Mathematical Operations of General Relativity

[GDavila]

How are the tensors of General Relativity simplified and operated? And can someone give me a mathematical example of General Relativity being done with just some basic values being plugged in for G=kT?

 

[WannabeNewton]

I'm sorry but what do you mean simplified and operated? And the most basic, non trivial solution for G = kT is actually when G = 0 and it is the Schwarzschild solution which you can solve if you assume spherical symmetry and time - like symmetry. You can find its derivation on all standard GR textbooks but also on Wikipedia http://en.wikipedia.org/wiki/Deriving_the_Schwarzschild_solution

 

[Mentz114]

How are the tensors of General Relativity simplified and operated? And can someone give me a mathematical example of General Relativity being done with just some basic values being plugged in for G=kT?

Not really sure what you mean. This is a simple one. This metric,

$$ds^2=-dt^2 + R(t)( dx^2+dy^2+dz^2)$$

after a 'straightforward but tedious' calculation gives the Einstein tensor whose non-zero components are

$$\begin{align*} G_{00}&=\frac{3\,{\left( \frac{d}{d\,t}\,R\right) }^{2}}{4\,{R}^{2}}\\ G_{11}=G_{22}=G_{33}&=-\frac{4\,R\,\left( \frac{{d}^{2}}{d\,{t}^{2}}\,R\right) -{\left( \frac{d}{d\,t}\,R\right) }^{2}}{4\,R} \end{align*}$$

( t=x⁰, x=x¹ etc ). This ( after dividing by $k=8\pi$ )corresponds to the energy momentum tensor of a perfect fluid.

 

[GDavila]

If some arbitrary valuse were plugged into general relativity, say those of the earth, how would they be operated on both sides of the equation?

 

[WannabeNewton]

I assume when you say into general relativity you mean the field equations. One usually constructs a general line element (ds^2 = ...) that, without loss of generality, closely identifies with the geometry of space - time for which one is solving, gets the components of the related tensors (Riemann, Ricci, Einstein) and using the appropriate mass - energy distribution sets up the energy - momentum tensor and finally goes about solving for the metric tensor components (this is of course just a process and solving the equations is usually very, very difficult for physically meaningful space - times). For the Earth you would simply assume a static, spherically symmetric line element (the Earth's rotation is negligible) in vacuum and when you solve this you will just end up with the aforementioned schwarzchild metric (look up Birkhoff's theorem if you want).