Einstein Tensor - Use in General Relativity

Use in General Relativity

The Einstein tensor allows the Einstein field equations (without a cosmological constant) to be written in the concise form:

which becomes in geometrized units,

From the explicit form of the Einstein tensor, the Einstein tensor is a nonlinear function of the metric tensor, but is linear in the second partial derivatives of the metric. As a symmetric 2nd rank tensor, the Einstein tensor has 10 independent components in a 4-dimensional space. It follows that the Einstein field equations are a set of 10 quasilinear second-order partial differential equations for the metric tensor.

The Bianchi identities can also be easily expressed with the aid of the Einstein tensor:

The Bianchi identities automatically ensure the conservation of the stress–energy tensor in curved spacetimes:

The geometric significance of the Einstein tensor is highlighted by this identity. In coordinate frames respecting the gauge condition

an exact conservation law for the stress tensor density can be stated:

.

The Einstein tensor plays the role of distinguishing these frames.