalar geodesic equation (equation of energy) of a mass-bearing particle.

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Note that cosmological mass-defect — an add-on to the mass of a particle according to the travelled distance — has never been considered prior to the present study. It is, therefore, a new effect predicted due to the General Theory of Relativity. A next step should logically be the calculation of the frequency shift of a photon according to the distance travelled by it. At first glance, this problem could be resolved very easily due to the similarity of the geodesic equations for mass-bearing particles and massless (light-like) particles (photons). However, this is not a trivial task. This is because massless particles travel in the isotropic space (home of the trajectories of light), which is strictly non-holonomic so that the lines of time meet the three-dimensional coordinate lines therein (hence the isotropic space rotates as a whole in each its point with the velocity of light). Therefore, all problems concerning massless (light-like) particles should be considered only by taking the strict non-holonomic condition of the isotropic space into account. I will focus on this problem, and on the calculation of the frequency shift of a photon according to the travelled distance, in the next paper (under preparation). Submitted on August 09, 2011 Corrected on October 25, 2012 P.S. A thesis of this presentation has been posted on desk of the 2011 Fall Meeting of the Ohio-Region Section of the APS, planned for October 14 –15, 2011, at Department of Physics and Astronomy, Ball State University, Muncie, Indiana.

1. Rabounski D. Hubble redshift due to the global non-holonomity of space. The Abraham Zelmanov Journal, 2009, vol. 2, 11– 28. 2. Rabounski D. On the speed of rotation of the isotropic space: insight into the redshift problem. The Abraham Zelmanov Journal, 2009, vol. 2, 208 –223. 3. Zelmanov A. L. Chronometric invariants and accompanying frames of reference in the General Theory of Relativity. Soviet Physics Doklady, 1956, vol. 1, 227–230 (translated from Doklady Academii Nauk USSR, 1956, vol. 107, no. 6, 815 – 818). 4. Zelmanov A. L. Chronometric Invariants: On Deformations and the Curvature of Accompanying Space. Translated from the preprint of 1944, American Research Press, Rehoboth (NM), 2006. 5. Zelmanov A. L. On the relativistic theory of an anisotropic inhomogeneous universe. The Abraham Zelmanov Journal, 2008, vol. 1, 33 – 63 (originally presented at the 6th Soviet Meeting on Cosmogony, Moscow, 1959). 6. Landau L. D. and Lifshitz E. M. The Classical Theory of Fields. 4th expanded edition, translated by M. Hammermesh, Butterworth-Heinemann, 1980.

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¨ 7. Schwarzschild K. Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der K¨ oniglich Preussischen Akademie der Wissenschaften zu Berlin, 1916, 189 –196 (published in English as: Schwarzschild K. On the gravitational field of a point mass according to Einstein’s theory. The Abraham Zelmanov Journal, 2008, vol. 1, 10 –19). ¨ 8. Reissner H. Uber die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie. Annalen der Physik, 1916, Band 50 (355), no. 9, 106 –120. 9. Nordstr¨ om G. On the energy of the gravitational field in Einstein’s theory. Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 1918, vol. XX, no. 9 –10, 1238 –1245 (submitted on January 26, 1918). 10. Kerr R. P. Gravitational field of a spinning mass as an example of algebraically special metrics. Physical Review Letters, 1963, vol. 11, no. 5, 237 – 238. 11. Boyer R. H. and Lindquist R. W. Maximal analytic extension of the Kerr metric. Journal of Mathematical Physics, 1967, vol. 8, no. 2, 265 –281. 12. Newman E. T. and Janis A. I. Note on the Kerr spinning-particle metric. Journal of Mathematical Physics, 1965, vol. 6, no. 6, 915 – 917. 13. Newman E. T., Couch E., Chinnapared K., Exton A., Prakash A., Torrence R. Metric of a rotating, charged mass. Journal of Mathematical Physics, 1965, vol. 6, no. 6, 918 – 919. 14. G¨ odel K. An example of a new type of cosmological solutions of Einstein’s field equations of gravitation. Reviews of Modern Physics, 1949, vol. 21, no. 3, 447 – 450. ¨ 15. Schwarzschild K. Uber das Gravitationsfeld einer Kugel aus incompressiebler Fl¨ ussigkeit nach der Einsteinschen Theorie. Sitzungsberichte der K¨ oniglich Preussischen Akademie der Wissenschaften zu Berlin, 1916, 424– 435 (published in English as: Schwarzschild K. On the gravitational field of a sphere of incompressible liquid, according to Einstein’s theory. The Abraham Zelmanov Journal, 2008, vol. 1, 20 – 32). 16. Borissova L. The gravitational field of a condensed matter model of the Sun: The space breaking meets the Asteroid strip. The Abraham Zelmanov Journal, 2009, vol. 2, 224 – 260. 17. Borissova L. De Sitter bubble as a model of the observable Universe. The Abraham Zelmanov Journal, 2010, vol. 3, 3 – 24. 18. De Sitter W. On the curvature of space. Koninklijke Nederlandsche Akademie van Wetenschappen Proceedings, 1918, vol. XX, no. 2, 229 – 243 (submitted on June 30, 1917, published in March, 1918). 19. De Sitter W. Einstein’s theory of gravitation and its astronomical consequences. Third paper. Monthly Notices of the Royal Astronomical Society, 1917, vol. 78, 3 – 28 (submitted in July, 1917). 20. Einstein A. Kosmologische Betrachtungen zur allgemeinen Relativit¨ atstheorie. Sitzungsberichte der K¨ oniglich preussischen Akademie der Wissenschaften zu Berlin, 1917, 142–152 (eingegangen am 8 February 1917). ¨ 21. Friedmann A. Uber die Kr¨ ummung des Raumes. Zeitschrift f¨ ur Physik, 1922, Band 10, No. 1, 377–386 (published in English as: Friedman A. On the curvature of space. General Relativity and Gravitation, 1999, vol. 31, no. 12, 1991– 2000).

