Benchmarks

22 CHRISTIAN KETTERER

speration function

$$\backslash tau\_\{B\backslash times\_fF\}((p,x),(q,y)):=\backslash sup\; L(\backslash gamma)$$

where the sup is w.r.t. all future directed causal curves $\backslash gamma$ that connect $(p,x)$ and $(q,y)$ . If there is no such curve, one sets $\backslash tau((p,x),(q,y))=0$ . $\backslash tau\_\{B\backslash times\_fF\}$ satisfies the reverse $\backslash triangle$ -inequality

$$\backslash tau\_\{B\backslash times\_fF\}((p,x),(q,y))\backslash ge\backslash tau\_\{B\backslash times\_fF\}((p,x),(r,z))+\backslash tau\_\{B\backslash times\_fF\}((r,z),(q,y)).$$

For all $(p,x),(q,y),(r,z)\backslash in\; I\backslash times\; F$ . In the following we write also $\backslash tau\_\{B\backslash times\_fF\}=\backslash tau$ .

The time seperation function has the following properties:

• $\backslash tau(y,y\text{'})=0$ if $y\text{'}$ and $y$ are not cuassally related,
• $\backslash tau(y,y\text{'})>0$ if $y\backslash ll\; y\text{'}$ .

The generalized Lorentzian cone w.r.t. $-I$ , the metric space $F$ and the function $f:I\backslash to(0,\backslash infty)$ is

$$(I\backslash times\; F,D,\backslash ll,\backslash le,\backslash tau)=:-I\backslash times\_fF.$$

If $F$ is an intrinsic metric space, then $-I\backslash times\_fF$ is a Lorentzian pre-length space in the sense of [KS18].

Here we collect some properties of $-I\backslash times\_fF$ . We always assume that $F$ is intrinsic.

• $-I\backslash times\_fF$ has the push-up property: if $(s,x)\backslash ll(t,y)$ if and only if there exists a future directed causal curve from $(s,x)$ to $(t,y)$ with positive length. Moreover $I^\{+/-\}((s,x))$ is open for all $(s,t)\backslash in\; I\backslash times\; F$ .
• If $F$ is geodesic, then $J^\backslash pm((s,x))$ is closed for all $(s,x)\backslash in\; I\backslash times\; F$ .

The next theorem is the Lorentzian analogue of Theorem 2.5 above.

Theorem 3.1 (Kunzinger, Saemann, Graf, Alexander). Let $X$ be a strictly intrinsic metric space and let $\backslash gamma=(\backslash alpha,\backslash beta)$ be a future directed, causal curve that is a maximizer w.r.t. $L$ in $-I\backslash times\_fF$ and parametrized proportional to arclength. Then

1. $\backslash beta$ is a minimizer in $F$ ;
2. (Fiber independence) $\backslash alpha$ is independent of $F$ , except for the total height, i.e. the length $L^F(\backslash beta)$ of $\backslash beta$ . More precisely, if $\backslash hat\{F\}$ is another strictly intrinsic metric space and $\backslash hat\{\backslash beta\}$ is a minimizing geodesic in $\backslash hat\{F\}$ with the same length and speed as $\backslash beta$ , then $(\backslash alpha,\backslash hat\{\backslash beta\})$ is a minimizer in $-I\backslash times\_f\backslash hat\{F\}$ .
3. If $\backslash gamma$ is timelike, then $\backslash beta$ has speed $\backslash frac\{c\_\backslash gamma\}\{2f^2\backslash circ\backslash alpha\}$ for a constant $c\_\backslash gamma$ .
4. If $\backslash gamma$ is timelike, then $-\backslash frac\{1\}\{2\}(\backslash alpha\text{'})^2+\backslash frac\{1\}\{2f^2\backslash circ\backslash alpha\}=E$ for a constant $E$ .

We stated above that a generalized cone $-I\backslash times\_fF$ over an intrinsic metric space $F$ is a Lorentzian pre-length space. If $F$ is locally compact, it was shown in [AGKS23] that $-I\backslash times\_fF$ is even a Lorentzian length space.

Theorem 3.2. Let $F$ be a locally compact length space. Then $-I\backslash times\_fF$ is a strongly causal Lorentzian length space.