Syllabus. May 3, Special relativity 1. 2 Differential geometry 3

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1 Syllabus May 3, 2017 Contents 1 Special relativity 1 2 Differential geometry 3 3 General Relativity Physical Principles Einstein s Equation Schwarzschild solution Lie derivative Stationary and static spacetimes Derivation of the Schwarzschild solution Geodesics in the Schwarzschild spacetime 18 6 Maximal extension of the Schwarzschild spacetime Motivating example Special relativity Postulates of special relativity 1. Principle of relativity 2. Invariance of the speed of light Notion of a spacetime Event is a specific point in space at specific time Inertial observer employs the coordinates x µ = (t, x, y, z); different observer employs coordinates x µ = (t, x, y, z ); µ = 0, 1, 2, 3 Spacetime is a set of all events Let dx µ = dt be the difference between coordinates of two close events Spacetime interval is defined by ds 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 (1) 1

2 Einstein s postulates are equivalent to statement ds 2 = invariant, (2) which means that for any two observers we have c 2 dt 2 + dx 2 + dy 2 + dz 2 = c 2 dt 2 + dx 2 + dy 2 + dz 2 (3) This is analogous to invariance of the spatial distance dr 2 = dx 2 + dy 2 + dz 2 (4) in the Euclidean 3D-space. Hence, we interpret ds 2 as the spacetime distance between two events. Difference from Euclidean case: ds 2 is not positive definite. Classification: ds 2 < 0: timelike interval ds 2 = 0: lightlike or, more often, null interval ds 2 > 0: spacelike interval Physical meaning of ds 2 In the rest frame of an observer with coordinates x µ ds 2 = c 2 dτ 2, (5) where τ is the proper time of the observer, i.e. time dτ is time elapsed on the clocks which are at rest in the observer s frame. Hence, up to factor c 2, spacetime interval is equal to the proper time of the observer. Let x µ describe the position of a particle (x, y, z) at time t. Then the speed is v = dr/dt, where dr is given by (4). Thus, ( ) ds 2 = c 2 1 v2 c 2 dt 2. (6) This implies the relation between the proper time of the particle and the coordinate time t: dτ = 1 v2 dt; for 0 < v < c we have dτ < dt; (7) c2 this is called the time dilation. Consider the propagation of light: dr = c dt: ds 2 = 0 (8) For superluminal velocity v > c we have ds 2 > 0 and proper time is imaginary which is impossible. Thus, for a physical observer we always have v < c. Summary. Subluminal velocities correspond to real proper time and ds 2 < 0 (timelike). Light propagates at the speed of light and ds 2 = 0 (null). Events which are separated by spacelike intervals ds 2 > 0 correspond to propagation with superluminal speed which is forbidden. There can be no causal relation between spacelike separated events. Light cone in units c = 1: 2

3 t ds 2 < 0 (future) ds 2 = 0 ds 2 > 0 r ds 2 > 0 ds 2 < 0 (past) 2 Differential geometry Manifolds. Manifold is a set of points M on which we can introduce coordinates. In general, we cannot cover entire manifold by a single coordinate system. Example: S 1 (see lecture notes). On a given portion of the manifold we can introduce many coordinate systems. Coordinates will be denoted by x = (x 1, x 2,..., x n ), or simply x i, i = 1, 2,..., n. (9) Given two coordinate systems x i and x i, we can express one in terms of the other: x i = x i (x ), x i = x i (x). (10) We require the existence of partial derivatives x i x j and The identity holds: x i x j x j x k = δi k. x i x j. (11) (12) When no confusion arises, we write i = x i. Chain rule for derivatives x i = x j x i x j, (13) dxi = xi x j dx j. (14) 3

