Singularities and Causal Structure in General Relativity

Transcription

1 Singularities and Causal Structure in General Relativity Alexander Chen February 16, What are Singularities? Among the many profound predictions of Einstein s general relativity, the prediction of the existence of a singularity in our space-time is by far the most intriguing and disturbing. The possibility of having some singular place in our space-time led to a lot of discussions about the meaning of physics and metaphysics. As now is commonly believed, the universe itself started from a singularity in the form of the Big Bang, so the question about singularity also concerns the origin of space-time itself. We cannot hope to answer all the intricate questions out there, but will suffice ourselves to study some restricted properties of space-time singularities and find the condition for them to occur. Before starting the discussion, we first need a working definition of the subject we are going to study. What is, precisely, a singularity in space-time? One way of thinking about singularities is the places where the metric blows up. One typical example is the r = 0 point in the Schwarzshild space-time ( ds 2 = 1 2GM ) ( dt GM ) 1 dr 2 + r 2 ( dθ 2 + sin 2 θdϕ 2) (1) r r In the limit r 0, the time and radial components of the metric blow up. However this is not a very good definition because it depends on our form of metric tensor, and it is hard to distinguish a true singularity of the space-time and the singularity caused by our coordinate choice. For example, at the point r = 2GM the radial component of the metric tensor also blows up, but we can do a standard Kruskal extension and redefine our coordinates r and t to totally avoid this singularity. Combining several steps of coordinate transformations, we have (4GM) 2 ( 1 The end result of Kruskal extension is r ) e r/2gm = T 2 R 2, 2GM ds 2 = 2GMer/2GM r ( ) t T/R = tanh 4GM ( dt 2 + dr 2 ) + r 2 (dθ 2 + sin 2 θdϕ 2 ) (3) Now the metric is free from any singularity at the Schwarzschild radius r = 2GM, and we can go across the horizon directly without any obstruction. Note however that the singularity at r = 0 can t be removed this way, as not only the metric blows up there, but also the scalar curvature R abcd R abcd. This suggests we use instead the blowing up of scalar curvature to define a singularity point. However we can simply remove that point from our space-time manifold, then the curvature is well-defined everywhere, but we know that there is a singular point somewhere on the boundary of the manifold. To remedy this 1 (2)

2 we could use some asymptotic behavior to define the singularity point even if it lies outside the manifold, by observing the behavior of the curvature tensor on a curve in space-time, but we could as well have a manifold where the curvature only blows up asymptotically when r, and we don t want to count that as a singularity. Finally there is also the typical example of a cosmic string, where the space-time is everywhere Minkowski except on a 1D straight line, and the string creates a deficit angle φ 0 < 2π and the manifold has a conical structure with φ = φ 0 and φ = 0 identified. We intuitively expect that the string represents a singular line in the space-time, but the curvature tensor obviously vanishes everywhere except at ρ = 0, so if we remove the string from our manifold then there is no way to use asymptotic behavior to locate the singularity. The most satisfactory idea proposed thus far is to use the holes left behind when we remove singularities from our manifold to characterize the singularities themselves. What happens when we remove a singularity from our manifold and leave a hole behind is that some geodesic which passes through the point will not be able to pass through and forced to terminate, and we call it incomplete. To be precise, an incomplete geodesic is one that has at most one endpoint, but has a finite range of affine parameter (or proper time, if the geodesic is timelike). We will give a formal definition of endpoints later, but it suffices to think of it as literally an endpoint. Note that we don t want to artificially remove some random point in the space-time, so we restrict our discussion on inextendible space-times, i.e. it is not a submanifold of some larger well-behaved manifold. We will call a space-time singular if there is at least one incomplete timelike or null geodesic. Physically it means that it is possible for one freely falling particle or photon to begin or end its existence within finite time (or affine parameter for the photon). This is the definition of singularity we will adopt in the following discussions. 