Lorentz signature | Wikipedia audio article
Channel: wikipedia tts
Date: 15/06/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Metric_signature


00:01:47 1 Definition
00:01:55 2 Properties
00:02:12 2.1 Signature and dimension
00:02:19 2.2 Sylvester's law of inertia: independence of basis choice and existence of orthonormal basis
00:02:29 2.3 Geometrical interpretation of the indices
00:03:23 3 Examples
00:03:32 3.1 Matrices
00:03:57 3.2 Scalar products
00:04:08 4 How to compute the signature
00:04:44 5 Signature in physics
00:04:55 6 Signature change
00:05:01 7 See also
00:05:15 8 Notes
00:05:52 Geometrical interpretation of the indices
00:06:04 The number v (resp.  p) is the maximal dimension of a vector subspace on which the scalar product g is positive-definite (resp. negative-definite), and r is the dimension of the radical of the scalar product g or the null subspace of symmetric matrix gab of the scalar product. Thus a nondegenerate scalar product has signature (v, p, 0), with v + p
00:06:58 Examples
00:07:07 Matrices
00:07:54 Scalar products
00:08:21 n and rank r
00:10:47 How to compute the signature
00:11:55 Signature in physics
00:14:31 Signature change
00:15:05 See also
00:15:20 Notes



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SUMMARY
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The signature (v, p, r) 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, zero, and negative eigenvalues of the real symmetric matrix gab  of the metric tensor with respect to a basis. In physics, the v represents for the time or virtual dimension, and the p for the space and physical dimension. 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. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers (v, p) implying r = 0 or as an explicit list of signs of eigenvalues such as (+, −, −, −) or (−, +, +, +) for the signature (1, 3, 0), respectively.The signature is said to be indefinite or mixed if both v and p are nonzero, and degenerate if r is nonzero. A Riemannian metric is a metric with a positive definite signature (v, p). The Lorentzian metric is a metric signature (v, p) with two eigenvalues.
There is another notion of signature of a nondegenerate metric tensor given by a single number s defined as (v−p), where v and p are as above, which is equivalent to the above definition when the dimension n = v + p is given or implicit.  For example, s = 1 − 3 = −2 for (+, −, −, −) and its mirroring s'  = −s = +2 for (−, +, +, +).