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I have a series of naive questions regarding the question in the title. Let $\mathcal{S}_1$ be a family of spacelike hypersurfaces $\{\Sigma_1\}$ in $\mathbb{M}$. Let $\Sigma$ be one of these spacelike hypersurfaces.

  1. Can I always find (infinitely many?) families of spacelike hypersurfaces which also contain $\Sigma$? Intuitively I want to say yes?
  2. Let $\mathcal{U}$ and $\mathcal{V}$ be spacelike-separated regions of $\mathbb{M}$, both of which lie in the causal past of $\Sigma$, and such that there exists a $\tilde{\Sigma} \in \mathcal{S}_1$ for which $\mathcal{U}$ is in the causal future of $\tilde{\Sigma}$ while $\mathcal{V}$ is in the causal past of $\tilde{\Sigma}$. If 1. holds, can I always find at least one family $\mathcal{S}_2$ of spacelike hypersurfaces which contains $\Sigma$ and which contains a spacelike hypersurface $\Sigma'$ for which $\mathcal{U}$ is now in the causal past of $\Sigma'$ while $\mathcal{V}$ is in the causal future of $\Sigma'$? Again, intuitively I want to say yes - since $\mathcal{U}$ and $\mathcal{V}$ are spacelike separated, there exists spacelike hypersurfaces for which one is in the causal future while the other is in the causal past, and vice-versa.
  3. Does the above still hold more generally in globally hyperbolic spacetimes?

Any proofs or references would be greatly appreciated! Thank you!

Samuel Fedida
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  • What is your definition of "family of hyperplanes"? What is a "globally hyperbolic spacetime"? – Nicholas Todoroff Dec 16 '24 at 17:18
  • Globally hyperbolic spacetime : https://en.wikipedia.org/wiki/Globally_hyperbolic_manifold; they have the topology of $\mathbb{R} \times \Sigma$ where $\Sigma$ is a Cauchy surface (spacelike hypersurface), thus defining a family of spacelike hypersurfaces. – Samuel Fedida Jan 07 '25 at 16:30

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