4 Vector fields. In the Euclidean space we think of a vector as an oriented line segment. This does not work for general manifold. For example, think of a sphere S 1, then vector tangent to S 1 does not lie on S 1 : v However, we want to express the idea that vector represents a direction. Curve on a manifold is a mapping S 1 γ : t x i (t). (15) Consider a function f = f(x) and a curve γ. We define the components of the vector tangent to the curve by X i = dxi dt. The derivative of f along the curve, i.e. in the direction of the tangent vector, is (16) df dt = f dx i x i dt = Xi i f. (17) Thus, we define vector X tangent to curve γ as the differential operator X = X i i. (18) Every vector is tangent to some curve. Differential operators i form a basis of vectors. Now consider two coordinate systems x i and x i. By the chain rule we have df dt = f dx i x i dt = f x j x j x i Xi (19) We define the components of X in the new coordinates as so that X j = X i x j x i, (20) X(f) = X i i f = X i if; (21) that is: action of a vector (as operator) on function f does not depend on the coordinate system, but we can express it in different coordinates. All vectors tangent to the manifold at given point P M form a tangent space T P M. We define a new operation commutator of vector fields: [X, Y ] i = X j j Y i Y j j X i. (22) Exercise: show that [X, Y ] transforms as a vector. Geometrical interpretation: see lecture notes. 4

5 Covector fields. Covectors are real-valued objects which linearly act on vectors. Such objects form a cotangent space TP M. Define the gradient of function f as a mapping df, which to any vector X assigns a real number df(x) = X(f), (23) i.e. derivative of function f in the direction of X. Hence df is an example of a covector, because it takes a vector and returns a real number. In given coordinates x µ, we can form the gradients of the coordinates dx i. (24) The identity holds: dx i ( j ) = j (x i ) = δj. i (25) We say that dx i is a dual basis for vector fields i. General covector is α = α i dx i. (26) In different coordinates we have α = α i x i x j dx j = α j dx j, (27) so that the components of a covector transform as α j = α i x i x j. (28) Metric tensor. We would like to measure the magnitudes of vectors and have an inner product (scalar product, dot product) on the space of vector fields. The role of the inner product will be played by the metric tensor. Motivation: length of the curve. In the Euclidean plane in the Cartesian coordinates, where l = B A dr = B A dx2 + dy 2 = B A ẋ2 + ẏ 2 dt = B A (X1 ) 2 + (X 2 ) 2 dt, (29) X i = dxi dt is the tangent vector to a curve. We can also write dr 2 = δ ij dx i dx j. (30) (31) In the polar coordinates x i = (r, θ) defined by x = r cos θ, y = r sin θ, (32) 5

6 we have dr 2 = dr 2 + r 2 dθ 2 g ij dx i dx j, (33) where g ij = ( ) r 2. (34) So, in general, the line element is given by dr 2 = g ij dx i dx j. (35) At the same time, if X i is tangent vector to the curve, its length is given by X 2 = g ij X i X j, (36) which reduces to the Cartesian coordinates to X 2 = (X 1 ) 2 + (X 2 ) 2. (37) The scalar product (dot product, inner product) is X Y = g ij X i Y j, (38) which reduces to the Cartesian product to X Y = X 1 Y 1 + X 2 Y 2. (39) Transformation of g ij : so that dr 2 = g ij dx i dx j = g ij g ij = g ij x i x k hence, g ij is a (0, 2) tensor. x i x k x j x l ddx k dx l, (40) x j ; (41) x l On a general manifold, metric tensor is a (0, 2) tensor g ij satisfying: Symmetry: g ij = g ji Non-degeneracy: det g 0 The line element is given by ds 2 = g ij dx i dx j. (42) The scalar product of two vectors is X Y = g ij X i Y j (43) Since g ij is non-degenerate, we can define the inverse g ij by g ij g jk = δ i k. (44) 6