2 Singularity Theorem, An Appetizer The question we want to answer is that, under what conditions do we expect to find singularities in our space-time? Or more specifically, do we expect to find space-time singularities in our universe? The partial answer, although somewhat abstract, is encapsulated in some of the theorems which states the sufficient conditions for the existence of singularities. Due to time constraint, I ll only have time to state and prove one of the many theorems Theorem 1. Let (M, g ab ) be a globally hyperbolic space-time with R ab ξ a ξ b 0 for all timelike ξ a, which will be the case if Einstein s equation is satisfied with the strong energy condition holding for matter. Suppose there exists a smooth spacelike Cauchy surface Σ for which the trace of the extrinsic curvature satiesfies K C < 0 everywhere, where C is a constant. Then no past directed timelike curve from Σ can have length greater than 3/ C. In particular, all past directed timelike geodesics are incomplete. I will explain the terminologies and sketch a proof of this theorem in remaining sections, but the message to be taken from the above statement is that: under some relatively reasonable assumptions for our universe, we can show that all particle worldlines are emanated from a singular point, i.e. our universe started with a singularity. 3 Causal Structure To understand the statement in Theorem 1, let s first study the causal structure of the universe and understand the notion of global hyperbolicity. Let (M, g ab ) be our space-time, with g ab our metric tensor 2

3 defined everywhere on M. Because of the manifold structure, the tangent space at each point p M can be identified with a Minkowski space and we can draw a lightcone centered at p with half identified to be the future of event p and the other half the past. It is physically reasonable to assume that we can make a continuous assignment of future and past for all points on M. Even if a space-time is not time-orientable, we can always unwind it and define its double cover M which is time-orientable. So we won t consider this as an obstruction. Since we can continuously assign future and past to every tangent space, a differentiable curve λ(t) is a future directed timelike curve if for each p λ the tangent vector t a of λ at p is a future directed timelike vector. A particle world-line is a typical example of a future directed timelike curve. Similarly we call a differentiable curve λ(t) a future directed causal curve if its tangent at any point p is either a timelike or null vector. We can define the chronological future of even p M as all the points that are connected to p by a future directed timelike curve. Formally we have I + (p) = {q M a future directed timelike curve from p to q} (4) Similar for I (p) which is just the chronological past of p in the same sense. causal, we can similarly define causal future and causal past of event p by Replacing timelike by J ± (p) = {q M a future/past directed causal curve from p to q} (5) Similarly we can define chronological and causal future and past for any set S M. Given a timelike or causal curve, we want to know if it ends somewhere or if it keeps going indefinitely. We say a point p M is a future endpoint of λ if for any open neighborhood O of p there exists t 0 such that for all t > t 0 we have λ(t) O. Such p can be interpreted as an end point in the sense that if λ is a worldline of a particle, then it will eventually be stuck at p when its proper time τ. Note if a curve has a future endpoint, we can smoothly join another curve to it such that we get a longer curve. So a curve is called future inextendible if it does not have a future endpoint. Similarly we can define past endpoints and past inextendibility. A subset S M is called achronal if there do not exist p, q S such that p and q are connected by a timelike curve. An achronal set can be thought of as a spacelike slice in the space-time and all points are causally independent of each other. Let S be a closed achronal set, we define the future domain of dependence of S, denoted D + (S) D + (S) = {p M Every past inextendible causal curve through p intersects S} (6) This set is important because if nothing can travel faster than light, then any signal sent to point p D + (S) must pass through some point on S. And by specifying the initial condition on S, in principle we should be able to predict anything inside D + (S). Similarly we define the past domain of dependence D (S), and the domain of dependence D(S) = D + (S) D (S) (7) A closed achronal set Σ for which D(Σ) = M is called a Cauchy surface. The name comes from Cauchy boundary condition, in the sense that if we specify everything on Σ as a Cauchy boundary condition, the entire past and future of the universe can be predicted. It can be shown that every Cauchy surface is an embedded C 0 submanifold of M, so it is proper to call it a surface. A Cauchy surface can be thought of as an instant of time in the evolution of the universe. If a spacetime (M, g ab ) possesses a Cauchy 3

