A poor man’s positive energy theorem: II. Null geodesics

Piotr T. Chruściel^{∗} Département de Mathématiques Faculté des Sciences Parc de Grandmont F37200 Tours, France

Abstract

We show that positivity of energy for stationary, or strongly uniformly Schwarzschildian, asymptotically flat, non-singular domains of outer communications can be proved using Galloway’s null rigidity theorem.

1 Introduction

In a recent note [4] we have shown that positivity of energy for uniformly Schwarzschildian, asymptotically flat, non-singular domains of outer communications can be proved using timelike lines together with the Lorentzian splitting theorem. The object of this paper is to show that a variation of the argument of [4], using null lines, can successfully be completed in a similar, “strongly uniformly Schwarzschildian”, setting. The extension of the result to more general metrics, as suggested in [13], still awaits a more complete justification.

For m ∈ ℝ, let g_{m} denote the n + 1 dimensional, n ≥ 3, Schwarzschild metric with mass parameter m; in isotropic coordinates [11],

g_{m} = (1 + \frac{m}{2|x|^{n−2}} )^{\frac{4}{n−2}} (∑_{1=1}^{n} dx_{i}^{2}) − (\frac{1 − m/2|x|^{n−2}}{1 + m/2|x|^{n−2}} )^{2} dt^{2} . (1.1)

∗Partially supported by a Polish Research Committee grant 2 P03B 073 24; email Piotr.Chrusciel@lmpt.univ-tours.fr, URL www.phys.univ-tours.fr/∼piotr

We shall say that a metric g on ℝ × (ℝ^{n} \ B(0, R)), R^{n−2} > m/2, is strongly uniformly Schwarzchildian if, in the coordinates of (1.1),

g − g_{m} = O(|m|r^{−(n−1)}) , ∂_{μ} (g − g_{m}) = O(|m|r^{−n}) ,  (1.2)
∂_{μ}∂_{ν} (g − g_{m}) = O(|m|r^{−n−1}) .  (1.3)

(Here O is meant at fixed g and m, uniformly in t and in angular variables, with r going to infinity.) What is meant in the case m = 0 is that g = g_{0}, i.e., g is flat^{1}, for r > R. We note that this condition is more restrictive than the “uniformly Schwarzschildian” one in [4].

We will use the symbol r to denote some fixed smooth function on ℳ which coincides with |x| in ℳ_{ext}.

The domain of outer communications ⋖ℳ_{ext}⋗ associated to ℳ_{ext} is defined as the intersection of the causal past J^{−}(ℳ_{ext}) of the asymptotic region

ℳ_{ext} = ℝ × (ℝ^{n} \ B(0, R))  (1.4)

with its causal future J^{+}(ℳ_{ext}).

We need a version of weak asymptotic simplicity [6] for uniformly Schwarzschildian spacetimes. Following [4], we shall say that such a spacetime (ℳ , g) is weakly asymptotically regular if every null line starting in the domain of outer communications ⋖ℳ_{ext}⋗ either crosses an event horizon (if any), or reaches arbitrarily large values of r in the asymptotically flat regions. Recall that a null line in (ℳ , g) is an inextendible null geodesic that is globally achronal.

The first main result of this paper is a simple proof of the following:

Theorem 1.1 Let (ℳ^{n+1} = ℳ , g) be a (n + 1)-dimensional weakly asymptotically regular space-time containing a strongly uniformly Schwarzschildian region ℳ_{ext} with

m ≤ 0 .

If the domain of outer communications associated to ℳ_{ext} has a Cauchy surface 𝒮, the closure of which is the union of one asymptotic end and of a compact interior region (with a differentiable boundary lying at the intersection of the future and past event horizons, if any), then ℳ contains a null line.

1The asymptotic conditions for the case m = 0 of our theorem are way too strong for a rigidity statement of real interest, even for stationary metrics, so our results only exclude m < 0 in practice. Nevertheless, in this context one should keep in mind an argument of Lohkamp [9] for initial data sets with trace of K = 0, which shows that the proof of positivity of mass is obtained in its full generality once it has been established for data which are flat outside of a compact set.

Theorem 1.1 implies non-existence of appropriately regular, uniformly Schwarzschildian space-times with negative mass in various situations of interest. For example, consider a uniformly Schwarzschildian fourdimensional, asymptotically simple, vacuum space-time (ℳ , g) with m ≤ 0. It then follows from Theorem 1.1 and from Galloway’s null rigidity theorem [5] that (ℳ , g) is the Minkowski space-time. This result is somewhat stronger than the one in [4], because no hypotheses on timelike geodesics are made (we will show below that, similarly to [4], in our context the hypothesis of asymptotic simplicity needed in [5] can be replaced by the weak version thereof). On the other hand, there is a restriction on space-time dimension, not made in [4].

The original idea in [12] was that space-times containing null lines cannot satisfy the so-called “genericity condition”. However, it is not clear how generic is “generic” when, e.g., vacuum equations are imposed, and therefore, it is not clear how much information is carried by theorems basing upon “genericity”. For this reason it appears useful to develop an argument which does not invoke this condition. As a first step towards this, assuming the null energy condition,

R_{μν}X^{μ}X^{ν} ≥ 0 for all null vectors X^{μ},  (1.5)

we prove the following:

Theorem 1.2 Under the hypotheses of Theorem 1.1, suppose moreover that the null energy condition (1.5) holds. Then

ℳ = ⋖ℳ_{ext}⋗ ,

and every maximally extended null geodesic is a line. Further, for each k⃗ ∈ S^{n−2} there exists a one-parameter family 𝒩k⃗ ≡ {𝒩_{k⃗,T} }_{T ∈ℝ}, with 𝒩k⃗′ ≠ 𝒩k⃗ if k⃗ ≠ k⃗′, of smooth, closed, null, totally geodesic, achronal hypersurfaces covering ℳ.

With a little more effort one can show that each family 𝒩k⃗ forms a foliation; we give no details as we will not pursue this line of approach. Now, each foliation of ℳ by null hypersurfaces as described in Theorem 1.2 defines a shear-free, rotation-free, and expansion-free congruence of null geodesics covering ℳ. Space-times admitting one such congruence are expected to be very special^{2}, while we actually have a family of congruences parameterized by S^{n−1}. We therefore expect that the only space-time satisfying the hypotheses of Theorem 1.2 is the Minkowski one, but we are not aware any results to this effect without further hypotheses (compare [10]). Assuming n equal to four, as well as existence of a smooth conformal completion, we will be able to reach this conclusion by an argument that does not proceed via the algebraically special structure of the Riemann tensor; this is the main result of this paper:

2See, e.g., [8, Section 7.6 and Chapter 31]. The results there are based upon a supplementary assumption on the vanishing of some components of the Ricci tensor. Note, however, that it follows from the Raychaudhuri equation that this condition will be satisfied in our case if the dominant energy condition is assumed.