7 Using the metric tensor we can raise and lower the indices on vectors and covectors: X i = g ij X j, X i = g ij X j. (45) The inner product is then X Y = g ij X i Y j = X j Y j = X i Y i. (46) At each tangent space T P M, we can choose a basis in which g ij is a diagonal matrix (by Gram-Schmidt orthonormalization process). In such a basis, g ij has a canonical form g ij = diag( 1, 1,..., 1, 1, 1,..., 1), r + s = n = dim M; (47) }{{}}{{} r s in that case, we say that the metric has signature (r, s). Euclidean space of dimension n has signature (0, n) and the metric tensor in canonical form is δ ij. In general relativity, we use the signature (1, 3), because the canonical form of the metric tensor is g ij = ( 1, 1, 1, 1). (48) In general, canonical form of metric g ij will be denoted by η ij and appropriate signature will be clear from the context. Tensor fields. By definition, tensor field of rank (p, q) is a set of components a1...ap tb 1... b q, (49) which transform as t a1...ap b 1... b q x = t c1...cp a1 d 1...d q x x ap c1 x cp x d1 x b1 xdq. (50) bq x Vectors are tensors of rank (1, 0), covectors have rank (0, 1), metric tensor is of rank (0, 2). Space of (p, q) tensor fields on manifold M will be denoted by T p q M. Connection. We need to differentiate tensor fields, but partial derivative does not work. For example, the partial derivative of a vector does not transform as the (1, 1) tensor: ix j = x k x i x j x l X l x k + xk x i Xl 2 x j x k x l. (51) The first term looks like correct tensorial term, but the second does not. Notice that partial derivative of a function, i f, transforms correctly as (0, 1) tensor (it s the gradient, after all). Geometrical reason why the partial derivative does not produce a well-behaved tensor can be expressed also in the following ways: taking partial derivatives involves subtracting two vectors from nearby points; but vectors from different points on the manifold belong to different tangent spaces: we simply cannot linearly combine them; values of the components at different points are actually components with respect to different bases, it does not make sense to compare them. In order to define a well-behaved tensorial derivative, we have to introduce a new structure called connection. 7

8 We define the connection or covariant derivative as a mapping m which acts on the components of any tensor field t i...j k...l of type (p, q) and produces a tensor field of type (p, q + 1), m : t i...j k...l mt i...j k...l. (52) The connection has to satisfy the following properties: on functions it reduces to the usual partial derivative, i f = i f; (53) on a product it acts according to the Leibniz rule, i (X j Y k ) = X j i Y k + Y k i X j (54) It can be shown (see the lecture notes) that connection is fully characterized by the Christoffel symbols Γ i jk. Covariant derivatives of vectors and covectors read i X j = i X j + Γ j ik Xk, i X j = i X j Γ k ij X k. (55a) When the manifold has no torsion, Christoffel symbols are symmetric, Γ i jk = Γ i kj. (56) and the connection itself is called symmetric connection, or torsion-free connection. We will consider only symmetric connections! Exercise: check, that for a symmetric connection we have i j f j i f = 0, (57) where f is arbitrary function (scalar). The operator on the left hand side is called commutator of covariant derivatives. It annihilates scalar quantities, but not general tensor fields. In general relativity we will need connection which is compatible with the metric in the sense that metric tensor is covariantly constant: i g jk = 0. (58) In terms of partial derivatives, this condition reads i g jk Γ l ij g lk Γ l ik g jl = 0. (59) Inverting this relation we find (assuming symmetry of the Christoffel symbols) Γ i jk = 1 2 gil ( j g lk + k g jl l g jk ). (60) This is called metric connection or Riemann Levi-Civita (RLC) connection or metric compatible connection. 8

