Metric Signature

The metric signature of a metric tensor on a smooth manifold records the numbers of positive, negative, and zero entries in its diagonal matrix representation in an orthogonal basis of each tangent space.

Most commonly, one identifies the signature of a metric tensor with the signature of the quadratic form induced by on any of the tangent spaces for points . Indeed, given an orthogonal basis for any tangent space , the associated symmetric bilinear form evaluated on arbitrary vectors and in is given by

(1)

whereby the signature of at is defined to be the signature of , i.e., the ordered triple counting the positive, negative, and zero diagonal values . This value is basis-independent by Sylvester's inertia law. For a nondegenerate metric tensor, its signature is locally constant and hence constant on each connected component of . For nondegenerate quadratic forms, the value will always satisfy , whereby the signature of will be the ordered pair .

Alternatively, one can view the signature of a metric tensor in terms of matrix signatures. For an -dimensional smooth manifold whose tangent space has vector basis , the tensor induces an matrix whose -entry is given by

(2)

Because the matrix signatures of the matrices are constant on each connected component for a nondegenerate metric tensor, one defines the signature of the metric tensor at to be the matrix signature of . Moreover, by rewriting on any pointwise tangent space , it follows that this definition is equivalent to the quadratic signature definition mentioned above.

In many contexts, one finds it beneficial to express the metric tensor itself as a diagonal matrix, usually denoted and whose components are sometimes called the "metric coefficients" associated to . In such circumstances, the signature of is precisely the matrix signature of . For example, in Minkowski space,

(3)

which corresponds to the fact that the metric tensor in four-dimensional Lorentzian space has signature (Misner et al. 1973). This viewpoint requires one to define a local vector basis for the action of , but by Sylvester's inertia law, this definition is well-defined independent of the choice of basis vectors.

In an -dimensional pseudo-Euclidean space, the metric tensor is often denoted and its signature is defined to be the ordered pair where , respectively , denotes the number of positive, respectively negative, terms in the expansion of :

(4)

The transition between the notation and the notation is summarized by the identity

(5)

where are the coordinate one-forms (Snygg 2012).

For -dimensional Euclidean spaces, the metric signature is . For -dimensional Lorentzian space , the metric signature is , e.g., (as above) for the Minkowski space of special relativity. Some authors instead report the negative count first, giving . Under this convention, the signature is for -dimensional Euclidean spaces and for -dimensional Lorentzian spaces.

Changing the reporting convention is independent of reordering the basis vectors. The diagonal matrices and each have three positive entries and one negative entry. Both therefore have signature in the positive-first convention and in the negative-first convention. Reordering the basis vectors preserves the signature by Sylvester's inertia law.

Quadratic forms of signature come about in the study of Clifford algebras.