The term metric signature refers to the signature of a metric tensor on a smooth manifold , a tool which quantifies the numbers of positive, zero, and negative infinitesimal distances of tangent vectors in the tangent bundle of and which is most easily defined in terms of the signatures of a number of related structures.
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 vector basis for any tangent space , the action of on arbitrary vectors and in is given by
whereby the signature of is defined to be the signature of any of the forms , i.e., the ordered triple of positive, negatives, and zero values for the inner products . This value is well-defined due to the fact that the signature of remains the same for all points in . For non-degenerate 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 differentiable manifold whose tangent space has basis , the tensor induces an matrix whose -entry is given by
Because the signatures of the matrices are the same for all , one may define the signature of the metric tensor to be the matrix signature of for any . 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,
which corresponds to the fact that the metric tensor in 4-dimensional Lorentzian space has signature (Misner et al. 1973). This viewpoint requires one to define a local basis for the action of , but by Sylvester's Law of Inertia, 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 positive, respectively negative, terms in the expansion of :
The transition between the notation and the notation is summarized by the identity
where are suitably-chosen basis vectors (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. Note that in (1) above, the order of the positive- and negative-squared terms is sometimes swapped, under which convention the signature would be given by , e.g., for -dimensional Euclidean spaces and for -dimensional Lorentzian spaces. This convention may also carry over to the case where is a matrix , e.g., in equation (2) above where may be replaced by .
General tensors of signature come about in the study of Clifford algebras.