Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

How can I go from the 'standard' Einstein equations $R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$ to these equations: $R_{\mu\nu} = \frac{8\pi G}{c^4}(T_{\mu\nu} - \frac{1}{2}g_{\mu\nu}T)$?

share|improve this question
add comment

1 Answer 1

up vote 7 down vote accepted

Take the trace of the equation by contracting it with $g^{\mu\nu}$:

$$ g^{\mu\nu}R_{\mu\nu}-\dfrac{1}{2}g^{\mu\nu}g_{\mu\nu}R=\dfrac{8\pi G}{c^4}g^{\mu\nu}T_{\mu\nu} $$

As $g^{\mu\nu}R_{\mu\nu} = R$, $g^{\mu\nu}T_{\mu\nu} \equiv T $ and $g^{\mu\nu}g_{\mu\nu} = 4$, the previous equation gives you $R = -\dfrac{8\pi G}{c^4}T$. Substituting this into Einstein's equation shall give you the result.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.