Chapter 7 Curved Spacetime and General Covariance

Chapter 7 Curved Spacetime and General Covariance In this chapter we generalize the discussion of preceding chapters to extend covariance to more general curved spacetimes. 145

146 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE 7.1 Covariance and Poincaré Transformations Lorentz covariance makes it manifest that the principles of special relativity (constant speed of light and invariance under Lorentz transformations) are obeyed by a set of equations. Poincaré transformations: (six Lorentz transformations plus the four possible uniform translations in space and time). Invariance under Poincaré transformations physics does not depend on choice of coordinate system origin, orientation,..., conservation laws such as those for energy and angular momentum. Covariance with respect to Poincaré transformations is still insufficient to deal with gravity. We seek a more general covariance that embraces the possibility of non-inertial coordinate systems. General Covariance: a physical equation holds in a gravitational field provided that It holds in the absence of gravity (agrees with the predictions of special relativity in flat spacetime). It maintains its form under a general coordinate transformation x x (possibly between accelerated frames).

7.2. CURVED SPACETIME 147 7.2 Curved Spacetime The deflection of light in a gravitational field suggests that gravity is associated with the curvature of spacetime. Thus, let us consider the more general issue of covariance in curved spacetime. 7.2.1 Curved Spaces and Gaussian Curvature Gauss demonstrated that for 2-surfaces there is a single invariant (Gaussian curvature) characterizing the curvature. For a 2-D coordinate system (x 1,x 2 ) having a diagonal metric with non-zero elements g 11 and g 22, the Gaussian curvature K is { 1 K = 2g 11 g 22 + 1 2g 22 which is generally 2 g 22 ( x 1 ) 2 2 g 11 ( x 2 ) 2 + 1 2g 11 [ g 11 x 2 g 22 x 2 + Position-dependent ( ) ]} 2 g22 x 1, [ g 11 x 1 g 22 x 1 + ( ) ] 2 g11 x 2 An intrinsic quantity expressed entirely in terms of the metric for the space and its derivatives

148 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE For orthogonal coordinates (x,y), 2K(x 0,y 0 ) = 1 R x (x 0 )R y (y 0 ), where R x (x 0 ) is the radius of curvature in the x direction and R y (y 0 ) is the radius of curvature in the y direction, both evaluated at a point (x 0,y 0 ). EXAMPLE: For the special case of a 2-sphere, R x = R y R and K = 1 R 2 where R is the radius of the sphere, which is constant.

7.2. CURVED SPACETIME 149 z r S C = 2πr θ R φ x y Figure 7.1: Measuring the circumference of a circle in curved space. Consider the 2-sphere of Fig. 7.1, defined by x 2 + y 2 + z 2 = R 2, Let us use the circumference of a circle relative to that for flat space to measure deviation from flatness. We may define a circle in the 2-dimensional space by marking a locus of points lying a constant distance S from a reference point, which we choose to be the north pole in Fig. 7.1 The angle subtended by S is S/R and r = Rsin(S/R). Then the circumference of the circle is C = 2πr = 2πRsin S ( ) R = 2πS 1 S2 6R 2 +....

150 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE z r S C = 2πr θ R φ x y If the space were flat, the circumference of the circle would just be 2πS, so the higher-order terms measure the curvature. But for S 0 we have that K = 1/R 2. Substituting R 2 = 1/K in C = 2πr = 2πRsin S ( ) R = 2πS 1 S2 6R 2 +.... and solving for the Gaussian curvature K gives K = Lim S 0 3 π ( 2πS C Thus, we may find the Gaussian curvature for an arbitrary 2-D surface by measuring the circumference of small circles. This example has been for 2-D space. Later we shall generalize the Gaussian curvature parameter for a two-dimensional surface to a set of parameters (elements of the Riemann curvature tensor) that describe the curvature of 4-dimensional spacetime. S 3 ).

7.2. CURVED SPACETIME 151 Notice in this and various other discussions that we often use the mental crutch of embedding a surface in a higherdimensional space in order to more easily visualize our arguments. It is important to emphasize that The intrinsic curvature properties of a space can be determined entirely by the properties of the space itself, without reference to a higher-dimensional embedding space.

