The null distance encodes causality

Sakovich, A.; Sormani, C.

doi:10.1063/5.0118979

A Lorentzian manifold, N, endowed with a time function, τ, can be converted into a metric space using the null distance, ${\hat{d}}_{τ}$ ⁠, defined by Sormani and Vega [Classical Quant. Grav. 33(8), 085001 (2016)]. We show that if the time function is a regular cosmological time function as studied by Andersson, Galloway, and Howard [Classical Quant. Grav. 15(2), 309–322 (1998)], and also by Wald and Yip [J. Math. Phys. 22, 2659–2665 (1981)], or if, more generally, it satisfies the anti-Lipschitz condition of Chruściel, Grant, and Minguzzi [Ann. Henri Poincare 17(10), 2801–2824 (2016)], then the causal structure is encoded by the null distance in the following sense: for any p ∈ N, there is an open neighborhood U_p such that for any q ∈ U_p, we have ${\hat{d}}_{τ} (p, q) = τ (q) - τ (p)$ if and only if q lies in the causal future of p. The local encoding of causality can be applied to prove the global encoding of causality in a variety of settings, including spacetimes N where τ is a proper function. As a consequence, in dimension n + 1, n ≥ 2, we prove that if there is a bijective map between two such spacetimes, F : M₁ → M₂, which preserves the cosmological time function, τ₂(F(p)) = τ₁(p) for any p ∈ M₁, and preserves the null distance, ${\hat{d}}_{τ_{2}} (F (p), F (q)) = {\hat{d}}_{τ_{1}} (p, q)$ for any p, q ∈ M₁, then there is a Lorentzian isometry between them, F_∗g₁ = g₂. This yields a canonical procedure allowing us to convert large classes of spacetimes into unique metric spaces with causal structures and time functions. This will be applied in our upcoming work to define spacetime intrinsic flat convergence.

I. INTRODUCTION

This paper is part of a series of papers developing the notion of Spacetime Intrinsic Flat (SIF) convergence of Lorentzian manifolds as suggested by Yau. The overarching plan is to convert the Lorentzian manifolds canonically into unique metric spaces and, then, to take the intrinsic flat limit of these metric spaces.¹ One method of converting a Lorentzian manifold (N, g) with a time function, τ, into a metric space is to use the null distance, ${\hat{d}}_{τ}$ ⁠, developed by Sormani and Vega in Ref. 2. This conversion process,

(N, g) \mapsto (N, {\hat{d}}_{τ}, τ),

(1)

is canonical for a Lorentzian manifold endowed with a regular cosmological time function, τ = τ_AGH, as defined by Andersson, Galloway, and Howard in Ref. 3.

In this paper, we prove that the conversion map in (1) is one-to-one from isometry classes of Lorentzian manifolds to time preserving isometry classes of the metric spaces (see Theorem 1.3). In addition, we prove that the causal structure of N is locally encoded by $(N, {\hat{d}}_{τ}, τ)$ (see Theorem 1.1). The local encoding can be applied to prove the global encoding of causality in a variety of settings, including spacetimes N where τ is also a proper function (see Theorem 4.1 within).

Given a Lorentzian manifold, (N, g), Andersson, Galloway, and Howard³ have defined the notion of a canonical time function,

τ_{AGH} (p) = \sup {L_{g} (C) : future timelike C : [0, 1] \to N, C (1) = p},

(2)

where

L_{g} (C) = \int_{0}^{1} | g (C^{'} (s), C^{'} (s)) |^{1 / 2} d s .

(3)

See also Wald and Yip.⁴ This τ_AGH is usually referred to as the cosmological time function. Ebrahimi⁵ has shown that on a Friedman–Robertson–Walker spacetime, τ_AGH may be viewed as the time elapsed since the big bang.

Andersson, Galloway, and Howard call this cosmological time regular if τ(p) < ∞ for all p ∈ M and τ → 0 along every inextensible past causal curve. While τ_AGH may not be differentiable, Sormani and Vega² showed that whenever τ_AGH is regular, it is at least locally anti-Lipschitz in the sense of Chruściel, Grant, and Minguzzi.⁶ Namely, for every point p ∈ N, there is a neighborhood U of p that has a Riemannian metric with a distance function d_U : U × U → [0, ∞) such that for all q, q′ ∈ U, we have

q in the causal future of q^{'} ⟹ τ (q) - τ (q^{'}) \geq d_{U} (q, q^{'}) .

(4)

Sormani and Vega² defined the notion of null distance between two events as the infimum of the null length over piecewise causal curves,

{\hat{d}}_{τ} (p, q) ≔ \inf {{\hat{L}}_{τ} (β) : β piecewise casual from p to q via x_{i} \in β},

(5)

so that either x_i is in the causal future of x_i+1 or x_i+1 is in the causal future of x_i, where the null length of the curve β as in (5) is

{\hat{L}}_{τ} (β) ≔ \sum_{i = 1}^{k} | τ (x_{i}) - τ (x_{i - 1}) | .

(6)

They observe that

{\hat{d}}_{τ} (p, q) \geq | τ (p) - τ (q) | for any p, q \in N .

(7)

Note that the time function, τ, here need only be a generalized time function: τ increases along causal curves but is not necessarily continuous.

Sormani and Vega² showed that the null distance converts the Minkowski space endowed with its standard time function into a metric space whose ${\hat{d}}_{τ}$ ball about a point p of radius R is a causal cylinder, whose top is the level set τ⁻¹(τ(p) + R) intersected with the point’s causal future, J⁺(p), as depicted in Fig. 1.

FIG. 1.

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In a Minkowski space with τ = t, the ${\hat{d}}_{τ}$ ball is a cylinder aligned with a light cone and a level set of τ so that (9) holds.

Sormani and Vega² proved that if τ is a regular cosmological time function, τ = τ_AGH, or, more generally, if τ is any other time function satisfying the anti-Lipschitz condition of Chruściel, Grant, and Minguzzi (4), then ${\hat{d}}_{τ}$ is definite,

q = q^{'} ⟺ {\hat{d}}_{τ} (q, q^{'}) = 0,

(8)

and induces the topology of the original manifold, N. Sormani and Vega conjectured that under these hypotheses, ${\hat{d}}_{τ}$ also encodes causality,

{\hat{d}}_{τ} (q, q^{'}) = τ (q) - τ (q^{'}) ⟺ q is in the causal future of q^{'},

(9)

and proved this for warped product spacetimes. For general spacetimes and time functions, it is immediate from the definition of ${\hat{d}}_{τ}$ that

q is in the causal future of q^{'} ⟹ {\hat{d}}_{τ} (q, q^{'}) = τ (q) - τ (q^{'}),

(10)

but the other direction was shown to be false without a stronger assumption on τ.² See Examples 2.1 and 2.2 within. The null distance has been studied further in the work of Allen and Burtscher,⁷ Vega,⁸ Kunzinger and Steinbauer,⁹ and Graf and Sormani.¹⁰

In this paper, we prove that the null distance ${\hat{d}}_{τ}$ locally encodes causality whenever τ satisfies the anti-Lipschitz condition of Chruściel, Grant, and Minguzzi.

Theorem 1.1.

Let (Nⁿ⁺¹, g) be a Lorentzian manifold of dimension n + 1, n ≥ 1. Suppose τ : N → [0, ∞) is a generalized time function that is locally anti-Lipschitz: about every point p ∈ N, there is a neighborhood U that has a Riemannian metric with a distance function d_U : U × U → [0, ∞) such that for all q, q′ ∈ U, we have

q in the causal future of q^{'} ⟹ τ (q) - τ (q^{'}) \geq d_{U} (q, q^{'}) .

(11)

Then,

{\hat{d}}_{τ}

locally encodes causality: about every point p ∈ N, there is a neighborhood W such that for all q ∈ W, we have

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p) ⟺ q is in the causal future of p

(12)

and

{\hat{d}}_{τ} (p, q) = τ (p) - τ (q) ⟺ q is in the causal past of p .

(13)

In particular, we have the following important corollary.

Corollary 1.2.

If τ is a regular cosmological time function, then ${\hat{d}}_{τ}$ locally encodes causality.

Theorem 1.1 is proven in Sec. III. An outline of the proof is provided at the beginning of that section. Note that there are examples of spacetimes where the null distance defined using cosmological time does not encode causality globally. See Example 2.2 within. Nevertheless, we prove global causality in Theorem 4.1 under the additional hypothesis that the time function is proper; see Sec. IV.

Once we know the causal structure on a Lorentzian manifold, we can recover the null structure, determine the null cones, and thus, also recover the Lorentzian metric up to its conformal class; cf. Ref. 11, Sec. 1.9. That is, if F : N₁ → N₂ is a smooth map that preserves causality,

p is in the causal future of q ⟺ F (p) is in the causal future of F (q),

(14)

one can rescale to see that F* preserves future causal vectors,

{v | F_{*} g_{1} (v, v) \geq 0} = {v | g_{2} (v, v) \geq 0} .

(15)

One can, then, deduce that F_*g₁ = ϕ²g₂, where ϕ : N₂ → (0, ∞) is a conformal factor. If we also have a pair of smooth cosmological time functions τ_i : N_i → [0, ∞) and we know τ₁ = τ₂◦F, then the fact that

g_{i} (\nabla τ_{i}, \nabla τ_{i}) = - 1 ⟹ ϕ = 1 everywhere.

(16)

Thus, F : N₁ → N₂ is a Lorentzian isometry.

In this paper, we prove the following theorem without assuming that F and τ_i are smooth and using only that ${\hat{d}}_{τ}$ encodes causality for a regular cosmological time, τ = τ_AGH.

