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A note on compact cauchy horizons
Volume 102A, number 5,6
PHYSICS LETTERS
21 May 1984
A NOTE ON COMPACT CAUCHY HORIZONS Arvind BORDE Department of Physics, University of British Columbia, Vancouver, BC Canada V6T 2A6 Received 17 February 1984
It is shown that space-times that obey the Hawking-Penrose generic condition and the weak energy condition cannot contain compact Cauchy horizons.
Some years ago Penrose proposed the strong cosmic censorship principle : a physically reasonable space-time always has a global Cauchy hypersurface [ 1]. Part of the motivation behind this proposal appears to have been the observation that the Cauchy horizons that are present in space-times that do not have global Cauchy surfaces generally possess instabilities of some form. Thus the Reissner-Nordstrom space-time has a non-compact Cauchy horizon that is unstable to small perturbations in that singularities develop on it under these perturbations [2]. Such singularities also develop on the compact Cauchy horizon in the T a u b NUT space-time when it is perturbed [2]. One line of attack on the strong cosmic censorship question might therefore be a study of the structure of Cauchy horizons to see if this structure is compatible with other "physically reasonable" conditions that we might expect to hold. In this note we will consider the case of compact Cauchy horizons. Such horizons have been studied before [3], notably by Moncrief and Isenberg [4] who have recently shown that compact Cauchy horizons are necessarily Killing horizons (under the assumption that the null generators of the horizon are closed) and hence may be expected to exist only in very special cases. Below we will obtain a rather simple result along the same lines (that compact Cauchy horizons can exist only in certain special cases). The notation and conventions that will be used are those of Hawking and Ellis [3].
Theorem: Let (M, g) be a space-time such that : 0) RabKaK b >1 0 for all null vectors K a (where Rab is 224
the Ricci tensor associated with the metric g), and (ii) the Hawking-Penrose generic condition holds [3,5]. Then, for any partial Cauchy surface S, if the Cauchy horizon H+(S) is not empty it must be non-compact [similary f o r H (S)]. This theorem does not yield the detailed information that the Moncrief-Isenberg theorem does, but on the other hand its assumptions (which are briefly discussed below) are very weak and very general. It should be noted that results on the non-existence of compact Cauchy horizons have other applications in addition to their relevance to the cosmic censorship question. For example, in the Hawking-Penrose singularity theorem [5] it is established, using the chronology condition, that H+(S) cannot be compact for certain surfaces S. This non-compactness of H+(S) then plays a crucial role in the singularity theorem. For compact S the results of this paper provide an alternative proof, without any causality assumptions, that H+(S) is not compact in reasonable space-times. The application of the theorem in this paper to singularity results is discussed in greater detail elsewhere [81. Conditions (i) and (ii) have both been discussed fairly extensively in the literature on singularity theorems in general relativity [3,5] and so an extended discussion will not be given here. Condition (i) follows if we assume the Einstein equations and that the energy density of matter as seen by any observer is non-negative (the weak energy condition). Condition (ii) essentially requires that every geodesic feel some non-zero 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 102A, number 5,6
PHYSICS LETTERS
gravitational tidal force at least at one point along it. This is generally regarded as a very weak requirement on any reasonable space-time. It does not hold in many known space-times that possess symmetries (e.g. radial null geodesics in the Schwarzschild spacetime do not obey it), but one would expect it to hold in the slightest deviation from these symmetries. Before proceeding with the proof of the theorem a comment on non-compact Cauchy horizons is in order. It appears to be impossible to rule out the existence of non-compact Cauchy horizons under only the very general assumptions that we are making in the compact case. This is because non-compact Cauchy horizons can arise even in globally hyperbolic spacetimes just because of a choice of Cauchy surface. For example, any asymptotically null partial Cauchy surface in Minkowski space-time (or, for that matter, in any asymptotically fiat space-time) would have a non-compact Cauchy horizon associated with it. We would, therefore, have to make additional assumptions about the choice of Cauchy surface being a "good" one, or equivalently the Cauchy horizon being "nontrivial". Such assumptions appear to be hard to pin down precisely without making them too strong. We might be tempted to require, for example, that the Cauchy surface be such that its future domain of dependence contain a future inextendible causal curve. Such an assumption would rule out asymptotically null partial Cauchy surfaces. It would, however, also exclude anti-DeSitter space-time [3] entirely from our considerations and hence is too strong an assumption to make. We will not discuss this question any further here. We now turn to the proof of the theorem. This follows almost immediately from the following two standard results:
Lemma 1: If H is a compact Cauchy horizon in a space-time obeying condition (i) of the theorem, then the null generators of H have zero divergence (ref. [3], pp. 295 - 298). Lemma 2: If ~, is a null geodesic in a space-time such that: (a) condition (i) of the theorem holds, (b) the generic condition holds at some point b on 7, and (c) the divergence 0 of a congruence of null geodesics neighboring 3, is zero at b. Then 0 becomes negative to the past o f b on 7 (ref. [3], pp. 9 8 - 1 0 1 ) .