Theorem 1.3 Let (ℳ^{3+1} = ℳ , g) be a four dimensional space-time satisfying the null energy condition (1.5), and suppose that ℳ contains a strongly uniformly Schwarzschildian region ℳ_{ext} = ℝ × (ℝ^{3} \ B(0, R)) which admits a smooth conformal completion at null infinity. Assume that (ℳ , g) is weakly asymptotically regular, and that the domain of outer communications of ℳ has a Cauchy surface 𝒮, the closure of which is the union of one asymptotic end and of a compact interior region (with a differentiable boundary lying at the intersection of the future and past event horizons, if any). Then

m > 0 ,

unless (ℳ , g) is the Minkowski space-time.

It is of interest to compare the results and proofs here to those in [4]. Recall that we are trying to implement the original idea of Penrose [12], that negative mass should imply existence of a null line in space-time, which then should be incompatible with energy conditions. In [4] we have shown, under the hypotheses there, that negative mass implies existence of a timelike line, and that this is compatible with the timelike convergence condition,

R_{μν}X^{μ}X^{ν} ≥ 0 for all timelike vectors X^{μ},  (1.6)

only if the space-time is the Minkowski one. Both in [4] and here we make the unsatisfactory hypothesis that the metric is uniformly Schwarzschildian. This is, however, a quite reasonable hypothesis for stationary space-times with m ≠ 0 (see, e.g., [14]). In the stationary case the only global hypothesis needed for both constructions is that of existence of an asymptotically flat Cauchy surface for the domain of outer communications, with conditionally compact interior. For stationary metrics the proof of existence of lines is completely elementary for timelike lines in [4], and marginally more complicated for null lines here.

It is not clear at all whether there exist non-stationary solutions of physically reasonable field equations that are uniformly Schwarzschildian, and therefore assuming that last condition without assuming stationarity is presumably an academic exercise. We have nevertheless carried this out, in order to test the limits of the techniques employed, and in the hope that the arguments will eventually generalise to a proof under less stringent asymptotic conditions. As already pointed out, in the non-stationary case the construction of a null line is only slightly more involved than that of a timelike one. On the other hand, the conclusion that the space-time must be flat requires considerably more work. However, the proof of existence of a timelike line needs a supplementary global hypothesis concerning timelike geodesics, which is avoided in the current setting. This last fact provides yet another motivation for the work here.

The next step in the positivity argument is to show that space-times containing lines and satisfying energy conditions are very special. In considerations involving timelike lines the natural energy condition is the unphysical inequality (1.6). From this point of view null lines are much more satisfactory, as their global causality properties are tied to the physically motivated null energy condition (1.5). This is the final motivation for our work.

This paper is organised as follows: In Section 2 we give the proofs of our results. Appendix A contains some technical results on geodesics in uniformly Schwarzschildian metrics, as needed in the proofs of Theorems 1.2 and 1.3.

2 Proofs

We start with a proof of Theorem 1.1. We will be sketchy in places, assuming that the reader is familiar with the argument in [4].

Let x^{μ} be the coordinates of (1.1) (x^{0} = t), and let the indices a, b, .. run from 1 to n − 1, where n is the space-dimension. Using (A.6)-(A.7) in the appendix below, one finds that the function

ρ := \sqrt{∑_{a=1}^{n−1}(x^{a})^{2}}

has convexity properties somewhat similar to those of r, as exploited in [4]:

Hess ρ = −\frac{mρ}{r^{n}} \{(n − 2)dt^{2} + ∑_{i=1}^{n}(dx^{i})^{2}\} + \frac{2m}{r^{n−1}} dρdr

+\frac{1}{ρ} \{∑_{a=1}^{n−1}(dx^{a})^{2} − dρ^{2}\} + ∑_{μ,ν=0}^{n} ∑_{a=1}^{n−1} \frac{x^{a}}{ρ} O(r^{−n})dx^{μ}dx^{ν} .(2.1)

On level sets of ρ all terms involving dρ drop out, then both expressions in braces are manifestly positive for negative m. In order to show that they dominate all the remaining terms when ρ ≥ R̊ for some R̊, one needs to check that

\frac{|m|ρ}{r^{n}} ≥ c(n)\frac{x^{a}}{ρ} O (\frac{1}{r^{n}} ) ⟺ |m|ρ^{2} ≥ c(n)x^{a}O(1) ,  (2.2)

where c(n) is a (large) dimension-dependent constant. Equation (2.2) will clearly hold for ρ large enough if m ≠ 0. It follows that Hess ρ, when restricted to {ρ = R}, with R ≥ R̊, is positive definite when m < 0. This shows, as in [4], for negative or vanishing m, that any geodesic segment Γ with initial point p and final point q such that ρ(p), ρ(q) < R̊ will satisfy

ρ ∘ Γ < R̊ .  (2.3)

We start with a lemma^{3}:

Lemma 2.1 For every p ∈ J^{−}(ℳ_{ext}) there exists an achronal, future inextendible, null geodesic ray

Γ_{p}^{+} : [0, ∞) → J^{−}(ℳ_{ext})

such that

Γ_{p}^{+}(0) = p , ρ ∘ (Γ_{p}^{+}) ≤ M ,

for some constant M = M(p).

Proof: Since p ∈ J^{−}(ℳ_{ext}) for any N ∈ ℕ there exists a causal curve from p to a point

p_{N}^{+} = (t_{N}^{+}, 0, ⋯, 0, N) ∈ ℳ_{ext} ,  (2.4)

for some t_{N}^{+}. Minimising the time of arrival t_{N}^{+} over all such causal curves one obtains, by global hyperbolicity, a null achronal geodesic segment Γ_{N}^{+} from p to perhaps a different point p_{N}^{+}, still of the form (2.4). By the convexity properties of the function ρ, discussed above, we have the bound ρ ∘ Γ_{N}^{+} ≤ M. Global hyperbolicity implies that there exists a subsequence of the sequence Γ_{N} accumulating at the desired inextendible null geodesic ray Γ_{p}^{+}. Achronality of Γ_{p}^{+} follows from the fact that an accumulation curve of achronal curves is achronal. □

It immediately follows that:

3I am grateful to G.Galloway for providing this simple proof.

Corollary 2.2 For every p ∈ ⋖ℳ_{ext}⋗ there exist achronal, future inextendible, null geodesic rays

Γ_{p}^{+} : [0, ∞) → ⋖ℳ_{ext} ⋗ and Γ_{p}^{−} : (−∞, 0] → ⋖ℳ_{ext}⋗

such that

Γ_{p}^{±}(0) = p , ρ ∘ (Γ_{p}^{±}) ≤ M ,

for some constant M = M(p).

Choose R̊ large enough so that the hypersurfaces {x^{n} = ±R̊} are closed boundaryless in ℳ_{ext}. Increasing R̊ we can also assume that ∂_{t} is timelike for r ≥ R̊, that the slopes of the light cones in the region r ≥ R̊ are between 1/2 and 2, and that Hess ρ is positive definite on the level sets {ρ = R} for all R ≥ R̊.

Choose τ ∈ ℝ, let p = (τ, 0, …, 0, −R̊), set

Γ^{+} := Γ_{p}^{+} , Γ^{−} := Γ_{p}^{−} ,

and define

𝒩 ^{+} := J̇^{−}(Γ^{+}) , 𝒩 ^{−} := J̇^{+}(Γ^{−}) .