Theorem 1.3.

Let (N₁, g₁, τ₁) and (N₂, g₂, τ₂) be two n + 1-dimensional Lorentzian manifolds, n ≥ 2, equipped with regular cosmological time functions τ_i, i = 1, 2, such that

{\hat{d}}_{τ_{i}}

encodes causality (for example, this will be the case if τ_i are proper). If there exists a bijection F : M₁ → M₂ that preserves null distances,

{\hat{d}}_{τ_{1}} (p, q) = {\hat{d}}_{τ_{2}} (F (p), F (q)) for any p, q \in N_{1},

(17)

and cosmological times,

τ_{1} = τ_{2} ◦ F,

(18)

then F is a diffeomorphism and is a Lorentzian isometry, F*g₂ = g₁.

This theorem is proven in Sec. V. It is natural to ask why we wish to prove Theorem 1.3 without assuming that F : M₁ → M₂ is differentiable. The short answer is that F is a map between metric spaces and differentiability is not defined between such spaces. The more serious answer is that we plan to apply this theorem to study limits of sequences of Lorentzian manifolds. Suppose (N_j, h_j) → (N_∞, h_∞) is defined by requiring that

(N_{j}, {\hat{d}}_{τ_{j}}, τ_{j}) \to (N_{\infty}, {\hat{d}}_{τ_{\infty}}, τ_{\infty})

(19)

in the intrinsic flat or Gromov–Hausdorff sense with control on the τ_j as well (to appear in Ref. 12), then we only expect the limit spaces to be defined uniquely up to a distance preserving and cosmological time preserving bijection. In order to guarantee that the limit space is unique when it happens to be a smooth Lorentzian manifold, we need Theorem 1.3 to be proven without assuming that F is a smooth map. The same idea arises when studying Gromov–Hausdorff and intrinsic flat limits of Riemannian manifolds, (N, g), viewed as metric spaces, (N, d_g), and in that setting, one shows that the limit spaces are unique up to a distance preserving bijection and that a distance preserving bijection between smooth Riemannian manifolds is, in fact, a smooth Riemannian isometry.

It would be interesting to explore to what extent these theorems hold on lower regularity spacetimes such as those studied by Harris,¹³ Alexander and Bishop,¹⁴ Chruściel and Grant,¹⁵ Burtscher,¹⁶ Kunzinger and Sämann,¹⁷ Alexander et al.,¹⁸ Graf and Ling,¹⁹ Cavalletti and Mondino,²⁰ McCann and Sämann,²¹ Hau, Cabrera Pacheco, and Solis,²² and Burtscher and García-Heveling.²³ Note that Kunzinger and Steinbauer have already extended the notion of the null distance to their notion of a Lorentz length space.⁹ Within, most of our proofs do not require much regularity as long as piecewise causal curves behave well enough. However, we do apply the results of Temple,²⁴ Levichev,²⁵ Hawking,²⁶ and Zeeman²⁷ that would need to be extended. See Remark 5.4.

II. EXAMPLES

Here, we present two examples. The first one is from Ref. 2, which we repeat here because it clarifies why our proofs involve the reverse Lipschitz property.

Example 2.1.

Consider τ = t³ on a Minkowski space that fails to satisfy the reverse Lipschitz property. In Ref. 2 , it was proven that for any two points p, q in the {t = 0} slice, we have ${\hat{d}}_{τ} (p, q) = 0$ . Thus, ${\hat{d}}_{τ}$ both is not definite and fails to encode causality both locally and globally.

To see why

{\hat{d}}_{τ} (p, q) = 0

, let c(s) = (0, x(s)) be the straight line from c(0) = p to c(D) = q parameterized by Euclidean arclength. Let β_j(s) be a piecewise null curve from β_j(0) = p to β_j(D) = q with 2j segments such that

β_{j} (i D / (2 j)) = (0, x (i D / (2 j))) for i even and

(20)

β_{j} (i D / (2 j)) = (D / (2 j), x (i D / (2 j))) for i odd .

(21)

Then,

{\hat{d}}_{τ} (p, q) \leq \sum_{i = 1}^{2 j} | τ (β_{j} (i D / (2 j))) - τ (β_{j} ((i - 1) D / (2 j))) |

(22)

= \sum_{i = 1}^{2 j} {(D / (2 j))}^{3} = (2 j) {(D / (2 j))}^{3} \to 0 .

(23)

Example 2.2.

Let N be a Minkowski upper-half space with a half-line removed,

N = {(t, x) : t \in (0, \infty), x \in R^{3}} \ {(t, 0, 0, 0) : t \in [2, \infty)},

(24)

endowed with the Minkowski metric. It is easy to see that the cosmological time τ = τ_AGH = t satisfies the reverse Lipschitz condition. However,

{\hat{d}}_{τ}

does not encode causality. To see this, consider the point p = (1, −1, 0, 0) and q = (3, 1, 0, 0). In a Minkowski space, these points are connected by a future causal curve C(s) = (1 + s, −1 + s, 0, 0) that runs from C(0) = p to C(2) = q; however, this curve runs through C(1) = (2, 0, 0, 0) ∉ N. In fact, q is not in the causal future of p in N.

Nevertheless, for every ϵ > 0, we have a piecewise causal curve that runs first past causal from p to p_ϵ = (1 − ϵ, −1, ϵ, 0) and then along a future causal curve,

C (s) = (1 - ϵ + s, - 1 + s, ϵ, 0) from C (0) = p_{ϵ} to C (2) = q_{ϵ},

(25)

and then future causal from q_ϵ = (3 − ϵ, 1, ϵ, 0) to q. See Fig. 2 . Thus,

{\hat{d}}_{τ} (p, q) \leq | τ (p) - τ (p_{ϵ}) | + | τ (p_{ϵ}) - τ (q_{ϵ}) | + | τ (q) - τ (q_{ϵ}) |

(26)

= ϵ + 2 + ϵ \to 2 = τ (q) - τ (p) as ϵ \to 0 .

(27)

Since

2 = τ (q) - τ (p) \geq {\hat{d}}_{τ} (p, q)

for all p, q ∈ M, and since the reverse inequality holds true [see (7)], we have

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p),

(28)

so

{\hat{d}}_{τ}

does not encode causality globally. It does encode causality locally if we take our neighborhoods to be small cylindrical blocks that avoid the missing half-line and are, thus, isometric to cylindrical blocks in a Minkowski space.

FIG. 2.

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In Example 2.2, N is a Minkowski upper-half space with a half-line (depicted as a black arrow) removed. Here, ${\hat{d}}_{τ}$ for τ = τ_AGH fails to encode causality globally: there exist p, q satisfying (28) such that q ∉ J⁺(p) because q lies in the shadow of the half-line for light rays from p.

III. THE NULL DISTANCE ENCODES CAUSALITY LOCALLY

In this section, we prove Theorem 1.1. In particular, we would like to show that around every point p, there is a neighborhood U such that if q ∈ U, then ${\hat{d}}_{τ} (p, q) = τ (q) - τ (p)$ implies q ∈ J⁺(p).

In the first part of Sec. III A, we do not yet restrict to neighborhoods. We show that if ${\hat{d}}_{τ} (p, q) = τ (q) - τ (p)$ holds, then for every ϵ > 0, there is a curve β from p to q as in (5), zigzaging backward and forward in time and such that the null length of its past directed part is less than ϵ; see Fig. 3 and Lemma 3.2. Subsequently, in Lemma 3.3, we prove a refined localized version of this general property under the assumption that the time function is locally anti-Lipschitz. However, note that the existence of a curve β as described above does not immediately imply that q ∈ J⁺(p), even though ϵ can be chosen to be arbitrarily small. In fact, we just presented a counterexample above; see Example 2.2.

FIG. 3.

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A piecewise causal curve β from p to q that almost achieves ${\hat{d}}_{τ} (p, q) = τ (q) - τ (p)$ has very short past causal segments.

In order to prove Theorem 1.1, we need a suitable “indicator function” of the causal future,

J^{+} (p) = {q | q is in the causal future of p},

(29)

at least within a neighborhood. As discussed in Sec. III B, it turns out that certain optical functions defined by Temple in 1938²⁴ are well suited for this purpose. In particular, their level sets are null hypersurfaces generated by null geodesics that can be used to set up a coordinate system capturing the local causal structure of the spacetime; see Theorem 3.4.

In Sec. III C, we apply this coordinate system, combined with the Lipschitzness of the optical function and the anti-Lipschitzness of the time function to complete the proof of Theorem 1.1.

A. Almost minimizing piecewise causal geodesics

Before establishing an implication of ${\hat{d}}_{τ} (p, q) = τ (q) - τ (p)$ for “almost minimizers” for the infimum in the definition of the null distance (5), we review a basic fact about lengths of piecewise causal curves. Recall that on any metric space (M, d), a curve is d-rectifiable if its d-rectifiable length is finite,

L_{d} (C [a, b]) = \sup \sum_{i = 1}^{m} d (C (s_{i}), C (s_{i - 1})),

(30)

where the supremum is taken over all partitions a = s₀ ≤ s₁ ≤ ⋯ ≤ s_m = b.

Lemma 3.1.

On any Lorentzian manifold with any time function τ, and for any piecewise causal curve, C : [a, b] → N, which is causal on segments [a_i−1, a_i], where a = a₀ ≤ a₁ ≤ ⋯ ≤ a_N = b, we see that the

{\hat{d}}_{τ}

-rectifiable length of the curve agrees with the null length,

L_{{\hat{d}}_{τ}} (C [a, b]) = {\hat{L}}_{τ} (C [a, b]) = \sum_{i = 1}^{N} | τ (C (a_{i})) - τ (C (a_{i - 1})) | = \int_{a}^{b} | d / d s (τ ◦ C) | d s .

