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Regular method to solve GR equations

  1. Jan 4, 2014 #1
    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 [itex]x^0[/itex], [itex]x^1[/itex], [itex]x^2[/itex], [itex]x^3[/itex] 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 [itex]m_1[/itex], [itex]m_2[/itex] are separated by distance [itex]a[/itex]. Obviously [itex]T_{\mu\nu}[/itex] 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 [itex]\delta(x^1 - a)[/itex] or [itex]\delta(\sqrt{(x^1)^2 + x^1 x^2} - a)[/itex] or ...? You may say - OK, choose the coordinates in such a way that [itex]\delta(x^1 - a)[/itex] 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?
  2. jcsd
  3. Jan 4, 2014 #2


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    Science Advisor
    Gold Member

    As starting point:


    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.
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