9 Geodesics. Motivation. Consider the stationary flow of a fluid in classical mechanics. The volume element at the position vector r moves with the velocity v( r). The change of the velocity d v along the streamline is given by d v dt = v dx i x i dt = v vi x i, which is the acceleration of the fluid element. If the fluid element moves with the constant velocity, its acceleration is zero and the streamlines are straight lines. Hence, the fact that the streamlines are straight lines can be expressed as the fact that the acceleration of the element is zero. In curved geometries we cannot define the straight lines, but we can define the curves with zero acceleration with respect to covariant derivatives. These are called geodesics and represent the curves which are, for given geometry, as straight as possible. Geodesic is a curve γ with the tangent vector X which satisfies (61) X i i X j = 0, (62) or, more explicitly, X i i X j + Γ j ik Xi X k = 0. (63) If we introduce a parameter t along the curve γ : t x i, we can write X i = dxi dt, and the geodesic condition acquires the form (64) dx j dt + Γ j ik Xj X k = 0, (65) or directly in terms of the coordinates d 2 x i dt 2 + Γi jk dx j dt dx k dt = 0. (66) Let us return to the analogy with the fluids. In fact, if the streamline is straight, the particles of the fluid are not necessarily moving with the constant speed. They can accelerate, but only along the streamline, their acceleration cannot have a normal part. That is, the acceleration of the fluid element can be proportional to the velocity itself. The condition for the straight streamlines is therefore, in general, d v dt v, where the proportionality factor can be arbitrary function. (67) If a vector field X satisfies equation (62), we say that the geodesic is affinely parametrized. Non-affinely parametrized geodesic has in general form X i i X j = f X j, (68) where f is arbitrary function. By appropriate rescaling of X we can always find affinely parametrized geodesic. Usually, it is easier and more natural to work with affinely parametrized geodesics. However, sometimes we have a natural non-affine parametrization of geodesics. For example, on the horizon of a black hole, the deviation from affine parametrization describes the surface gravity of a black hole. 9

10 Curvature. Standard motivating example: transport of the vector over the manifold can depend on path. In the flat space, it does not depend on path: B path 1 A path 2 Here we paralelly transport vector from point A to point B either along the path 1 or along the path 2. In both cases we obtain the same result. On the sphere, however: path 1 A path 2 A B C B First, we transport the vector from the point A along the quarter of the meridian to the point B, then along the quarter of the equator to the point C, and back to point A. We see that the resulting vector is different from the initial one. On the other hand, we can transport the initial vector from the point A along the full meridian, and we end up with the same vector. Hence: the result of the parallel transport depends on the path. Infinitesimal version: X i ε C X i D ε ε X i X i X i A ε B 10

11 Choose a 2d-surface and coordinates x and y. Choose a vector X i at point a. Transport X i along the coordinate line of x by ε to B. Then along y by ε to C, by ε along x to D, finally along y by ε back to A. Resulting vector is X i. In the flat space X i = X i and the vector is the same as the initial one. In curved space, the variation of the vector will be i δx i = ε 2 Rjkl ( x ) j ( y k ) X l. (69) Here, tensor R i jkl is the Riemann tensor defined by i j X k j i X k = R ijkl X l. (70) and it describes the curvature of the connection m. In coordinates, Symmetries: l Rijk = jγ l ik i Γ l jk + Γ m ik Γ l jm Γ m jk Γ l im (71) R ijkl = R [ij]kl = R ij[kl] = R klij (72) Bianchi identities 1st Bianchi identity (algebraic) R [ijk]l = 0, i.e. R ijkl + R jkil + R kijl = 0 (73) 2nd Bianchi identity (differential) [m R ij]kl = 0 (74) Ricci tensor and scalar curvature (Ricci scalar) R ij = R k ikj, R = g ij R ij (75) Symmetry: R ij = R ji Contracted Bianchi identities i R ij = 1 2 jr (76) Riemann normal coordinates Metric in the diagonal form, g ij = η ij Christoffel symbols vanish at given point Then Γ i jk P = 0 (77) g ij = η ij 1 3 R ikjl x k x l + O ( x 3) (78) This shows that Γ i jk is a coordinate effect, but curvature R ijkl cannot be eliminated by the choice of the coordinates. 11