152 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE 7.2.2 Distance Intervals in Curved Spacetime In curved spacetime the interval between two events may be expressed as ds 2 = g αβ (x)dx α dx β. The metric tensor g αβ (x) in a curved spacetime generally has a more complicated form than that for Minkowski space, and is a function of the spacetime coordinates.

7.3. COVARIANT DERIVATIVES AND PARALLEL TRANSPORT 153 Tangent point Tangent point Figure 7.2: Tangent planes and vectors in curved spaces. 7.3 Covariant Derivatives and Parallel Transport Covariant derivatives have a geometrical interpretation associated with comparison of vectors located at two different spacetime points. The comparison issue becomes critical for the calculation of derivatives because, by definition, constructing the derivative of a vector requires taking the difference of vectors at two different points within the space. In curved spaces we must first ask the general question of how to define a vector at some spacetime point. The answer is that we do not define a vector on the curved space itself, but rather on the plane tangent to the point on the curved surface (called the tangent space), as illustrated in Fig. 7.2 for the simple case of a sphere.

154 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE Since we are dealing with Riemannian spaces for which a locally Euclidean coordinate system can be constructed around an arbitrary point, it is always possible to define such a tangent space. Tangent point Tangent point The image of the preceding figure is conceptually useful, but defining the tangent space by a local flat coordinate system at a point is an intrinsic process with respect to the original manifold and does not require embedding in a higher-dimensional manifold. Directional derivatives evaluated in the intrinsic manifold may be used to define the tangent space.

7.3. COVARIANT DERIVATIVES AND PARALLEL TRANSPORT 155 Tangent point Tangent point Parallel transport of vectors is necessary to compare two vectors at different points (e.g., to define derivatives). For a flat space the tangent space corresponds with the space itself and we can just move one vector, keeping its orientation fixed with respect to a global set of coordinate axes, to the position of the other vector and compare. On a curved surface this issue is more complicated because the tangent plane also rotates between two points (see the figure above).

156 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE B A (a) Rotation of vector by parallel transport (b) Path dependence of parallel transport Figure 7.3: (a) Parallel transport of a vector in a closed path on a curved surface. (b) Demonstration that parallel transport is path dependent. Natural notion of parallel transport: keep the vector parallel to itself in infinitessimal steps; see Fig. 7.3 (space locally euclidean). As Fig. 7.3 illustrates, parallel transport of vectors on a curved surface is generally path-dependent. Hence parallel transport in curved spaces is not unique and requires a prescription. The apparent rotation of a vector when parallel transported around a closed path is a measure of the curvature of the space. (On a 2D surface the rotation is proportional to the Gaussian curvature.)

7.3. COVARIANT DERIVATIVES AND PARALLEL TRANSPORT 157 Vµ Vµ Vµ Vµ δvµ = 0 Vµ + δvµ Parallel transport in flat space Parallel transport in curved space Figure 7.4: Illustration of comparing vectors transported on flat and curved surfaces. In comparing vectors V µ (x) and V µ (y) at two spacetime points, there are two contributions to any difference V µ (see Fig. 7.4): V µ = dv }{{} µ + δv }{{} µ, Diff in same coordinates Change of tangent space For flat space δv µ = 0 and we obtain the familiar V µ = dv µ = ( α V µ )dx α (covariant deriv = partial deriv). For curved space we may expect δv µ between infinitesimally separated x and y to be linear in V µ and dx α, δv µ = Γ ν µα V νdx α, where Γ ν µα is termed the affine connection. Therefore, V µ = dv µ + δv µ = ( α V µ + Γ ν µα V ν) dx α }{{} Covariant derivative (curved space) = α V µ dx α (flat space)

158 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE We will see later that Γ ν µα is called the affine connection or the connection coefficient, or the metric connection, or just the connection. The equivalence of notation for the affine connection and the Christoffel symbol introduced earlier is deliberate. Later we shall discuss that, up to differences that don t matter, they are equivalent to each other. Γ ν µα can be constructed from the metric tensor and its derivatives. Γ ν µα can be chosen to vanish in a space with constant metric. Γ ν µα does not follow from the differential geometry of the manifold but is an additional imposed structure that specifies how tangent spaces at different points are related (connected). The Riemann curvature tensor describing the local intrinsic curvature of the spacetime may be constructed from the affine connection. Thus the affine connection is central to defining the covariant derivative, implementing parallel transport of tensors, and measuring quantitatively the curvature of a manifold. It will play a central role in general relativity.

