Theory of relativity

Differential geometry and spacetime curvature

The metric encodes distances and proper time; the Riemann tensor describes spacetime curvature.

ds^2 = g_{\mu\nu}dx^\mu dx^\nu

Definition: Key quantity

The metric and field tensors have coordinate components, but physical predictions must be invariant. The central relation is $ds^2 = g_{\mu\nu}dx^\mu dx^\nu$ .

Drag to rotate the view and vary parameters to explore geometry and motion.

Reading the equation

The metric encodes distances and proper time; the Riemann tensor describes spacetime curvature.

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: Curvature from geodesic deviation

In a small region, two nearby freely falling particles with separation vector $\xi^\mu$ obey geodesic deviation: $D^2\xi^\mu/D\tau^2=-R^\mu{}_{\nu\alpha\beta}u^\nu\xi^\alpha u^\beta$ . Their relative acceleration depends on the Riemann tensor, not on the coordinate choice.

Solution

If $R^\mu{}_{\nu\alpha\beta}=0$ in the region, separation vectors are parallel-transported without intrinsic tidal acceleration. Conversely, measuring relative accelerations of a particle cloud probes curvature components: this gives curvature its operational meaning.

In general coordinates, Christoffel symbols $\Gamma^\rho_{\mu\nu}$ describe how the basis changes, but they are not a tensor and can be made to vanish at one point. The Riemann tensor, built from derivatives of $\Gamma$ and quadratic $\Gamma\Gamma$ terms, cannot be removed where curvature is genuinely nonzero. Contractions give the Ricci tensor and scalar; the Einstein tensor $G_{\mu\nu}=R_{\mu\nu}-\tfrac12Rg_{\mu\nu}$ has vanishing covariant divergence. This hierarchy separates coordinate effects from measurable geometry.

Parallel-transporting a vector around a closed loop can change its direction; the accumulated rotation depends on the curvature enclosed. On a flat manifold the result is path-independent, whereas on a curved manifold it measures holonomy. This illustrates that curvature belongs to the metric and local connection, not to a fold in an external space.

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