12 The volume element is ( dv = R ik x i x k + O ( x 3)) dv 0, (79) where dv 0 = dx 1... dx n is the Euclidean volume element. That is: Ricci tensor describes the change of the volume, either contraction or the expansion V 0 V Example of volume contraction by the presence of the Ricci curvature. Interpretation of the Ricci scalar R. For 2D surfaces, scalar curvature is related to the Gauss curvature (up to factor 2) At given point P of surface S, erect a normal vector n Any plane containing n and P is called normal plane and its intersection with S is normal section Choose a specific normal plane and corresponding normal section it is a curve, say, γ = γ(s) Define the unit tangent vector t = γ(s), and unit vector N orthogonal to curve, lying in the normal plane Then, t is necessarily orthogonal to t (since t 2 = 1, we have t t = 0) t = κ(s) N, (80) the scalar κ(s) is called the curvature of the curve. In this way we define the curvature of any normal section of a surface. For different normal plane, corresponding curvature will be, in general, different. The maximal and the minimal values of the curvature of a normal section at given point P are called principal curvatures κ 1 and κ 2. The Ricci scalar is then R = 2 κ 1 κ 2 2 R 1 R 2, (81) where R = 1/κ is the radius of the corresponding curvature. Weyl tensor is the trace-free part of the Riemann tensor so that R ijkl = C ijkl + 2 ( ) 2 gi[k R l]j g k[i R j]l n 2 (n 1)(n 2) R g i[k g l]j, (82) g ik C ijkl = 0 (83) Symmetries like the Riemann tensor C ijkl = C [ij]kl = C ij[kl] = C klij, C [ijk]l = 0. (84) 12

13 Weyl tensor describes the deformations preserving the volume, the so-called shear (we will discuss the shear later) 3 General Relativity 3.1 Physical Principles Weak Equivalence Principle (WEP) in different formulations 1. The ratio of the inertial and gravitational mass (m i and m g, respectively) is constant (can be set equal to 1) for all particles and objects F = m i a m i measures the property of an body to resist the change of motion due to an external force m i is a universal property of bodies independent og the nature of the force in case of gravity: F = mg U = m g g, where U is the gravitational potential m g is specific to the gravitational force it is the gravitational charge of the body object in gravitational field with no other forces present, the WEP implies a = g (85) 2. A local experiment cannot distinguish between a gravity and uniform acceleration local : no tidal effects ( picture) and no communication of the experiment with the outside world picture 3. The trajectory of a test particle (no internal structure, e.g., no spin, no electric charge) depends only on its initial position and velocity Why bother with curved space? Bending of light curved trajectories gravity is locally just an acceleration 13

14 3.2 Einstein s Equation Up to now kinematical approach? Do not know what curves the spacetime. Description of matter in GR via the energy momentum tensor T µν contains energy density, en Einstein Equations R µν 1 2 g µνr = κt µν (86) 4 Schwarzschild solution Schwarzschild solution represents a static, spherically symmetric black hole. It is a vacuum solution of Einstein s equations. In order to derive it, we have to specify what the staticity and spherical symmetry mean and how these symmetries can us derive the solution. In general, we have to discuss how we describe symmetries in general relativity and differential geometry. Similarly to Noether theorems, also in this context the symmetries imply some conservation laws. 4.1 Lie derivative Consider vector field X i. The orbits of vector field X i are curves such that X i is everywhere tangent to the curve. In coordinates, curve is given by x i = x i (t), and it is tangent to X i, if X i = dxi dt. (87) X i y i t = x i (t) x i (0) Flow φ t is a mapping which moves points along the orbit, φ t : x i (0) y i t = x i (t), i.e. maps the point with coordinates x i on the orbit to another point on the orbit which lies in the parametric distance t. If we fix x i (0), φ t (x(0)) for varying t is the orbit itself. In what follows we suppress the argument x i (0) and write simply x i. Now, consider a general vector field Y i defined everywhere, in particular at points x i and y i t. Y i (x) Y i (y t ) y i t = x i (t) x i (0) 14

