The Einstein Equation

The Einstein equation asserts a local relation between spacetime curvature and the stress–energy of matter. Its geometric side is not merely the Ricci tensor: the trace adjustment that forms the Einstein tensor is what makes the equation compatible with covariant matter dynamics. The equation fixes neither a coordinate system nor a global spacetime by itself, and the equivalence principle motivates metric gravity without uniquely deriving this field law.

Required background. We use the Riemann curvature tensor for Ricci contractions and the Bianchi identity, Hilbert stress–energy for the meaning of the source, and the scope of the equivalence principles to separate local metric physics from a choice of dynamics.

Helpful background. Minkowski geometry supports the weak-field limit, natural tensor operations clarify covariance, and presymplectic potentials prepare the later action derivation.

Geometry coupled to matter

For a four-dimensional Lorentzian metric with its Levi-Civita connection, define

$$G_{\mu\nu}=R_{\mu\nu}-\frac12 Rg_{\mu\nu}.$$

The field equation is

$$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G\,T_{\mu\nu}.$$

Reall states the same normalization and the cosmological extension in Reall 2013, § 6.7, printed pp. 64–65, Eqs. (6.32) and (6.35), PDF.

Every term is a symmetric rank-two tensor. Metric compatibility and the contracted Bianchi identity give $\nabla^\mu G_{\mu\nu}=0$ and $\nabla^\mu g_{\mu\nu}=0$. Thus a constant $\Lambda$ makes the geometric side identically divergence-free, while a viable source must satisfy $\nabla^\mu T_{\mu\nu}=0$ when the matter equations hold. The conservation statement is developed on the Bianchi-identity page.

The equation is local and nonlinear. Curvature contains second derivatives of $g_{\mu\nu}$ and terms quadratic in its first derivatives; $T_{\mu\nu}$ can itself depend on the metric and matter fields. With $c=\hbar=1$, curvature and $\Lambda$ have dimension $L^{-2}$, while $G T_{\mu\nu}$ has the same dimension. This checks the coupling but does not determine its numerical coefficient.

The tensor equation also packages distinct physical projections. An observer with unit four-velocity $u^\mu$ measures energy density $T_{\mu\nu}u^\mu u^\nu$; one timelike and one spatial projection gives momentum density; two spatial projections give pressure and shear stress. Einstein’s equation relates the corresponding projections to Ricci curvature. It does not say that energy density alone curves spacetime. In particular, pressure contributes to relativistic gravity even when a Newtonian slow-matter approximation suppresses it.

The free indices matter as well. A scalar equation involving only $R$ and $T$ would retain just one contraction and discard trace-free source information. The symmetric rank-two equation couples all observer projections consistently, while its divergence identity prevents them from being chosen independently of matter dynamics. This gives a useful conceptual check before any component calculation: both sides must transform tensorially, have identical index symmetry, and obey compatible differential identities.

Fixing the normalization

The factor $8\pi G$ is fixed by the Newtonian limit, not by covariance alone. Take a weak, stationary field and slowly moving matter,

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

with $|\Phi|\ll1$ and negligible stresses. Write the coupling temporarily as $\kappa$. In four dimensions, tracing and substituting back with $\Lambda=0$ gives

$$R_{\mu\nu}=\kappa\left(T_{\mu\nu}-\frac12 Tg_{\mu\nu}\right).$$

Consequently $R_{00}\simeq\kappa\rho/2$. A direct linearized-curvature calculation in the stationary metric gives

$$R_{00}\simeq\nabla^2\Phi.$$

Agreement with $\nabla^2\Phi=4\pi G\rho$ therefore requires $\kappa=8\pi G$. Carroll gives this normalization calculation, including the assumptions that discard time derivatives, stresses, and terms quadratic in the perturbation, in Carroll 1997, § 4, printed pp. 109–112, Eqs. (4.35)–(4.52), PDF.

This check also fixes a sign: positive mass density produces the usual Poisson equation with the inherited curvature convention. With non-negligible $\Lambda$, the same limit instead yields $\nabla^2\Phi\simeq4\pi G\rho-\Lambda$.

What the equation does and does not assert

The left-hand side describes Ricci curvature. Weyl curvature is not fixed algebraically at a point by $T_{\mu\nu}$; it is constrained and propagated through the differential equations and boundary or initial information. Conversely, stress–energy includes energy density, momentum density, pressure, and shear stress, not mass density alone.

Covariance also does not make all ten displayed component equations independent evolution laws. Four differential identities and four coordinate functions organize part of their content into constraints and gauge freedom, leaving two local gravitational polarizations in the linearized vacuum theory. That structural count is explained on Constraints, Evolution, and Gauge Content; it is not a proof of nonlinear well-posedness.

Finally, the equation is a postulate of classical Einstein gravity. Extra fields, higher derivatives, torsion, nonlocality, or dimensions above four enlarge the possible dynamics. Even the four-dimensional uniqueness result requires precise hypotheses, stated on Lovelock Uniqueness and Its Hypotheses.

A solution also requires more than writing down the differential equation. One must specify the matter model, its equations, and suitable initial or boundary data, then identify which coordinate descriptions represent the same geometry. A named metric that satisfies the vacuum equation locally may have several coordinate patches and distinct global extensions. Those questions are deliberately outside this page; the present result is the local dynamical law and its normalization.

Exercises

Suppose the field equation were $G_{\mu\nu}=\kappa T_{\mu\nu}$ in four dimensions. Recover $\kappa$ from the weak, stationary, pressureless limit without assuming its value.

Solution

Tracing gives $-R=\kappa T$, hence $R_{\mu\nu}=\kappa(T_{\mu\nu}-\tfrac12Tg_{\mu\nu})$. For pressureless slow matter, $T_{00}\simeq\rho$, $T\simeq-\rho$, and $g_{00}\simeq-1$, so $R_{00}\simeq\kappa\rho/2$. The weak stationary metric has $R_{00}\simeq\nabla^2\Phi$. Comparing with Poisson’s equation gives $\kappa/2=4\pi G$, or $\kappa=8\pi G$.

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.