7.4. ABSOLUTE DERIVATIVES 159 7.4 Absolute Derivatives Absolute derivatives (intrinsic derivatives) closely related to covariant derivatives. Covariant derivatives defined over an entire manifold in terms of ordinary partial derivatives plus correction terms to cancel non-tensorial character. Absolute derivatives defined only along paths in manifold in terms of ordinary derivatives plus correction terms to cancel non-tensorial character. Using DA/Du to denote the absolute derivative along a path parameterized by u DA α Du = da α du dx γ Γβ αγa β du DA α Du = daα du + Γα dxγ βγaβ du (Covariant vectors) (Contravariant vectors) (Generalizations for higher-order tensors similar to that discussed earlier for covariant derivatives.)

160 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE Parallel Transport of Vectors: For euclidean (pseudoeuclidean) manifold, parallel transport of a vector along a path parameterized by u means that length and direction of vector (referenced to a universal coordinate system) don t change, implying that components satisfy da µ du = 0 (Flat space). For a more general Riemannian (pseudo-riemannian) manifold this generalizes to DA µ Du = 0 (Curved space),

7.5. GRAVITATIONAL FORCES 161 7.5 Gravitational Forces Free Particle: A particle moving solely under the influence of gravitation is commonly termed a free particle in general relativity, because we shall find that the effect of the classical gravitational force will be replicated by particles propagating with no forces acting on them, but in a curved spacetime. Equivalence Principle: in a freely-falling coordinate system labeled by coordinates ξ µ, the special theory of relativity is valid and the equation of motion is given by (assume unit mass) d 2 ξ µ dτ 2 = 0, (the special relativistic generalization of Newton s second law). The proper time interval dτ is dτ 2 = η µν dξ µ dξ ν and the Minkowski metric is defined by η µν = diag( 1,1,1,1), which by equivalence is valid in the coordinates ξ µ.

162 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE Introduce another arbitrary coordinate system x µ (not necessarily inertial). The freely-falling coordinates ξ µ are functions of the new coordinates, ξ µ = ξ µ (x), and by the chain rule Chain rule d 2 ξ α dτ 2 = d ( dξ α ) = d {}}{ ξ α dx µ dτ dτ dτ x µ dτ = 0 ξ α d 2 x µ x µ dτ 2 + x ν }{{} Derivative of product ξ α d 2 x µ x µ dτ 2 + d ( ξ α ) dx µ dτ x µ dτ = 0 ( ξ α ) dx ν dx µ dτ = 0 } x {{ µ dτ } Chain rule ξ α d 2 x µ x µ dτ 2 + 2 ξ α dx µ x µ x ν dτ Multiply by x λ / ξ α (note implied sum on α) x λ ξ α ξ α x }{{ µ } δµ λ d 2 x µ dτ 2 + xλ ξ α to obtain the geodesic equation, dx ν dτ = 0. 2 ξ α dx µ dx ν x µ x ν dτ dτ = 0 } {{ } Γ λ µν d 2 x λ dτ 2 +Γλ µν dx µ dτ dx ν dτ = 0 Γλ µν xλ 2 ξ α ξ α x µ x ν (affine connection)

7.5. GRAVITATIONAL FORCES 163 The identical notation Γ λ µν for the affine connection and the Christoffel symbol is deliberate. Their difference is a tensor (though neither is a tensor alone) that vanishes in an inertial coordinate system and thus in all coordinate systems. The proper time interval in this coordinate system is dτ 2 = η αβ dξ α dξ β ξ α ξ β = η αβ dxµ x µ }{{ x ν dx ν } chain rule ξ α ξ β = η αβ } x {{ µ x ν dx µ dx ν. } g µν Thus, the proper time interval may be written as dτ 2 = g µν dx µ dx ν, where the metric tensor is defined by g µν = η αβ ξ α x µ ξ β x ν. Clearly g µν η µν, and generally g µν = g µν (x) is a function of the spacetime coordinates because ξ α /x µ are generally spacetime dependent.

