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number of positive, negative and zero eigenvalues of a metric tensor

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In mathematics, the signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix gab of the metric tensor with respect to a basis. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis and thus can be used to classify the metric. It is denoted by three integers (v, p, r), where v is the number of positive eigenvalues, p is the number of negative ones and r is the number of zero eigenvalues of the metric tensor. It can also be denoted (v, p) implying r = 0, or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signatures (1, 3, 0) and (3, 1, 0), respectively. The choice of the variable names v and p reflects the convention in relativistic physics that v represents the number of time or virtual dimensions, and p the number of space or physical dimensions.

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