result it is impossible to prove the invariance of the causal relations defined in section 9.1 under the equivalence relation on boundary sets.
9.3.1 Lemma. Let M be a space-time, let φ : M → Mφ be a boundary strongly
causal envelopment, let γ : [0,1) → M be a future directed, timelike curve with no limit points in M so thatlimt→1γ0 exists and is non-zero and, lastly, let B ⊂∂φM
so that γ approaches B. Then φ(γ(0))∈Ij−(B;φM).
Proof. Since γ approaches B, there exists x ∈Wφγ, the set of limit points of φγ in
Mφ, where x ∈ B. Suppose there also exists y ∈ Wφγ, where x 6= y. It is clear
that y ∈∂φM. Choose disjoint open neighbourhoodsUx ∈ Nφ(x) and Uy ∈ Nφ(y).
As φ satisfies the boundary strong causality condition we know that there exists
Vy ∈ Nφ(y),Vy ⊂Uy, so thatγ does not exit and then return toVy. This contradicts
that both x and y are limit points of φγ. It follows that Wφγ ={x}.
By lemma 8.2.11 and the assumption, φγ|(0,1) is a future directed, timelike, join- ing curve from φ(γ(0)) to x, or φ(γ(0)) ∈ Ij−(x;φM) and , since Ij−(x;φM) ⊂
Ij−(B;φM), we have our result.
We now continue to show that, with the assumption of the Abstract Boundary strong causality condition, we can give a definition of causality for the Abstract Boundary.
9.4 Conclusions and future work 155 9.3.2 Lemma. Let M be a space-time and let φ : M → Mφ and ϕ : M → Mϕ
be two boundary strongly causal envelopments. Let Bφ ⊂∂φM and Bϕ ⊂∂ϕM. If
BφBϕ then φ−1 Ij+(Bφ;φM)∩φM ⊃ϕ−1I+ j (Bϕ;ϕM)∩ϕM . Proof. Lety∈ϕ−1Ij+(Bϕ;ϕM)∩ϕM
, so that there exists a past directed timelike joining curve, ϕγ, from y to x ∈Bϕ. That is γ approaches Bϕ. Since BφBϕ we
see that γ approaches Bφ, hence by lemma 9.3.1 y ∈ φ−1
Ij+(Bφ;φM)∩φM
, as required.
9.3.3 Theorem. Let M be a space-time and let φ :M → Mφ and ϕ :M → Mϕ
be two boundary strongly causal envelopments. Let Bφ ⊂∂φM and Bϕ ⊂∂ϕM. If
Bφ ≡Bϕ, then φ−1 Ij+(Bφ;φM)∩φM =ϕ−1Ij+(Bϕ;ϕM)∩ϕM . Proof. This is clear from lemma 9.3.2.
Theorem 9.3.3 proves the invariance of our definition of causality on the boundary under the equivalence on boundary sets. Hence we can make the following defini- tion.
9.3.4 Definition. Let [B] be an Abstract Boundary set where B ⊂ ∂φM and φ
is boundary strongly causal, then define Ij+([B]) = φ−1I+
j (B;φM)∩φM
. By theorem 9.3.3 this definition is well defined.
Although it took an assumption, which restricts the sort of limiting behaviour that causal curves are allowed, we have shown that it is possible to define an analog of causality for the Abstract Boundary.
Note that the definition above applies to all Abstract Boundary sets, not just Ab- stract Boundary points and that the past/future of an Abstract Boundary set in- volves just points in the manifold, not other Abstract Boundary sets.
Note also that care must be taken when using the joining causal relations, since they do not satisfy the same conditions as the usual causality conditions, for example they are not transitive.
9.4
Conclusions and future work
We have given a definition of causality on the boundary of a space-time in some larger embedding, and shown how this can be generalised to the Abstract Bound-
ary. We have had to restrict to a specific class of envelopments in order to achieve this. This work is important because it gives us additional physical interpretations for the Abstract Boundary, more tools to analyse the Abstract Boundary and, we believe, will lead to important results. In particular, there are direct applications to the development of weak cosmic censorship results that assume no boundary con- ditions on the space-time. This chapter takes the first small steps towards giving the mathematical structure necessary to define things like future timelike infinity for general space-times.
