Now, that said, you can play embedding space games in GR, and some space-times can be visualized as curved hypersurfaces in a higher dimensional, flat, Minkowski like space-time.

Indeed, dcl's hypersphere has some utility here. DeSitter space-time (empty space with a positive Lambda) can be visualized neatly as an expanding hypershphere in a 1,4 space-time. 1-4 means 1 time-like dimension and 4 space-like dimensions.

Imagine an observer in this 1-4 space-time at the "center". Let the radius of a hypersphere around him expand, and expand in a Rindler hyperbolic accelerating world line. The subspace of the surface of that hypersphere is the curved, 1-3 deSitter space-time.

[Because of the non-positive definite behavior here, hyperbolas behave like spheres. And you can see the space-like part of the hyperbolic structures can be nicely spherical. Anyway, this hyperbolic sub-hypersurface is one of constant space-time curvature. That is given by Lambda, and that relates directly to the proper acceleration of the radius. So you can think of Lambda as how fast the hypersphere is accelerating.]

All well and good and neat. But throw matter in the mix for a LCDM universe and it doesn't work so neatly anymore. We've just let the air out of our little hyperballoon.

And that's the problem. Many valid curved 1-3 space-times (and by valid I mean they are solutions of the EFE and so can represent some plausible space-time) cannot be embedded in a 1-4 flat space-time. In fact, some of them will require 2 or more time-like dimensions to do so.

I just get off the train at the thought of 2 or more time dimensions. Indeed, the opposite of deSitter space, one with constant *negative* curvature, requires an embedding space of 2 time and 3 space. A certain subsurface gives you a 1-3 sub-space-time that is anti-deSitter space-time (and that sucker is very weird, indeed).

And guess what. Even simple Schwarzschild cannot be embedded in a 1-4 higher space-time! Oddly, if you supress the tangential directions and just have a 1-1 space-time using r and t, *that* can be emdedded in a 1-2 flat space-time. And you can draw pictures of that. However, add the two additional space dimensions, and it no longer works.


-Richard