(31)

So the curve can always be reparametrized proportional to

{\hat{d}}_{τ}

-length to

C : [0, 1] \to N such that | d / d s (τ ◦ C) | = {\hat{L}}_{τ} (C [a, b]) .

(32)

Proof.

Given any partition, a = s₀ ≤ s₁ ≤ ⋯ ≤ s_m = b, we can take a subpartition

a = s_{0}^{'} \leq s_{1}^{'} \leq \dots \leq s_{m}^{'} = b

such that

{s_{0}^{'}, s_{1}^{'}, \dots, s_{m}^{'}} = {s_{0}, s_{1}, \dots, s_{m}} \cup {a_{0}, a_{1}, \dots, a_{N}},

(33)

and by the triangle inequality, we have

L_{{\hat{d}}_{τ}} (C [a, b]) \geq \sum_{i = 1}^{m^{'}} {\hat{d}}_{τ} (C (s_{i}^{'}), C (s_{i - 1}^{'})) \geq \sum_{i = 1}^{m} {\hat{d}}_{τ} (C (s_{i}), C (s_{i - 1})) .

(34)

Since C is causal on each segment

C [s_{i - 1}^{'}, s_{i}^{'}]

⁠, the middle term is

\sum_{i = 1}^{m^{'}} {\hat{d}}_{τ} (C (s_{i}^{'}), C (s_{i - 1}^{'})) = \sum_{i = 1}^{m^{'}} | τ (C (s_{i}^{'})) - τ (C (s_{i - 1}^{'})) | = {\hat{L}}_{τ} (C) .

(35)

Plugging this back in the middle and taking the supremum over partitions, a = s₀ ≤ s₁ ≤ ⋯ ≤ s_m = b, we see that

L_{{\hat{d}}_{τ}} (C [a, b]) \geq {\hat{L}}_{τ} (C) \geq \sup \sum_{i = 1}^{m} {\hat{d}}_{τ} (C (s_{i}), C (s_{i - 1})) = L_{{\hat{d}}_{τ}} (C [a, b]) .

(36)

Let C be a piecewise causal curve as in the formulation of the lemma, and let

L = L_{{\hat{d}}_{τ}} (C [a, b])

⁠. Without loss of generality, we may assume that C : [0, 1] → N and that it is causal on the segments [s_i, s_i+1], i = 0, …, m − 1, where 0 = s₀ ≤ s₁ ≤ ⋯ ≤ s_m = 1. In this case, on each segment [s_i, s_i+1], the function s ↦ τ(C(s)) is monotone. If C is future causal on [s_i, s_i+1], we may parameterize it so that

τ (C (s)) = L (s - s_{i}) + τ (C (s_{i})) for s \in [s_{i}, s_{i + 1}],

(37)

and if C is past causal on [s_i, s_i+1], we may parameterize it so that

τ (C (s)) = - L (s - s_{i}) + τ (C (s_{i})) for s \in [s_{i}, s_{i + 1}] .

(38)

In this case, |d/ds(τ◦C)| = L, and we see that C is parameterized proportional to

{\hat{d}}_{τ}

-rectifiable length because the segments are piecewise causal, so within the segments,

| τ (C (s)) - τ (C (s^{'})) | = {\hat{d}}_{τ} (C (s)), (C (s^{'})) = L | s^{'} - s | .

(39)

□

Now, we consider “almost minimizers” of the infimum in the definition of the null distance (5).

Lemma 3.2.

Suppose

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p)

⁠, where τ is a time function, then for any ϵ > 0, there exists a piecewise causal curve β : [0, 1] → N such that β(0) = p and β(1) = q with

0 = s_{0} \leq s_{1} \leq s_{2} \leq \dots \leq s_{2 k} \leq s_{2 k + 1} = 1

(40)

such that if s_2i ≠ s_2i+1, then β is future causal on the interval [s_2i, s_2i+1] for i = 0, …, k and such that if s_2i+1 ≠ s_2i+2, then β is past causal on the interval [s_2i+1, s_2i+2] for i = 0, …, k − 1. Furthermore, we have

\sum_{i = 0}^{k} (τ (β (s_{2 i + 1})) - τ (β (s_{2 i}))) < {\hat{d}}_{τ} (p, q) + ϵ

(41)

and

\sum_{i = 0}^{k - 1} | τ (β (s_{2 i + 2})) - τ (β (s_{2 i + 1})) | < ϵ .

(42)

Moreover, we can parameterize β proportional to

{\hat{d}}_{τ}

-arclength (see Lemma 3.1). Thus,

τ (β (s)) - τ (β (s^{'})) = (s - s^{'}) {\hat{L}}_{τ} (β) for any s, s^{'} \in [s_{2 i}, s_{2 i + 1}],

(43)

for i = 0, …, k, and for i = 0, …, k − 1,

τ (β (s)) - τ (β (s^{'})) = - (s - s^{'}) {\hat{L}}_{τ} (β) for any s, s^{'} \in [s_{2 i + 1}, s_{2 i + 2}] .

(44)

Keep in mind that Example 2.1 satisfies the hypotheses of this lemma.

Proof.

By the definition of

{\hat{d}}_{τ} (p, q)

⁠, we know that there exists a piecewise causal curve β : [0, 1] → N such that β(0) = p and β(1) = q, where

{\hat{L}}_{τ} (β) < {\hat{d}}_{τ} (p, q) + ϵ

⁠. By allowing β to have segments, where s_i = s_i+1, we can ensure that β is as in the formulation of the lemma. In this case, we have

τ (β (s_{2 i + 1})) - τ (β (s_{2 i})) \geq 0

(45)

and

τ (β (s_{2 i + 2})) - τ (β (s_{2 i + 1})) \leq 0 .

(46)

On each interval [s_i, s_i+1], where s_i ≠ s_i+1, the function s ↦ τ(β(s)) is monotone. Consequently, we can parameterize β starting with the first interval so that it satisfies () and then continuing along the second interval satisfying () and so on up and down each interval until, at the end, we reach

q = β ({\hat{L}}_{τ} (β) / {\hat{L}}_{τ} (β)) = β (1)

⁠.

By the definition of

{\hat{L}}_{τ} (β)

and (), we have

{\hat{L}}_{τ} (β) ≔ \sum_{i = 0}^{k - 1} | τ (β (s_{2 i + 2})) - τ (β (s_{2 i + 1})) | + \sum_{i = 0}^{k} (τ (β (s_{2 i + 1})) - τ (β (s_{2 i}))) < {\hat{d}}_{τ} (p, q) + ϵ .

(47)

Dropping the first sum, which is non-negative, we have ().

Telescoping our sum and then applying (), we obtain

\begin{matrix} τ (q) - τ (p) & = & τ (β (s_{2 k + 1})) - (τ (β (s_{0}))) \\ = & \sum_{i = 0}^{k - 1} (τ (β (s_{2 i + 2})) - τ (β (s_{2 i + 1}))) + \sum_{i = 0}^{k} (τ (β (s_{2 i + 1})) - τ (β (s_{2 i}))) \\ \leq & \sum_{i = 0}^{k} (τ (β (s_{2 i + 1})) - τ (β (s_{2 i}))) . \end{matrix}

Plugging this and the hypothesis of our lemma,

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p)

⁠, into (), we get

\sum_{i = 0}^{k - 1} | τ (β (s_{2 i + 2})) - τ (β (s_{2 i + 1})) | + τ (q) - τ (p) < τ (q) - τ (p) + ϵ,

(48)

which gives ().□

We now prove a local consequence of this result under the assumption that the time function satisfies the locally anti-Lipschitz condition in the sense of Chruściel, Grant, and Minguzzi.⁶ Note that the result below holds if we use a different Riemannian metric on U, up to rescaling the right hand side.

Lemma 3.3.

Given a point p ∈ N and a neighborhood U ⊆ N about p that has a Riemannian metric with a distance function d_U : U × U → [0, ∞), suppose that τ is a generalized time function satisfying the anti-Lipschitz condition (4) for all q, q′ ∈ U.

Let r_p > 0 be such that

B_{{\hat{d}}_{τ}} (p, 2 r_{p}) \subset U

and take any ϵ ∈ (0, r_p). For any

q \in W_{p} = B_{{\hat{d}}_{τ}} (p, r_{p}) \subset U

such that

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p),

(49)

there exists a piecewise causal curve β : [0, 1] → U ⊆ N such that β(0) = p and β(1) = q, where β satisfies all the properties of Lemma 3.2, and we have the following estimate for the past causal intervals:

\sum_{i = 0}^{k - 1} | s_{2 i + 2} - s_{2 i + 1} | < ϵ / {\hat{d}}_{τ} (p, q) .

(50)

Keep in mind that Example 2.2 satisfies the hypotheses of this lemma where the neighborhood U can be taken to be the entire space.

Proof.

Taking p, q as in the statement, there is a piecewise causal curve β defined in Lemma 3.2 and parameterized so that () and () hold. Consequently, within each causal interval of β that lies within U, we have

| s - s^{'} | {\hat{L}}_{τ} (β) = | τ (β (s)) - τ (β (s^{'})) | \geq d_{U} (β (s), β (s^{'}))

(51)

by hypothesis () and the fact that β(s) and β(s′) are causally related.