It is clear that more work needs to be done. In particular: the consequences of restricting the class of envelopments needs to be considered; conjecture 9.2.2 should be examined and, if not true, counter examples should be produced and applied to proposition 8.2.15; a thorough review of the properties of Ij+(x;M) and ≤j should
be conducted by trying to generalise the results in [84]; lastly, further development of concepts related to causality and global techniques for the Abstract Boundary is needed.
Part IV
Physical predictions from
singularity theorems
Chapter 10
Consequences of the
Penrose-Hawking singularity
theorems
Ashley and Scott have advocated, [5], applying the Abstract Boundary singularity theorem, theorem 3.3.3, to singularity results by linking ‘physical’ tensors and the Penrose-Hawking singularity theorems.
The Penrose and Hawking singularity theorems prove the existence of incomplete, inextendible geodesics. Little is said about the properties of these geodesics. The few conclusions that can be made, e.g., that they are conjugate point free, say nothing about the behaviour of the metric or Riemann tensor. The same is true for most singularity theorems. Those that do make physical predictions about such geodesics are usually highly specialised, applying to a small class of space-times, [101]. The assumptions of the singularity theorems, particularly the existence of a closed, trapped surface can be equated with gravitational collapse; see for example [8] or [96]. Indeed Hawking has commented in [58] that,
The main, indeed the only, reason for believing that singularities occur in gravitational collapse is the singularity theorems.
Thus results linking the behaviour of ‘physical’ tensors to the geodesics predicted by singularity theorems are of great importance. First, they would make predictions about the behaviour of known objects in our universe. These predictions can then be compared to observation, e.g., of suspected black holes. Second, they would dis- cuss the points where general relativity fails to apply, so would provide additional inspiration to more general theories, e.g., loop quantum gravity, string theory etc.
Third, they would also provide additional data for investigation of any topic involv- ing either an assumption of gravitational collapse or of the existence of singularities, e.g., the Penrose inequality [80] or cosmic censorship [72].
Unfortunately, it is not easy to prove such results. The predictions of the most general singularity theorems are vague and the assumptions, because of their math- ematical form, are hard to link to physical conditions. Much, much work has been done on the study of singularities with varying degrees of success (see [101] for a review). Yet there has been no conclusive answer. This problem remains one of the longest standing unsolved problems in general relativity. Here we review the Penrose-Hawking singularity theorems with the intention of outlining what facts can be concluded for the predicted incomplete, inextendible geodesics. The original papers are [57], [60] and [83] for our analysis we have relied heavily on [7] and [59]. The chapter is divided into six sections, one for each of the theorems, a summary of our conclusions and a discussion of additional work. Since this chapter contains a great many standard definitions and to include them would obscure the results below we have given them in the appendix. Please refer, in particular to sections A.4, A.5, A.6 and A.7.
10.1
Theorem 1 of [59]
10.1.1 Theorem ([59, page 163]). Space-time (M, g) cannot be null geodesically complete if:
1. For all null vectors, K, RabKaKb ≥0,
2. There is a non-compact cauchy surface H in M,
3. There is a closed, trapped surface T in M.
Outline of proof. Suppose that M is null geodesically complete. Since there is a cauchy surface, M is globally hyperbolic and therefore ∂J+(T) = I+(T)\I+(T). Conditions 1 and 3 imply that every geodesic orthogonal toT contains a focal point to T. This fact can be used to construct a map β :T ×[0, c]× {0,1} → M so that its image contains ∂J+(T), where cis an upper bound on the affine distance along the geodesics orthogonal to T for the focal point to T. Hence ∂J+(T) is compact. From this a contradiction can be derived by using the non-compactness of H.
10.2 Theorem 2 of [59] 161