Do these causality conditions correspond to ones in our universe? In other words, given that our universe appears to be causal and that information transfer is limited to the speed of light, can 3+1 curved spacetime be embedded in 5 dimensions?
I think the theorem is for isometric embeddings only, however the little part you reference is fairly intuitive when you think about it. Adding extra spatial dimensions to Minkowski space, whilst keeping the metric flat and the topology R[sup]]n[/sup] will not enable CTCs (closed timelike curves) to exist within it, so any spacetime with a CTC can’t be isometrically embedded in a pseudo-Euclidean space of index 1.
Imposing the condition that any physical spacetime could be embedded in a 4+1 spacetime pseudo-Euclidean spacetime would be far too restrictive, however a condition that it can be embedded in 87+1 dimensional pseudo-Eucildean spacetime is a slightly weaker condition than is often imposed for the physicality of spacetime.
Sorry just to add as to my previous answer as I didn’t pick up on two points: casual means there are no closed timelike curves, however this in itself doesn’t even mean you need only one time dimension (for the embedding space), you need the stronger condition of stable causality. Information transfer being limited to the speed of light is a local condition all spacetimes have.
What about something like 3+2 (or n+2) dimensional spacetime? I can (vaguely) imagine some phenomena we witness with GR being possible if there were a second dimension of time perpendicular to the arrow we move along.
Say something like, the closer you reach c, the more you would begin to turn on that “plane of time”, into the second axis, since what theoretically happens to things moving at c, is from the observer they are frozen in time, but to the traveler, the universe would speed up toward the infinite. Their timeline would look point-like to us.
I’m not saying that’s valid at all, just painting a narrow illustration of more macro-dimensions, but once you introduce more temporal dimensions (as well as spacial) certain things seem to make more sense to what our models tell us.
Actually, that’s already fully described within 3+1 dimensions. Accelerating along the x direction is equivalent to “rotating” in the xt plane, which is why it screws with both length in the x direction and time.
And as an aside, that’s special relativity, not general.
Interesting. I mean, I’m well aware of this. Also the visual aid of a 2-dimensional graph with time on the vertical axis and space on the horizontal; where you’re always “traveling” at c either through space or time, and the graph shows the relativistic trade off when you accelerate.
Is this what you mean?
That’s… a lot of dimensions! Is there any easy way to characterize where the dimensions come from? Something to do with the coefficients of the GR metric tensor, perhaps?
Not a :), but a 
Holy shit, Chronos has never heard of some theorems relevant to general relativity. Now I’m nervous.
You should be more nervous if there weren’t anything in GR I hadn’t heard of. There’s a lot of this stuff. And I was never at the more mathematical end of the field to begin with (well, OK, it’s all pretty darned mathematical from the layman’s point of view, but some is more mathematical than others).
Well, I bet neither you or any other physicists here were aware of Trapezium Giza Pyramid Artificial Black Hole Theory, by Frank M. Conaway Jr. Or am I wrong?
I just happened on it today. His references from Wiki are awesome. And he’s a member of the Maryland state legislature, so he knows what he’s talking about it.