Since the Γ^{±}’s are achronal we have Γ^{±} ⊂ 𝒩 ^{±}, thus the 𝒩 ^{±}’s are nonempty, closed, achronal, locally Lipschitz submanifolds of ℳ. Denote by

𝒫^{±} := 𝒩 ^{±} ∩ {x^{n} = −R̊} .

The manifolds {x^{0} = ±R̊} with the induced metric are globally hyperbolic submanifolds of (ℳ , g), with the 𝒫^{±} non-empty (as Γ^{±} ∩ {r = ±R̊} = p ∈ 𝒫^{±}), which shows that the 𝒫^{±}’s are closed, achronal, locally Lipschitz submanifolds of {x^{n} = −R̊}, and can thus be written as graphs

𝒫^{±} = {t = s^{±}(x^{a})} ,

for some locally Lipschitz functions s^{±}.

In the coordinates of (1.1) the null geodesics Γ^{±} can be written as

Γ^{±}(s) = (t^{±}(s), γ^{±}(s)) .

For k ∈ ℕ let p_{k} = Γ^{+}(k), set

q_{k} = J̇^{−}(p_{k}) ∩ (ℝ × {γ^{−}(−k)}) .

By global hyperbolicity there exists a null achronal geodesic Γ_{k} from q_{k} to p_{k}. By Corollary 2.2 there exists a constant M such that ρ(q_{k}), ρ(p_{k}) ≤ M, and by the argument presented in [4] we then have ρ∘Γ_{k} ≤ M for all k large enough. Let

r_{k} = Γ_{k} ∩ {x^{n} = −R̊} .

We claim that r_{k} satisfies

r_{k} ∈ \underbrace{\{(x^{0}, x^{a}) : s^{−}(x^{a}) ≤ x^{0} ≤ s^{+}(x^{a}) , ρ(x^{a}) ≤ M\}}_{=:K} ⊂ \{x^{n} = −R̊\} . (2.5)

The upper bound on x^{0} follows immediately from r_{k} ∈ J^{−}(Γ^{+}). To obtain the lower one we show that r_{k} ∈ J^{+}(Γ^{−}): Indeed, since achronal null geodesics maximise the time of arrival (see, e.g., [1, Proposition A.2]) we have

t(q_{k}) = sup_{p∈J^{−}(p_{k})∩{x⃗=γ^{−}(−k)}} t(p) ≥ t^{−}(−k) = t(Γ^{−}(−k)) .

This shows that a future directed causal curve from Γ^{−}(−k) to r_{k} is obtained by first following the coordinate line

[t^{−}(−k), t(r_{k})] ∋ t → (t, γ^{−}(−k)) ,

and then following Γ_{k} until it meets {x^{n} = −R̊}.

We have thus shown that all the achronal geodesic segments Γ_{k} pass through the compact set K defined in (2.5), and the existence of a null line Γ at which the Γ_{k}’s accumulate follows from standard results in Lorentzian geometry. □

Proof of Theorem 1.2: Let Γ be the line constructed in the proof of Theorem 1.1. Writing Γ ∩ ℳ_{ext} in coordinates on ℳ_{ext} as x^{μ}(s), by Corollary A.4 there exists k, T and β such that (A.3)-(A.4) hold. Since ρ is bounded along Γ we must have k = (1, 0, …, 0, 1).

The construction of Γ applies for every choice of asymptotic direction k = (1, k⃗), we shall denote by Γ̊_{k⃗,T} the resulting line. By Corollary A.4 we have

Γ̊_{k⃗,T} = Γ_{(1,k⃗),T,β(k⃗,T)}  (2.6)

for some β(k⃗, T) ∈ ℝ^{n−1}. Note that Γ depends upon a parameter τ ∈ ℝ, so this equation defines a function T (τ). The construction of Γ, together with Corollary A.4, similarly shows that

Γ̊_{k⃗,T} = Γ_{(−1,−k⃗),T′,β′(k⃗,T)} ,  (2.7)

for some T′ ∈ ℝ and β′ ∈ ℝ^{n−1}. Here ℝ^{n−1} is understood as the plane orthogonal to k⃗ with respect to the Euclidean metric on ℝ^{n}.

Let 𝒩̃_{k⃗,T} denote the collection of points in J̇^{−}(Γ̊_{k⃗,T}) ∩ J̇^{+}(Γ̊_{k⃗,T}) which lie on a generator of J̇^{−}(Γ̊_{k⃗,T}) which is past complete, future complete, and entirely contained in ⋖ℳ_{ext}⋗. Then 𝒩̃_{k⃗,T} is non-empty, as it contains Γ̊_{k⃗,T}. We define

𝒩_{k⃗,T} = {the connected component of 𝒩̃_{k⃗,T} containing Γ̊_{k⃗,T} }.

Assuming the null energy condition, we claim that 𝒩_{k⃗,T} is a smooth null hypersurface. This results from the discussion in [5], and can be seen as follows: Let p ∈ Γ̊_{k⃗,T}, as ℳ_{ext} is open there exists a neighborhood 𝒪 of p entirely contained in ℳ_{ext}. Consider the generators of J̇^{−}(Γ̊_{k⃗,T}) passing through 𝒪. As p is an interior point of such a generator, all the null tangents in T 𝒪 can be made as close to the null tangent at p as desired if 𝒪 is made small enough (see, e.g., [3]). It follows from continuous dependence of geodesics upon initial data that all those generators have asymptotic behavior as described in Proposition A.1 when 𝒪 is chosen small enough (with perhaps an asymptotic direction vector k⃗′ ≠ k⃗), in particular they will be future complete. By Lemma 4.2 in [5] the divergence of J̇^{−}(Γ̊_{k⃗,T}) (as measured towards the future, defined in the sense of support hypersurfaces) is non-negative at those points of J̇^{−}(Γ̊_{k⃗,T}) which lie on generators meeting 𝒪. Similarly, reducing 𝒪 if necessary, the divergence of generators of J̇^{+}(Γ̊_{k⃗,T}) (as measured again towards the future) is non-positive at those points of J̇^{−}(Γ̊_{k⃗,T}) which lie on generators meeting 𝒪. By Theorem 3.4 of [5] J̇^{−}(Γ̊_{k⃗,T})∩𝒪 is a smooth null hypersurface which coincides with J̇^{+}(Γ̊_{k⃗,T}) ∩ 𝒪. It is then easily seen that the collection of points of J̇^{−}(Γ̊_{k⃗,T}) which lie on generators meeting 𝒪 is a smooth null hypersurface contained in J̇^{+}(Γ̊_{k⃗,T}), with all the generators there being both future and past complete, contained in ℳ_{ext}, extending arbitrary far in the asymptotic region both to the future and to the past.

