## 1. Preliminaries

The definition of a sliced space, which one can read in Reference [

1], is a continuation of a study in References [

2] and [

3] on systems of Einstein equations.

Let

$V=M\times I$, where

M is an

n-dimensional smooth manifold, and

I is an interval of the real line,

$\mathbb{R}$. We equip

V with a

$n+1$-dimensional Lorentz metric

g, which splits in the following way:

where

${\theta}^{0}=dt$,

${\theta}^{i}=d{x}^{i}+{\beta}^{i}dt$,

$N=N(t,{x}^{i})$ is the

lapse function,

${\beta}^{i}(t,{x}^{j})$ is the shift function and

${M}_{t}=M\times \left\{t\right\}$, spatial slices of

V, are spacelike submanifolds equipped with the time-dependent spatial metric

${g}_{t}={g}_{ij}d{x}^{i}d{x}^{j}$. Such product space

V is called a sliced space.

Throughout the paper, we consider $I=\mathbb{R}$.

The author in Reference [

1] considered sliced spaces with uniformly bounded lapse, shift, and spatial metric; by this hypothesis, it is ensured that parameter

t measures up to a positive factor bounded (below and above) the time along the normals to spacelike slices

${M}_{t}$, the

${g}_{t}$ norm of the shift vector

$\beta $ is uniformly bounded by a number, and the time-dependent metric

${g}_{ij}d{x}^{i}d{x}^{j}$ is uniformly bounded (below and above) for all

$t\in I(=\mathbb{R})$, respectively.

Given the above hypothesis, in the same article, the following theorem was proved.

**Theorem** **1** **(Cotsakis).** Let $(V,g)$ be a sliced space with uniformly bounded lapse N, shift β and spatial metric ${g}_{t}$. Then, the following are equivalent:

- 1.
$({M}_{0},\gamma )$ a complete Riemannian manifold.

- 2.
Spacetime $(V,g)$ is globally hyperbolic.

In this article, we review global hyperbolicity of sliced spaces in terms of the product topology defined on space $M\times \mathbb{R}$ for some finite dimensional smooth manifold M.

## 2. Strong Causality of Sliced Spaces

Let $(V=M\times \mathbb{R},g)$ be a sliced space. Consider product topology ${T}_{P}$ on V. Since M is finite-dimensional, a base for ${T}_{P}$ consists of all sets of form $A\times B$, where $A\in {T}_{M}$ and $B\in {T}_{\mathbb{R}}$. Here, ${T}_{M}$ denotes the natural topology of manifold M where, for an appropriate Riemann metric h, it has a base consisting of open balls ${B}_{\u03f5}^{h}\left(x\right)$, and ${T}_{\mathbb{R}}$ is the usual topology on the real line, with a base consisting of open intervals $(a,b)$. For trivial topological reasons, we can restrict our discussion on ${T}_{P}$ to basic-open sets ${B}_{\u03f5}^{h}\left(x\right)\times (a,b)$, which can intuitively be called “open cylinders” in V.

We remind that the Alexandrov topology

${T}_{A}$ (see Reference [

4]) has a base consisting of open sets of the form

$<x,y>={I}^{+}\left(x\right)\cap {I}^{-}\left(y\right)$, where

${I}^{+}\left(x\right)=\{z\in V:x\ll z\}$ and

${I}^{-}\left(y\right)=\{z\in V:z\ll y\}$, where ≪ is the chronological order defined as

$x\ll y$ iff there exists a future-oriented timelike curve joining

x with

y. By

${J}^{+}\left(x\right)$, one denotes the topological closure of

${I}^{+}\left(x\right)$, and by

${J}^{-}\left(y\right)$ that one of

${I}^{-}\left(y\right)$.

We use the definition of global hyperbolicity from Reference [

4], where one can read about global causality conditions in more detail, as well as characterizations for strong causality. In particular, a spacetime is strongly causal iff it possesses no closed timelike curves, and global hyperbolicity is an important causal condition in a spacetime related to major problems such as spacetime singularities and cosmic cencorship.

**Definition** **1.** A spacetime is globally hyperbolic iff it is strongly causal and the “causal diamonds” ${J}^{+}\left(x\right)\phantom{\rule{3.33333pt}{0ex}}\cap \phantom{\rule{3.33333pt}{0ex}}{J}^{-}\left(y\right)$ are compact.

We prove the following theorem:

**Theorem** **2.** Let $(V,g)$ be a Hausdorff sliced space. Then, the following are equivalent.

- 1.
V is strongly causal.

- 2.
${T}_{A}\equiv {T}_{P}$.

- 3.
${T}_{A}$ is Hausdorff.

**Proof.** Here, 2. implies 3. is obvious and that 3. implies 1. can be found in Reference [

4].

For 1. implies 2., we consider two events $X,Y\in V$, such that $X\ne Y$; we note that each $X\in V$ has two coordinates, say $({x}_{1},{x}_{2})$, where ${x}_{1}\in M$ and ${x}_{2}\in \mathbb{R}$. Obviously, $X\in {M}_{x}=M\times \left\{x\right\}$ and $Y\in {M}_{y}=M\times \left\{y\right\}$. Then, $<X,Y>={I}^{+}\left(X\right)\cap {I}^{-}\left(Y\right)\in {T}_{A}$. Let also $A\in {M}_{a}=M\times \left\{a\right\}$, where $a<x$ (< is the natural order on $\mathbb{R}$) and $B\in {M}_{b}=M\times \left\{b\right\}$, where $y<b$. Consider some $\u03f5>0$, such that ${B}_{\u03f5}^{h}\left(A\right)\in M$. Obviously, ${B}_{\u03f5}^{h}\left(A\right)\times (a,b)\in {T}_{P}$ and, for $\u03f5>0$ sufficiently large enough, $<X,Y>\subset {B}_{\u03f5}^{h}\left(A\right)\times (a,b)$. Thus, $<X,Y>\in {T}_{P}$.

For 2. implies 1., we consider $\u03f5>0$, such that ${B}_{\u03f5}^{h}\left(A\right)\in {T}_{M}$, so that ${B}_{\u03f5}^{h}\left(A\right)\times (a,b)=B\in {T}_{P}$. We let strong causality hold at an event P and consider $P\in B\in {T}_{P}$. We show that there exists $<X,Y>\in {T}_{A}$, such that $P\in <X,Y>\subset B$. Now, consider a simple region R in $<X,Y>$ which contains P and $P\in Q$, where Q is a causally convex-open subset of R. Thus, we have $U,V\in Q$, such that $P\in <U,V>\subset Q$. Finally, $P\in <U,V>\subset Q\subset B$, and this completes the proof. ☐

## 3. Global Hyperbolicity of Sliced Spaces, Revisited

For the following theorem, we use Nash’s result that refers to finite-dimensional manifolds (see Reference [

5]).

**Theorem** **3.** Let $(V,g)$ be a Hausdorff sliced space, where $V=M\times R$, M is an n-dimensional manifold and g the $n+1$ Lorentz metric in V. Then, $(V,g)$ is globally hyperbolic iff ${T}_{P}={T}_{A}$, in V.

**Proof.** Given the proof of Theorem 2, strong causality in V holds iff ${T}_{P}={T}_{A}$ and, given Nash’s theorem, the closure of ${B}_{\u03f5}^{h}\left(x\right)\times (a,b)$ is compact. ☐

We note that neither in Theorem 2 nor in Theorem 3 did we make any hypothesis on the lapse function, shift function, or spatial metric.