Mathematics of General Relativity - Tensor Fields in General Relativity

Tensor Fields in General Relativity

Tensor fields on a manifold are maps which attach a tensor to each point of the manifold. This notion can be made more precise by introducing the idea of a fibre bundle, which in the present context means to collect together all the tensors at all points of the manifold, thus 'bundling' them all into one grand object called the tensor bundle. A tensor field is then defined as a map from the manifold to the tensor bundle, each point being associated with a tensor at .

The notion of a tensor field is of major importance in GR. For example, the geometry around a star is described by a metric tensor at each point, so at each point of the spacetime the value of the metric should be given to solve for the paths of material particles. Another example is the values of the electric and magnetic fields (given by the electromagnetic field tensor) and the metric at each point around a charged black hole to determine the motion of a charged particle in such a field.

Vector fields are contravariant rank one tensor fields. Important vector fields in relativity include the four-velocity, which is the coordinate distance travelled per unit of proper time, the four-acceleration and the four-current describing the charge and current densities. Other physically important tensor fields in relativity include the following:

• The stress-energy tensor, a symmetric rank-two tensor.
• The electromagnetic field tensor, a rank-two antisymmetric tensor.

Although the word 'tensor' refers to an object at a point, it is common practice to refer to tensor fields on a spacetime (or a region of it) as just 'tensors'.

At each point of a spacetime on which a metric is defined, the metric can be reduced to the Minkowski form using Sylvester's Law of Inertia.