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¨ 22. Friedmann A. Uber die M¨ oglichkeit einer Welt mit konstanter negativer Kr¨ ummung des Raumes. Zeitschrift f¨ ur Physik, 1924, Band 21, No. 1, 326 – 332 (published in English as: Friedmann A. On the possibility of a world with constant negative curvature of space. General Relativity and Gravitation, 1999, vol. 31, no. 12, 2001– 2008). 23. Lemaˆıtre G. Note on de Sitter’s universe. Journal of Mathematical Physics, 1925, vol. 4, 188 –192. 24. Lemaˆıtre G. Un Univers homog` ene de masse constante et de rayon croissant rendant compte de la vitesse radiale des n´ ebuleuses extra-galactiques. Annales de la Societe Scientifique de Bruxelles, ser. A, 1927, tome 47, 49–59 (published in English, in a substantially shortened form — we therefore strictly recommend to go with the originally publication in French, — as: Lemaˆıtre G. Expansion of the universe, a homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulæ. Monthly Notices of the Royal Astronomical Society, 1931, vol. 91, 483 – 490). 25. Robertson H. P. On the foundations of relativistic cosmology. Proceedings of the National Academy of Sciences of the USA, 1929, vol. 15, no. 11, 822 – 829. 26. Walker A. G. On Milne’s theory of world-structure. Proceedings of the London Mathematical Society, 1937, vol. 42, no. 1, 90 –127. 27. Kamke E. Differentialgleichungen: L¨ osungsmethoden und L¨ osungen. Chelsea Publishing Co., New York, 1959.

Vol. 4, 2011

ISSN 1654-9163

THE

ABRAHAM ZELMANOV JOURNAL The journal for General Relativity, gravitation and cosmology

TIDSKRIFTEN

ABRAHAM ZELMANOV Den tidskrift f¨ or allm¨ anna relativitetsteorin, gravitation och kosmologi

Editor (redakt¨ or): Dmitri Rabounski Secretary (sekreterare): Indranu Suhendro The Abraham Zelmanov Journal is a non-commercial, academic journal registered with the Royal National Library of Sweden. This journal was typeset using LATEX typesetting system. Powered by BaKoMa -TEX. The Abraham Zelmanov Journal ¨ ar en ickekommersiell, akademisk tidskrift registrerat hos Kungliga biblioteket. Denna tidskrift ¨ ar typsatt med typs¨ attningssystemet LATEX. Utf¨ ord genom BaKoMa -TEX. c The Abraham Zelmanov Journal, 2011 Copyright All rights reserved. Electronic copying and printing of this journal for non-profit, academic, or individual use can be made without permission or charge. Any part of this journal being cited or used howsoever in other publications must acknowledge this publication. No part of this journal may be reproduced in any form whatsoever (including storage in any media) for commercial use without the prior permission of the publisher. Requests for permission to reproduce any part of this journal for commercial use must be addressed to the publisher. Eftertryck f¨ orbjudet. Elektronisk kopiering och eftertryckning av denna tidskrift i icke-kommersiellt, akademiskt, eller individuellt syfte ¨ ar till˚ aten utan tillst˚ and eller kostnad. Vid citering eller anv¨ andning i annan publikation ska k¨ allan anges. M˚ angfaldigande av inneh˚ allet, inklusive lagring i n˚ agon form, i kommersiellt syfte ¨ ar f¨ orbjudet utan medgivande av utgivarna. Beg¨ aran om tillst˚ and att reproducera del av denna tidskrift i kommersiellt syfte ska riktas till utgivarna.

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