Doing General Relativity with Cartesian coordinates?

[kochanskij]

Is it possible to do general relativity but avoid the difficult mathematics of generalized coordinates, tensors, and computing the metric of a space-time manifold by using ordinary cartesian coordinates in a 5 dimensional space?

We can picture a curved 4 dimensional spacetime as being embedded in a Euclidean 5 dimensional space. Cartesian retangular coordinates would work in this Euclidean space. Then you could add a constraint that only points in that 5-D space that fall on the 4-D curved "hypersurface" are allowed.

 

[atyy]

See George Jones's post #5 at https://www.physicsforums.com/showthread.php?t=290098.

 

[pervect]

If you have a basic knowledge of SR, there are a few educational papers that describe how you can draw Lorentzian space-time diagrams on a 3D surface to model part of the space-time around a massive nonrotating body (specifically, the R-T plane of Schwarzschild solution to a black hole).

See for instance Marolf's paper at http://arxiv.org/abs/gr-qc/9806123

To actually get any qunatitative results out of this, though, will be messy. Having the embedding diagram doesn't really make it much simpler to calculate geodesic curves, for instance.

The idea is that you just have to draw the normal space-time diagrams of SR on the surface of a curved sheet of paper rather than a flat one is somewhat useful conceptually, though.

This particular embedding isn't a euclidian one, though - if you look at a small, flat piece of the curved paper, the space-time diagram on that flat piece transforms via the Lorentz transform, so it will only be useful to people who understand SR well enough to know what that means.

NOt quite what you asked for, but you might find it interesting.