15 The components of Y i at x i are Y i (x), at point y i t we write Y i (y t ). Since y i t lies on the orbit with the tangent X i, we can write y i t = x i + t X i + O ( t 2), (88) where we will omit the higher order terms. The latter relation also implies y i t x j = δi j + t j X i, x i y j t = δ i j t j X i. (89) The derivative i is always understood with respect to x i, because in the limit t 0 it coincides with the derivative with respect to y i t and we will be interested in this limit only. Having the coordinate transformation x i y i t, we can pull the vector Y i from point y i t back to point x i. The pull-back φ t (with respect to the flow φ t of vector field X i ) is (φ t Y i )(x) = xi y j Y j (y t ). (90) t and it is a vector at x i. In general it is different from original value of vector Y i (x) at this point. The Lie derivative of a vector field Y i along X is then defined by X Y i = d dt φ t (Y i ). (91) t=0 In components we get (see the lecture notes for the derivation!) X Y i = X j j Y i Y j j X i, (92) which is exactly the commutator of vector fields: X Y i = [X, Y ] i. (93) For covectors we get (see lecture notes again) X α i = X j j α i + α j i X j. (94) For a general tensor field, we have X τ i... j... = X m m τ i... j... τ m... j... m X i... Tensor field is Lie constant if its Lie derivative is zero. + τ i... m... j X m (95) In all formulas for the Lie derivative, partial derivatives can be replaced by covariant derivatives. Symmetry of a metric is such vector field K i that metric is Lie constant along K i. This condition is given by the Killing equations X g ij = 0, or X m m g ij + g mj i X m + g im j X m = 0. (96) In terms of covariant derivatives, Killing equations read i X j + j X i = 0. (97) Solution of Killing equations is called Killing vector. In general, manifold has no Killing vectors, but it can have at most n(n + 1)/2 independent Killing vectors, in this case it is called maximally symmetric. 15

16 4.2 Stationary and static spacetimes Now we return to general relativity. Spacetime is said to be stationary if there exists a timelike Killing vector K µ. This means that the metric is Lie constant in some timelike direction. One can always find the coordinates x µ = (t, x 1, x 2, x 3 ) in which K µ = (1, 0, 0, 0). In these coordinates, stationary metric does not depend on time. For a timelike observer, hypersurface of constant time, t = constant, is in general not orthogonal to K µ. We say that vector K µ is twisting. K µ Spacetime is static if it is stationary and, in addition, the Killing vector K µ is hypersurface orthogonal, i.e. orthogonal to a family of spacelike hypersurfaces. Condition for K µ to be hypersurface orthogonal is given by the Frobenius criterion: K [µ ν K λ] = 0. How this formula emerges? Sketch: consider hypersurface t = constant. Then, the gradient µ t is always orthogonal to this hypersurface, so the static Killing vector must be a multiple of the gradient, K µ = f µ t, where f is some function. In order to obtain a universal criterion on K µ solely in terms of K µ, we want to get rid of t and f. Differentiate: ν K µ = ( ν f)( µ t) + f ν µ t. (100) Multiply by K α : K α ν K µ = f( α t)( µ t)( ν f) + f 2 ( α t)( ν µ t). (101) Now, the first term on the right hand side is symmetric in α, µ, while the second term is symmetric in µ, ν. Taking the totally antisymmetric part, we get K [α ν K µ] = 0. (98) (99) (102) The Frobenius theorem asserts that also the converse is true, and hence the last condition is also sufficient for K µ to be hypersurface orthogonal. The proof can be done, for example, in the language of differential forms. 16

