On sliced spaces: Global Hyperbolicity revisited

TL;DR

This paper establishes a topological criterion for when a sliced space is globally hyperbolic, removing the need for assumptions on lapse, shift, or spatial metric functions.

Contribution

It introduces a new topological condition that guarantees global hyperbolicity in sliced spaces without relying on metric-specific hypotheses.

Findings

Provides a topological criterion for global hyperbolicity

Eliminates the need for lapse, shift, and spatial metric assumptions

Enhances understanding of the structure of sliced spaces

Abstract

We give a topological condition for a generic sliced space to be globally hyperbolic, without any hypothesis on the lapse function, shift function and spatial metric.

Equations

$$g=-N^2({\theta }^0{)}^2+g_{ij}{\theta }^i{\theta }^j,$$

Full text

1 Preliminaries.

We begin with the definition of a sliced space, that one can read in [3], as a continuation of a study in [1] and [2] 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:

[TABLE]

where $\theta^{0}=dt$, $\theta^{i}=dx^{i}+\beta^{i}dt$, $N=N(t,x^{i})$ is the lapse function, $\beta^{i}(t,x^{j})$ is the shift function and $M_{t}=M\times\{t\}$, spatial slices of $V$, are spacelike submanifolds equipped with the time-dependent spatial metric $g_{t}=g_{ij}dx^{i}dx^{j}$. Such a product space $V$ is called a sliced space.

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

The author in [3] considered sliced spaces with uniformly bounded lapse, shift and spatial metric; by this hypothesis, it is ensured that parametre $t$ 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 $g_{ij}dx^{i}dx^{j}$ 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 is proved.

Theorem 1.1** (Cotsakis).**

Let $(V,g)$ be a sliced space with uniformly bounded lapse $N$, shift $\beta$ and spatial metric $g_{t}$. Then, the following are equivalent:

$(M_{0},\gamma)$* a complete Riemannian manifold.* 2. 2.

The spacetime $(V,g)$ is globally hyperbolic.

In this article we review global hyperbolicity of sliced spaces, in terms of the product topology defined on the 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 the product topology $T_{P}$, on $V$. A base for $T_{P}$ consists of all sets of the form $A\times B$, where $A\in T_{M}$ and $B\in T_{\mathbb{R}}$. Here $T_{M}$ denotes the natural topology of the manifold $M$ where, for an appropriate Riemann metric $h$, it has a base consisting of open balls $B_{\epsilon}^{h}(x)$ 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 $B_{\epsilon}^{h}(x)\times(a,b)$, which can be intuitively called as “open cylinders” in $V$.

We remind the Alexandrov topology $T_{A}$ (see [4]) has a base consisting of open sets of the form $<x,y>=I^{+}(x)\cap I^{-}(y)$, where $I^{+}(x)=\{z\in V:x\ll z\}$ and $I^{-}(y)=\{z\in V:z\ll y\}$, where $\ll$ is the chronological order defined as $x\ll y$ iff there exists a future oriented timelike curve, joining $x$ with $y$. By $J^{+}(x)$ one denotes the topological closure of $I^{+}(x)$ and by $J^{-}(y)$ that one of $I^{-}(y)$.

We use the definition of strong causality and global hyperbolicity from [4]; global hyperbolicity is an important causal condition in a spacetime related to major problems such as spacetime singularities, cosmic cencorship etc.

Definition 2.1**.**

A spacetime is globally hyperbolic, iff it is strongly causal and the “causal diamonds” $J^{+}(x)\cap J^{-}(y)$ are compact.

We prove the following theorem.

Theorem 2.1**.**

Let $(V,g)$ be sliced space. Then, the following are equivalent.

$V$* is strongly causal.* 2. 2.

$T_{A}\equiv T_{P}$. 3. 3.

$T_{A}$* is Hausdorff.*

Proof.

2. implies 3. is obvious and that 3. implies 1. can be found in [4].

For 1. implies 2.: That $T_{A}$ is coarser than $T_{P}$ is trivial. Now, we consider $\epsilon>0$, such that $B_{\epsilon}^{h}(A)\in T_{M}$, so that $B_{\epsilon}^{h}(A)\times(a,b)=B\in T_{P}$. 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.

Theorem 3.1**.**

Let $(V,g)$ be a sliced space, where $V=M\times\mathbb{R}$, $M$ is an $n$-dimensional manifold ($n\geq 2$) and $g$ the $n+1$ Lorentz metric in $V$. Let $T_{A}$ be the Alexandrov spacetime topology on $V$ and $T_{P}$ the product topology on $V$. Then, $(V,g)$ is globally hyperbolic, iff for every basic-open set in $D$ in $T_{A}$ there exists a basic-open set $B$ in $T_{P}$, such that $D\subset B$.

Proof.

If $V$ is globally hyperbolic, then it is strongly causal (see [5]) and from Theorem 2.1 we get that $T_{P}\equiv T_{A}$. Consider a basic-open set (an “open diamond”) $D$, in $T_{A}$; then, $D\in T_{P}$. Choose an open diamond $B\in T_{P}$, such that $D\subset B$; the result follows.

Now, consider $D=I^{+}(x)\cap I^{-}(y)$ be a basic-open set in $T_{A}$. Then, there exists $B^{h}(x)$ in the manifold topology on $M$ and $(a,b)$ in the usual topology of $\mathbb{R}$, such that $D\subset B^{h}(x)\times(a,b)=B$. But, The closure of $B$ is compact (as a product of compact sets) and so the closure of $D$ will be compact too, as a closed subset of a compact set.

∎

We note that neither in Theorem 2.1 nor in Theorem 3.1 we made any hypothesis on the lapse function, shift function or on the spatial metric.