We have thus shown that 𝒩_{k⃗,T} has all the properties listed in the statement of the theorem except perhaps for being closed. In order to establish that last property, consider a sequence of points p_{n} ∈ 𝒩_{k⃗,T} converging to p ∈ ℳ, and let Γ_{n} denote the generator of J̇^{−}(Γ̊_{k⃗,T}) passing through p_{n}. Set

q_{n} = Γ_{n} ∩ 𝒮 ,

where 𝒮 is the Cauchy surface for ⋖ℳ_{ext}⋗, then a subsequence, still denoted by q_{n}, converges to a point q ∈ \overline{𝒮} . Let Γ denote an accumulation curve of the Γ_{n}’s passing through q, then Γ is an achronal, maximally extended in \overline{⋖ℳ_{ext}⋗} , geodesic passing through q. Suppose that q ∈ ∂𝒮, then the portion of Γ_{n} which lies to the future of 𝒮 accumulates to a generator of ∂𝒟^{+}(𝒮), while the portion of Γ_{n} which lies to the past of 𝒮 accumulates to a generator of ∂𝒟^{−}(𝒮), resulting in an accumulation curve Γ which is not differentiable at q. This, however, contradicts the fact that Γ is a geodesic. Hence q ∈ 𝒮, and Γ is both future and past complete. It follows that q ∈ 𝒩_{k⃗,T}, with p lying on Γ. This shows that p ∈ 𝒩_{k⃗,T}, and closedness of 𝒩_{k⃗,T} follows.

Note that so far T was an arbitrarily chosen fixed number. If the domain of outer communications is stationary one can move 𝒩_{k⃗,T} by isometries to obtain a one-parameter family 𝒩_{k⃗,t} := ϕ_{t−T} (𝒩_{k⃗,T}), t ∈ ℝ, of such hypersurfaces. A similar construction can be done in the stationary-rotating case.

In general, we start by noting that:

Lemma 2.3 Let s_{0} ∈ ℝ, k⃗ ∈ S^{n−1}, a = 1, 2. For T_{a} ∈ ℝ and β_{a} ∈ ℝ^{n−1} set Γ_{a} = Γ_{(1,k⃗),T_{a},β_{a}}|_{[s_{0},∞)}. Then

T_{1} < T_{2} ⟹ Γ_{1} ⊂ I^{−}(Γ_{2}; ℳ_{ext}) .

Proof: We write x_{a}^{μ} for Γ_{a} in the asymptotically Minkowskian coordinates on ℳ_{ext}, and use an affine parameter as in (A.4). By Proposition A.3 for any λ > 0 we have

x_{2}^{μ}(s + λ) − x_{1}^{μ}(s) →_{s→∞} (T_{2} − T_{1} + λ, β_{2} − β_{1}, λ) .  (2.8)

Consider the coordinate-line segment

[0, 1] ∈ t → γ_{s}^{μ}(t) = tx_{2}^{μ}(s + λ) + (1 − t)x_{1}^{μ}(s) .

Calculating with respect to the Minkowski metric η, the quantity η_{μν}γ̇_{s}^{μ}γ̇_{s}^{ν} (a dot denotes a t-derivative) approaches, as s tends to infinity, the Minkowskian length of the vector appearing at the right-hand-side of (2.8), which is

−(T_{2} − T_{1} + λ)^{2} + |β_{2} − β_{1}|_{δ}^{2} + λ^{2} = −2(T_{2} − T_{1})λ + |β_{2} − β_{1}|_{δ}^{2} .

When T_{2} > T_{1} this is negative for all λ sufficiently large. It follows that γ_{s} is a timelike curve with respect to the Minkowski metric for all λ and s large. Since the metric g uniformly tends to the Minkowski one along γ_{s} as s tends to infinity, γ_{s} will be a timelike curve with respect to g for all λ and s large enough. □

In what follows R̊ is chosen at least as large as in the proof of Theorem 1.1, and in (A.22):

Corollary 2.4 Let T′, β′ be such that the geodesic Γ_{p}^{+}, p = (τ, 0, …, 0, −R̊), of Lemma 2.1 is a subset of Γ_{(1,k⃗),T′,β′} (when both are understood as point sets in ℳ), and let T = T (τ) be as defined by (2.6). Then

T′ = T .

Proof: Clearly T ≤ T′ by construction. Suppose T < T′, then Γ̊_{k⃗,T} would lie to the timelike past of Γ_{p}^{+} by Lemma 2.3. But, by construction, Γ̊_{k⃗,T} ⊂ J̇^{−}(Γ_{p}^{+}), whence the result. □

Corollary 2.5 The map which to the parameter τ assigns T, as defined by (2.6), is a diffeomorphism from ℝ to ℝ.

Proof: The map τ → T′(τ) is the inverse of the map Φ defined after Equation (A.21), Appendix A, which is shown to be a diffeomorphism of ℝ there, and the result follows from Corollary 2.4. □

We continue with

Proposition 2.6 𝒩_{k⃗,T} = ∪_{β∈ℝ^{n}}Γ_{(1,k⃗),T,β}.

Proof: Let p ∈ 𝒩_{k⃗,T}, then there exists a causal line Γ_{p} passing through p contained in ⋖ℳ_{ext}⋗. By weak asymptotic regularity Γ_{p} satisfies (A.18), and thus Γ_{p} = Γ_{(1,k⃗′),T′,β′} for some Γ_{(1,k⃗′),T′,β′} by Corollary A.4.

We want, first, to show that

k⃗′ = k⃗ .  (2.9)

Since 𝒩_{k⃗,T} is a smooth hypersurface its field of null tangents is also smooth. This, together with connectedness of 𝒩_{k⃗,T}, allows one to reduce the proof of (2.9) to a situation where k⃗′ is close to k⃗. We shall therefore assume that k⃗ = (0, …, 0, 1) and that k^{′n} > \sqrt{15}/4. It follows from (A.4) that choosing R̊ and s_{0} large enough one can also assume that Γ_{(1,k⃗′),T′,β′}(s) ∈ {x^{n} ≥ R̊} for s ≥ s_{0}.

Let s be an affine parameter as in (A.3)–(A.5), and for i ∈ ℕ let p_{i} = p_{i}(s) ∈ ℳ_{ext} denote a point with coordinates (x^{0}(s) − 1/i, x⃗(s)). Since p_{i} ∈ I^{−}(Γ_{(1,k⃗′),T′,β′}) we have p_{i} ∈ I^{−}(Γ̊_{k⃗,T}), thus there exists a future directed causal curve γ_{i} from p_{i} to a point q_{i} ∈ Γ̊_{k⃗,T}.

Consider, now, the optical function S^{+} defined in Appendix A.2, see (A.22). As ∇S^{+} is lightlike, the function S^{+} is strictly increasing on every timelike curve contained within the domain of definition of S^{+}. Suppose, first, that γ_{i} is entirely contained in {x_{n} ≥ R̊}. Then S^{+}(p_{i}) < S^{+}(q_{i}). By construction of S^{+} we have S^{+}(q_{i}) = T. On the other hand, the asymptotic behavior (A.23) of S^{+} and of Γ_{(1,k⃗′),T′,β′} gives, to leading order,

S^{+}(p_{i}) = x^{0}(s) − x^{n}(s) + o(s) = s − k^{′n}s + o(s) →_{s→∞} ∞

except if (2.9) is satisfied. Thus, if (2.9) does not hold, then γ_{i} crosses the boundary of the set {x^{n} ≥ R̊}; let r_{i} denote the first crossing point, and let t_{i} denote the last one. Note that J̇^{−}(Γ̊_{k⃗,T}; {x^{n} ≥ R̊}) ∩ {x^{n} = R̊} can be represented as a graph:

J̇^{−}(Γ̊_{k⃗,T}; {x^{n} ≥ R̊}) ∩ {x^{n} = R̊} = {x^{0} = s^{−}(x^{a}) , x^{a} ∈ ℝ^{n−1}} .