17 4.3 Derivation of the Schwarzschild solution Vacuum Einstein equations R µν = 0. (103) We assume: Existence of the static Killing vector K a, i.e. timelike Killing vector K a satisfying K [µ α K ν] = 0. (104) Rotational symmetry, i.e. the group of symmetries whose orbits are 2-dimensional spheres. The metric on the sphere is of the form (2) ds 2 = f(r)(dθ 2 + sin 2 θ dφ 2 ). (105) Function f does not depend on t by stationarity, and on θ, φ by rotational symmetry. The are of a sphere is S = π 0 2π dθ dφ f 2 (r) sin θ (106) 0 For Euclidean sphere, we have S = 4 π r 2. We can define a new coordinate r by r = f(r), (107) so that the metric on the sphere becomes (2) ds 2 = r 2 (dθ 2 + sin 2 θ dφ 2 ). (108) Since now we omit the prime. Since the spacetime is also static, K µ must be orthogonal to spheres, so these spheres lie in the foliation of the spacetime by hypersurfaces Σ t orthogonal to K µ. Construction of the coordinates: Choose one particular sphere S 0 in particular hypersurface Σ. Introduce spherical coordinates on S 0 such that (2) ds 2 = r 2 (dθ 2 + sin 2 θdφ 2 ) (109) To propagate coordinates θ and φ to all spheres in Σ by geodesics. Propagate all coordinates along the orbits of the Killing vector. Then, the metric acquires the form ds 2 = A(r) 2 dt 2 + B(r) 2 dr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ). (110) Now we calculate the Ricci tensor and set it to zero (details will be added). The solution is A = B 1 = 1 C r (111) 17

18 In the Newtonian limit, g tt = (1 + 2U), where U is the Newtonian potential. For a point mass M, U = M r. (112) For r, our solution must reduce to the Newtonian one (field is weak), so A = 1 C r U = 1 2 M r (113) which implies C = 2 M. Final solution is ( ds 2 = 1 2 M r ) ( dt M ) 1 r 2 dr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ). (114) This is the Schwarzschild metric. Birkhoff theorem: the only static, vacuum solution of Einstein s equations with spherical symmetry is the Schwarzschild solution. Singularities: Coordinate singularity at r S = 2 M (Schwarzschild radius). Curvature singularity at r = 0. The Kretschmen scalar R αβµν R αβµν = 48 M 2 blows up for r 0. r 6 (115) Schwarzschild describes the exterior of a spherically symmetric star, not the interior (there is no vacuum). The Schwarzschild radius is r S = 2 M G c 2 = 2.93 km M M. (116) For star, the singularity r = 2M is in the interior. Full Schwarzschild solution describes a spherically symmetric black hole. Then we have to worry about singularities. Causal structure of Schwarzschild in these coordinates. 5 Geodesics in the Schwarzschild spacetime Geodesic equation u µ µ u ν = 0. (117) Christoffel symbols inside: difficult to solve directly. 18

19 6 Maximal extension of the Schwarzschild spacetime 6.1 Motivating example Consider metric [Wald(2010)] ds 2 = dt 2 T 4 + dx2. This is just flat metric: set t = 1/T, then ds 2 = dt 2 + dx 2. (118) (119) However, the original metric has singularity at T = 0. For this reason, T is not valid for every T (, ) and we have to choose, say, T > 0. But new coordinate t has no singularity, so it covers the region t (, ). By coordinate transformation we extended the original spacetime. How can we see that the original spacetime can be extended? The only non-vanishing Christoffel symbol is Γ 0 00 = 2 T, (120) and the time component of the geodesic equations has solution T (s) = T 2 0 T 0 s v 0, (121) where T 0 and v 0 are integration constants. Geodesics is not defined for s = T 0 /v 0, which is called geodesic incompleteness. Hence, extension of the spacetime means that we add some points to the original spacetime, so that the new spacetime is geodesically complete. 6.2 Kruskal-Szekeres extension of Schwarzschild References [Wald(2010)] R. Wald, General Relativity (University of Chicago Press, 2010), ISBN , URL 19