Metric Tensor - Components of The Metric

Components of The Metric

This section assumes some familiarity with coordinate vectors.

The components of the metric in any basis of vector fields, or frame, f = (X₁, …, X_n) are given by

(4)

The n2 functions g_ij form the entries of an n×n symmetric matrix, G. If

are two vectors at p ∈ U, then the value of the metric applied to v and w is determined by the coefficients (4) by bilinearity:

Denoting the matrix (g_ij) by G and arranging the components of the vectors v and w into column vectors v and w,

where vT and wT denote the transpose of the vectors v and w, respectively. Under a change of basis of the form

for some invertible n×n matrix A = (a_ij), the matrix of components of the metric changes by A as well. That is,

or, in terms of the entries of this matrix,

For this reason, the system of quantities g_ij is said to transform covariantly with respect to changes in the frame f.