Since S^{+} ≤ T on J̇^{−}(Γ̊_{k⃗,T}; {x^{n} ≥ R̊}) ∩ {x^{n} = R̊} we have, by (A.23),

s^{−} = O(ln(2 + ρ)) .  (2.10)

Somewhat similarly, J̇^{+}(p_{1}; {r ≥ R̊})∩{x^{n} = R̊} is a graph of a function s^{+}. With our choice of R̊ the slopes of the light cones in ℳ_{ext} are bounded from below by one half, which implies

s^{+}(x^{a}) ≥ \underbrace{x^{0}(s) − 1 + \frac{x^{n}(s) − R̊}{2}}_{I} + \underbrace{\frac{1}{2}\sqrt{∑_{a}(x^{a} − x^{a}(s))^{2}}}_{II}

≥ \frac{5}{4}s + \frac{1}{2}\sqrt{∑_{a}(x^{a} − x^{a}(s))^{2}} − C ,  (2.11)

for some s–independent constant C. Here the contribution denoted as I is obtained by following a coordinate line, of slope one half lying in a x^{0} − x^{n} plane, from p_{1} to a point r ∈ {x^{n} = R̊}, while the contribution denoted as II is obtained from a cone of slope one half issued from r.

Now

ρ = \sqrt{∑_{a}(x^{a})^{2}} ≤ \sqrt{∑_{a}(x^{a} − x^{a}(s))^{2}} + \sqrt{∑_{a}(x^{a}(s))^{2}}

≤ \sqrt{∑_{a}(x^{a} − x^{a}(s))^{2}} + \frac{1}{2}s + C′ ,

so that

ρ − \frac{1}{2}s − C′ ≤ \sqrt{∑_{a}(x^{a} − x^{a}(s))^{2}} ,

and (2.11) gives

s^{+}(x^{a}) ≥ s + \frac{1}{2}ρ − \frac{1}{2}C′ − C .  (2.12)

It follows that for any future directed causal curve starting from one of the points p_{i} and entirely contained in {r ≥ R̊}, at each crossing point (2.12) holds.

Suppose that γ_{i} never exits {r ≥ R̊}. Then at the last point t_{i} = (x_{i}^{0}, x_{i}^{a}) at which γ_{i} crosses {x^{n} = R̊} we must have

s^{+}(x_{i}^{a}) ≤ s^{−}(x_{i}^{a}) ,

which is not possible by (2.10) and (2.12) if s is chosen sufficiently large. Thus, either (2.9) holds, or γ_{i} enters and exits {r ≥ R̊} (perhaps more than once). Let r̂_{i} denote the first exit point. Let ŝ^{−} be the graphing function of J̇^{−}(Γ̊_{k⃗,T}; ℳ) ∩ {r = R̊} (exceptionally, for emphasis, we write J̇^{−}(Γ̊_{k⃗,T}; ℳ) for J̇^{−}(Γ̊_{k⃗,T})):

J̇^{−}(Γ̊_{k⃗,T}; ℳ) ∩ {r = R̊} = {x^{0} = ŝ^{−}(v) , v ∈ S^{n−1}(R̊) ⊂ ℝ^{n}} .

Since r̂_{i} = (x̂_{i}^{0}, v̂_{i}) ∈ J^{−}(Γ̊_{k⃗,T}; ℳ) we have x̂_{i}^{0} ≤ ŝ^{−}(v_{i}) ≤ sup ŝ^{−} < ∞, the last inequality following from compactness and from the fact that

J^{−}(Γ̊_{k⃗,T}; ℳ)∩{r = R̊} ⊃ Γ̊_{k⃗,T} ∩{r = R̊} ⟹ J^{−}(Γ̊_{k⃗,T}; ℳ)∩{r = R̊} ≠ ∅ ,

so that ŝ^{−} ≢ ∞. However, x^{0} is increasing along γ_{i}, so that x_{i}^{0} ≥ x^{0}(s) − 1 →_{s→∞} ∞. It follows that an appropriate choice of s guarantees that all the γ_{i}’s are entirely contained within {r ≥ R̊}. Consequently, (2.9) must be true.

From (2.9) and the definition of the p_{i}’s we have S^{+}(p_{i}) →_{i→∞} T′, which implies T′ ≤ T. Lemma 2.3 shows that no point of the form Γ_{(1,k⃗),T′,β′}(s), with T′ < T and s large, can be in J̇^{−}(Γ̊_{k⃗,T}), hence

T′ = T .

We have thus shown

𝒩_{k⃗,T} ⊂ ∪_{β∈ℝ^{n}}Γ_{(1,k⃗),T,β} .  (2.13)

Equality ensues now from the fact that the intersections of both sets with the region {x^{n} > R̊} are smooth closed hypersurfaces. □

Corollary 2.7 Each maximally extended geodesic Γ in ⋖ℳ_{ext}⋗ such that r(Γ(s)) →_{s→∞} ∞ is a line.

Proof: By Corollary A.4 every such future directed Γ coincides with some Γ_{(1,k⃗),T,β}. But Corollary 2.5 and Proposition 2.6 show that each Γ_{(1,k⃗),T,β} is a line. The result for past-directed geodesics follows by changing time orientation. □

Returning to the proof of Theorem 1.2, we claim that each point p in ⋖ℳ_{ext}⋗ lies on some Γ_{(1,k⃗),T,β}. Indeed, Lemma 2.1 with the direction

(0, …, 0, 1) rotated into k⃗ provides a geodesic ray through p with asymptotic direction k⃗, and the result follows from Proposition A.3.

We have thus proved that each of the distinct families of null hypersurfaces

𝒩k⃗ := {𝒩_{k⃗,T} }_{T ∈ℝ}

covers ⋖ℳ_{ext}⋗. Suppose that ⋖ℳ_{ext}⋗ ≠ ℳ, then there exists a point

p ∈ I^{−}(∂𝒮) ∩ I^{−}(ℳ_{ext}) \ ⋖ℳ_{ext}⋗ ⊂ J^{−}(ℳ_{ext})

and, by Lemma 2.1, a future directed null half-line from p extending to infinity in ℳ_{ext}. This is, however, impossible by Corollary 2.7, and the proof is complete. □

Proof of Theorem 1.3: By Theorem 1.2 every maximally extended geodesic is a line. In view of the Raychaudhuri equation this is only possible if for all null vectors k we have

Ric(k, k) = 0 .  (2.14)

Elementary algebra implies then that the Ricci tensor is proportional to the metric. For future reference we note a somewhat stronger result, which does not assume that (2.14) holds for all k:

Lemma 2.8 Suppose that there exists an open set Ω of null vectors k at p ∈ ℳ for which we have

R_{μν}k^{μ}k^{ν} = 0 .

