Geometry in General Relativity: Round Marble on a Box

[berra]

Say you have a box with a round marble on it. Classically I would have no problem describing that, but how is it done in general relativity? Do one just assume that the marble is round and that the box is ... boxy... in the 3D subspace at a certain foil of time? (Foliation is the correct term right?) Or does one iteratively perturb/deform the (materials) geometry, until Einsteins laws with the stress energy tensor etc is satisfied? I'm asking because I have a gut feeling that a round marble isn't round in general relativity. And also because every single textbook I have read have been void of examples.

Thanks!

 

[A.T.]

Classically I would have no problem describing that, but how is it done in general relativity?

It sounds like you are asking more generally about Euclidean vs. non-Euclidean geometry:
http://en.wikipedia.org/wiki/Non-Euclidean_geometry

 

[berra]

It sounds like you are asking more generally about Euclidean vs. non-Euclidean geometry:
http://en.wikipedia.org/wiki/Non-Euclidean_geometry

I am asking how you go from what you see with your eyes or a measurement device that is to my understanding the best representation of a time-foliation of spacetime to making a mathematical model of it in spacetime.

I will read the wiki page now and see if it addresses my question.

Nope, I think rectangles have right angles so the wiki page doesn't help. I am asking how you interpolate your measurements into a bunch of positions in spacetime. Does one assume that the 3D subspace is euclidian and that all points came from the same time? Or something else?

 

[DEvens]

One relates geodesics to motion. That gives the entire thing. Or, as stated in _Gravitation_ by Misner, Thorne, and Wheeler, time is defined to make motion look simple.

 

[pervect]

Say you have a box with a round marble on it. Classically I would have no problem describing that, but how is it done in general relativity? Do one just assume that the marble is round and that the box is ... boxy... in the 3D subspace at a certain foil of time? (Foliation is the correct term right?) Or does one iteratively perturb/deform the (materials) geometry, until Einsteins laws with the stress energy tensor etc is satisfied? I'm asking because I have a gut feeling that a round marble isn't round in general relativity. And also because every single textbook I have read have been void of examples.

Thanks!

Take a look at http://www.eftaylor.com/pub/chapter2.pdf if you get a chance, an excerpt from Taylor and Wheeler's book "Exploring black holes". They have an example there of how you could measure the shape of a rowboat by driving nails into it, and precisely measuring the distances between nails with strings running along the surface of the rowboat.

The mathematical object in General Relativity that gives you distances between points (and also the more general space-time equivalent, the Lorentz interval) is called a metric.

Einstein's field equations are basically equations that the metric must satisfy, given the matter distribution. So the short answer to your question is that you define the shapes by the metric, which can be regarded as a tool that gives you the distance between every pair of events on the space-time manifold, which are analogous to the nails that you drove into the hull of the rowboat.

 

[Mentz114]

Say you have a box with a round marble on it. Classically I would have no problem describing that, but how is it done in general relativity? Do one just assume that the marble is round and that the box is ... boxy... in the 3D subspace at a certain foil of time? (Foliation is the correct term right?) Or does one iteratively perturb/deform the (materials) geometry, until Einsteins laws with the stress energy tensor etc is satisfied? I'm asking because I have a gut feeling that a round marble isn't round in general relativity. And also because every single textbook I have read have been void of examples.

Thanks!

You do not need GR to describe your setup. According to GR we live in a locally minkowskian geometry that is 'carried' with us as we go through time ( and possibly space).

Your local frame in your lab is as close enough to Euclidean/Minkowski over a large enough volume, to use those models.

The deviation of the local coordinates from flatness is calculable ( actual numbers !).