What is physically a spacelike hypersurface?

Question

Let $(M,g)$ be spacetime. A spacelike hypersurface $\Sigma\subset M$ is a submanifold of dimension $3$ with a timelike normal vector field. They have many uses: to define conserved quantities by integrating over them, to define an inner product on the solutions to the Klein-Gordon equation, and so forth.

I have the impression though that those submanifolds are a generalization of the idea of "space at a fixed time" and hence of simultaneous events. I just don't know how to justify that and that's the main point of the question.

The situation seems simpler when we have a family of hypersurfaces obtained by a timelike future-directed vector field $\xi$ satisfying the integrability condition, i.e., letting $\omega$ the physically equivalent one-form $\omega_\mu = g_{\mu\nu}\xi^ \nu$, the condition $\omega\wedge d\omega = 0$.

In that case each integral line of $\xi$ is an observer and, if I'm not wrong, each hypersurface would consist of 'space with fixed time' for the reference frame of those observers.

But this is a family of hypersurfaces, not one alone.

What is the physical meaning of a spacelike hypersurface in General Relativity? Is it really a generalization of the idea of "space with time fixed"? If so, how can we justify this way to understand this mathematical object?

Answer 1

The edited question seems clearer, and maybe the following will address your question.

The short physics answer to your question is that GR doesn't have a global notion of simultaneity or a notion of a global frame of reference. Therefore a spacelike hypersurface is not a surface of simultaneity. What is true is that such a surface locally defines a surface of simultaneity. This works because locally, GR becomes SR, curved spacetime becomes flat spacetime (Minkowski space), a smooth spacelike curve becomes a spacelike hyperplane, and in flat spacetime any spacelike hyperplane defines a notion of simultaneity.

It may help to think about what we need to do operationally in order to establish simultaneity of events in SR. For example, inertial observers Alice and Bob send each other flashes of light, and if they find that the time between sending their flash and receiving the other person's flash is the same as the time measured similarly by the other person, they know that they sent the flashes simultaneously. This clearly won't work in GR. For example, Alice could be inside the event horizon of a black hole and Bob outside it.

Sometimes we do have a family of observers who are somehow preferred, and then this establishes a notion of simultaneity. For example, in a cosmological spacetime we can have an observer who is at rest with respect to the local matter, and this observer has a preferred status. The existence of these preferred observers defines a preferred time coordinate, which is the proper time of such an observer, measured from the Big Bang.

Typically the reason we would care about a spacelike hypersurface in GR is that it could be an appropriate place on which to define initial conditions. Given these initial conditions, we typically expect that we can use the laws of physics to evolve conditions forward in time. When we use a surface for this purpose, it doesn't matter if it represents any reasonable notion of simultaneity. It does matter that it's spacelike, although that's not sufficient. (In general, we need it to be a Cauchy surface, which exists in globally hyperbolic spacetimes.)

Answer 2

What is the physical meaning of a spacelike hypersurface in General Relativity?

A spacelike hypersurface means (physically) a set of events which are all pairwise causally disconnected (spacelike),
and which constitutes a hypersurface, i.e. an $n - 1$-dimensional manifold embedded in the spacetime manifold of dimension $n$ by virtue of its ("inherited") subspace topology.

In other (perhaps even more physical) words: Each event of a spacelike hypersurface is "outside the light cone" of any other;
and together they constitute a (generally discontinuously curved) 3-dimensional space.

[...] But this is a family of hypersurfaces, not one alone.

Any one event of a given spacelike hypersurface, or indeed any number of its events with given geometric (spatial) relations between each other, are not sufficient for uniquely determining the geometric (spatial) relations to and between all remaining events. Instead, all events are merely constrained to (a suitable sense of) Lipschitz continuity among each other.

Is it really a generalization of the idea of "space with time fixed"?

Perhaps rather some idea of "spacetime with time fixed concurrently".