Einstein Field Equation

The Einstein field equation relates Ricci curvature to matter in classical metric general relativity. Its precise form depends on the curvature convention, the definition of stress–energy, the spacetime dimension, and whether the cosmological constant is written geometrically or as an effective source. This record uses the site’s $(-,+,+,+)$ convention and retains $G$ explicitly.

Four-dimensional field equation

For a Lorentzian metric $g_{\mu\nu}$ with Levi–Civita connection,

$$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,T_{\mu\nu}, \qquad G_{\mu\nu}=R_{\mu\nu}-\frac12 Rg_{\mu\nu}.$$

The matter normalization is

$$T_{\mu\nu} =-\frac{2}{\sqrt{-g}} \frac{\delta S_{\mathrm m}}{\delta g^{\mu\nu}}.$$

With $c=\hbar=1$, $R_{\mu\nu}$ and $\Lambda$ have dimension $L^{-2}$, while $GT_{\mu\nu}$ has the same dimension. Reall states the equation, normalization, and cosmological extension in Reall 2013, § 6.7, printed pp. 63–65, especially Eqs. (6.32) and (6.35), PDF.

Quantity	Meaning and domain
$g_{\mu\nu}$	Dynamical Lorentzian metric
$R_{\mu\nu}$, $R$	Ricci tensor and scalar formed with the site’s curvature convention
$\Lambda$	Constant cosmological term; dimension $L^{-2}$
$T_{\mu\nu}$	Hilbert stress–energy of the specified matter action
$G$	Newton’s constant in four dimensions

The equation does not choose coordinates, matter equations, initial data, boundary conditions, or a global extension. Those data are required to define a solution rather than merely its local field law.

Trace reversal and conservation

In $D\gt 2$ dimensions, write the coupling as $8\pi G_D$ and let $T=g^{\mu\nu}T_{\mu\nu}$. Taking the trace and substituting back gives

$$R =\frac{2D\Lambda-16\pi G_D T}{D-2},$$

and

$$R_{\mu\nu} =8\pi G_D \left(T_{\mu\nu}-\frac{T}{D-2}g_{\mu\nu}\right) +\frac{2\Lambda}{D-2}g_{\mu\nu}.$$

The division by $D-2$ is essential: in two dimensions the Einstein tensor vanishes identically, so this trace-reversed form is unavailable. In four-dimensional vacuum with $\Lambda=0$, the equation reduces to $R_{\mu\nu}=0$; Ricci flatness does not imply vanishing Weyl curvature.

Metric compatibility and the contracted Bianchi identity imply

$$\nabla^\mu G_{\mu\nu}=0, \qquad \nabla^\mu T_{\mu\nu}=0$$

for a constant $\Lambda$ and matter satisfying its equations of motion. The first equality is a geometric identity. The second is a compatibility condition normally obtained from diffeomorphism invariance together with the matter equations; it is not a derivation of every matter equation from gravity alone.

Normalization check

In a weak stationary field with slowly moving matter,

$$g_{00}=-(1+2\Phi), \qquad T_{00}\simeq\rho, \qquad T\simeq-\rho.$$

Setting $\Lambda=0$, the trace-reversed equation gives $R_{00}\simeq4\pi G\rho$, while the linearized metric gives $R_{00}\simeq\nabla^2\Phi$. The result is Poisson’s equation $\nabla^2\Phi=4\pi G\rho$. This fixes both the factor $8\pi G$ and its sign for the adopted conventions; the assumptions and component calculation appear in Carroll 1997, § 4, printed pp. 109–112, Eqs. (4.35)–(4.52), PDF.

The full classical treatment is organized in Einstein Field Equations. For the derivation and its limitations, continue to The Einstein Equation. The cosmological-constant treatment distinguishes moving $\Lambda g_{\mu\nu}$ across the equality from changing the physical vacuum contribution, and Bianchi Identity and Stress–Energy Conservation develops the conservation statement.

This record is limited to classical two-derivative metric general relativity. Higher-curvature theories, torsionful or metric-affine theories, semiclassical expectation values, and quantum-gravity corrections change the dynamical equation or the meaning of its source and require their own domain statements.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv record.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), § 6.7, printed pp. 63–65. PDF.