4 surface, we call it globally hyperbolic. So in a globally hyperbolic spacetime we have the nice property that the future and past of the universe can be predicted from information at an instant of time. Of course this can t be done in reality, as our space-time may well not be globally hyperbolic, and the other difficulty is that there is no way we can get all the information on a complete Cauchy surface at one instant of time. One thing to note for a globally hyperbolic space-time is that it can t possess a closed timelike curve. For if a closed timelike curve exists, then it either intersects the Cauchy surface Σ or not. If it intersects Σ, then acronality is violated because we have a timelike curve connecting p to p. If it does not intersect, then starting at any point p on the curve we can follow it around and around to get an inextendible curve which never intersects Σ, henve violating the assumption that Σ is a Cauchy surface. In fact, we have the following lemma Lemma 1. Let (M, g ab ) be a globally hyperbolic space-time, then (M, g ab ) is strongly causal, i.e. for any p M and every neighborhood O of p, there exists a neighborhood V of p with V O such that no causal curve intersects V more than twice. Strong causality ensures that any causal curve does not come infinitely close to intersecting itself, such that a small perturbation of the metric will result in violation of causality. This behavior is desirable in physically reasonable space-times. In fact we will prove an even stronger result for globally hyperbolic space-times which makes their topology very simple (or according to Hawking, very dull). 4 Topology of Causal Curves In order to study causal curves in a strongly causal space-time and identify the condition in which some causal curves are incomplete, it is convenient to consider the set of causal curves between two points as a topological space. Let (M, g ab ) be a strongly causal space-time and p, q M. We define C(p, q) to be the set of continuous, future directed causal curves from p to q. Note that we need to expand our definition of causal curves to continuous curves. A continuous future directed causal (or timelike) curve λ is such that for any p λ there is a convex normal neighborhood U of p such that if λ(t 1 ), λ(t 2 ) U with t 1 < t 2 then there is a future directed differentiable causal curve from λ(t 1 ) to λ(t 2 ) which lies entirely in U. A convex normal neighborhood is an open set U with p U such that for all q, r U there exists a unique geodesic γ connecting q and r which stays entirely in U. For an arbitrary space-time (M, g ab ) one can show that a convex normal neighborhood exists for any point p M. We can define a natural topology on C(p, q) as follows. Let U M be open and p, q U, we define O(U) C(p, q) by O(U) = {λ C(p, q) λ U} (8) and call O(U) as open sets in C(p, q). We define the topology T on C(p, q) as the topology generated by the basis {O(U)}. It can be proved that T is Hausdorff and second countable. We define convergence as follows: a sequence of curves λ n λ if for every open set U M with λ U, there exists N such that λ n U for all n > N. One of the most important theorems about the set C(p, q) is stated as follows: Theorem 2. Let (M, g ab ) be a globally hyperbolic space-time and let p, q M, then C(p, q) is compact. Sketch of proof. We need to prove that every sequence {λ n } in C(p, q) has a limit curve. The proof makes use of a technical lemma which states that, if {λ n } is a sequence of future inextendible causal curves passing through p, then there exists a future inextendible causal curve λ passing through p which is a limit curve 4

5 of {λ n }. Now suppose p, q D (Σ) where Σ is a Cauchy surface, then if we remove the point q from the space-time, then all curves in C(p, q) become future inextendible, and from the lemma we see they have a limit curve λ. If we add the point q back then it has to be a future end point of λ, so we found a limit curve in C(p, q). Similar for p, q D + (Σ). Now if p and q are on different sides of Σ, then we can find a point r in I + (Σ) and a future directed causal curve λ from p to r such that a subsequence {λ n} converges to λ pointwise. Reversing the process we can find a limit curve λ of {λ n} from q to r. Joining the two curves we get a limit curve for the sequence {λ n }. From the compactness of C(p, q) we can