General relativity distinguishes coordinate effects from genuine gravitational geometry through curvature.

Riemann curvature

The failure of covariant derivatives to commute is encoded by the Riemann curvature tensor. Acting on a vector field,

$$[\nabla_\mu,\nabla_\nu]V^\rho =R^\rho_{\ \sigma\mu\nu}V^\sigma.$$

The same curvature determines how a vector changes after parallel transport around an infinitesimal loop.

Tidal gravity

Curvature has direct physical meaning. Neighboring freely falling particles can accelerate relative to one another even though each follows a geodesic. This geodesic deviation is governed by the Riemann tensor and represents tidal gravity that cannot be removed by a coordinate choice over an extended region.

Ricci curvature and scalar curvature

Contracting the Riemann tensor produces the Ricci tensor

$$R_{\mu\nu}$$

and a further contraction gives the Ricci scalar

$$R=g^{\mu\nu}R_{\mu\nu}.$$

These combine into the Einstein tensor

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

Its covariant divergence vanishes identically, matching the local conservation structure required of matter and energy.

Stress-energy tensor

Matter and nongravitational fields are summarized by the stress-energy tensor $T_{\mu\nu}$. Its components describe quantities such as energy density, momentum density, pressure and stresses as measured in a chosen frame.

Einstein field equations

With cosmological constant $\Lambda$, the field equations are

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

The left side is constructed from spacetime geometry; the right side describes matter and energy.

These are nonlinear differential equations for the metric. The geometry determines how matter and light move, while the distribution of matter and energy constrains the geometry.

Recovering known physics

A viable relativistic solution must reproduce special relativity locally and Newtonian gravity in the appropriate weak-field, low-speed limit. Particular solutions describe phenomena such as gravitational time dilation, orbital corrections, black holes and cosmological expansion.

The equation is therefore not a slogan that “mass bends space.” It is a precise relation between a dynamical spacetime metric, its curvature and the stress-energy content of spacetime.