164 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE Much of the mathematics of general relativity lies in the discipline of differential geometry. However, The affine connection is not strictly part of differential geometry since it is not a natural consequence of differential structure on the manifold, and it is in fact not even a tensor. The affine connection is an augmentation of differential geometry that gives shape and curvature to a manifold; it is a defined rule for parallel transport on curved surfaces. The affine connection generally is not unique because we can define many notions of parallel transport for a curved surface. Nevertheless, we shall see that Under conditions that are assumed to be satisfied in general relativity the affine connection is uniquely determined by the metric tensor.

7.6. THE LOCAL INERTIAL COORDINATE SYSTEM 165 7.6 The Local Inertial Coordinate System We now demonstrate explicitly that the affine connection Γ λ µν at a fixed point X in an arbitrary coordinate system xµ defines the local inertial coordinates ξ α at the point X. Multiply Γ λ µν = xλ 2 ξ α ξ α x µ x ν. through with ξ β / x λ and utilize x λ ξ α ξ α x µ = δ µ λ to give the differential equation 2 ξ β x µ x ν = ξ β x λ Γλ µν. This has a power series solution near the point X ξ α (x) = ξ α (X)+ ξ α (X) x µ (x µ X µ ) + 1 2 ξ α (X) x λ Γ λ µν(x µ X µ )(x ν X ν )+... Thus, Γ λ µν and the partial derivatives ξ/ x at the point X determine the local inertial coordinates ξ α up to order (x X) 2. Sufficient, since the inertial coordinates are valid only in the vicinity of the point X.

166 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE Strictly, the local inertial coordinates ξ µ are determined only up to an arbitrary Poincaré transformation. Since in the absence of gravitational forces the laws of physics are invariant with respect to Poincaré transformations, no physical observables are influenced by this ambiguity.

7.7. THE AFFINE CONNECTION AND THE METRIC TENSOR 167 7.7 The Affine Connection and the Metric Tensor We have seen from the preceding derivation that The affine connection Γ λ µν determines the gravitational force through the geodesic equation; thus, the affine connection may be viewed as the gravitational field. The metric tensor determines the properties of the interval dτ through dτ 2 = g µν dx µ dx ν, Now we show that, in fact, g µν alone determines the full effect of gravitation because Γ λ µν can be expressed in terms of the metric tensor and its derivatives. Thus, we shall show that The metric tensor may be viewed as the gravitational potential.

168 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE Let us first differentiate with respect to x λ : g µν x λ g µν = η αβ ξ α x µ ξ β x ν. = 2 ξ α ξ β x λ x µ x ν η αβ + ξ α 2 ξ β x µ x λ x ν η αβ. But we have already shown that the inertial coordinates ξ α obey the differential equation 2 ξ β x µ x ν = ξ β x λ Γλ µν. Inserting this in the preceding equation we obtain g µν x λ = ξ α ξ β Γρ λ µ x ρ x ν η ρ αβ +Γ }{{} g ρν = Γ ρ λ µ g ρν + Γ ρ λν g ρµ, λν ξ β ξ α x ρ x µ η αβ }{{} g ρµ where g µν = η αβ ξ α x µ ξ β x ν. has been used.

