Lecture: General Theory of Relativity

Transcription

1 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

2 178 CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY 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), 2 ϕ = 4πGρ î x + ĵ y + ˆk z The Newtonian equation of motion is then d 2 x i dt 2 = Fi = ϕ x i, where F is the gravitational force. For a point-like mass M the potential is ϕ = GM r

3 179 Earlier we showed that d 2 x λ dτ 2 + Γλ µν dx µ dτ dx ν dτ = 0 (geodesic equation) If there is no gravity 1. The metric is flat g µν (x) = η µν = constant, in which case g µν (x)/ x α = The affine connection vanishes ( Γλ σ µ = 2 1 gµν gνσ x λ + g λν x µ g ) µλ x ν = Covariant derivatives equal partial derivatives. 4. The equation of motion becomes that of a free particle in Minkowski space: d 2 x λ dτ 2 = 0. Generally though, space is curved by mass, which produces gravity, and the second term in the geodesic equation does not vanish.

4 180 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 g µν x 0 = 0 }{{} Slowly varying dx i dτ << 1 }{{} Weak The geodesic equation of motion becomes d 2 x µ dτ 2 + Γµ 00 and the connection reduces to Γ µ 00 = 2 1 gµρ g 0ρ x 0 + g 0ρ x 0 } {{ } Neglect dx 0 dτ 1 ( dt dτ). }{{} v<<c ( dx 0) 2 = 0, dτ g 00 x ρ = g 1 2 gµρ 00 x ρ.

5 Since the field is weak, expand the metric around the flat-space one, 181 g µν = η µν + ε µν where ε µν is a small correction. Then, g 00 / x ρ = ε 00 / x ρ and to lowest order in ε µν Γ µ 00 = 1 2 η µρ ε 00 x ρ. From the metric η µν = diag( 1,1,1,1) the connection components are explicitly Γ 0 00 = 1 ε 00 2 x 0 = 0 Γi 00 = ε x i. and we thus obtain (restore the cs) d 2 x 0 dτ 2 = 0 d 2 x i dt 2 = 1 2 c2 ε 00 x i. Comparing with the Newtonian equation d 2 x i dt 2 = Fi = ϕ x i, we conclude that ε 00 = 2ϕ/c 2 and thus that ( g 00 = η 00 + ε 00 = 1+ 2ϕ ) c 2.

6 182 CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY This implies a scalar-field source for weak gravity having the form ϕ = 1 2 c2 (g ). 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 g µν as its source. At the surface of spherical gravitating object of mass M and radius R the potential is ϕ = GM/R and ( g GM ) Rc 2. 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.

7 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 x µ 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.

8 184 CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY Matter Distribution Introduce the stress energy tensor T µν, where 1. T 00 = ρc 2 (energy density) 2. T ii = P i (pressure) 3. ct 0i (energy flow per unit area in direction i) 4. ct i0 (momentum density in direction i) 5. T i j (i j) (shear of the pressure P i in the j direction) Physical arguments: T µν is a symmetric rank-2 tensor 10 independent components. Most general form consistent with Lorentz invariance: T µν = (ε + P)u µ u ν + Pη µν (flat spacetime), η µν is the Minkowski metric ε = ρc 2 is the energy density, u µ = dx µ /ds is the velocity x µ (s) describes the world line of a particle with τ = s/c the proper time.

9 185 Conservation of 4-momentum may then be expressed by T µν,ν = 0 (flat spacetime). The generalization of the stress energy tensor to curved spacetime is T µν = (ε + P)u µ u ν + Pg µν (curved spacetime), where g µν is the (position-dependent) metric. Conservation of 4- momentum in curved spacetime corresponds to replacing partial derivatives with covariant derivatives T µν,ν = 0 }{{} flat T µν;ν = 0. }{{} 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.

10 186 CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY 8.1 A Fully Covariant Theory of Gravitation Combining the Poisson equation 2 ϕ = 4πGρ with the density expressed in terms of the time time component of the stress energy tensor, ρ = 1 c 2T 00, and the weak-gravity scalar field, ϕ = 1 2 c2 (g ), gives 2 g 00 = 8πG c 4 T 00 (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.

11 8.1. A FULLY COVARIANT THEORY OF GRAVITATION Conjectured Form of the Covariant Theory 1. The right side of 2 g 00 = 8πG c 4 T 00 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 T 00 T µν. 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 g 00, 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 T µν on the right side. It must be divergenceless with respect to covariant differentiation since T µν;ν = 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.

12 188 CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY The Riemann Curvature Let us introduce the Riemann curvature tensor, R σ µνλ = Γσ µλ,ν Γσ µν,λ + Γσ αν Γα µλ Γσ αλ Γα µν, which has the following symmetries: R σ µνλ = R µσνλ = R σ µλν R σ µνλ = R νλσµ R σ µνλ + R σλ µν + R σνλ µ = 0 and also satisfies the Bianchi Identity: R σ µνλ;ρ + Rσ µρν;λ + Rσ µλρ;ν = 0. Because of the symmetries, only 20 of the 4 4 = 256 components of the Riemann tensor are independent in 4-D spacetime. In 2-D, 15 symmetry relations on 2 4 = 16 components 1 independent component (Gaussian curvature). All components of the Riemann tensor vanish in flat spacetime. Conversely, if R σ µνλ vanishes the geometry of spacetime is flat. The 20 independent components of the Riemann curvature tensor generalize Gaussian curvature to 4-D spacetime.

13 8.1. A FULLY COVARIANT THEORY OF GRAVITATION 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 R µν by contracting the Riemann tensor, R µν = R νµ = g λσ R λ µσν = R σ µσν, = Γ λ µν,λ Γλ µλ,ν + Γλ µνγ σ λσ Γσ µλ Γλ νσ (Ricci tensor), and the Ricci scalar, R, by a further contraction of the Ricci tensor, R = g µν R µν (Ricci scalar). Finally multiply the Bianchi identity by g µν and g σρ and do the implied sums to give R σ µνλ;ρ + Rσ µρν;λ + Rσ µλρ;ν = 0 R µν 1 2 gµν R }{{} Einstein tensor ;ν = 0 where the Einstein tensor is defined by the quantity in parentheses, G µν R µν 1 2 gµν R (Einstein tensor). and has vanishing convariant 4-divergence: G µν ;ν = 0.

14 190 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, G µν ( R µν 1 2 g µνr ) = 8πG c 4 T µν. or in geometrized units G µν = 8πT µν. 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.

15 8.1. A FULLY COVARIANT THEORY OF GRAVITATION 191 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) R µν = 8πG c 4 (T µν 2 1 g µνtα α ), where the full contraction Tα α represents the trace of the stress energy tensor expressed as a mixed rank-2 tensor. Vacuum Solutions: The stress energy tensor vanishes in the region where the solution is valid and Einstein equations reduce to R µν = 0. Requires only the Riemann curvature tensor, not the full Einstein tensor.

16 192 CHAPTER 8. LECTURE: GENERAL THEORY OF RELATIVITY Limiting Cases of the Einstein Tensor It is relatively easy to show that the Einstein tensor G µν R µν 1 2 g µνr has the following limiting behavior For weak, non-relativistic fields, G 00 2 g 00, as required in our previous derivation of the weak-field limit. If spacetime is flat (no curvature), G µν 0. If there were no matter, energy, or pressure in the universe, then G µν 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.