I am trying to write the Einstein field equations(adsbygoogle = window.adsbygoogle || []).push({});

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

in such a way that the Ricci curvature tensor $$R_{\mu\nu}$$ and scalar curvature $$R$$ are replaced with an explicit expression involving the metric tensor $$g_{\mu\nu}$$ using the equations $$R=g^{\mu\nu}R_{\mu\nu}$$ (relating the scalar curvature to the trace of the Ricci curvature tensor) and $$R_{\mu\nu}=R^\lambda_{\mu\lambda\nu}$$ (relating the Ricci curvature tensor to the trace of the Riemann curvature tensor). Would anyone be willing to give recommendations on how to proceed, or already know the equation?

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# Stress-energy tensor explicitly in terms of the metric tensor

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