21 May 1984
Now consider a null generator ~t of some compact future Cauchy horizon H. If~t has no endpoint on H, then the fact that the generic condition holds at some point ofju will force a contradiction between lemmas 1 and 2. (If,u had a future endpoint p on H a n d if the generic condition held on ~t to the future o f p we would not necessarily obtain such a contradiction). The existence o f g is shown by:
Lemma 3: Let S be a partial Cauchy surface such that the future Cauchy horizon H+(S) is not empty and is compact. Then there exists a generator of H+(S) with no future or past endpoint o n H ÷ (S). Proof: By the definition of a partial Cauchy surface, S has no edge. Therefore H = H+(S) also has no edge and so its null generators have no past endpoints (ref. [3], proposition 6.5.3). Let 7 be any null generator of H. If it is closed then it has no future endpoint on H and is the required generator. If it is not closed we may choose a sequence of points {bi} on ')" such that bi+ 1 @J-(bi)for each i and no compact segment of 7 contains an infinite number of the b i. Since H is compact, the sequence £bi} will have a limit point b E H . The portions of')' to the future of each b i will then yield a sequence o f causal curves with limit point b. We will show that the causal limit curve through b to this sequence is the required generator. To do this, pick a point c on H a s follows: ifb E3', then c is any point on 7 to the past of b; ifb ~ 7, then c is b 2. In either case only a finite number of the b i will lie to the future of c. Let/a i stand for the portion of 7 between those b i that lie to the past of c, and c. Then in H ' = H - {c}, the point b is a limit of the sequence of future inextendible null geodesics {/ai}. There is, therefore, a future inextendible (in H ' ) limit causal curve to this sequence through b. Call this curve ~t./.t must be a null generator of H. Its only possible future endpoint on H is c itself. However, suppose that c is the future endpoint of/a o n H . If b ~ "/then c was b 2 and so we have b E J - ( c ) and d E J - ( b l ) and hence b E J - ( b l ) . But the causal curve from c to b 1 was 7 on which b does not lie and so b E I - ( b l), violating the achronality of H. And if b E 7 then c was chosen so that c ~ J - ( b ) . But now b E J - ( e ) and so 3' must be a closed null geodesic. Thus c cannot be the future endpoint of p and hence /.t is the required generator of H. 225
Volume 102A, number 5,6
PHYSICS LETTERS
I would like to thank J. Isenberg and W. U n r u h for useful discussions and R. Riley for e n c o u r a g e m e n t . This research was supported by the National Sciences and Engineering Research Council o f Canada.
References [ I I R. Penrose, Singularities and time asymmetry, in: General relativity: an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge Univ. Press, London, 1979), and references therein.
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21 May 1984
[2] F. Tipler, C. Clarke and G. Ellis, Singularities and horizons: a review article, in: General relativity and gravitation, ed. A. Held (Plenum, New York, 1980), and references therein on Cauchy horizon stability; S. Chandrasekhar and J. Hartle, On crossing the Cauchy horizon of a black hole, Univ. of Chicago preprint ( 1982). [3] S.W. Hawking and G.F.R. Ellis: The large scale structure of space-time (Cambridge Univ. Press, London, 1973). [4] V. Moncrief and J. Isenberg, Comm. Math. Phys. 89 (1983) 387. [5] S.W. Hawking and R. Penrose, Proc. R. Soc. A314 (1970) 529. [6] A. Borde, Singularities in closed sgace-times, U.B.C. preprint (1984).