We claim that β lies entirely in U. Indeed, suppose that β leaves U so that

s_{max} = \sup {s : β [0, s] \subset U} < 1 .

(52)

Let β₀ be the restriction of β to the interval [0, s_max]. In this case, we have

{\hat{L}}_{τ} (β) \geq {\hat{L}}_{τ} (β_{0})

⁠, which is straightforward to show by introducing an artificial breaking point at s = s_max; see (). Consequently, we have

{\hat{L}}_{τ} (β) \geq {\hat{L}}_{τ} (β_{0}) \geq {\hat{d}}_{τ} (p, β (s_{max})) \geq 2 r_{p} .

(53)

On the other hand, following the proof of Lemma 3.2, we have chosen β so that

{\hat{L}}_{τ} (β) < {\hat{d}}_{τ} (p, q) + ϵ \leq r_{p} + ϵ

⁠, a contradiction.

By Lemma 3.2, β is past causal on intervals [s_2i+1, s_2i+2] unless s_2i+1 = s_2i+2 for i = 0, …, k − 1. Since these intervals lie within U, we can apply both () and (), obtaining

\sum_{i = 0}^{k - 1} | s_{2 i + 2} - s_{2 i + 1} | {\hat{L}}_{τ} (β) = \sum_{i = 0}^{k - 1} | τ (β (s_{2 i + 2})) - τ (β (s_{2 i + 1})) | < ϵ .

(54)

Since

{\hat{d}}_{τ} (p, q) \leq {\hat{L}}_{τ} (β)

⁠, we have ().□

B. Optical functions

In general, an optical function on a spacetime (N, g) is a solution ω of the eikonal equation g(∇ω, ∇ω) = 0. Its level sets are null hypersurfaces generated by null geodesic segments. Optical functions are important in the study of spacetimes; in particular, they are used in the proof of stability of Minkowski spacetime by Christodoulou and Klainerman.²⁸ Many recent results in mathematical general relativity are proven using double null coordinates that are constructed using incoming and outgoing level sets of an optical function.

In this paper, we apply two coordinate systems introduced in a 1938 paper by Temple²⁴ that we will call his future null coordinate chart and his past null coordinate chart. The future null coordinate system is depicted in Fig. 4.

FIG. 4.

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Temple’s future null coordinate chart along a timelike curve.

Theorem 3.4

(Ref. 24). Given any p, let η : (−ϵ, ϵ) → N be a unit speed future (respectively, past) timelike geodesic through η(0) = p. Let

{\hat{e}}_{0} = η^{'} (0)

⁠, and let

{\hat{e}}_{1}, \dots, {\hat{e}}_{n} \in T_{p} N

be an orthonormal collection of spacelike vectors such that

{\hat{e}}_{i} + {\hat{e}}_{0}

is future (respectively, past) null. We extend this frame by parallel transport along η, noting that since η is a geodesic,

η^{'} (t) = {\hat{e}}_{0}

at η(t). Note that for any

x \in R^{m}

,

\sum_{i = 1}^{n} x_{i} {\hat{e}}_{i} + | x | η^{'} (t) is a null vector in T_{η (t)} N .

(55)

We define a future (respectively, past) null chart

Φ : {\tilde{U}}_{η} \to U_{η} \subset N

, by

Φ (t, x_{1}, \dots, x_{n}) = \exp_{η (t)} (\sum_{i = 1}^{n} x_{i} {\hat{e}}_{i} + | x | η^{'} (t)),

(56)

which is continuous and invertible on a neighborhood

{\tilde{U}}_{η}

of (−ϵ, ϵ) × {0}ⁿ and is smooth away from η. In this chart, we define the future (respectively, past) optical function

ω : U_{η} \to R

by

ω (Φ (t, x_{1}, \dots, x_{n})) = t

(57)

and a radial function

λ : U_{η} \to R

by

λ (Φ (t, x_{1}, \dots, x_{n})) = \sqrt{x_{1}^{2} + \dots + x_{n}^{2}} .

(58)

Given a unit speed future timelike geodesic η : (−ϵ, ϵ) → N through η(0) = p, we can reverse the parametrization and define both the future and past null charts,

Φ_{+} : {\tilde{U}}_{+} \to U_{η} and Φ_{-} : {\tilde{U}}_{-} \to U_{η}, respectively,

(59)

together onto the same domain, U_η, after possibly shrinking their domains. We let

ω_{+} : U_{η} \to R and ω_{-} : U_{η} \to R

(60)

be the future optical function and past optical function, respectively.

Example 3.5.

Consider the standard Minkowski spacetime

R^{n, 1}

with coordinates (x₀, x₁, …, x_n) and the metric

g_{Mink} = - d x_{0}^{2} + d x_{1}^{2} + \dots + d x_{n}^{2}

. Let η(t) = (t, 0, 0, 0),

t \in R

. Then, in the notations of Theorem 3.4, we have

λ = r ≔ \sqrt{x_{1}^{2} + \dots + x_{n}^{2}},

(61)

and the future optical function is given by

ω_{+} (x_{0}, x_{1}, \dots, x_{n}) = x_{0} - r .

(62)

The level sets of ω₊,

ω_{+}^{- 1} (c) = \{(x_{0}, x_{1}, \dots, x_{n}) : x_{0} = c + \sqrt{x_{1}^{2} + \dots + x_{n}^{2}}\},

(63)

are the future null cones of (c, 0, …, 0). Note that ω₊ is non-negative on J⁺((0, 0, …, 0)) and negative elsewhere. Taking, instead, η(t) = (−t, 0, 0, 0),

t \in R

, we find that the past optical function is

ω_{-} (x_{0}, x_{1}, \dots, x_{n}) = - x_{0} - r .

(64)

The level sets of ω₋,

ω_{-}^{- 1} (c) = \{(x_{0}, x_{1}, \dots, x_{n}) : x_{0} = - c - \sqrt{x_{1}^{2} + \dots + x_{n}^{2}}\},

(65)

are the past null cones of (−c, 0, …, 0). Clearly, ω₋ is non-negative on J⁻((0, 0, …, 0)) and negative elsewhere. Note that both ω₊ and ω₋ are differentiable away from the x₀-axis.

We will also use the following key property of these charts that we have discovered.

Lemma 3.6.

Given p ∈ N, let U_η be the image of Temple’s past and future null coordinate charts centered at p. Let ω₊ be the future optical function. For any q ∈ U_η, we have

ω_{+} (q) \geq 0 ⟹ q \in J^{+} (p) .

(66)

Furthermore, if q, q′ ∈ U_η, then

q^{'} \in J^{+} (q) ⟹ ω_{+} (q^{'}) \geq ω_{+} (q) .

(67)

Similarly, if ω₋ is the past optical function, then

ω_{-} (q) \geq 0 ⟹ q \in J^{-} (p),

(68)

and

q^{'} \in J^{-} (q) ⟹ ω_{-} (q^{'}) \geq ω_{-} (q)

(69)

whenever q, q′ ∈ U_η.

Proof.

We present the proof in the case when ω = ω₊ is a future optical function; the necessary modifications in the case of a past optical function ω = ω₋ are straightforward: one essentially needs to replace “future” by “past” in the arguments below. Suppose that ω(q) ≥ 0. If q ∈ η, then q = η(t) for t = ω(q) ≥ 0. Since η is timelike future directed, it follows that q ∈ J⁺(p). If q ∉ η, then there exists a future lightlike geodesic γ_q such that γ_q(0) = η(t), where t = ω(q) ≥ 0 and γ_q(1) = q. Consequently, q ∈ J⁺(η(t)) and η(t) ∈ J⁺(p), which implies that q ∈ J⁺(p).

Let γ be a piecewise smooth future causal geodesic such that γ(0) = q and γ(1) = q′. Without loss of generality, we may assume that γ intersects η finitely many times. Recalling that ω is differentiable away from η with g(∇ω, ∇ω) = 0, we get

ω (q^{'}) - ω (q) = \int_{0}^{1} \frac{d}{d σ} ω (γ (σ)) d σ

(70)

= \int_{0}^{1} g (\nabla ω (γ (σ)), γ^{'} (σ)) d σ

(71)

= \int_{0}^{1} - g (\nabla (- ω) (γ (σ)), γ^{'} (σ)) d σ

(72)

\geq \int_{0}^{1} | \nabla (- ω) (γ (σ)) |_{g} | γ^{'} (σ) |_{g} d σ

(73)

= 0 .

(74)

Here, we have used the fact that ∇(−ω) is future null and the reverse Cauchy–Schwartz inequality (see, e.g., O’Neill,²⁹ Chap. 5). Thus, ω(q′) ≥ ω(q) as claimed.□

Using either of Temple’s coordinate charts, we can define a Riemannian metric on the neighborhood covered by the chart.

Lemma 3.7.

Let

Φ : {\tilde{U}}_{η} \to U_{η}

be the future or past null coordinate chart as in Theorem 3.4, with the respective optical function ω. We can define a continuous Riemannian metric on U_η by

g_{R} (V, W) = \frac{2}{| g (X, X) |} g (X, V) g (X, W) + g (V, W),

(75)

where X is a continuous vector field such that X = η′(t) along η and X = ∂_t, where t is as in Theorem 3.4 outside η. Furthermore, if we define the Riemannian gradient ∇_Rω of the optical function ω by

g_{R} (\nabla_{R} ω, V) = V (ω) for any vector field V on U_{η},

(76)

then

| \nabla_{R} ω |_{g_{R}} = \sqrt{g_{R} (\nabla_{R} ω, \nabla_{R} ω)} = \sqrt{2 | g (X, X) |^{- 1}} < 2

(77)

away from η, up to shrinking U_η if necessary.

