Lorentzian manifold - Knowlpedia

Definition

An $n$ -dimensional Lorentzian manifold is a pseudo-Riemannian manifold $(M,g)$ whose metric has signature $(1,n-1)$ . In the convention used here, a nonzero tangent vector $v$ is timelike, null, or spacelike according as

g(v,v)<0,\qquad g(v,v)=0,\qquad\text{or}\qquad g(v,v)>0.

The timelike and null directions determine the causal cones of $M$ .

Time orientation

At each point the timelike cone has two connected components. A time orientation is a continuous choice of one component as future-pointing; equivalently, it is represented by a continuous timelike vector field. A Lorentzian manifold need not be time-orientable, and the Lorentzian metric alone does not choose a time orientation. In relativity, a spacetime commonly means a connected, time-oriented four-dimensional Lorentzian manifold, sometimes with additional causality hypotheses.

Geometry and morphisms

The metric determines a Levi–Civita connection, geodesics, curvature, and a volume density. A Lorentzian isometry $F:(M,g)\to(N,h)$ is a diffeomorphism satisfying $F^*h=g$ . If time orientations or space orientations are part of the objects, their morphisms are normally required to preserve that extra structure.

Examples

Minkowski spacetime is flat and has a canonical time orientation once the coordinate vector $\partial_t$ is declared future-pointing. The product metric

g=-dt^2+h_t

on $I\times\Sigma$ , where each $h_t$ is Riemannian, is Lorentzian and time-oriented by $\partial_t$ . Curved examples include de Sitter and anti-de Sitter spacetimes.

Sign warning

Sources that list positive directions first call the same $(-+\cdots+)$ metric's signature $(n-1,1)$ , while its timelike vectors still have negative squared length. Other sources reverse the metric to $(+-\cdots-)$ and call vectors with positive squared length timelike; in this collection's negative-first ordering, that opposite metric also has signature $(n-1,1)$ . Formulas for the wave operator and Clifford multiplication must be translated consistently when the overall sign changes.

References

Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, 1983. Publisher record. Relevant: Chapters 5 and 14.
John K. Beem, Paul E. Ehrlich, and Kevin L. Easley, Global Lorentzian Geometry, 2nd ed., Marcel Dekker, 1996. Publisher record. Relevant: Chapters 1–3.