PHYSICS LETTERS
21 May 1984
A NOTE ON COMPACT CAUCHY HORIZONS Arvind BORDE Department of Physics, University of British Columbia, Vancouver, BC Canada V6T 2A6 Received 17 February 1984
It is shown that space-times that obey the Hawking-Penrose generic condition and the weak energy condition cannot contain compact Cauchy horizons.
Some years ago Penrose proposed the strong cosmic censorship principle : a physically reasonable space-time always has a global Cauchy hypersurface [ 1]. Part of the motivation behind this proposal appears to have been the observation that the Cauchy horizons that are present in space-times that do not have global Cauchy surfaces generally possess instabilities of some form. Thus the Reissner-Nordstrom space-time has a non-compact Cauchy horizon that is unstable to small perturbations in that singularities develop on it under these perturbations [2]. Such singularities also develop on the compact Cauchy horizon in the T a u b NUT space-time when it is perturbed [2]. One line of attack on the strong cosmic censorship question might therefore be a study of the structure of Cauchy horizons to see if this structure is compatible with other "physically reasonable" conditions that we might expect to hold. In this note we will consider the case of compact Cauchy horizons. Such horizons have been studied before [3], notably by Moncrief and Isenberg [4] who have recently shown that compact Cauchy horizons are necessarily Killing horizons (under the assumption that the null generators of the horizon are closed) and hence may be expected to exist only in very special cases. Below we will obtain a rather simple result along the same lines (that compact Cauchy horizons can exist only in certain special cases). The notation and conventions that will be used are those of Hawking and Ellis [3].
Theorem: Let (M, g) be a space-time such that : 0) RabKaK b >1 0 for all null vectors K a (where Rab is 224
the Ricci tensor associated with the metric g), and (ii) the Hawking-Penrose generic condition holds [3,5]. Then, for any partial Cauchy surface S, if the Cauchy horizon H+(S) is not empty it must be non-compact [similary f o r H (S)]. This theorem does not yield the detailed information that the Moncrief-Isenberg theorem does, but on the other hand its assumptions (which are briefly discussed below) are very weak and very general. It should be noted that results on the non-existence of compact Cauchy horizons have other applications in addition to their relevance to the cosmic censorship question. For example, in the Hawking-Penrose singularity theorem [5] it is established, using the chronology condition, that H+(S) cannot be compact for certain surfaces S. This non-compactness of H+(S) then plays a crucial role in the singularity theorem. For compact S the results of this paper provide an alternative proof, without any causality assumptions, that H+(S) is not compact in reasonable space-times. The application of the theorem in this paper to singularity results is discussed in greater detail elsewhere [81. Conditions (i) and (ii) have both been discussed fairly extensively in the literature on singularity theorems in general relativity [3,5] and so an extended discussion will not be given here. Condition (i) follows if we assume the Einstein equations and that the energy density of matter as seen by any observer is non-negative (the weak energy condition). Condition (ii) essentially requires that every geodesic feel some non-zero 0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 102A, number 5,6
PHYSICS LETTERS
gravitational tidal force at least at one point along it. This is generally regarded as a very weak requirement on any reasonable space-time. It does not hold in many known space-times that possess symmetries (e.g. radial null geodesics in the Schwarzschild spacetime do not obey it), but one would expect it to hold in the slightest deviation from these symmetries. Before proceeding with the proof of the theorem a comment on non-compact Cauchy horizons is in order. It appears to be impossible to rule out the existence of non-compact Cauchy horizons under only the very general assumptions that we are making in the compact case. This is because non-compact Cauchy horizons can arise even in globally hyperbolic spacetimes just because of a choice of Cauchy surface. For example, any asymptotically null partial Cauchy surface in Minkowski space-time (or, for that matter, in any asymptotically fiat space-time) would have a non-compact Cauchy horizon associated with it. We would, therefore, have to make additional assumptions about the choice of Cauchy surface being a "good" one, or equivalently the Cauchy horizon being "nontrivial". Such assumptions appear to be hard to pin down precisely without making them too strong. We might be tempted to require, for example, that the Cauchy surface be such that its future domain of dependence contain a future inextendible causal curve. Such an assumption would rule out asymptotically null partial Cauchy surfaces. It would, however, also exclude anti-DeSitter space-time [3] entirely from our considerations and hence is too strong an assumption to make. We will not discuss this question any further here. We now turn to the proof of the theorem. This follows almost immediately from the following two standard results:
Lemma 1: If H is a compact Cauchy horizon in a space-time obeying condition (i) of the theorem, then the null generators of H have zero divergence (ref. [3], pp. 295 - 298). Lemma 2: If ~, is a null geodesic in a space-time such that: (a) condition (i) of the theorem holds, (b) the generic condition holds at some point b on 7, and (c) the divergence 0 of a congruence of null geodesics neighboring 3, is zero at b. Then 0 becomes negative to the past o f b on 7 (ref. [3], pp. 9 8 - 1 0 1 ) .
