Theory of relativity

The Einstein field equations

Einstein’s field equations relate spacetime curvature to matter’s energy, momentum, and stress.

G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}

Definition: Key quantity

The metric and field tensors have coordinate components, but physical predictions must be invariant. The central relation is $G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$ .

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Reading the equation

Einstein’s field equations relate spacetime curvature to matter’s energy, momentum, and stress.

Example: Limit and interpretation

Consider a weak field, low speeds, and a region far from the source. General relativity should approach the appropriate Newtonian description; near a horizon or in a strong field this approximation fails.

Solution

This is the correspondence principle: the newer theory recovers tested results of the older one in its domain of validity.

Example: Checking stress-energy conservation

The Einstein tensor obeys the contracted Bianchi identity, $\nabla_\mu G^{\mu\nu}=0$ . Apply the covariant derivative to the full field equation and use $\nabla_\mu g^{\mu\nu}=0$ . What condition follows for the stress-energy tensor $T^{\mu\nu}$ ?

Solution

For constant cosmological constant, the left-hand side has zero divergence, so $\nabla_\mu T^{\mu\nu}=0$ . This is covariant local energy-momentum conservation: geometry and sources must be mutually consistent.

The field equation represents ten nonlinear differential equations for the metric, constrained by Bianchi identities. In vacuum with $\Lambda=0$ , $T_{\mu\nu}=0$ and the equations reduce to $R_{\mu\nu}=0$ , yet the Riemann tensor can remain nonzero: exterior fields and gravitational radiation still curve spacetime. There is no local, invariant tensor energy density for gravity analogous to $T_{\mu\nu}$ ; gravitational energy is often defined through boundary quantities or asymptotic observables. This makes initial-value problems and global energy subtler than in Newtonian mechanics.

Finding a solution requires boundary conditions or initial data and a gauge choice; many metric components are not independent physical degrees of freedom. The weak-field limit around flat spacetime yields Poisson’s equation, $\nabla^2\Phi=4\pi G\rho$ , recovering Newtonian gravity. Global solutions can nevertheless contain horizons, singularities, or wave structure that linearized theory misses.

Quick check

What does the central expression in this topic describe?

Which is the most appropriate interpretation?

References

Sean Carroll (2019). Spacetime and Geometry
Misner, Thorne, Wheeler (1973). Gravitation