[jfengel]

1. You're correct. It's most often phrased in terms of a spacetime interval, ds^2=dx^2-cdt^2, which is the "true", observer-independent distance between two events. You can divide by dt and rearrange to get a constant velocity c through spacetime. A different rearrangement lets you derive the gamma term from special relativity.

2. The dimensions are still independent. There are a total of five terms in the equation I just gave: ds (spacetime interval), dt (time), and dx (which is the three terms of space). You can express any one in terms of the others, leaving you with four independent variables.

The holographic view is a different phenomenon, derived from a basic integration of QM with GR (semiclassical gravity). It relies on the way black holes are complete information traps, so that only the surface area (and not the volume) are meaningful. (All of the information in it has to pass through the surface and nothing inside matters.) It also requires that energy be quantized; otherwise you get the dimension back via the infinite degrees of freedom of each particle.

So yes, you do get 3D spacetime, but only by completely rearranging everything. It doesn't pop directly out of general relativity. GR still requires 4 numbers to place any event.

[karmakaze]

Thanks for the short explanation that I can look further into. After rethinking it, it makes sense.

Could it be handwavingly said that a single observer only experiences spacetime in 3D?