Then R_{μν}(p) is proportional to g_{μν}.

Proof: Let e^{μ} be an ON-frame at p, (changing e^{0} to −e^{0} if necessary) in this frame we write k = (|k⃗|_{δ}, k⃗), and there exists an open cone 𝒰 ⊂ ℝ^{n−1} of k⃗’s such that

α_{ij}k^{i}k^{j} + 2β_{i}k^{i}|k⃗|_{δ} = 0 ,  (2.15)

where β_{i} = R_{0i}(p), α_{ij} = R_{ij}(p) + R_{00}(p)δ_{ij}. Let k⃗_{0} ∈ 𝒰 , after rescaling we can assume that |k⃗_{0}|_{δ} = 1. Let m⃗ be any vector such that m⃗ · k⃗_{0} = 0, |m⃗|_{δ} = 1. Setting k⃗ = k⃗_{0} +xm⃗ in (2.15) one obtains an equation of the form

f(x) = a + bx + cx^{2} + (d + ex)\sqrt{1 + x^{2}} = 0 ,

for all x in a neighborhood of zero. As f is analytic in a neighborhood of zero, all the coefficients of its power series there vanish, leading to a = b = c = d = e = 0, which is equivalent to

α_{ij}k_{0}^{i}k_{0}^{j} = α_{ij}k_{0}^{i}m^{j} = α_{ij}m^{i}m^{j} = β_{i}k_{0}^{i} = β_{i}m^{i} = 0 .

Since m is arbitrary we obtain β_{i} = 0, and α_{ij} = 0 follows by polarisation.□

Returning to the proof of Theorem 1.3, Lemma 2.8 shows that the Ricci tensor is proportional to the metric. By a Bianchi identity the proportionality factor must be a constant, and asymptotic flatness shows that the constant is zero. Therefore (ℳ , g) is null geodesically complete and vacuum, and we conclude by [5]. □

A Null geodesics extending to infinity in uniformly Schwarzschildian space-times

In this appendix we study null geodesics in ℳ_{ext}, as needed in the argument above. We start by showing existence of null geodesics Γ(s) with freely prescribable asymptotic direction, which can without loss of generality be assumed to be e_{0} + e_{n}, as well as freely prescribable values

β^{a} := lim_{s→∞} Γ^{a}(s) , a = 1, …, n − 1 .

We will assume that there exist constants C_{0} > 0 and ϵ > 0 such that, in the coordinates of (1.1) on ℳ_{ext},

|g − g_{m}| ≤ C_{0}r^{−1−ϵ} , |∂_{μ} (g − g_{m}) | ≤ C_{0}r^{−2−ϵ} ,  (A.1)
|∂_{μ}∂_{ν} (g − g_{m}) | ≤ C_{0}r^{−3−ϵ} .  (A.2)

When m = 0 and g is flat in the asymptotic region all the results established in this appendix hold trivially, and therefore from now on we assume that m ≠ 0. On the other hand, for the purposes of the analysis of the behavior of the null geodesics in this appendix, the separation of g into a Schwarzschild part and a remainder is only relevant in dimension n = 3, so to avoid unnecessary discussions we assume here that

0 < ϵ ≤ max(1, n − 3) .

A.1 Asymptotic behavior of null geodesics

Let

α = (α^{μ}) = (T, β = (β^{a}), R) ∈ ℝ × ℝ^{n−1} × ℝ ,  (A.3)
b = |β|_{δ} ,

we want to construct geodesics

[0, ∞) ∋ s → x^{μ}(s) ∈ ℳ_{ext}

of the form

x^{μ}(s) = α^{μ}+sk^{μ}+\underbrace{ψ_{0}^{μ}(s) + δψ^{μ}(s)}_{ψ^{μ}(s)} ,  k = (k^{μ}) = (1, k⃗) , |k⃗|_{δ} = 1 . (A.4)

The function ψ_{0}^{μ} is chosen so as to eliminate one of the leading order terms in the geodesic equation for n = 3 and μ = 0. So we set ψ_{0}^{μ} ≡ 0 unless n = 3 and μ = 0, in which case we set

ψ_{0}^{0}(s) = 2m ln (\frac{R + s + \sqrt{(R + s)^{2} + b^{2}}}{2} ) ,

so that

ψ̇_{0}^{0}(s) = \frac{2m}{\sqrt{(R + s)^{2} + b^{2}}} .

Proposition A.1 Let n ≥ 3 and let g satisfy (A.1)-(A.2) on ℳ_{ext} with some 0 < ϵ ≤ max(1, n − 3). There exist constants R̊ and C̊ such that for every k⃗ ∈ S^{n−1} ⊂ ℝ^{n}, (T, β) ∈ ℝ × ℝ^{n−1} and R ≥ R̊ there exists an affinely parameterized null geodesic of the form (A.3)-(A.4) satisfying

|δψ| + \sqrt{(R + s)^{2} + |β|_{δ}^{2}} |δψ̇| ≤ \frac{C̊}{((R + s)^{2} + |β|_{δ}^{2})^{ϵ/2}} .  (A.5)

Remark A.2 If the metric has a smooth, or polyhomogeneous, conformal completion, one immediately obtains a polyhomogeneous expansion for δψ in terms of s.

Proof: x^{μ}(s) will be an affinely parameterized geodesic if and only if

\frac{d^{2}δψ^{μ}}{ds^{2}} = F^{μ}(δψ, δψ̇, s) ,

where a dot denotes an s–derivative, and

F^{μ}(χ, λ, s) = − (Γ_{νρ}^{μ} ∘ (α + sk + ψ_{0} + χ)) (k^{ν}+ψ̇_{0}^{ν}+λ^{ν})(k^{ρ}+ψ̇_{0}^{ρ}+λ^{ρ})−\frac{d^{2}ψ_{0}^{μ}}{ds^{2}} .

From (1.1) and (A.1)-(A.2) we have Γ_{νρ}^{μ} = O(r^{−2−ϵ}) except for

Γ_{00}^{k} = Γ_{k0}^{0} = Γ_{0k}^{0} = \frac{(n−2)m}{r^{n−1}} \frac{x^{k}}{r} + O(r^{−2−ϵ}) ,  (A.6)
Γ_{ij}^{k} = \frac{m}{r^{n−1}} (δ_{ij} \frac{x^{k}}{r} − δ_{jk} \frac{x^{i}}{r} − δ_{ik} \frac{x^{j}}{r} ) + O(r^{−2−ϵ}) .  (A.7)

We will write δΓ_{νρ}^{μ} for all the remainder terms not explicitly listed above. In dimension three one has

F^{0} = −\frac{2m}{r^{2}} (( ∑_{i} \frac{x^{i}}{r} (1 + ψ̇_{0}^{0} + λ^{0})(k^{i} + λ^{i})) − \frac{r^{2}(R + s)}{((R + s)^{2} + b^{2})^{3/2}} )
−δΓ_{μν}^{0}(k^{μ} + ψ̇_{0}^{μ} + λ^{μ})(k^{ν} + ψ̇_{0}^{ν} + λ^{ν}) ,  (A.8)
F^{i} = −\frac{m}{r^{2}}(\frac{x^{i}}{r} (|k⃗ + λ|_{δ}^{2} + (1 + ψ̇_{0}^{0} + λ^{0})^{2}) − 2(k^{i} + λ^{i})( ∑_{j} \frac{x^{j}}{r} (k^{j} + λ^{j})))
−δΓ_{μν}^{i}(k^{μ} + ψ̇_{0}^{μ} + λ^{μ})(k^{ν} + ψ̇_{0}^{ν} + λ^{ν}) .  (A.9)

Here, and in all the calculations that follow, x^{μ} has to be replaced by α^{μ} + sk^{μ} + ψ_{0}^{μ} + χ^{μ}.

