On Sliced Spaces: Global Hyperbolicity Revisited

Abstract

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

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:

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

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 $B_{ϵ}^h\left(x\right)$, 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 $B_{ϵ}^h\left(x\right)\times (a,b)$, 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” $J^{+}\left(x\right)\cap J^{-}\left(y\right)$ 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 $ϵ>0$, such that $B_{ϵ}^h\left(A\right)\in M$. Obviously, $B_{ϵ}^h\left(A\right)\times (a,b)\in T_P$ and, for $ϵ>0$ sufficiently large enough, $<X,Y>\subset B_{ϵ}^h\left(A\right)\times (a,b)$. Thus, $<X,Y>\in T_P$.

For 2. implies 1., we consider $ϵ>0$, such that $B_{ϵ}^h\left(A\right)\in T_M$, so that $B_{ϵ}^h\left(A\right)\times (a,b)=B\in T_P$. 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 $B_{ϵ}^h\left(x\right)\times (a,b)$ 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.