A few neat bits from Geroch and Horowitz

One of my favorite GR paper is "Global structure of spacetimes", by Robert Geroch and Gary Horowitz, included in Hawking and Israel's "General relativity : an Einstein centenary survey". It contains quite a few proofs and ideas rarely repeated elsewhere, as well as the following few interesting passages :

First : a brief description of the general methods used for proofs in the study of spacetime global structures in the paper :

The role of key theorems has been assumed instead by 'methods of proof', for indeed there are a limited number of these, which do occur over and over. In fact, the following eight methods of proof will suffice for most results in this subject : 'Introduce a timelike vector field (and, usually, then consider its integral curves)', 'Carry information about closed curves and compare with what one had at the beginning', 'Connect timelike or null curves together, and smooth the corners, to obtain new curves', 'Find a sequence of points with some property and ask for an accumulation point', 'Choose a sequence of points which approach the point in question (usually from a timelike direction) and try to carry some property to the limit', 'Ask for the timelike curve of maximal length between two points, or between a point and a surface', 'Take the domain of dependence, and study the Cauchy horizon', and 'Find a sequence of timelike curves with some property, and ask for an accumulation curves'.

General tips for constructing spacetime examples :

The examples, although they are particularly numerous in this subject, are in practice rather easy to obtain. Indeed, the following seven techniques (or, more often, combinations of two or more) will normally produce the desired example : 'Check the known solutions', 'Tip the light-cones in some way', 'Take a covering space or product', 'Isolate what makes an example fail in a local region, and push that region off to infinity', 'Introduce a conformal factor', 'Patch spacetimes together across boundary surfaces' and 'Cut holes of various types in given spacetimes'.

And this heap of random theorems presented without proofs as a quiz (Answers included):

Prove or find a counterexample to the following assertions. Take all spacetimes to be four-dimensional, and, for assertions involving causal structure, time-oriented. All manifolds are to be Hausdorff, paracompact, connected, and without boundary.

Manifolds, metrics, orientation

  1. A manifold admits a Lorentz metric if and only if its universal covering manifold does. (FALSE)
  2. Every flat spacetime admits a nonzero Killing field. (FALSE)
  3. Call two Lorentz metrics on $M$ equivalent if there is a diffeomorphism from $M$ to $M$ which sends one to the other. Then each of two equivalent metrics can be continuously deformed to the other. (FALSE)
  4. For $M$ a, a manifold, and $p$, a point of $M$, $M$ with $p$ removed is not diffeomorphic with $M$. (FALSE)
  5. If spacetime $M, g_{ab}$ has vanishing Weyl tensor, then $\Omega^2 g_{ab}$ is flat for some $\Omega$. (FALSE)
  6. If, for $C$ a closed subset of $M$, $M \setminus C$ admits a Lorentz metric, then so does $M$. (FALSE)
  7. For $S$, a two-dimensional manifold, and $p$, a point of $S$, $S \setminus p$ is not simply connected. (FALSE)
  8. The product of two simply connected manifolds is simply connected. (TRUE)
  9. If, for $S$ and $S'$, three-dimensional manifolds, $S \times \mathbb{R}$ and $S' \times \mathbb{R}$ are diffeomorphic then $S$ and $S'$ are diffeomorphic. (FALSE)
  10. No compact spacetime is extendible. (TRUE)
  11. Every simply connected $4$-manifold which is a product admits a Lorentz metric, except $S^2 \times S^2$. (TRUE)
  12. A spacetime is extendible if and only if its universal covering spacetime is. (FALSE)
  13. A non-simply connected manifold which admits a Lorentz metric admits one which is neither time-orientable nor space-orientable. (FALSE)
  14. A flat spacetime is time-orientable. (FALSE)
  15. If each of two spacetimes can be embedded isometrically as an open subset of the other, then the spacetimes are isometric. (FALSE)
  16. The sum of two complete vector fields is complete. (FALSE)

Domain of influence

  1. If the futures of two points do not intersect, then neither do their pasts. (FALSE)
  2. A non-empty subset of a spacetime equal to its own future and its own past is the entire spacetime. (TRUE)
  3. A non-compact manifold admits a stably causal Lorentz metric. (TRUE)
  4. A stably causal spacetime admits a time-function $t$ with all sub-manifolds $t = \text{constant}$ connected. (FALSE)
  5. A manifold which admits a Lorentz metric admits a stably causal Lorentz metric. (FALSE)
  6. Every compact spacetime admits a closed timelike curve through every point. (FALSE)
  7. A spacetime admitting a closed timelike curve admits a closed null curve. (TRUE)
  8. For $p$ a point of a spacetime, the past of the future of the past... of $p$ is the entire spacetime. (TRUE)
  9. Fix a point $p$ of a spacetime. We say that $p$ has index $n$ if the past of the future of the past... ($n$ times) of $p$ is the entire spacetime, while this fails for $(n-1)$. Every spacetime has finite index. For every positive integer, there exists a spacetime with that integer as index. Every compact spacetime has finite index. (FALSE, TRUE, TRUE)
  10. No spacetime has exactly one closed timelike curve; exactly one closed null curve. (TRUE, FALSE)
  11. Every local Maxwell field in the example of figure $5.9$ can be extended to a global field. (TRUE)
  12. The set of points a spacetime through which there pass closed timelike curves is open. (TRUE)
  13. If there passes a closed timelike curve through $p$, and $q$ precedes $p$, then there passes a closed timelike curve through $q$. (FALSE)
  14. There exists a spacetime with points $p$ and $q$ such that these can be joined simultaneously by timelike, null and spacelike geodesics. (TRUE)
  15. If the past of $p$ intersects the future of $q$, then the past of $p$ contains $q$. (TRUE)
  16. Any two points of a spacetime can be joined by a spacelike curve. (TRUE)
  17. For any timelike vector field $t_a$ in Minkowski spacetime, $g_{ab} - t_at_b$ admits no closed timelike curves. (FALSE)
  18. In a stably causal spacetime, two points with the same future are the same. (TRUE)
  19. If the future of $p$ is the entire spacetime, then through every point of that spacetime there passes a closed timelike curve. (FALSE)
  20. If a spacetime violates stable causality, then some open subset of that spacetime with compact closure violates stable causality. (FALSE)
  21. If $M, g_{ab}$ is stably causal then, for some timelike $t_a$, $M, g_{ab} - t_at_b$ is stably causal. (TRUE)
  22. If the past of $p$ contains the future of $q$, then there is a closed timelike curve through $p$. (TRUE)
  23. If a stably causal spacetime has an extension, then it has a stably causal one. (TRUE)