On the other hand, in higher dimension it will suffice to estimate directly from

F^{α} = −Γ_{μν}^{α}(k^{μ} + λ^{μ})(k^{ν} + λ^{ν}) .

We choose

0 < σ < ϵ ,

and we let X ⊂ C([0, ∞)) × C([0, ∞)) denote the set of pairs of maps into ℝ^{n+1} such that the following norm is finite:

∥(χ, λ)∥_{R,b} = ∥(R + s + b)^{σ}χ∥_{L^{∞}([0,∞))} + \frac{2∥(R + s + b)^{1+σ}λ∥_{L^{∞}([0,∞))}}{1 + σ} .

(A.10)

Let B(1) ⊂ X be the unit closed ball in X and define the map

B(1) ∋ (χ, λ) → 𝒯 (χ, λ) = (− ∫_{s}^{∞} λ(u)du , − ∫_{s}^{∞} F^{μ}(χ, λ, u)du) .

(A.11)

Clearly a fixed point (χ_{0}, λ_{0}) of 𝒯 provides an affinely parameterized geodesic of the form (A.4), with |δψ(s)| = |χ_{0}| ≤ (R + s + b)^{−σ}, and with

ẋ^{μ} →_{s→∞} k^{μ} .  (A.12)

In order to check that 𝒯 does indeed have a fixed point, note first the elementary inequality: for R + s ≥ 0, b ≥ 0,

\sqrt{(R + s)^{2} + b^{2}} ≤ R + s + b ≤ \sqrt{2}\sqrt{(R + s)^{2} + b^{2}} .  (A.13)

It follows that for (χ, λ) ∈ B(1) and for R sufficiently large we have

\frac{1}{2}(R + s + b) ≤ r ≤ 2(R + s + b) ,  (A.14)

|ρ − b| ≤ \frac{C}{(R + s + b)^{σ}} ,  (A.15)
|x^{n} − R − s| = |χ^{n}(s)| ≤ \frac{1}{(R + s + b)^{σ}} .  (A.16)

This implies

|r − \sqrt{(R + s)^{2} + b^{2}}| = |\frac{r^{2} − (R + s)^{2} − b^{2}}{r + \sqrt{(R + s)^{2} + b^{2}}} |

= |\frac{(x^{n} − (R + s))(x^{n} + (R + s)) + (ρ − b)(ρ + b)}{r + \sqrt{(R + s)^{2} + b^{2}}} |
≤ \frac{1 + C}{(R + s + b)^{σ}} .  (A.17)

Equation (A.14) immediately gives an estimate

|δΓ_{μν}^{σ}(k^{μ} + ψ̇_{0}^{μ} + λ^{μ})(k^{ν} + ψ̇_{0}^{ν} + λ^{ν})| ≤ \frac{C_{2}}{(R + s + b)^{2+ϵ}} .

Writing

F^{μ}(χ, λ) = χ ∫_{0}^{1} \frac{∂F^{μ}}{∂χ} (sχ, λ)ds + λ ∫_{0}^{1} \frac{∂F^{μ}}{∂λ} (0, sλ)ds + F^{μ}(0, 0) ,

with a little more work a similar estimate is obtained for the whole of F^{μ} using (A.15)-(A.17). This leads to

|∫_{s}^{∞} F^{μ}(χ, λ, u)du| ≤ C_{3}(R + s + b)^{−1−ϵ} ≤ C_{3}R^{σ−ϵ}(R + s + b)^{−(1+σ)} , for some C_{3} = C_{3}(C_{1}, |m|, ϵ), with C_{3} also depending on the constant dominating the error terms in (A.1). Increasing R_{1} if necessary one finds that 𝒯 maps B(1) into B(1) for all R ≥ R_{1}.

One similarly checks that the map 𝒯 defined in (A.11) is a contraction for R sufficiently large, uniformly in b, and the existence of a geodesic of the form (A.4), with δψ(s) →_{s→∞} 0, ensues. Once we know that the solution just described exists, it is straightforward to show, using the geodesic equation and the estimates already available, that δψ satisfies (A.5). □

We will need to know that all geodesics extending to infinity are as above:

Proposition A.3 Let g satisfy (A.1)-(A.2) on ℳ_{ext}. Then every null geodesic Γ(s) such that

r(Γ(s)) →_{s→∞} ∞  (A.18)

satisfies (A.3)-(A.5).

Proof: The existence of k^{μ} is provided by Proposition B.1 in [2]. From that last proposition one also has a uniform bound on dx^{μ}/ds, as well as the estimate (A.14), and the remaining claims easily follow by inspection of the geodesic equation. □

Corollary A.4 The map which to every maximally extended, null geodesic satisfying (A.18) assigns k, β and T is bijective.

Proof: It remains to prove injectivity. But Proposition A.3 implies that geodesics satisfying (A.18) are solutions of the fixed point problem considered in the proof of Proposition A.1, and are therefore unique. □

Let Γ_{k,T,β}(s) denote the unique affinely parameterized maximally extended geodesic provided by Corollary A.4. The differentiability properties of Γ_{k,T,β} with respect to k, T and β are best studied by considering Jacobi fields along Γ. The method of proof of Proposition A.1 applies to give:

Proposition A.5 Let Γ : [0, ∞) be a null affinely parameterized geodesic satisfying (A.18). Then for every A, B ∈ ℝ^{n+1} there exists a solution Z of the Jacobi equation,

\frac{D^{2}Z}{ds^{2}} (s) = Riem(γ̇, Z)γ̇ ,  (A.19)

where Riem denotes the Riemann tensor, such that

Z^{μ} = \{ \begin{matrix}(R + s)A^{μ} + O(ln(R + s + \sqrt{(R + s)^{2} + b^{2}})), & A^{μ} ≠ 0, n = 3; \\ (R + s)A^{μ} + O((R + s + b)^{−ϵ}), & A^{μ} ≠ 0, n > 3; \\ B^{μ} + O((R + s + b)^{−(1+ϵ)}), & A^{μ} = 0, \end{matrix}

(A.20)

with the error terms being uniform in b.

Similarly to Proposition A.3, every Jacobi field along Γ as in Proposition A.5 is a linear combination of solutions of the form (A.20); this is proved by usual techniques.