Note that in this lemma, the Riemannian metric, $g_{R}^{+}$ ⁠, defined by the future null chart does not necessarily agree with the Riemannian metric, $g_{R}^{-}$ ⁠, defined by the past null chart. We will only use one at a time anyway.

Proof.

Given a spacetime (N, g) and a continuous timelike vector field X, it is straightforward to check using an orthonormal frame

\{e_{0} = \frac{X}{| g (X, X) |^{1 / 2}}, e_{1}, \dots, e_{n}\}

(78)

that () defines a continuous Riemannian metric on N. The challenge here is to prove that the vector field X as in the formulation of the theorem is continuous on U_η, even though the optical function ω is not differentiable along η and the vector field ∂_t is a priori not defined on η.

We claim that X is continuous along null geodesics emanating from η, up to η. Fix a unit vector

u \in R^{n}

⁠, and consider a family of null geodesics

{γ_{(t, u)}}_{t}

⁠, where γ_(t,u) is a null geodesic starting at η(t) and defined for λ ≥ 0 sufficiently small by

γ_{(t, u)} (λ) = \exp_{η (t)} (λ (\sum_{i = 1}^{n} u_{i} {\hat{e}}_{i} + η^{'} (t))) .

(79)

Note that when λ ≠ 0, this can be written as γ_(t,u)(λ) = Φ(t, λu). Observe that taking a variation in t, we obtain a Jacobi field J_(t,u)(λ) such that

J_{(t, u)} (0) = η^{'} (t) = X and J_{(t, u)} (λ) = \partial_{t} = X for λ > 0 .

(80)

Since Jacobi fields are continuous along their geodesic, we conclude that X is continuous along γ_(t,u)(λ), including λ = 0.

We will now apply the fundamental theorem of ordinary differential equations, to show that X is continuous in all directions. Observe that the initial value J_(t,u)(0) = η′(t) and the initial covariant derivative

\nabla_{γ_{(t, u)}^{'} (0)} J_{(t, u)} (0) = 0

(81)

are smooth along η(t); thus, the Jacobi fields vary continuously in the t direction for any fixed u and for λ ≥ 0. If we vary u, then we are just rotating our initial direction within the null cone, and the variation of the null geodesics is smooth and the Jacobi fields vary continuously as well.

Finally, we verify (). Let {e₀, e₁, …, e_n} be an orthonormal frame for the Lorentzian metric g on U_η as in (); in particular, e₀ is timelike and e₁, …, e_n are spacelike. In what follows, we work on U_η\η. Using (), it is straightforward to verify that {e₀, e₁, …, e_n} is also an orthonormal frame for the Riemannian metric g_R so that

\nabla_{R} ω = e_{0} (ω) e_{0} + e_{1} (ω) e_{1} + \dots + e_{n} (ω) e_{n}

(82)

and

g_{R} (\nabla_{R} ω, \nabla_{R} ω) = {(e_{0} (w))}^{2} + {(e_{1} (w))}^{2} + \dots + {(e_{n} (w))}^{2} .

(83)

Since ω satisfies the eikonal equation g(∇ω, ∇ω) = 0 away from η, we have

{(e_{0} (ω))}^{2} = {(e_{1} (ω))}^{2} + \dots + {(e_{n} (ω))}^{2},

(84)

so g_R(∇_Rω, ∇_Rω) = 2e₀(ω)². Recalling that

e_{0} = \frac{X}{| g (X, X) |^{1 / 2}}

⁠, we obtain

g_{R} (\nabla_{R} ω, \nabla_{R} ω) = \frac{2 X {(ω)}^{2}}{| g (X, X) |} = \frac{2}{| g (X, X) |}

(85)

and () follows, since by the continuity of X, we may assume that |g(X, X)| > 1/2 in U_η, as g(X, X) = −1 on η.□

Remark 3.8.

Note that in the proof above, we have only claimed that X is continuous and not smooth. This is because we never pass through η and only check that the Jacobi fields are varying continuously for λ ≥ 0. Even though we could show the Jacobi fields are differentiably continuous for λ ≥ 0, that does not prove they are differentiable on η. Perhaps, they are, or perhaps, they are not.

Lemma 3.9.

If g_R is the continuous Riemannian metric on U_η as defined in Lemma 3.7 using the future (respectively, past) null coordinate chart, and

d_{g_{R}}

is the Riemannian distance with respect to g_R defined by

d_{g_{R}} (q, q^{'}) = \inf {L_{g_{R}} (C) : C piecewise smooth curve from q to q^{'}},

(86)

then

\sup \{\frac{| ω (q) - ω (q^{'}) |}{d_{g_{R}} (q, q^{'})} : q \neq q^{'} \in U_{η}\} < 2,

(87)

where ω is the future (respectively, past) optical function.

Proof.

Let γ_i : [0, 1] → U_η be (piecewise smooth) curves from γ_i(0) = q to γ_i(1) = q′ such that

\lim_{i \to \infty} L_{g_{R}} (γ_{i}) \to d_{g_{R}} (q, q^{'}) .

(88)

We can assume that γ_i hits the image of η at most finitely many times so that ω is differentiable along γ_i away from those times, with

| \nabla_{R} ω |_{g_{R}} \leq 2

by Lemma 3.7. Then,

| ω (q) - ω (q^{'}) | \leq \int_{0}^{1} | \frac{d}{d σ} ω (γ_{i} (σ)) | d σ

(89)

= \int_{0}^{1} | g_{R} (\nabla_{R} ω (γ_{i} (σ)), γ_{i}^{'} (σ)) | d σ

(90)

\leq \int_{0}^{1} | \nabla_{R} ω (γ_{i} (σ)) |_{g_{R}} | γ_{i}^{'} ((σ)) |_{g_{R}} d σ

(91)

\leq 2 \int_{0}^{1} | γ_{i}^{'} ((σ)) |_{g_{R}} d σ

(92)

= 2 L_{g_{R}} (γ_{i}) .

(93)

Taking the limit as i → ∞, we have

\frac{| ω (q) - ω (q^{'}) |}{d_{g_{R}} (q, q^{'})} \leq 2 for all q \neq q^{'} \in U_{η} .

(94)

□

C. Proof that the null distance encodes causality locally

In this section, we prove Theorem 1.1.

Proof.

Let p ∈ N. We will first prove that there is a neighborhood W of p such that for all q ∈ W, we have

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p) ⟹ q is in the causal future of p .

(95)

For this, take any timelike future unit speed geodesic η through p and define a neighborhood U_η ⊂ U and the future optical function

ω : U_{η} \to R

as in Theorem 3.4. Let g_R be the continuous Riemannian metric on U_η as in Lemma 3.7. Furthermore, let r_p be such that

B_{{\hat{d}}_{τ}} (p, 2 r_{p}) \subset U_{η}

and set

U_{p} = B_{{\hat{d}}_{τ}} (p, r_{p} / 2)

⁠. Then, for all p′ ∈ U_p, we have

U_{p} = B_{{\hat{d}}_{τ}} (p, r_{p} / 2) \subset B_{{\hat{d}}_{τ}} (p^{'}, r_{p}) \subset B_{{\hat{d}}_{τ}} (p, 2 r_{p}) \subset U_{η} .

(96)

Consequently, by choosing r_p′ = r_p, we can ensure that

for any p^{'} \in U_{p} we have U_{p} \subset W_{p^{'}} of Lemma 3.3 .

(97)

We will prove that for all q ∈ U_p, we have

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p) ⟹ q is in the causal future of p .

(98)

Fix an arbitrary ϵ > 0, and let β : [0, 1] → U ⊂ M such that β(0) = p and β(1) = q be a piecewise causal curve as in Lemma 3.3. By telescoping sums, we have

ω (q) - ω (p) = \sum_{i = 0}^{k - 1} (ω (β (s_{2 i + 2})) - ω (β (s_{2 i + 1}))) + \sum_{i = 0}^{k} (ω (β (s_{2 i + 1})) - ω (β (s_{2 i}))) .

(99)

The second sum is non-negative by Lemma 3.6, as β(s_2i+1) is in the causal future of β(s_2i) as long as s_2i ≠ s_2i+1. Since ω(p) = 0, we, thereby, have

ω (q) \geq \sum_{i = 0}^{k - 1} (ω (β (s_{2 i + 2})) - ω (β (s_{2 i + 1}))),

(100)

where β is past causal on [s_2i+1, s_2i+2] if s_2i+1 ≠ s_2i+2. Each term in the right hand side is non-positive but controlled in view of Lemma 3.9 and (),

\begin{aligned} ω (β (s_{2 i + 2})) - ω (β (s_{2 i + 1})) & \geq - 2 d_{g_{R}} (β (s_{2 i + 2}), β (s_{2 i + 1})) \\ \geq - 2 C_{p} d_{U} (β (s_{2 i + 2}), β (s_{2 i + 1})) \\ \geq - 2 C_{p} {\hat{L}}_{τ} (β) | s_{2 i + 2} - s_{2 i + 1} | \end{aligned},

where C_p is a constant such that

C_{p}^{- 1} d_{g_{R}} (q, z) \leq d_{U} (q, z) \leq C_{p} d_{g_{R}} (q, z) for all q, z \in U_{p} .

(101)

Finally, applying (), we arrive at

ω (q) \geq - 2 C_{p} {\hat{L}}_{τ} (β) \sum_{i = 0}^{k - 1} | s_{2 i + 2} - s_{2 i + 1} | \geq - 2 C_{p} ϵ .

