I see two definitions of a Cauchy surfaces $\Sigma$ of a spacetime $M$.
- a closed achronal subset such that the domain of dependence is $M$
i.e. a closed subset where each inextensible time-like curve intersects exactly once and each inextensible causal curve intersects - a subset where inextensible time-like curve intersects exactly once
The first one implies the second. Are they really equivalent? Can someone give some reference or proof?
Thanks in advance.
Some definitions:
$M$ is a smooth manifold (Hausdorff, connected, no boundary, paracompact)
Lorentzian metric : $g$ a continuous psedo-Riemannian metric on $M$, and at each point, $g$ has signs $(-1,1,\cdots,1)$
time-like: $v\in T_pM$ is time-like $\iff$ $g(v,v)<0$
causal: $v\in T_pM$ is causal $\iff$ $g(v,v)\leq0$
time-orientable: $(M,g)$ is time-orientable $\iff$ there exists a continuous field of time-like vectors.
spacetime: $(M,g)$ with $M$ smooth manifold and $g$ continuous Lorentzian metric and $(M,g)$ is time-orientable
curve: piecewise differentiable, i.e. $\gamma\colon I\to M$ continuous such that except finite many points in $I$, $\gamma'$ exists in the classical sense.
inextensible curve: $\gamma\colon(A,B)\to M$ without the existence of $\lim_{t\to A+}\gamma$ and $\lim_{t\to B-}\gamma$