Slices, domain of dependence

  1. Every compact slice in a spacetime with Cauchy surface is itself a Cauchy surface.35 (TRUE)
  2. No compact spacetime admits an achronal slice. (FALSE)
  3. Let $S$ be a three-dimensional manifold, and set $M = S times \mathbb{R}$. Then there exists a Lorentz metric on $M$ which admits a Cauchy surface. (TRUE)
  4. The future Cauchy horizon is achronal. (TRUE)
  5. Through a point of the future Cauchy horizon there passes a unique past-directed null geodesic remaining in that horizon. (FALSE)
  6. All Cauchy surfaces for a spacetime are diffeomorphic. (TRUE)
  7. No spacetime with non-compact Cauchy surface admits a compact slice. (TRUE)
  8. Through a point of the future Cauchy horizon there passes a future-directed null geodesic remaining, at least for a while, in that horizon. (FALSE)
  9. If all connected slices in a spacetime are achronal, then it admits a Cauchy surface. (FALSE)
  10. In a spacetime with Cauchy surface, every connected slice is achronal. (FALSE)
  11. Let $S$ be an achronal slice not a Cauchy surface. Then the past of the future Cauchy horizon of $S$ includes $S$. (FALSE)
  12. Let $S$ be an achronal slice. Then the future domain of dependence of any slice in $D^+(S)$ is itself in $D^+(S)$. (TRUE)
  13. Every maximally extended, past-directed null geodesic from a point of $D^+(S)$ meets $S$. (FALSE)
  14. For $p$ and $q$ in $D^+(S)$, with $p$ preceding $q$, there is a timelike geodesic joining $p$ and $q$. (FALSE)
  15. If the non-empty future Cauchy horizon of $S$ is compact, so is $S$. (TRUE)
  16. Two slices having the same domain of dependence are the same. (FALSE)
  17. If two points of a stably causal spacetime can be joined by neither a timelike nor null curve, then some connected achronal slice contains them both. (FALSE)
  18. If $p$ precedes $q$ in a spacetime with Cauchy surface, then there is a Cauchy surface having $p$ in its past and $q$ in its future. (TRUE)
  19. If $S$ is a Cauchy surface, then for no $p$ is $S$ in $I^-(p)$. (TRUE)
  20. If through every point of a spacetime there passes an achronal slice, then that spacetime is stably causal. (FALSE)
  21. A simply connected spacetime which has a closed timelike curve through every point admits no slices. (TRUE)
  22. Every maximally extended null geodesic meets a Cauchy surface. (TRUE)
  23. No null curve can meet an achronal slice more than once. (TRUE)
  24. For $S$ a Cauchy surface, every point of the spacetime is on $S$, in its future, or in its past. (TRUE)
  25. If the closure of $I^-(p)$ meets achronal slice $S$ compactly, then $p$ is in the future domain of dependence of $S$. (FALSE)
  26. A spacetime with orientable Cauchy surface is space-orientable. (TRUE)

Singular spacetimes

  1. Every spacetime is conformally related to one which is timelike and null geodesically complete. (FALSE)
  2. Two points, one of which precedes the other, in a geodesically complete spacetime can be joined by a timelike geodesic. (FALSE)
  3. If a spacetime has a compact Cauchy surface with converging normal, and satisfies the energy condition, then every timelike geodesic is incomplete. (TRUE)
  4. Every non-compact manifold admits a geodesically complete Lorentz metric. (TRUE)
  5. Every compact spacetime is geodesically complete. (FALSE)
  6. In a geodesically complete spacetime, every maximally extended timelike curve has infinite length.(FALSE)


(35) Budic, R. et al. (1978). Commun. Math. Phys., 61, 87

These all sound like fairly easy theorems with non-trivial solutions (a lot of those are indeed true in Minkowski space or other non-pathological spacetimes). I will try to prove these later on in future posts.

Posted on 2019-02-27 11:20:54
Tags : general-relativity , physics