(102)

Since this is true for all ϵ > 0, we have ω(q) ≥ 0 and, thus, q is in the causal future of p.

Finally, we describe how the above argument needs to be modified to show that there is a neighborhood W of p such that for all q ∈ W, we have

{\hat{d}}_{τ} (p, q) = τ (p) - τ (q) ⟹ q is in the causal past of p .

(103)

We let η be a timelike past unit speed geodesic through p, U_η be the domain of the past null chart, and ω be the past optical function. We define U_p as before, and note that in this case, p ∈ W_q for any q ∈ U_p. As a consequence, for any ϵ > 0, there is a piecewise causal curve β : [0, 1] → U ⊂ N such that β(0) = q and β(1) = p with small past segments in the sense of Lemma 3.3. Equivalently, there is a piecewise causal curve β : [0, 1] → U ⊂ M such that β(0) = p and β(1) = q with small future segments. We may now repeat the above argument using the past optical function in place of future one to conclude that q ∈ J⁻(p).□

IV. THE NULL DISTANCE ENCODES CAUSALITY WHEN τ IS PROPER

In this section, we prove that ${\hat{d}}_{τ}$ globally encodes causality for spacetimes with τ : N → (0, T) that is proper. Recall that a function is proper if preimages of compact sets are compact. So we are assuming that

τ^{- 1} [T_{1}, T_{2}] is compact whenever 0 < T_{1} \leq T_{2} < T,

(104)

where T = sup_N τ ∈ (0, ∞]. We also assume that N is path connected.

Theorem 4.1.

Let (N, g) be a path connected spacetime, and let

τ : N \to R

be a proper time function locally satisfying the reverse Lipschitz condition of (4). Then,

{\hat{d}}_{τ}

globally encodes causality in N; that is for all p, q ∈ N,

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p) ⟹ q \in J^{+} (p) .

(105)

We can also prove this theorem for more general classes of manifolds, but the proofs are much longer, so we will postpone these theorems for the future.

Remark 4.2.

Recall that in Example 2.2, we gave an example of a path connected spacetime, (N, g), with a generalized time function, τ, locally satisfying the reverse Lipschitz condition that had a pair of points p, q such that ${\hat{d}}_{τ} (p, q) = τ (q) - τ (p)$ ⁠, but q was not in the causal future of p. The key obstruction in this example was that a ray was missing, which led to the nonexistence of the causal curve from p to q. In this example, the level sets of τ were not compact, and in particular, the sequence of piecewise causal curves β_j from p to q such that ${\hat{L}}_{τ} (β_{j}) \to {\hat{d}}_{τ} (p, q)$ did not have a converging subsequence.

A. Finding ${\hat{d}}_{τ}$ minimizing curves

In this section, we prove that under the appropriate hypotheses on N and τ, every pair of points p, q ∈ N such that ${\hat{d}}_{τ} (p, q) = τ (q) - τ (p)$ has a curve C_p,q whose ${\hat{d}}_{τ}$ -rectifiable length achieves ${\hat{d}}_{τ} (p, q)$ ⁠; see Lemma 4.3 below. This is not true, in general, as was seen in Example 2.2.

Lemma 4.3.

Let (N, g) be a spacetime with a proper time function τ. Suppose that p, q ∈ N are such that

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p) .

(106)

Then, there exists a curve C_p,q = C ⊂ N such that C(0) = p, C(1) = q, and for all 0 ≤ s < s′ ≤ 1, we have

{\hat{d}}_{τ} (C (s), C (s^{'})) = τ (C (s^{'})) - τ (C (s)) .

(107)

Proof.

By the definition of null distance, for any j > 0, there exists a piecewise causal curve β_j ⊂ N from p to q such that

{\hat{d}}_{τ} (p, q) \leq L_{j} = {\hat{L}}_{τ} (β_{j}) < {\hat{d}}_{τ} (p, q) + 1 / j .

(108)

Recall that by Lemma 3.6,² we have

{\hat{L}}_{τ} (β_{j}) \geq \max_{β_{j}} τ - \min_{β_{j}} τ .

(109)

As a consequence of (), (), and (), we have

\begin{matrix} τ (q) - τ (p) + 1 / j & \geq & \max_{β_{j}} τ - τ (p), \\ τ (q) - τ (p) + 1 / j & \geq & τ (q) - \min_{β_{j}} τ . \end{matrix}

It follows that

τ (p) - 1 / j \leq \min_{β_{j}} τ < \max_{β_{j}} τ \leq τ (q) + 1 / j .

(110)

Since τ(p), τ(q) ∈ (0, T), by taking j to be sufficiently large, we see that all β_j are contained in a compact set τ⁻¹([T₁, T₂]) for some 0 < T₁ ≤ T₂ < T.

Since β_j is piecewise causal, by Lemma 3.1, we can parameterize each β_j proportional to

{\hat{d}}_{τ}

-rectifiable length. In this case, we have β_j : [0, 1] → N, and for any s, s′ ∈ [0, 1], we have

L_{{\hat{d}}_{τ}} (β_{j} |_{[s, s^{'}]}) = L_{j} | s - s^{'} | \geq | τ (β_{j} (s)) - τ (β_{j} (s^{'})) | .

(111)

Since

L_{j} \to L = {\hat{d}}_{τ} (p, q)

and since all the images of β_j are contained in the compact set, τ⁻¹([T₁, T₂]), we can apply the Arzela–Ascoli theorem (cf. Ref. 30, Theorem 2.5.14) to see that there is a subsequence of {β_j}, denoted by the same notation, that C⁰ converges to a curve C : [0, 1] → N that is contained in τ⁻¹([T₁, T₂]). By the lower semicontinuity of

L_{{\hat{d}}_{τ}}

(cf. Ref. 30, Proposition 2.3.4), we have

L_{{\hat{d}}_{τ}} (C) \leq {lim inf}_{j \to \infty} L_{{\hat{d}}_{τ}} (β_{j}) = {lim inf}_{j \to \infty} {\hat{L}}_{τ} (β_{j}) = {\hat{d}}_{τ} (p, q) .

(112)

On the other hand, by the triangle inequality and definition of rectifiable length,

L_{{\hat{d}}_{τ}} (C) \geq {\hat{d}}_{τ} (p, q) .

(113)

Combining this with our hypothesis, we have

L_{{\hat{d}}_{τ}} (C) = {\hat{d}}_{τ} (p, q) = τ (q) - τ (p) .

(114)

Note also that for any 0 ≤ s < s′ ≤ 1, we have

L_{{\hat{d}}_{τ}} (C [s, s^{'}]) \geq {\hat{d}}_{τ} (C (s), C (s^{'})) \geq | τ (C (s) - τ (C (s^{'}))) | \geq τ (C (s) - τ (C (s^{'})))

(115)

by the definition of rectifiable length, triangle inequality, and ().

We now turn to the proof of the last claim. Assume, on the contrary, that there is a segment [s, s′] ⊆ [0, 1] such that

{\hat{d}}_{τ} (C (s), C (s^{'})) > τ (C (s)) - τ (C (s^{'})) .

(116)

Applying () to [0, s], [s, s′], and [s′, 1], we would have

\begin{matrix} L_{{\hat{d}}_{τ}} (C [0, 1]) & = & L_{{\hat{d}}_{τ}} (C [0, s]) + L_{{\hat{d}}_{τ}} (C [s, s^{'}]) + L_{{\hat{d}}_{τ}} (C [s^{'}, 1]) \\ \geq & {\hat{d}}_{τ} (C (0), C (s)) + {\hat{d}}_{τ} (C (s), C (s^{'})) + {\hat{d}}_{τ} (C (s^{'}), C (1)) \\ > & (τ (C (0)) - τ (C (s))) + (τ (C (s)) - τ (C (s^{'}))) + (τ (C (s^{'})) - τ (C (1))) \\ = & τ (C (0)) - τ (C (1)), \end{matrix}

which contradicts ().□

As seen from this proof, the curve C_p,q such that its ${\hat{d}}_{τ}$ -rectifiable length achieves ${\hat{d}}_{τ} (p, q)$ will exist if we replace (106) and the hypothesis of properness of τ by any other assumption that ensures that almost ${\hat{d}}_{τ}$ -minimizing curves β_j are contained in a compact set. Note that, in general, such a curve C_p,q need not be piecewise causal even though it is a limit of piecewise causal curves β_j as we see in the following example.

Example 4.4.

Consider, for instance, p = (0, 0, 0, 0) and q = (0, 1, 0, 0) in a Minkowski space with τ = t. Let β_j be piecewise causal curves running from C(0) = p = p₀ to C(1) = q = p_2j via the points

p_{2 i} = (0, i / j, 0, 0) and p_{2 i + 1} = (1 / (2 j) + (1 / j^{2}), i / j + 1 / (2 j), 0, 0)

(117)

with

{\hat{L}}_{τ} (β_{j}) = \sum_{i = 1}^{j} (| τ (p_{2 i - 1}) - τ (p_{2 i - 2}) | + | τ (p_{2 i}) - τ (p_{2 i - 1}) |)

(118)

= \sum_{i = 1}^{j} 2 (1 / (2 j) + (1 / j^{2})) = j (2 (1 / (2 j) + (1 / j^{2}))) = 1 + 2 / j .

(119)

As seen in the Minkowski space example in Ref. 2 ,

{\hat{d}}_{τ} (p, q) = 1

, so as j → ∞, we see

{\hat{L}}_{τ} (β_{j}) \to {\hat{d}}_{τ} (p, q) .

