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A Lorentzian metric is a metric with signature (p, 1), or (1, p). 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.
A Lorentzian manifold is an important special case of a pseudo-Riemannian manifold in which the signature of the metric is (1, n−1) (equivalently, (n−1, 1); see Sign convention). Such metrics are called Lorentzian metrics. They are named after the Dutch physicist Hendrik Lorentz.
Moreover, the metric is required to be nondegenerate with signature (− + + +). A manifold equipped with such a metric is a type of Lorentzian manifold. Explicitly, the metric tensor is a symmetric bilinear form on each tangent space of that varies in a smooth (or differentiable) manner from point to point.
If M is four-dimensional with signature (1, 3) or (3, 1), then the metric is called Lorentzian. More generally, a metric tensor in dimension n other than 4 of signature (1, n − 1) or (n − 1, 1) is sometimes also called Lorentzian. If M is 2n-dimensional and g has signature (n, n), then the metric is called ultrahyperbolic.
The lattice is positive definite, Lorentzian, and so on if its vector space is. The signature of a lattice is the signature of the form on the vector space. Examples
In mathematics and physics, n-dimensional anti-de Sitter space (AdS n) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory.
The n-dimensional model is the celestial sphere of the (n + 2)-dimensional Lorentzian space R n+1,1. Here the model is a Klein geometry: a homogeneous space G/H where G = SO(n + 1, 1) acting on the (n + 2)-dimensional Lorentzian space R n+1,1 and H is the isotropy group of a fixed null ray in the light cone.
Here we use the (, +, +, +,) metric signature. We say that a tangent vector is non-spacelike if it is null or timelike. The canonical Lorentzian manifold is Minkowski spacetime, where = and is the flat Minkowski metric. The names for the tangent vectors come from the physics of this model.