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Curvature and the Einstein Field Equations

Measure curvature with the Riemann tensor, recognise tidal forces as geodesic deviation, and assemble the equation that ties spacetime geometry to the matter within it.

Measuring curvature: the Riemann tensor

Guide 3 hinted that parallel-transporting a vector around a closed loop rotates it when space is curved. The Riemann curvature tensor R^{\rho}{}_{\sigma\mu\nu} is precisely the machine that reports that rotation, per unit area of the loop. It is the complete, coordinate-independent description of curvature — built from Christoffel symbols and their derivatives:

R^{\rho}{}_{\sigma\mu\nu} = \partial_\mu \Gamma^{\rho}{}_{\nu\sigma} - \partial_\nu \Gamma^{\rho}{}_{\mu\sigma} + \Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}_{\nu\sigma} - \Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma}

The Riemann tensor. It is a genuine tensor: if it vanishes in one frame it vanishes in all, so 'flat' is an honest, observer-independent statement.

Tidal forces are geodesic deviation

Recall from Guide 1 that the truly irremovable part of gravity is the tidal force — the relative acceleration of two nearby freely-falling bodies. Mathematically that relative acceleration is governed by the Riemann tensor through the geodesic deviation equation, where \xi^\mu is the separation between two neighbouring geodesics and u^\alpha their four-velocity:

\frac{D^2 \xi^\mu}{d\tau^2} = -\,R^{\mu}{}_{\alpha\nu\beta}\,u^\alpha \xi^\nu u^\beta

Nearby free-fallers accelerate toward or away from each other in proportion to the Riemann tensor. This is the physical meaning of curvature.

This closes the loop opened in Guide 1. The two balls dropped above Earth that drift together, and the head-and-foot pair that drift apart, are literally neighbouring geodesics converging and diverging — and the amount is set by R^{\mu}{}_{\alpha\nu\beta}. Tidal force is curvature. It is why the equivalence principle is only local: you can zero the Christoffel symbols at a point, but you can never zero the Riemann tensor over a region.

From Riemann to Ricci to Einstein

The full Riemann tensor is more than the field equation needs. Contracting a pair of its indices gives the Ricci tensor R_{\mu\nu} (roughly, how the volume of a small cloud of free-fallers shrinks); contracting again gives the Ricci scalar R, a single number measuring average curvature:

R_{\mu\nu} = R^{\lambda}{}_{\mu\lambda\nu}, \qquad R = g^{\mu\nu}R_{\mu\nu}

Successive contractions distil the 20-component Riemann tensor down to the pieces that couple to matter.

The right combination to put on the geometry side of the field equation is the Einstein tensor G_{\mu\nu}. It is engineered to be automatically divergence-free (\nabla^\mu G_{\mu\nu}=0, the contracted Bianchi identity), which is essential — it will let the equation enforce local conservation of energy and momentum for free:

G_{\mu\nu} = R_{\mu\nu} - \tfrac{1}{2}\,R\,g_{\mu\nu}

The Einstein tensor: the unique combination of curvature that is symmetric, built from the metric and its first two derivatives, and identically conserved.

The Einstein field equations

Now match geometry to its source. The matter and energy content of spacetime — mass density, pressure, momentum flux, all of it — is packaged in the stress-energy tensor T_{\mu\nu}. Setting geometry proportional to matter gives the Einstein field equations, the heart of the theory:

Curvature (geometry, left) is set equal to the stress-energy of matter (right). The two sides speak different languages — geometry and matter — and Einstein's constant translates between them.

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

Einstein's field equations. Ten coupled nonlinear PDEs for the metric; the cosmological constant Lambda represents the energy of empty space (dark energy).

Why that constant, and that sign

The proportionality constant 8\pi G/c^4 is not arbitrary — it is fixed by demanding that the weak, slow, static limit reproduces Newton. In that limit the time-time component of the field equations reduces to Poisson's equation, the field form of Newtonian gravity relating the potential to mass density \rho:

\nabla^2 \Phi = 4\pi G\,\rho

Newtonian gravity as a field equation. Matching this fixes Einstein's coupling constant — the new theory must contain the old.

So the tiny-looking c^4 in the denominator makes gravity's coupling extraordinarily weak: it takes a planet's worth of mass to curve spacetime perceptibly. The same equations, applied to different T_{\mu\nu}, generate every prediction of the theory. In Guide 5 we solve them for the simplest interesting case — the empty space around a spherical star — and out fall black holes and the classic tests.