(120)

It is easy to see that these β_j converge in the C⁰ sense to

C_{p, q} (s) = (0, s, 0, 0),

(121)

which is spacelike from p to q and not piecewise null.

B. Local-to-global with a proper time function

We can now prove Theorem 4.1.

Proof.

Let p, q ∈ N be such that

{\hat{d}}_{τ} (p, q) = τ (q) - τ (p) .

(122)

In this case, Lemma 4.3 implies that there is a curve C = C_p,q : [0, 1] → N such that C(0) = p, C(1) = q, and for all 0 ≤ s < s′ ≤ L, we have

L_{{\hat{d}}_{τ}} (C |_{[s, s^{'}]}) = {\hat{d}}_{τ} (C (s), C (s^{'})) = τ (C (s^{'})) - τ (C (s)) .

(123)

For each point C(t), t ∈ [0, L], we have an open neighborhood U_C(t), where

{\hat{d}}_{τ}

locally encodes causality. Since C([0, L]) is compact, we have finitely many

U_{i} = U_{C (s_{i})}

required to cover it. Taking

0 = t_{0} < s_{1} < t_{1} < \dots < s_{N} < t_{N} = L

(124)

so that C(t_i), C(s_i+1), C(t_i+1) ∈ U_i, for all i = 0, …, N − 1, we have

{\hat{d}}_{τ} (C (s_{i + 1}), C (t_{i})) = τ (C (s_{i + 1})) - τ (C (t_{i})),

(125)

and for all i = 1, …, N, we have

{\hat{d}}_{τ} (C (s_{i}), C (t_{i})) = τ (C (t_{i})) - τ (C (s_{i})) .

(126)

By local causality on U_i, for all i = 0, …, N − 1, we have

C (s_{i + 1}) is in the causal future of C (t_{i}),

(127)

and for all i = 1, …, N, we have

C (t_{i}) is in the causal future of C (s_{i}) .

(128)

Thus, q = C(L) is in the causal future of p = C(0).□

V. THE ISOMETRY THEOREM

In this section, we will prove Theorem 1.3. This theorem concerns a pair of spacetimes, (N₁, g₁) and (N₂, g₂), equipped with regular cosmological time functions, τ₁ and τ₂, and a bijection F : N₁ → N₂ that preserves null distances,

{\hat{d}}_{τ_{1}} (p, q) = {\hat{d}}_{τ_{2}} (F (p), F (q)) for any p, q \in N_{1},

(129)

and cosmological times,

τ_{1} = τ_{2} ◦ F .

(130)

We assume that the cosmological time functions τ_i are proper so that the causality is encoded by the associated null distances ${\hat{d}}_{τ_{i}}$ by Theorem 4.1.

As explained in the Introduction, if F was known to be a diffeomorphism, then the fact that it preserves the causal structure would immediately imply that it is a conformal isometry. However, we do not know that F is even differentiable.

To prove that F is a conformal isometry, we will apply the following theorem of Levichev,²⁵ which builds upon a lemma in Hawking’s 1966 Adams Prize Essay (that appears as Lemma 19 in the reprint²⁶), which, in turn, builds upon the work of Zeeman.²⁷ See also the paper by Hawking, King, and McCarthy³¹ and the work of Malament.³² See Minguzzi’s recent survey¹¹ for an overview of these results.

Theorem 5.1

(Levichev). Let (N₁, g₁) and (N₂, g₂) be two n + 1-dimensional distinguishing spacetimes, n ≥ 2, and let F : N₁ → N₂ be a causal bijection, i.e., a bijection such that

q \in J^{+} (p) ⟺ F (q) \in J^{+} (F (q)) .

(131)

Then, F is a smooth conformal isometry; i.e., there exists a smooth function ϕ > 0 such that F*g₂ = ϕ²g₁.

Anderson, Galloway, and Howard³ proved that a regular cosmological time function is continuous, which implies that the spacetimes (N_i, g_i) are distinguishing, see, e.g., Ref. 11, Theorem 4.5.8 (v′). Thus, Levichev’s theorem can be applied on manifolds with regular cosmological time functions. In particular, we can apply it in our proof of Theorem 1.3.

Proof of Theorem 1.3.

The assumptions of the theorem together with Theorem 4.1 imply that F : N₁ → N₂ is a causal bijection. Since our spacetimes have regular cosmological time functions and are thus distinguishing, we may apply Theorem 5.1, to conclude that F is a conformal isometry: there exists a smooth positive function

ϕ : N_{1} \to R

so that

F^{*} g_{2} = ϕ^{2} g_{1} on N_{1} .

(132)

We will show that ϕ ≡ 1 so that F : N₁ → N₂ is an isometry. Assume, without loss of generality, that there is q ∈ N₁ such that that ϕ(q) > 1 [the case ϕ(q) < 1 is treated similarly]. Let U ⊆ N₁ be a precompact neighborhood of q such that ϕ > 1 in U, and let V be the closure of U. We will compute the volume of V in two different ways to reach a contradiction. On the one hand, we have

\int_{F (V)} d μ^{g_{2}} = \int_{V} d μ^{F^{*} g_{2}} = \int_{V} d μ^{ϕ^{2} g_{1}} = \int_{V} ϕ^{n} d μ^{g_{1}} .

(133)

Suppose that τ₁(V) = [τ_min, τ_max], then we also have τ₂(F(V)) = [τ_min, τ_max]. Since the cosmological time functions τ_i, i = 1, 2, are Lipschitz with

| \nabla τ_{i} |_{g_{i}} = 1

almost everywhere, we can compute

\begin{aligned} \int_{F (V)} d μ^{g_{2}} & = \int_{τ_{\min}}^{τ_{max}} (\int_{τ_{2}^{- 1} (τ)} {d H}_{n - 1}^{g_{2}}) d τ, by the coarea formula, \\ = \int_{τ_{\min}}^{τ_{max}} (\int_{F^{- 1} (τ_{2}^{- 1} (τ))} {d H}_{n - 1}^{F^{*} g_{2}}) d τ, by change of variables, \\ = \int_{τ_{\min}}^{τ_{max}} (\int_{τ_{1}^{- 1} (τ)} {d H}_{n - 1}^{F^{*} g_{2}}) d τ, by τ_{2} ◦ F = τ_{1}, \\ = \int_{τ_{\min}}^{τ_{max}} (\int_{τ_{1}^{- 1} (τ)} ϕ^{n - 1} {d H}_{n - 1}^{g_{1}}) d τ, by (132) applied in dimension (n - 1), \\ = \int_{V} ϕ^{n - 1} d μ^{g_{1}}, by the coarea formula again. \end{aligned}

If we subtract this from (), we have

\int_{V} ϕ^{n - 1} (ϕ - 1) d μ^{g_{1}} = \int_{V} ϕ^{n} d μ^{g_{1}} - \int_{V} ϕ^{n - 1} d μ^{g_{1}} = 0,

(134)

which contradicts ϕ > 1 in V. Consequently, ϕ ≡ 1 and F*g₂ = g₁ on N.□

Note that in the above proof, we strongly use that τ_i are cosmological time functions.

Example 5.2.

Let (N₁, g₁) be any Lorentzian manifold with any time function τ₁. If we let N₂ = N₁ and g₁ = ϕ²g₂ and τ₁ = τ₂, then

{\hat{d}}_{τ_{1}} (p, q) = {\hat{d}}_{τ_{2}} (p, q)

(135)

because the same curves, β, are piecewise causal with respect to both g₁ and g₂ and

{\hat{L}}_{τ_{1}} (β) = \sum_{i = 1}^{k} | τ_{1} (x_{i}) - τ_{1} (x_{i - 1}) | = \sum_{i = 1}^{k} | τ_{2} (x_{i}) - τ_{2} (x_{i - 1}) | = {\hat{L}}_{τ_{2}} (β) .

(136)

Thus, the identity map F : N₁ → N₂ is an isometry between

(N_{1}, {\hat{d}}_{τ_{1}})

and

(N_{2}, {\hat{d}}_{τ_{2}})

⁠, which preserves the time functions, τ₁ = τ₂◦F, but F is not a Lorentzian isometry.

Remark 5.3.

Note the dimensional restriction in Theorem 1.3 is a consequence of the respective restriction in Levichev’s Theorem 5.1²⁵ and Hawking’s 1966 work that Levichev builds upon. In Hawking’s proof (cf. Ref. 26 ), the dimension condition is essential for proving the differentiability of F. We do not know of a non-example demonstrating that their dimension condition is necessary to prove that F is differentiable. Nor do we know of a non-example demonstrating that the dimension condition is necessary to prove our isometry theorem.

Remark 5.4.

It should also be noted that we have deliberately proven our theorem using the coarea formula so that we may, then, imitate this proof on a lower regularity space such as the integral current spaces of Sormani and Wenger³³ in the future. Levichev’s theorem is very much proven in the style of Alexandrov geometry and should extend easily. Hawking’s work is harder to dissect. More recent work studying Lorentz spaces of lower regularity might be applied to extend the work of Levichev and Hawking to these settings and, thus, extend our isometry theorem. See the work of Harris,¹³ Alexander and Bishop,¹⁴ Chruściel and Grant,¹⁵ Burtscher,¹⁶ Kunzinger and Sämann,¹⁷ Alexander et al.,¹⁸ Graf and Ling,¹⁹ Cavalletti and Mondino,²⁰ McCann and Sämann,²¹ Hau, Cabrera Pacheco, and Solis,²² and Burtscher and García-Heveling.²³

ACKNOWLEDGMENTS

The authors began this research project while in residence at the Mathematical Sciences Research Institute (MSRI), funded by NSF Grant No. 0932078000. This research was also funded, in part, by Sormani’s NSF Grant No. DMS-1612409 and PSC-CUNY funding. A. Sakovich was partly funded by the Swedish Research Council dnr. 2016-04511.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

A. Sakovich: Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). C. Sormani: Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

REFERENCES

1.