show that the set J + (p) J (q) is compact in the manifold topology. In fact we have the more general result Theorem 3. Let (M, g ab ) be a globally hyperbolic space-time and let K M be compact, then J + (K) is closed. Similar for J (K). This result is intuitive for a Minkowski space-time, as the causal future for any set is bounded by the lightcone. This is its generalization to any physically nice space-time. Using these results we can prove a very strong theorem describing the exact behavior of globally hyperbolic space-times Theorem 4. Let (M, g ab ) be a globally hyperbolic space-time, then it is stably causal, i.e. there exists a differentiable global time function f on M such that a f is a past directed timelike vector field. Furthermore, we can choose an f such that every surface of constant f is a Cauchy surface and the topology of M reduces to R Σ. Sketch of proof. We can introduce a volume measure µ on M such that µ(m) is finite, and define the required function f by f (p) = µ[i (p)] (9) So this is a monotonic function along any timelike curve. By using the fact that J + (p) and J (p) are closed, one can show that this function is continuous by considering a sequence of points {x n } on λ which converges to r. We can then deform it infinitesimally to smooth out and get a differentiable global time function. To find the f which foliates M, we define f + (p) = µ[i + (p)] and take the quotient f(p) = f (p)/f + (p) (10) We need to prove that f (p) goes to zero along every past inextendible causal curve and f + (p) goes to zero along every future inextendible causal curve, so that f can take any value from 0 to. Suppose λ starts at q and is past inextendible, but f (p) does not go to 0. Because the volume of I (p) never goes to zero, there must be some point p I (r) for all r λ. So λ must be contained within J + (p) J (q). But by the compactness and strong causality λ must have a past endpoint in the set, which contradicts with past inextendible. Same for f +. If we let p run over the Cauchy surface Σ and take the union of C(p, q), we get the set C(Σ, q). Slightly generalizing the above result we know that C(Σ, q) is compact. We will use this result to find the condition when there is a maximum length curve in the set C(Σ, q). We can define a length function τ on any smooth causal curve in C(p, q), by the following natural formula τ[λ] = ( T a T a ) 1/2 dt (11) 5

6 where t is the affine parameter. This length function is defined on a subset of C(p, q), which we denote C(p, q), the subset of smooth timelike curves. Now we can extend this definition to all continuous causal curves in C(p, q) for q I + (p), by taking the limit τ[µ] = lim n τ[λ n], where lim n λ n = µ (12) However it is obvious that this function is not continuous even on C(p, q), as we can infinitesimally deform any smooth timelike curve into null almost everywhere, and the length will be arbitrarily close to zero. However we have the following proposition Proposition 1. Let (M, g ab ) be a strongly causal spacetime and p, q M with q I + (p). Then the length function τ as defined above is upper semicontinuous on C(p, q), i.e. for any λ C(p, q) and ɛ > 0 there exists an open neighborhood O of λ such that for all λ in O we have τ[λ ] τ[λ] + ɛ. Idea of proof. The idea is to define a time function F for some neighborhood U M of λ such that a surface of constant F is a spacelike hypersurface. We normalize a F a F = 1 on λ and by continuity we can control its variation in the neighborhood. For any neighboring curve ρ U we can parametrize ρ by F and compare its tangent v a with a F and obtain an upper bound for the length ( v a v a ) 1/2. The upper semicontinuity then follows from the continuity of a F a F. Now given that τ is upper semicontinuous on C(p, q) we can extend it to C(p, q). Essentially everything can be generalized to C(Σ, p), so we know the length function τ is upper semicontinuous on C(Σ, p). An upper semicontinuous function on a compact set always reaches its maximum, so our final result for this section is the following theorem which follows from the compactness of C(Σ, p) and upper semicontinuity Theorem 5. Let (M, g ab ) be a globally hyperbolic spacetime, p M and Σ be a Cauchy surface. Then there exists a curve γ C(Σ, p) for which τ takes the maximum value on C(Σ, p). 