I have read that the Einstein Field Equations (http://en.wikipedia.org/wiki/Einstein_field_equations) can be expressed as a series of differential equations. Some say 16, others say 10 (The disparity seems to stem from a simplification involving the Bianchi identities). However, no source actually lists them.

What are they?

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    $\begingroup$ Well the page you linked gives EFEs in terms of the Ricci scalar, $R$ and the Ricci tensor, $R_{\mu\nu}$. The pages I just liked give these quantities in terms of the Christoffel symbols,$\Gamma_{ab}^c$ which can be written in terms of the metric tensor, $g_{\mu\nu}$. So all you have to do is substitute all of that in and you will understand exactly why writes this out. $\endgroup$ Jun 15, 2015 at 0:41
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    $\begingroup$ I think @BySymmetry neglected the "no one" in that last sentence (seems like it should be "...you will understand exactly why no one writes this out."). $\endgroup$
    – Kyle Kanos
    Jun 15, 2015 at 1:44
  • $\begingroup$ I remember trying some of that in my general relativity class, but I gave up very quickly. Mathematica, Maple or one of the other computer algebra tools should be able to do this easily... of course, there will be nothing to see but pages and pages of partial derivatives. $\endgroup$
    – CuriousOne
    Jun 15, 2015 at 4:02
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    $\begingroup$ It's pretty rare to write them all out because for the most part, nobody is gonna solve a non-linear system of 10 equations by hand. Usually people will use the symmetries to lighten it up. I don't have the full equations on me, tho I do have the Ricci scalar in full : $R = \frac{1}{2} g_{ab,c}g_{de,f} (g^{ac}g^{de}g^{bf} - g^{ab}g^{de}g^{cf} + 2 g^{dc}g^{be}g^{af} + 2 g^{df}g^{be}g^{ac}) + g_{ab,cd} (g^{ac}g^{bd} - g^{ab}g^{cd})$. $\endgroup$
    – Slereah
    Jun 15, 2015 at 7:19
  • $\begingroup$ Hmm. I guess that makes sense. I didn't realize the expansion would be so large; I guess I assumed the 16 equations would each be smaller. For what it's worth, my motivation is that I'm attempting to write a physics engine, so I won't be solving it by hand. $\endgroup$
    – kd8azz
    Jun 15, 2015 at 11:39

1 Answer 1


As asked in the comments, here is one answer :

One formalism where it is somewhat common to expand the Einstein equations into a full set of equations is the Newman-Penrose formalism. Not quite common as it uses both spinors instead of tensors and the coordinates are weird complex null-vectors, but it should give an idea of the whole thing.



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