[contravariant]

That is a diagram of empty space, figuring out what that horizontal line represents in places with strong gravity is a bit tricky. In fact I'm not 100% sure there's a way to define it uniquely.

You could define the global time for a position to be the time at which light it sends reaches the earth, minus the time it took for the light to travel, but with things like gravitational lensing this can have some weird results, and may in fact not be well defined.

[imh]

I get that it might be hard/impossible to uniquely define a single correct 3D subspace of the full 4D spacetime we're in as simultaneous to a given point in a given frame, but surely simultaneity is a 3D notion, right? This 3D spacelike subset of spacetime is simultaneous in some frame (or set of frames)?

I imagine that if you define everything on the boundary of your past light cone as simultaneous to you, it requires that everything on your future light cone to be simultaneous too, which would mean that everything between them is also simultaneous. That kind of definition defines a 4D subset of spacetime as simultaneous. In that set are points in each others future and past light cones, which we can unambiguously say are not simultaneous to each other. I think that's the heart of the argument everyone's having. I never took GR, so maybe you can shed some light on it (heh) from that perspective?