C.

Sormani

, “

Oberwolfach report 2018: Spacetime intrinsic flat convergence

,” Oberwolfach Report No. 36/2018,

2018

.

Google Scholar

2.

C.

Sormani

and

C.

Vega

, “

Null distance on a spacetime

,”

Classical Quant. Grav.

33

(

8

),

085001

(

2016

).

https://doi.org/10.1088/0264-9381/33/7/085001

Google Scholar

Crossref

3.

L.

Andersson

,

G. J.

Galloway

, and

R.

Howard

, “

The cosmological time function

,”

Classical Quant. Grav.

15

(

2

),

309

–

322

(

1998

).

https://doi.org/10.1088/0264-9381/15/2/006

Google Scholar

Crossref

4.

R. M.

Wald

and

P.

Yip

, “

On the existence of simultaneous synchronous coordinates in spacetimes with spacelike singularities

,”

J. Math. Phys.

22

,

2659

–

2665

(

1981

).

https://doi.org/10.1063/1.524844

Google Scholar

Crossref

5.

N.

Ebrahimi

, “

Some observations on cosmological time functions

,”

J. Math. Phys.

54

(

5

),

052503

(

2013

).

https://doi.org/10.1063/1.4807429

Google Scholar

Crossref

6.

P. T.

Chruściel

,

J. D. E.

Grant

, and

E.

Minguzzi

, “

On differentiability of volume time functions

,”

Ann. Henri Poincare

17

(

10

),

2801

–

2824

(

2016

).

https://doi.org/10.1007/s00023-015-0448-3

Google Scholar

Crossref

7.

B.

Allen

and

A.

Burtscher

, “

Properties of the null distance and spacetime convergence

,”

Int. Math. Res. Not.

2022

(

10

),

7729

–

7808

.

https://doi.org/10.1093/imrn/rnaa311

Crossref

8.

C.

Vega

, “

Spacetime distances: An exploration

,” arXiv:2103.01191 (

2021

).

Google Scholar

9.

M.

Kunzinger

and

R.

Steinbauer

, “

Null distance and convergence of Lorentzian length spaces

,”

Ann. Henri Poincare

23

(

12

),

4319

–

4342

(

2022

).

https://doi.org/10.1007/s00023-022-01198-6

Google Scholar

Crossref

PubMed

10.

M.

Graf

and

C.

Sormani

, “

Lorentzian area and volume estimates for integral mean curvature bounds

,”

International Meeting on Lorentzian Geometry

(

Springer

,

2022

), pp.

105

–

128

.

Google Scholar

Crossref

11.

E.

Minguzzi

, “

Lorentzian causality theory

,”

Living Rev. Relativ.

22

,

3

(

2019

).

https://doi.org/10.1007/s41114-019-0019-x

Google Scholar

Crossref

12.

A.

Sakovich

and

C.

Sormani

, “

Spacetime intrinsic flat convergence of cosmological integral current spacetimes

,” (unpublished).

13.

S. G.

Harris

, “

A triangle comparison theorem for Lorentz manifolds

,”

Indiana Univ. Math. J.

31

(

3

),

289

–

308

(

1982

).

https://doi.org/10.1512/iumj.1982.31.31026

Google Scholar

Crossref

14.

S. B.

Alexander

and

R. L.

Bishop

, “

Lorentz and semi-Riemannian spaces with Alexandrov curvature bounds

,”

Commun. Anal. Geom.

16

(

2

),

251

–

282

(

2008

).

https://doi.org/10.4310/cag.2008.v16.n2.a1

Google Scholar

Crossref

15.

P. T.

Chruściel

and

J. D. E.

Grant

, “

On Lorentzian causality with continuous metrics

,”

Classical Quant. Grav.

29

(

14

),

145001

(

2012

).

https://doi.org/10.1088/0264-9381/29/14/145001

Google Scholar

Crossref

16.

A. Y.

Burtscher

, “

Length structures on manifolds with continuous Riemannian metrics

,”

New York J. Math.

21

,

273

–

296

(

2015

).

Google Scholar

17.

M.

Kunzinger

and

C.

Sämann

, “

Lorentzian length spaces

,”

Ann. Global Anal. Geom.

54

(

3

),

399

–

447

(

2018

).

https://doi.org/10.1007/s10455-018-9633-1

Google Scholar

Crossref

18.

S. B.

Alexander

,

M.

Graf

,

M.

Kunzinger

, and

C.

Sämann

, “

Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems

,”

Comm. Anal. Geom.

(to be published); arXiv:1909.09575.

19.

M.

Graf

and

E.

Ling

, “

Maximizers in Lipschitz spacetimes are either timelike or null

,”

Classical Quant. Grav.

35

(

8

),

087001

(

2018

).

https://doi.org/10.1088/1361-6382/aab259

Google Scholar

Crossref

20.

F.

Cavalletti

and

A.

Mondino

, “

Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications

,” arXiv:2004.08934 (

2020

).

Google Scholar

21.

R. J.

McCann

and

C.

Sämann

, “

A Lorentzian analog for Hausdorff dimension and measure

,”

Pure Appl. Anal.

4

(

2

),

367

–

400

(

2022

).

https://doi.org/10.2140/paa.2022.4.367

Google Scholar

Crossref

22.

L. A.

Hau

,

A. J.

Cabrera Pacheco

, and

D. A.

Solis

, “

On the causal hierarchy of Lorentzian length spaces

,”

Classical Quant. Grav.

37

(

21

),

215013

(

2020

).

https://doi.org/10.1088/1361-6382/abb25f

Google Scholar

Crossref

23.

A.

Burtscher

and

L.

Garcia-Heveling

, “

Time functions on Lorentzian length spaces

,” arXiv:2108.02693 (

2021

).

Google Scholar

24.

G.

Temple

, “

New systems of normal co-ordinates for relativistic optics

,”

Proc. R. Soc. London, Ser. A

168

(

932

),

122

–

148

(

1938

).

https://doi.org/10.1098/rspa.1938.0164

Google Scholar

Crossref

25.

A. V.

Levichev

, “

The causal structure of a Lorentzian manifold determines its conformal geometry

,”

Dokl. Akad. Nauk SSSR

293

(

6

),

1301

–

1305

(

1987

).

Google Scholar

26.

S.

Hawking

, “

Singularities and the geometry of spacetime

,”

Eur. Phys. J. H

39

,

413

–

503

(

2014

).

https://doi.org/10.1140/epjh/e2014-50013-6

Google Scholar

Crossref

27.

E. C.

Zeeman

, “

Causality implies the Lorentz group

,”

J. Math. Phys.

5

,

490

(

1964

).

https://doi.org/10.1063/1.1704140

Google Scholar

Crossref

28.

D.

Christodoulou

and

S.

Klainerman

, “

The nonlinear stability of the Minkowski metric in general relativity

,” in

Nonlinear Hyperbolic Problems

(

Springer

,

1989

), pp.

128

–

145

.

Google Scholar

Crossref

29.

B.

O’Neill

,

Semi-Riemannian Geometry with Applications to Relativity

(

Academic Press

,

1983

).

Google Scholar

30.

D.

Burago

,

Y.

Burago

, and

S.

Ivanov

,

A Course in Metric Geometry

(

American Mathematical Society

,

2022

), Vol. 33.

Google Scholar

31.

S. W.

Hawking

,

A. R.

King

, and

P. J.

McCarthy

, “

A new topology for curved space–time which incorporates the causal, differential, and conformal structures

,”

J. Math. Phys.

17

(

2

),

174

–

181

(

1976

).

https://doi.org/10.1063/1.522874

Google Scholar

Crossref

32.

D. B.

Malament

, “

The class of continuous timelike curves determines the topology of spacetime

,”

J. Math. Phys.

18

(

7

),

1399

–

1404

(

1977

).

https://doi.org/10.1063/1.523436

Google Scholar

Crossref

33.

C.

Sormani

and

S.

Wenger

, “

The intrinsic flat distance between Riemannian manifolds and other integral current spaces

,”

J. Differ. Geom.

87

(

1

),

117

–

199

(

2011

).

https://doi.org/10.4310/jdg/1303219774

Google Scholar

Crossref

2023

Author(s)

The null distance encodes causality

I. INTRODUCTION

II. EXAMPLES

III. THE NULL DISTANCE ENCODES CAUSALITY LOCALLY

A. Almost minimizing piecewise causal geodesics

B. Optical functions

C. Proof that the null distance encodes causality locally

IV. THE NULL DISTANCE ENCODES CAUSALITY WHEN τ IS PROPER

A. Finding ${\hat{d}}_{τ}$ minimizing curves

B. Local-to-global with a proper time function

V. THE ISOMETRY THEOREM

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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The null distance encodes causality Open Access

I. INTRODUCTION

II. EXAMPLES

III. THE NULL DISTANCE ENCODES CAUSALITY LOCALLY

A. Almost minimizing piecewise causal geodesics

B. Optical functions

C. Proof that the null distance encodes causality locally

IV. THE NULL DISTANCE ENCODES CAUSALITY WHEN τ IS PROPER

A. Finding d̂τ minimizing curves

B. Local-to-global with a proper time function

V. THE ISOMETRY THEOREM

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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The null distance encodes causality

A. Finding ${\hat{d}}_{τ}$ minimizing curves