5 Conjugate Points We defined a length function in the last section on the set of causal curves C(p, q), and gave the condition under which it attains its maximum in the set. Now we want to ask to what degree can we identify a maximum in the length function as a timelike geodesic. To answer this question, we need to introduce the concept of conjugate points. Up to now all our discussion is global and not even differential geometry is explicitly used. Here we will do some local calculations using covariant derivatives and applying Einstein s equations. Let M be any manifold where a connection is defined, and let λ s be a smooth family of geodesics with v a as the tangent vector field. We define the deviation vector field η a as ( / s) a. The deviation vector satisfies the geodesic deviation equation a v c c (v b b η a ) = Rcbd ηb v c v d (13) Now if we fix a geodesic γ with tangent vector v a, then we call the solution η a to the above equation a Jacobi field on γ. A pair of points p, q on γ are called conjugate points if there exists a Jacobi field η a which is not identically zero, but vanishes at p and q. Roughly it means that there exists a geodesic infinitesimally nearby which intersects γ at points p and q. For example, the south and north poles of the earth is a pair of conjugate points for all the longitudinal geodesics. The absence of conjugate points means geodesics are curves maximizing the proper time 6

7 Theorem 6. Let γ be a smooth timelike curve connecting two points p, q M. The necessary and sufficient condition that γ locally maximizes the proper time between p and q is that γ be a geodesic with no point conjugate to p between p and q. Now we want to find the condition when conjugate points exist. Before doing that we need some quantities and terminologies. Given an open set O in M, we will be considering is a family of timelike geodesics such that each point p O is passed by exactly one geodesic. It is obvious that if we take the tangent vectors at each point p then we have a smooth local vector field ξ a in the open set O. We normalize this vector field such that ξ a ξ a = 1 everywhere inside O. Such a family of curves is called a congruence. We define the following tensor field B ab = b ξ a (14) The physical meaning of B ab is that it measures the failure of the geodesic deviation vector η a to be parallelly transported. So an observer going along a curve γ 0 will see the geodesics around him twist and deform under the effect of B ab. One noteworthy property of B ab is that it is purely spatial, and B ab ξ a = B ab ξ b = 0. We can define a spatial metric h ab for the subspace perpendicular to ξ a by the following h ab = g ab + ξ a ξ b (15) It is defined such that h a b ξb = 0, so h a b is a projection operator onto the subspace. We also have Tr h ab = 3 which is in accord with the interpretation. Now using this metric we can decompose B ab into the trace part, the traceless symmetric part and the antisymmetric part B ab = 1 3 θh ab + σ ab + ω ab (16) where θ = B ab h ab is called the expansion of the congruence. To get the rate of change of θ along a geodesic, we can just use the standard equation dθ dτ = ξc c B a a = 1 3 θ2 σ ab σ ab + ω ab ω ab R ab ξ a ξ b (17) This is the key equation we will need in our study. The last term on the right hand side can be rewritten using the Einstein s equations R ab ξ a ξ b = 8πG (T ab ξ a ξ b + 12 ) T (18) If the matter in the universe satisfy the strong energy condition T ab ξ a ξ b 1 2 T (19) for all the unit timelike vectors ξ a, then the last term in equation (17) will be negative. If we diagonalize the energy-momentum tensor T ab into the form of a perfect fluid, then the strong energy condition is equivalent to 3 ρ + p i 0, and ρ + p i 0 (20) i=1 The third term on the right hand side of equation (17) vanishes if the geodesic congruence is hypersurface orthogonal, which means there is a family of hypersurfaces orthogonal to ξ a at every point. This is ensured 7

8 by Frobenius theorem. The second term on the right is σ ab σ ab which is nonpositive. So from the equation we get an inequality dθ dτ θ2 0 = θ 1 (τ) θ τ (21) This inequality directly leads to the following lemma Lemma 2. Let ξ a be the tangent field of a hypersurface orthogonal timelike geodesic congruence and suppose R ab ξ a ξ b 0. If the expansion θ takes negative value θ 0 at any point on a geodesic in the congruence, then θ goes to along that geodesic within proper time τ 3/ θ 0. In fact if we consider a congruence of geodesics starting from p, it can be shown that θ at some point q in the congruence is a necessary and sufficient condition for q to be conjugate to p. Thus the above lemma really says that there exists a conjugate point to p within proper time τ 3/ θ 0 from the point where θ takes the negative value θ 0. Now we generalize the concept of conjugacy to a spacelike hypersurface Σ. Let ξ a be the unit tangent field of the congruence of timelike geodesics orthogonal to Σ, we can define the extrinsic curvature K ab of Σ by K ab = a ξ b = B ba (22) From the definition of the congruence