7.7. THE AFFINE CONNECTION AND THE METRIC TENSOR 169 We may solve this equation for the connection Γ ρ by observing that λ µ if we switch the indices λ µ and λ ν g µν x λ = Γρ λ µ g ρν +Γ ρ λν g ρµ (original) g λν x µ = Γρ µλ g ρν +Γ ρ µνg ρλ (λ µ) g µλ x ν = Γρ νµg ρλ + Γ ρ νλ g ρµ (λ ν) Taking the sum of the first two and the difference of the last g µν x λ + g λν x µ g µλ x ν = g ρνγ ρ λ µ + g ρµγ ρ λν + g ρνγ ρ µλ +g ρλ Γ ρ µν g ρλ Γ ρ νµ g ρµ Γ ρ νλ. But both Γ ρ µν and g µν are symmetric in their lower indices so this reduces to g µν x λ Multiplying this by g νσ, where + g λν x µ g µλ x ν = 2g ρνγ ρ λ µ. g νσ = g σν αβ xν x σ = η ξ α ξ β and g νσ g ρν = δ σ ρ, we obtain finally ( Γλ σ µ = 1 gµν 2 gνσ x λ + g λν x µ g ) µλ x ν = 1 2 gνσ ( g µν,λ + g λν,µ g µλ,ν ).

170 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE Therefore the connection, and hence the gravitational field, are entirely determined by the metric tensor and its derivatives: in general relativity, the metric tensor is the source of the gravitational field.

7.7. THE AFFINE CONNECTION AND THE METRIC TENSOR 171 Table 7.1: Equations of geodesic motion Case Variational principle Equation of motion Flat spacetime δ ( η µν dx µ dx ν ) 1/2 = 0 General spacetime δ ( g µν (x)dx µ dx ν ) 1/2 = 0 d 2 x µ dτ = 0 2 d 2 x µ dτ = dx ν dx λ 2 Γµ νλ dτ dτ The equations of geodesic motion are contrasted for flat spacetime and for a general curved spacetime in Table 7.1.

172 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE 7.8 Uniqueness of the Affine Connection The affine connection is an additional feature imposed on the differential structure of a manifold through a definition that generally is not unique. However, If a manifold has both a metric and a connection defined for it, one usually makes certain compatibility demands that constrain the connection. If the manifold has a metric tensor, the divergence of a vector field and the metric are said to be compatible if the inner (scalar) product of two arbitrary vectors is preserved under parallel transport. In proving preceding results we have assumed symmetry of the connection in its lower indices. The torsion tensor is defined by tensor { }} { Tµν λ = Γ λ µν }{{} Γ λ νµ }{{} not tensor not tensor and measures the deviation from symmetry in the lower indices.

7.8. UNIQUENESS OF THE AFFINE CONNECTION 173 The connection defined on a manifold with metric g µν is unique and determined completely by the metric if 1. The manifold is torsion-free: T λ µν = 0. 2. The covariant derivative (defined in terms of the connection) of the metric tensor vanishes everywhere, D α g µν = 0, which is sufficient to ensure that the scalar product of vectors is preserved under parallel transport. In this case the connection is termed a metric connection. The previous result that ( Γλ σ µ = 1 gµν 2 gνσ x λ + g λν x µ g ) µλ x ν determines the affine connection uniquely in terms of the metric tensor is a consequence of the assumptions (1) and (2) above. These assumptions are then justified after the fact by the concordance of the resulting theory and observations.

174 CHAPTER 7. CURVED SPACETIME AND GENERAL COVARIANCE 7.9 Weak-Field Limit of the Connection In various applications such as ascertaining the relationship between general relativity and classical Newtonian gravity, or in the investigation of gravitational waves, it is useful to have approximate expressions for the connection coefficients that are valid in the limit of weak, slowlyvarying fields. For example, if we assume that the metric in that case differs only slightly from the constant Minkowski metric η µν and can be expressed as g µν = η µν + h µν (x), then you are asked to show in an Exercise that ( Γλ σ µ = 1 gµν 2 gνσ x λ + g λν x µ g ) µλ x ν gives Γ σ 0µ 1 ( h0ν 2 ηνσ x µ h ) 0µ x ν for weak, slowly-varying fields. Γ σ 00 1 2 ηνσ h 00 x ν

7.10. ONWARD TO EINSTEIN S THEORY OF GRAVITATION 175 7.10 Onward to Einstein s Theory of Gravitation Einstein s theory of gravitation is a covariant relationship among mass, density, curvature that implements the principles of equivalence and general covariance. We now have the tools to construct that theory.