Chapter 8 Lecture: General Theory of Relativity We shall now employ the central ideas introduced in the previous two chapters: • The metric and curvature of spacetime • The principle of equivalence • The principle of general covariance to construct the general theory of relativity and the corresponding theory of gravitation.
We know that the weak-field, low-velocity limit of this theory must be Newtonian gravitation, so we begin by asking how Newtonian gravity can be generalized to respect the principles of special and then general relativity.
177
178
CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY
8.0.1 Weak Field Limit
We begin by considering the weak field limit of Einstein’s theory as a guide to what the full theory should look like. We may expect that in the weak field limit we should recover Newton’s gravitational theory.
The Newtonian gravitational field may be derived from a scalar potential ϕ that obeys the Poisson equation (assume unit mass), ∇2ϕ = 4π Gρ
∇ ≡ iˆ
∂ ∂ ∂ + jˆ + kˆ ∂x ∂y ∂z
The Newtonian equation of motion is then d 2xi ∂ϕ i = F = − , dt 2 ∂ xi where F is the gravitational force. For a point-like mass M the potential is
ϕ =−
GM r
179
Earlier we showed that µ ν d 2xλ λ dx dx =0 + Γµν dτ 2 dτ dτ
(geodesic equation)
If there is no gravity 1. The metric is flat gµν (x) = ηµν = constant, in which case ∂ gµν (x)/∂ xα = 0. 2. The affine connection vanishes
Γλσ µ
=
1 νσ 2g
∂ gµν ∂ gλ ν ∂ gµλ + − ∂ xµ ∂ xν ∂ xλ
= 0.
3. Covariant derivatives equal partial derivatives. 4. The equation of motion becomes that of a free particle in Minkowski space: d 2xλ = 0. dτ 2
Generally though, space is curved by mass, which produces gravity, and the second term in the geodesic equation does not vanish.
180
CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY
Assume for the moment gravitational fields that • are slowly varying in time • are weak • have v << c, implying the conditions
∂ gµν =0 0 ∂ x | {z }
dxi << 1 d τ | {z }
Slowly varying
Weak
dx0 ≃ 1 (→ dt ≃ d τ ). d τ | {z } v<
The geodesic equation of motion becomes
0 2 d 2xµ dx µ = 0, + Γ 00 dτ dτ 2 and the connection reduces to
∂ g0ρ ∂ g0ρ ∂ g00 µ 1 µρ ∂ g00 Γ00 = 21 gµρ ∂ x0 + ∂ x0 − ∂ xρ = − 2 g ∂ xρ . | {z } Neglect
181
Since the field is weak, expand the metric around the flat-space one, gµν = ηµν + εµν where εµν is a small correction. Then, ∂ g00 /∂ xρ = ∂ ε00 /∂ xρ and to lowest order in εµν µ
Γ00 = − 21 η µρ
∂ε00 . ∂ xρ
From the metric ηµν = diag (−1, 1, 1, 1) the connection components are explicitly Γ000 =
1 ∂ε00 2 ∂ x0
Γi00 = − 21
=0
∂ε00 . ∂ xi
and we thus obtain (restore the cs) d 2x0 =0 dτ 2
d 2xi 1 2 ∂ε00 = 2c . dt 2 ∂ xi
Comparing with the Newtonian equation d 2xi ∂ϕ = Fi = − i , 2 dt ∂x we conclude that ε00 = −2ϕ /c2 and thus that
2ϕ g00 = η00 + ε00 = − 1 + 2 c
.
182
CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY
This implies a scalar-field source for weak gravity having the form
ϕ = − 12 c2 (g00 + 1).
Thus we obtain in the weak-gravity limit a clear manifestation of the Einstein conjecture that gravity derives from the geometry of spacetime, with the metric tensor gµν as its source.
At the surface of spherical gravitating object of mass M and radius R the potential is ϕ = −GM/R and 2GM . g00 ≃ − 1 − Rc2
Second term measures the strength of the gravitational field at the surface of the object. • It is about 10−6 for the Sun. • It is only of order 10−4 even for a white dwarf. • It is about 0.3 for the surface of a neutron star (invalidating the assumptions of the weak-gravity derivation).
Neutron star densities signal the onset of significant effects from the curvature of spacetime and non-negligible general relativistic corrections to Newtonian gravity.