we can see it is manifestly hypersurface orthogonal. The trace of the extrinsic curvature is just the expansion K = K a a = h ab K ab = θ (23) We can define a point p to be conjugate to Σ in the congruence if there is an orthogonal deviation vector η a which is nonzero on Σ but vanishes at p. This is a natural generalization of conjugate point of another point. Again p is conjugate to Σ if and only if θ approaches at p. The generalization of lemma 2 reads Lemma 3. Let (M, g ab ) by a spacetime satisfying R ab ξ a ξ b 0 for all timelike ξ a. Let Σ be a spacelike hypersurface with K = θ < 0 at some point q Σ, then within propertime τ 3/ K there exists a point p conjugate to Σ along the geodesic γ orthogonal to Σ and passing through q. The generalization of theorem 6 reads Theorem 7. Let (M, g ab ) be a strongly causal spacetime. Let p M and Σ be an achronal, smooth spacelike hypersurface and define the length function τ on C(Σ, p). A necessary condition for τ to attain its maximum value at γ C(Σ, p) is that γ be a geodesic orthogonal to Σ with no point conjugate to Σ between Σ and p. 6 Proof of the Singularity Theorem, and Comments Now we have all the ingredients to prove theorem 1. Proof of Theorem 1. Suppose there were a past directed timelike curve λ from Σ with length greater than 3/ C. Let p on λ be a point lying beyond length 3/ C. By theorem 5 there exists a maximum length curve γ from p to Σ which obviously also has length greater than 3/ C. By theorem 7 γ must be a geodesic with no point conjugate to Σ between p and Σ, but this contradicts with lemma 3 which states that there is a conjugate point within length 3/ C. 8

9 We have come a long way to prove this seemingly very technical theorem. But along the way we have also developed a systematic way to treat the global structure of space-time. We have seen that global hyperbolicity is a rather strong requirement on our space-time manifold, and it could well not be true that our universe is globally hyperbolic. In fact there are some known solutions to the Einstein equation that are not globally hyperbolic. For example the Anti de-sitter space and the Taub-NUT space. There is little evidence a priori for our universe to be globally hyperbolic. However, the Roberson-Walker universe is manifestly globally hyperbolic as we can choose the surface of constant t as a Cauchy surface. It is in this sense that theorem 1 is useful, because very compelling evidences suggest that our universe can be described by the FRW metric. In fact we can have singularities when we drop the condition of global hyperbolicity. There is a variation of the above theorem due to Hawking which removes the assumption that our universe is globally hyperbolic, but requires that Σ be compact. The statement is as follows Theorem 8. Let (M, g ab ) be a strongly causal spacetime with R ab ξ a ξ b 0 for all timelike ξ a. Suppose there exists a compact, edgeless, achronal, and smooth spacelike hypersurface S such that for the past directed normal geodesic congruence from S we have K C < 0 everywhere on S. Then at least one inextendible past directed timelike geodesic from S has length no longer than 3/ C. The proof can be found in either Hawking and Ellis, or Wald. The strong causality condition in the statement can also be dropped, but requires harder techniques for the proof. Another kind of singularity theorems concerns the singularity introduced by gravitational collapse, and is relevant to the existence of black holes. We introduce the concept of a trapped surface. By this we mean a 2 dimensional spacelike surface S M such that the two families of null geodesics orthogonal to S have expansion θ < 0 on S. One way to think about this kind of surface is that the gravitational field is so strong that even the outgoing light rays are dragged back and converging. In the Kruskal extended Schwarzschild solution, all the 2-spheres inside the horizon are trapped surfaces. The following theorem was proved by Hawking and Penrose in 1970 and we shall use it to conclude our discussion Theorem 9. A spacetime (M, g ab ) must contain at least one incomplete timelike or null geodesic if it satisfies the following four conditions: (1) R ab ξ a ξ b 0 for all timelike and null ξ a. (2) The generic conditions are satisfied, i.e. each nonspacelike geodesic has some point where R abcd ξ a ξ d 0. (3) No closed timelike curves exist. (4) At least one of the following is true: (a) (M, g ab ) has a compact achronal set without edge, (b) (M, g ab ) has a trapped surface, or (c) there is a point p M such that the expansion θ of past (or future) directed null geodesics emanating from p becomes negative along each geodesic in this congruence. 9