= Einstein field equations
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{title2=$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G T_{\mu\nu}/c^4$}
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= Einstein equation
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With signature $(-,+,+,+)$ and vanishing cosmological constant, the Einstein field equations are
$$
R_{\mu\nu}-\frac12 g_{\mu\nu}R=\kappa T_{\mu\nu}.
$$
In four spacetime dimensions, contraction with $g^{\mu\nu}$ gives the useful trace relation $R=-\kappa T$.
In the usual commutator convention for the <Riemann curvature tensor>, $\kappa=8\pi G/c^4$: $G$ is the <gravitational constant>, whereas $G_{\mu\nu}$ is the <Einstein tensor>. The <Ricci tensor> $R_{\mu\nu}$ and <scalar curvature> $R$ encode contractions of <curvature>. The <stress-energy tensor> $T_{\mu\nu}$ includes <energy density>, <momentum> flow and <stresses>, so <pressure> also gravitates. A <cosmological constant> adds $\Lambda g_{\mu\nu}$ to the left-hand side. See https://www.damtp.cam.ac.uk/user/tong/gr/grhtml/S4.html[Tong's Einstein equations and matter coupling].
For a concrete consequence, set $T_{\mu\nu}=0$ and $\Lambda=0$. The trace gives $R=0$, and substitution gives $R_{\mu\nu}=0$. Yet the exterior <Schwarzschild spacetime> still has the <tidal forces> computed under <geodesic deviation>: vanishing <Ricci tensor> does not imply vanishing <Riemann curvature tensor>. Geometry carries gravitational effects through empty space. Einstein presented the final matter equations on 25 November 1915, including the essential trace correction, in https://d-nb.info/1379427029/34[Die Feldgleichungen der Gravitation, pages 844-847, especially equation (2a)]. His symbols and sign conventions differ from modern ones; the displayed formula uses modern notation.
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