183
8.0.2 General Recipe for Motion in a Gravitational Field The preceding discussion suggests a general recipe for writing the equations of motion in a gravitational field. • Invoke the equivalence principle to justify a local Minkowski coordinate system ξ µ and formulate the appropriate equations of motion for flat Minkowski spacetime in tensor form. • Replace the Minkowski coordinates ξ µ by general curvilinear coordinates xµ in all equations. • Replace all derivatives by the corresponding covariant derivatives. • Replace all integral volume elements by invariant volume elements. The resulting equations describe physics in a gravitational field.
Because of the structure of the covariant derivatives, this procedure clearly implies a relationship gravity
↔
spacetime curvature
↔
mass/energy.
that we will now exploit.
The first step is to describe the distribution of matter (and energy and pressure) covariantly.
184
CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY
8.0.3 Matter Distribution Introduce the stress–energy tensor Tµν , where 1. T00 = ρ c2 (energy density) 2. Tii = Pi (pressure) 3. cT0i (energy flow per unit area in direction i) 4. cTi0 (momentum density in direction i) 5. Ti j (i 6= j) (shear of the pressure Pi in the j direction)
Physical arguments: Tµν is a symmetric rank-2 tensor → 10 independent components. Most general form consistent with Lorentz invariance: Tµν = (ε + P)uµ uν + Pηµν
(flat spacetime),
• ηµν is the Minkowski metric • ε = ρ c2 is the energy density, • uµ = dxµ /ds is the velocity • xµ (s) describes the world line of a particle with τ = s/c the proper time.
185
Conservation of 4-momentum may then be expressed by Tµν ,ν = 0
(flat spacetime).
The generalization of the stress–energy tensor to curved spacetime is Tµν = (ε + P)uµ uν + Pgµν
(curved spacetime),
where gµν is the (position-dependent) metric. Conservation of 4momentum in curved spacetime corresponds to replacing partial derivatives with covariant derivatives Tµν ,ν = 0 | {z } flat
−→
Tµν ;ν = 0 . | {z } curved
The stress–energy tensor in curved spacetime implies a fundamental difference between Einstein and Newton: Lorentz transformations mix the components of T µν , so all components of the stress– energy tensor (energy, mass, and pressure) will contribute to the curvature and hence to the source of the gravitational field. That increased pressure enhances the strength of gravity has implications for stability of massive objects, suggesting a mass limit beyond which even increasing the pressure without bound cannot stop gravitational collapse.
186
CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY
8.1 A Fully Covariant Theory of Gravitation Combining the Poisson equation ∇2ϕ = 4π Gρ with the density expressed in terms of the time–time component of the stress–energy tensor,
ρ=
1 T00 , c2
and the weak-gravity scalar field,
ϕ = − 12 c2 (g00 + 1), gives ∇2 g00 =
8π G T00 c4
(First stab at covariant gravity)
This expression is clearly not yet satisfactory: • Not covariant: it is expressed in tensor components, not tensors. • Not generally valid: It was derived assuming weak, slowly-varying fields. But the limit is correct, so let’s use it as a guide to guessing the form of a fully covariant gravitational theory.
8.1. A FULLY COVARIANT THEORY OF GRAVITATION
187
8.1.1 Conjectured Form of the Covariant Theory 1. The right side of ∇2g00 =
8π G T00 c4
is not a tensor, but since the Newtonian limit is proportional to one component of the stress–energy tensor, assume that the right side should be modified by the replacement T00 → Tµν . 2. The right side now transforms as a rank-2 tensor (the constants are scalars), so covariance requires that the left side be replaced by something having the same transformation properties. We surmise the following general requirements on the new left side: • The weak-field limit is proportional to a curvature ∇2 g00 , so it should be a covariant measure of spacetime curvature. • It must be a rank-2 covariant tensor to match the right side. • It must be symmetric in its lower indices to match the corresponding property of Tµν on the right side. • It must be divergenceless with respect to covariant differentiation since Tµν ;ν = 0. 3. The candidate field equations must reduce to the Poisson equation describing Newtonian gravitation in the limit limit of weak, slowly-varying fields and non-relativistic velocities. Before we can implement these ideas quantitatively we must generalize Gaussian curvature to 4-dimensional spacetime.
