Finally we collect here some useful properties of the metric tensor.

This equation has a different, but still diagonal, metric tensor.

The solution is a metric tensor field, rather than a wavefunction.

The first fundamental form is often written in the modern notation of the metric tensor.

The metric tensor requires one transformation for each of its indices.

In general relativity, the gravitational potential is replaced by the metric tensor.

The possibility of generalizing the metric tensor has been considered by many, including Einstein and others.

Thus the metric tensor gives the infinitesimal distance on the manifold.

Here, is the metric tensor on the manifold M.

Using conditions 1 and the symmetry of the metric tensor g we find:

The metric tensor is commonly written as a 4 by 4 matrix.

The signature (p,q,r) of the metric tensor gives these numbers, shown in the same order.

The determinant of the metric tensor is represented by (with no indices).

The signature + would correspond to the following metric tensor:

In outline, general relativity is based on a metric tensor concerning space-time.

Orthogonal coordinates never have off-diagonal terms in their metric tensor.

For example, the metric tensor can be expressed as:

The metric tensor is often just called 'the metric'.

The metric tensor in units where the speed of light is one is:

The reverse is possible by contracting with the (matrix) inverse of the metric tensor.

The first metric tensor, describes the geometry of space-time and thus the gravitational field.

It is more properly termed a metric tensor.

It transforms the Metric tensor in the following way:

These parallel vector fields give rise to the metric tensor as a by-product.

Guarantee local covariant conservation of energy-momentum for any metric tensor.