It is standard to show now

Proposition A.6 The map

S^{n−1} × ℝ × ℝ^{n−1} × ℝ ⊃ 𝒰 ∋ (k⃗, T, β, s) → Γ_{(1,k⃗),T,β}(s)

is differentiable on the open set 𝒰 on which it is defined.

Fix β, R, with R ≥ R_{1}, and let

Φ : ℝ → ℝ  (A.21)

be the map which to T ∈ ℝ assigns x^{0}(0), the coordinate time at which the null geodesic, as constructed in the proof of Proposition A.1, intersects {x^{n} = R}. Proposition A.5 (with A = 0) and standard ODE considerations show that Φ is differentiable, with strictly positive derivative (perhaps increasing R_{1} if necessary) and hence strictly increasing, in particular τ is continuous. As the constant C̊ in (A.5) is uniform in T, one clearly has

τ(T) →_{T →±∞} ±∞ .

It follows that Φ is a smooth bijection from ℝ to itself.

A.2 The optical functions S^{±}

There exists R̊_{1} such that, using the implicit function theorem, one can define a function

S^{+} : {x^{n} ≥ R̊_{1}} → ℝ  (A.22)

by assigning T to each point lying on Γ_{k,T,β}. Then S^{+} is differentiable, with null gradient, and satisfies

S^{+} = x^{0} − x^{n} + R̊ − \{ \begin{matrix}2m ln (^{\frac{x^{n}+\sqrt{(x^{n})^{2}+ρ^{2}}}{2}} ) + o(1), & n = 3; \\ o(1), & n ≥ 4, \end{matrix} (A.23)

with the error term tending to zero as r(x) goes to infinity, uniformly with respect to (x^{1}, …, x^{n−1}) ∈ ℝ^{n−1}. We give some details of the construction, because the function S^{+} plays an important role in Section 2; furthermore, the proof of existence of S^{+} (and its time-reverse counterpart S^{−}), with the required properties, is the missing step in the argument in [13]. Let

(y^{μ}) = (T, β, s) ,

then (A.4) with k^{μ} = (1, 0, …, 0, 1) and R = R̊ (as given by Proposition A.1) defines a map x^{μ}(y^{α}). We have

1. ∂x^{μ}/∂y^{0} at the point x^{μ}(y^{α}) is given by the value of the Jacobi field Z^{μ} which asymptotes to δ_{0}^{μ} as s = y^{n} tends to infinity; from the proof of Proposition A.5 we thus have

\frac{∂x^{μ}}{∂y^{0}} = δ_{0}^{μ} + O((R̊ + y^{n} + \sqrt{∑_{a}(y^{a})^{2}})^{−1}) .

2. Similarly ∂x^{μ}/∂y^{a} at x^{μ}(y^{α}) is given by the value of the Jacobi field Z^{μ} which asymptotes to δ_{a}^{μ} as s = y^{n} tends to infinity; from the proof of Proposition A.5 one finds

\frac{∂x^{μ}}{∂y^{a}} = δ_{a}^{μ} + O((R̊ + y^{n} + \sqrt{∑_{a}(y^{a})^{2}})^{−1}) .

3. Directly from its definition, ∂x^{μ}/∂y^{n} = ∂x^{μ}/∂s equals k^{μ} + dψ^{μ}/ds, so that

\frac{∂x^{μ}}{∂y^{n}} = (1, 0, …, 0, 1) + O((R̊ + y^{n} + \sqrt{∑_{a}(y^{a})^{2}})^{−1}) .

Increasing R̊ if necessary, it follows that y^{μ} → x^{μ} is a smooth local diffeomorphism for r(y) ≥ R̊.

To prove bijectivity with the image we need the following, certainly well known, result:

Proposition A.7 Let 𝒪 ⊂ ℝ^{n} be an open convex set and let Φ : 𝒪 → ℝ^{n} be continuously differentiable. Then there exists ϵ = ϵ(n) > 0 such that if

|\frac{∂Φ^{μ}}{∂y^{ν}} − δ_{ν}^{μ}| < ϵ ,  (A.24)

then Φ is injective. If 𝒪 = ℝ^{n} then Φ(𝒪) = ℝ^{n}.

Proof: The calculation follows those in [7, Theorem 3.1]. We have

|Φ(x) − Φ(y) − (x − y)| = | ∫_{0}^{1} (Φ′(tx + (1 − t)y) − Id)(x − y)dt|
≤ C(n)ϵ|x − y| .

This implies

|x − y| ≤ |Φ(x) − Φ(y)| + |Φ(x) − Φ(y) − (x − y)|
≤ |Φ(x) − Φ(y)| + C(n)ϵ|x − y| .

If C(n)ϵ < 1 we obtain

|x − y| ≤ \frac{1}{1 − C(n)ϵ}|Φ(x) − Φ(y)| ,  (A.25)

and injectivity follows. By the implicit function theorem Φ is open. It further follows from (A.25) that if the sequence (Φ(x_{i}))_{i∈ℕ} is Cauchy, then so is the sequence (x_{i})_{i∈ℕ}. This shows that if 𝒪 = ℝ^{n} then Φ(𝒪) is closed, proving surjectivity. □

For R large enough consider the map Φ_{R} which to (T, β) ∈ ℝ^{n−1} assigns x^{μ}(s), where s = s(T, β) is chosen so that x^{n}(s) = R. By arguments similar to the ones already given one finds that Φ_{R} satisfies the hypotheses of Proposition A.7 for R ≥ R̊_{1}, increasing R̊_{1} if necessary. Hence the Φ_{R}’s are bijective for R ≥ R̊_{1}, which implies that every point in the region {x^{n} ≥ R̊_{1}} lies on precisely one null geodesic with asymptotic velocity (1, 0, …, 0, 1). It follows that the map y^{μ} → x^{μ}(y^{α}) is injective onto {x^{n} ≥ R̊_{1}}, and therefore is a global diffeomorphism from some set 𝒰 ⊂ [0, ∞) × ℝ^{n} to {x^{n} ≥ R̊_{1}}. Inverting this map one obtains smooth functions y^{μ}(x^{α}) on {x^{n} ≥ R̊_{1}}. The function S^{+} is then set to be equal to y^{0}(x^{μ}). Equation (A.23) is obtained by eliminating s and β from the definition

S^{+}(x^{μ}) = T (x^{μ}) = x^{0}(s) − s − ψ^{0}(s)

using equations (A.4).

As already pointed out, the function S^{−} is obtained by a time-reverse of the construction just carried out.

Because the estimates above are uniform in r(y) for r(y) large, it should be clear that the domain of definition of S^{+} can be extended to a region of the form {r ≥ R̊_{1} , x^{n} ≥ 0}, increasing R̊_{1} if necessary. Similarly S^{−} can be defined on a region of the form {r(x) ≥ R̊_{1} , x^{n} ≤ 0}. Those extensions are relevant to the original Penrose-Sorkin-Woolgar argument.

Acknowledgements: I am grateful to Gregory Galloway and Eric Woolgar for useful discussions, and to Malcolm MacCallum for bibliographical advice.

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