21 May 1984
Now consider a null generator ~t of some compact future Cauchy horizon H. If~t has no endpoint on H, then the fact that the generic condition holds at some point ofju will force a contradiction between lemmas 1 and 2. (If,u had a future endpoint p on H a n d if the generic condition held on ~t to the future o f p we would not necessarily obtain such a contradiction). The existence o f g is shown by:
Lemma 3: Let S be a partial Cauchy surface such that the future Cauchy horizon H+(S) is not empty and is compact. Then there exists a generator of H+(S) with no future or past endpoint o n H ÷ (S). Proof: By the definition of a partial Cauchy surface, S has no edge. Therefore H = H+(S) also has no edge and so its null generators have no past endpoints (ref. [3], proposition 6.5.3). Let 7 be any null generator of H. If it is closed then it has no future endpoint on H and is the required generator. If it is not closed we may choose a sequence of points {bi} on ')" such that bi+ 1 @J-(bi)for each i and no compact segment of 7 contains an infinite number of the b i. Since H is compact, the sequence £bi} will have a limit point b E H . The portions of')' to the future of each b i will then yield a sequence o f causal curves with limit point b. We will show that the causal limit curve through b to this sequence is the required generator. To do this, pick a point c on H a s follows: ifb E3', then c is any point on 7 to the past of b; ifb ~ 7, then c is b 2. In either case only a finite number of the b i will lie to the future of c. Let/a i stand for the portion of 7 between those b i that lie to the past of c, and c. Then in H ' = H - {c}, the point b is a limit of the sequence of future inextendible null geodesics {/ai}. There is, therefore, a future inextendible (in H ' ) limit causal curve to this sequence through b. Call this curve ~t./.t must be a null generator of H. Its only possible future endpoint on H is c itself. However, suppose that c is the future endpoint of/a o n H . If b ~ "/then c was b 2 and so we have b E J - ( c ) and d E J - ( b l ) and hence b E J - ( b l ) . But the causal curve from c to b 1 was 7 on which b does not lie and so b E I - ( b l), violating the achronality of H. And if b E 7 then c was chosen so that c ~ J - ( b ) . But now b E J - ( e ) and so 3' must be a closed null geodesic. Thus c cannot be the future endpoint of p and hence /.t is the required generator of H. 225
Volume 102A, number 5,6
PHYSICS LETTERS
I would like to thank J. Isenberg and W. U n r u h for useful discussions and R. Riley for e n c o u r a g e m e n t . This research was supported by the National Sciences and Engineering Research Council o f Canada.
References [ I I R. Penrose, Singularities and time asymmetry, in: General relativity: an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge Univ. Press, London, 1979), and references therein.
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21 May 1984
[2] F. Tipler, C. Clarke and G. Ellis, Singularities and horizons: a review article, in: General relativity and gravitation, ed. A. Held (Plenum, New York, 1980), and references therein on Cauchy horizon stability; S. Chandrasekhar and J. Hartle, On crossing the Cauchy horizon of a black hole, Univ. of Chicago preprint ( 1982). [3] S.W. Hawking and G.F.R. Ellis: The large scale structure of space-time (Cambridge Univ. Press, London, 1973). [4] V. Moncrief and J. Isenberg, Comm. Math. Phys. 89 (1983) 387. [5] S.W. Hawking and R. Penrose, Proc. R. Soc. A314 (1970) 529. [6] A. Borde, Singularities in closed sgace-times, U.B.C. preprint (1984).