Conformally Embedded Spacetimes and the Space of Null Geodesics

Abstract

It is shown that the space of null geodesics of a causally simple spacetime is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature conditions. Examples of causally simple spacetimes are given not conformally embeddable into globally hyperbolic ones.

Appendix A. The Index Form of a Null Geodesic

Here we recall the definition and the properties of the index form of a null geodesic for the convenience of the reader. The material with proofs and additional explanations can be found in [1, Chapter 10] and the references therein.

Let (M, g) be a spacetime and $\gamma :[a,b]\rightarrow M$ be a null geodesic. Denote with

$$\begin{aligned} \gamma ^\perp :=\{v\in \gamma ^*TM|\; g(v,{\dot{\gamma }})=0\} \end{aligned}$$

the orthogonal bundle to $\gamma$. Define a equivalence relation $\sim$ on $\gamma ^\perp$ by setting $v\sim w$ if $w-v\in \text {span}({\dot{\gamma }})$. Denote with $\overline{v}$ the equivalence class of $v\in \gamma ^\perp$. The quotient bundle

$$\begin{aligned} \overline{\gamma ^\perp }:=\gamma ^\perp /\sim \end{aligned}$$

is a smooth bundle over [a, b]. Since $\gamma$ is null one has $g(v_1,w_1)=g(v_2,w_2)$ for all $v_1,v_2,w_1,w_2\in \gamma ^\perp$ with $\overline{v_1}=\overline{v_2}$ and $\overline{w_1}=\overline{w_2}$. The same is true for the curvature endomorphism $R(.,{\dot{\gamma }}){\dot{\gamma }}$, i.e.

$$\begin{aligned} R(v,{\dot{\gamma }}){\dot{\gamma }}=R(w,{\dot{\gamma }}){\dot{\gamma }}\in \gamma ^\perp \end{aligned}$$

if $v-w\in \text {span}({\dot{\gamma }})$. Thus both the metric g and the curvature endomorphism $R(.,{\dot{\gamma }}){\dot{\gamma }}$ descend to a well defined metric $\overline{g}$ on $\overline{\gamma ^\perp }$ with

$$\begin{aligned} \overline{g}(\overline{v},\overline{w}):=g(v,w) \end{aligned}$$

and a well defined endomorphism field $\overline{R}(.,{\dot{\gamma }}){\dot{\gamma }}$ on $\overline{\gamma ^\perp }$ with

$$\begin{aligned} \overline{R}(\overline{v},{\dot{\gamma }}){\dot{\gamma }}:=\overline{R(v,{\dot{\gamma }}){\dot{\gamma }}}. \end{aligned}$$

If $X\in \Gamma (\gamma ^\perp )$ the covariant derivative is again a smooth section of $\gamma ^\perp$. Further if $X-Y\in \text {span}({\dot{\gamma }})$ everywhere, then $\nabla _{{\dot{\gamma }}}(X-Y)\in \text {span}({\dot{\gamma }})$ everywhere as well. Therefore the covariant derivative $\nabla _{{\dot{\gamma }}}$ descends to a covariant derivative on $\overline{\gamma ^\perp }$. Abbreviate the covariant derivative by a prime, i.e.

$$\begin{aligned} V':=\overline{\nabla _{{\dot{\gamma }}} X}, \end{aligned}$$

where V denotes the quotient section of $X\in \Gamma (\gamma ^\perp )$, i.e. $\overline{X}_t=V_t$ for all $t\in [a,b]$.

Denote with $\mathfrak {X}(\gamma )$ the piecewise smooth sections of $\overline{\gamma ^\perp }$ and let

$$\begin{aligned} \mathfrak {X}_0(\gamma ):=\{V\in \mathfrak {X}(\gamma )|\; V_a=0_a\text { and }V_b=0_b\}, \end{aligned}$$

where $0_t$ denotes the zero vector in $\overline{\gamma ^\perp }_t$.

Definition A.1

A smooth section $V\in \mathfrak {X}(\gamma )$ is said to be a Jacobi class in $\gamma ^\perp$ if V satisfies the Jacobi equation

$$\begin{aligned} V''+\overline{R}(V,{\dot{\gamma }}){\dot{\gamma }}=0. \end{aligned}$$

Lemma A.2

Let W be a Jacobi class in $\mathfrak {X}(\gamma )$. Then there exists a Jacobi field $Y\in \Gamma (\gamma ^\perp )$ with $\overline{Y_t}=W_t$ for all $t\in [a,b]$. Conversely if Y is a Jacobi field in $\gamma ^\perp$, then $t\mapsto W_t:=\overline{Y_t}$ is a Jacobi class in $\overline{\gamma ^\perp }$.

Lemma A.3

Let $W\in \mathfrak {X}(\gamma )$ be a Jacobi class with $W_a=0_a$ and $W_b=0_b$. Then there is a unique Jacobi field $Z\in \Gamma (\gamma ^\perp )$ with $\overline{Z_t}=W_t$ for all $t\in [a,b]$ that vanishes at a and b.

Definition A.4

For $s\ne t\in [a,b]$, s and t are said to be conjugated along$\gamma$ if there exists a Jacobi class $W\ne 0$ in $\mathfrak {X}(\gamma )$ with $W_{s}=0_s$ and $W_{t}=0_t$. Also $t\in (a,b]$ is said to be a conjugate point of$\gamma$ if $s=a$ and t are conjugate along $\gamma$.

Definition A.5

The index form$\overline{I}:\mathfrak {X}(\gamma )\times \mathfrak {X}(\gamma )\rightarrow {\mathbb {R}}$ is given by

$$\begin{aligned} \overline{I}(V,W)=-\int _a^b [\overline{g}(V',W')-\overline{g}(\overline{R}(V,{\dot{\gamma }}){\dot{\gamma }},W)]dt. \end{aligned}$$

Theorem A.6

Let $\gamma :[a,b]\rightarrow M$ be a null geodesic segment. Then the following are equivalent:

• (a) The segment $\gamma$ has no conjugate points to $s=a$ in (a, b].

• (b) $\overline{I}(W,W)<0$ for all $W\in \mathfrak {X}_0(\gamma )$, $W\ne 0$.