What Mathematical Knowledge is Needed to Understand Einstein's Field Equations?

[implicit]

Hello,

As with a lot of people, I have been excited and fascinated by the field equations Einstein described, revealing the curvature of spacetime. I would like to create a computer simulation which simulates the effects of the Einstein Field Equations, in other words, the curvature of spacetime by objects of a certain mass (stars, black holes, binary star systems, etc...). I have the knowledge and the tools to program such a simulation, however I am not familiar with the EFE. I would like someone to help me point out the mathemetical and physical knowledge I have to have in order to understand them. I am already somewhat familar with tensors, and some 3D geometry. Can someone give me a list of required mathematical theorems and tools which I should study in order to understand th EFE?

Thank you

 

[implicit]

No reply...

 

[olgranpappy]

Check out:

"Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity" by Steven Weinberg.

 

[implicit]

Thank you!

 

[A.T.]

I would like to create a computer simulation which simulates the effects of the Einstein Field Equations, in other words, the curvature of spacetime by objects of a certain mass (stars, black holes, binary star systems, etc...).

You mean, you want to numerically solve the equations, for general cases? There schould be some material on the net about it. I have done a visualization of a simple case, the Schwarzschild Solution:
http://www.adamtoons.de/physics/gravitation.swf

 

[pervect]

You mean, you want to numerically solve the equations, for general cases? There schould be some material on the net about it. I have done a visualization of a simple case, the Schwarzschild Solution:
http://www.adamtoons.de/physics/gravitation.swf

Interesting diagram, but what does it mean?

Being a space-time diagram, when you say it preserves distance, do you mean it preserves the Lorentz interval? And what are the equations of the embedding?

There are some other interesting embeddings of the Schwarzschild geometry that I could post links to, if this is the sort of thing the OP is interested in.

http://arxiv.org/abs/gr-qc/9806123

I found it a bit hard to follow, so the plots and equations in:

in this thread..

might help in understanding the paper.

But I'm not really clear on what the Original Poster (OP) is interested in - stuff like the above may be what he's really after, but it's not at all about solving the EFE, it's only about demonstrating how a specific known solution of the EFE (the Schwarzschild geometry) works. Solving the EFE would be very difficult (for instance computing how black holes collide would require this) - finding the orbits of planets by treating them as geodesics is a much more realistic task for someone without a PHD.

 

[A.T.]

Interesting diagram, but what does it mean?

It shows how the observed movement of free fallers translates to geodesics on curved space-eigentime.

And what are the equations of the embedding?

The idea is simple: The radius of the rotational surface, is proportional to the gravitational time dilatation. The distances along the meridians represent the relationship between the radial coordinates and proper distances along the space dimension. A free faller is simulated by following a geodesic on this rotational surface.

Similar embeddings are derived in this papers for the standard space-time:
http://fy.chalmers.se/~rico/Webarticles/2001GRG-Jonsson33p1207.pdf
http://fy.chalmers.se/~rico/Webarticles/2005AJP-Jonsson73p248.pdf

 

[implicit]

Thank you for the replies!

First of all, no, it is not my aim to find general solutions to Einstein's Equations, I believe that would be quite a difficult task. Instead, I would like to simulate the curvature of spacetime in a Minkowski sytem. In short, imagine a ball of mass m (a star), and what my simulation would try to show, is the way the space is curved in the surroundings. Then extend the program for more complicated systems.

Here is an image to show you what I mean :

http://upload.wikimedia.org/wikipedia/commons/2/22/Spacetime_curvature.png

Thank you

 

[A.T.]

And what are the equations of the embedding?

The embedding of the space-eigentime I used for my visualization is described here:
http://fy.chalmers.se/~rico/Theses/licx.pdf
in chapter 6.

 

[quetzalcoatl9]

you may also find this thread helpful:

https://www.physicsforums.com/showthread.php?t=168995

be sure to pickup a copy of Wald's "General Relativity". you will also might likely search the literature on recent numerical studies; the references that they contain will point you in the right direction.

 

[implicit]

Thank you!