Lorentzian Signature

The signature of the metric tensor g at the point P of the D-dimensional smooth (connected Hausdorff C ^∞) manifold is the number of positive eigenvalues of the matrix (g _ab ) at P, minus the number of negative ones[1]. A metric whose signature is (D − 2) is called a metric of Lorentzian signature or simply a Lorentz metric . The canonical form is

If the metric on has a Lorentzian signature then each nonzero vector X with coordinates X ^a at point P can be divided into three classes according to whether the scalar product g _mn X ^m X ⁿ is negative, zero, or positive. Correspondingly the vector X is said to be timelike, null, or spacelike at P[2].