Geodesics and Curvature

Non-accelerated observers follow geodesics in spacetime. In Special Relativity, spacetime is flat, so these geodesics are simply straight worldlines. In General Relativity, the effect of matter on space (gravity) is taken into account. Gravity is measured as a curvature of spacetime, which is represented in several ways.

Geodesics

Geodesics follow a rule called the Geodesic Equation. The Geodesic Equation is:

+= 0

This equation gives the shape of the geodesic in an arbitrary coordinate system in terms of a parameter (such as the proper time).

Curvature

In General Relativity, gravity is seen as a distortion of spacetime, which is referred to as a curvature (because of its similarities to a curved 2-dimensional surface in 3-dimensional space). However, in the case of gravity, a 4-dimensional spacetime is being curved, and, as far as is known, there is no "super-space" in which it resides.

The Metric Tensor

One representation of curvature is the Metric Tensor,

g

The metric tensor explains how distances relate to changes in coordinates, by

ds=gdxdx

The Riemann Curvature Tensor

Another, more direct representation, is the Riemann Curvature Tensor. The curvature tensor is given by

R=-+ -

and is the four-dimensional equivalent of the curvature of a two-dimensional surface. The Riemann tensor measures how quickly geodesics separate from one another.

The Einstein Tensor

Some part of the curvature of spacetime is generated by the local presence of matter and energy. That part is measured by the Einstein Tensor, G.

G= R-gR=8T

The Einstein Tensor, therefore, is only affected by local considerations, and is unaffected by gravitational waves from distant sources (which result in a change to the Riemann Tensor).