Solve GR Equations with Regular Method

[quZz]

Hi everyone,

I don't fully understand what is the regular method to state and solve problems in GR when no handy hints like spherical symmetry or homogeneity of time are assumed. If I find myself in arbitrary reference frame with coordinates $x^0$, $x^1$, $x^2$, $x^3$ the meaning of which is unknown beforehand (or known only locally), how do I proceed with boundary conditions and stress-energy tensor?

Consider the following simple problem: two point masses $m_1$, $m_2$ are separated by distance $a$. Obviously $T_{\mu\nu}$ is a sum of two delta functions. Suppose the first mass is at the origin but where then is the second mass? Should I write $\delta(x^1 - a)$ or $\delta(\sqrt{(x^1)^2 + x^1 x^2} - a)$ or ...? You may say - OK, choose the coordinates in such a way that $\delta(x^1 - a)$ would be valid. But there are only 4 possible coordinate transformations, what if I had thousands of point masses?

So it seems that in general a statement of a problem in GR is interconnected with its very solution, which is confusing: are all problems solvable? uniquely? what would a "real mccoy" equation look like?

 

[PAllen]

Hi everyone,

I don't fully understand what is the regular method to state and solve problems in GR when no handy hints like spherical symmetry or homogeneity of time are assumed. If I find myself in arbitrary reference frame with coordinates $x^0$, $x^1$, $x^2$, $x^3$ the meaning of which is unknown beforehand (or known only locally), how do I proceed with boundary conditions and stress-energy tensor?

Consider the following simple problem: two point masses $m_1$, $m_2$ are separated by distance $a$. Obviously $T_{\mu\nu}$ is a sum of two delta functions. Suppose the first mass is at the origin but where then is the second mass? Should I write $\delta(x^1 - a)$ or $\delta(\sqrt{(x^1)^2 + x^1 x^2} - a)$ or ...? You may say - OK, choose the coordinates in such a way that $\delta(x^1 - a)$ would be valid. But there are only 4 possible coordinate transformations, what if I had thousands of point masses?

So it seems that in general a statement of a problem in GR is interconnected with its very solution, which is confusing: are all problems solvable? uniquely? what would a "real mccoy" equation look like?

As starting point:

http://en.wikipedia.org/wiki/ADM_fo...Numerical_Solutions_of_the_Einstein_Equations

Read here about the ADM formalism in general, then its use for initial value problems in numerical relativity. Of course, start with wikipedia with a grain of salt, and study the references.