Metric Tensor (general Relativity) - Local Coordinates and Matrix Representations

Local Coordinates and Matrix Representations

Physicists usually work in local coordinates (i.e. coordinates defined on some local patch of M). In local coordinates (where is an index which runs from 0 to 3) the metric can be written in the form

The factors are one-form gradients of the scalar coordinate fields . The metric is thus a linear combination of tensor products of one-form gradients of coordinates. The coefficients are a set of 16 real-valued functions (since the tensor g is actually a tensor field defined at all points of a spacetime manifold). In order for the metric to be symmetric we must have

giving 10 independent coefficients. If we denote the symmetric tensor product by juxtaposition (so that ) we can write the metric in the form

If the local coordinates are specified, or understood from context, the metric can be written as a 4×4 symmetric matrix with entries . The nondegeneracy of means that this matrix is non-singular (i.e. has non-vanishing determinant), while the Lorentzian signature of g implies that the matrix has one negative and three positive eigenvalues. Note that physicists often refer to this matrix or the coordinates themselves as the metric (see, however, abstract index notation).

With the quantity being an infinitesimal coordinate displacement, the metric acts as an infinitesimal invariant interval squared or line element. For this reason one often sees the notation for the metric:

In general relativity, the terms metric and line element are often used interchangeably.

The line element imparts information about the causal structure of the spacetime. When, the interval is timelike and the square root of the absolute value of ds2 is an incremental proper time. Only timelike intervals can be physically traversed by a massive object. When, the interval is lightlike, and can only be traversed by light. When, the interval is spacelike and the square root of ds2 acts as an incremental proper length. Spacelike intervals cannot be traversed, since they connect events that are out of each other's light cones. Events can be causally related only if they are within each other's light cones.

The metric components obviously depend on the chosen local coordinate system. Under a change of coordinates the metric components transform as