188
CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY
8.1.2 The Riemann Curvature Let us introduce the Riemann curvature tensor, σ σ Rσµνλ = Γσµλ ,ν − Γσµν ,λ + Γαν Γαµλ − Γαλ Γαµν ,
which has the following symmetries: Rσ µνλ = −Rµσ νλ = −Rσ µλ ν Rσ µνλ = Rνλ σ µ
Rσ µνλ + Rσ λ µν + Rσ νλ µ = 0
and also satisfies the Bianchi Identity: Rσµνλ ;ρ + Rσµρν ;λ + Rσµλ ρ ;ν = 0. • Because of the symmetries, only 20 of the 44 = 256 components of the Riemann tensor are independent in 4-D spacetime. In 2-D, 15 symmetry relations on 24 = 16 components → 1 independent component (Gaussian curvature). • All components of the Riemann tensor vanish in flat spacetime. • Conversely, if Rσµνλ vanishes the geometry of spacetime is flat.
The 20 independent components of the Riemann curvature tensor generalize Gaussian curvature to 4-D spacetime.
8.1. A FULLY COVARIANT THEORY OF GRAVITATION
189
8.1.3 The Einstein Equations Having now a covariant description of matter, energy, pressure, and curvature, we possess the tools to implement a covariant theory of gravitation.
First form the Ricci tensor Rµν by contracting the Riemann tensor, Rµν = Rν µ = gλ σ Rλ µσ ν = Rσµσ ν , = Γλµν ,λ − Γλµλ ,ν + Γλµν Γλσ σ − Γσµλ Γλνσ
(Ricci tensor),
and the Ricci scalar, R, by a further contraction of the Ricci tensor, R = gµν Rµν
(Ricci scalar).
Finally multiply the Bianchi identity by gµν and gσ ρ and do the implied sums to give
Rσµνλ ;ρ + Rσµρν ;λ + Rσµλ ρ ;ν = 0 −→ Rµν − 12 gµν R = 0 | {z } Einstein tensor
;ν
where the Einstein tensor is defined by the quantity in parentheses, Gµν ≡ Rµν − 12 gµν R
(Einstein tensor). µν
and has vanishing convariant 4-divergence: G;ν = 0.
190
CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY
The Einstein tensor is in fact the tensor that we have been seeking to replace the left side of the weak-field equation: • It is a covariant measure of spacetime curvature since it is formed by contractions of the Riemann curvature tensor. • It is a rank-2 tensor. • It is symmetric in its lower indices. • It has vanishing covariant 4-divergence.
Thus, we may express the covariant theory of gravitation in terms of the Einstein equation, Gµν ≡ Rµν − 12 gµν R = 8π4G Tµν . c or in geometrized units Gµν = 8π Tµν . The tensors are symmetric so this deceptively compact expression represents 10 coupled, partial, non-linear differential equations that must be solved to determine the effect of gravitation.
8.1. A FULLY COVARIANT THEORY OF GRAVITATION
It is not too difficult to show (Exercise) that in the weakfield, slowly-varying, low-velocity limit, the Einstein field equations reduce to the Poisson field equation of Newtonian gravity.
By contraction with the metric tensor the Einstein equation can also be written in the alternative form (Exercise) 8π G (Tµν − 21 gµν Tαα ), 4 c where the full contraction Tαα represents the trace of the stress–energy tensor expressed as a mixed rank-2 tensor. Rµν =
Vacuum Solutions: The stress–energy tensor vanishes in the region where the solution is valid and Einstein equations reduce to Rµν = 0. Requires only the Riemann curvature tensor, not the full Einstein tensor.
191
192
CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY
8.1.4 Limiting Cases of the Einstein Tensor It is relatively easy to show that the Einstein tensor Gµν ≡ Rµν − 12 gµν R has the following limiting behavior • For weak, non-relativistic fields, G00 → ∇2 g00 , as required in our previous derivation of the weak-field limit. • If spacetime is flat (no curvature), Gµν → 0. • If there were no matter, energy, or pressure in the universe, then Gµν → 0.
These are exactly the properties that we expect from a theory of gravitation in which • curved spacetime is responsible for gravity, • mass–energy–pressure is responsible for curving spacetime, and that reduces to Newtonian gravitation in the weakfield limit.
