Causality relation

In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.

In modern physics (especially general relativity) spacetime is represented by a Lorentzian manifold. The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events.

Minkowski spacetime is a simple example of a Lorentzian manifold. The causal relationships between points in Minkowski spacetime take a particularly simple form since the space is flat. See Causal structure of Minkowski spacetime for more information.

The causal structure of an arbitrary (possibly curved) Lorentzian manifold is made more complicated by the presence of curvature. Discussions of the causal structure for such manifolds must be phrased in terms of smooth curves joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships.

If ${\displaystyle \,(M,g)}$ is a Lorentzian manifold (for metric ${\displaystyle g}$ on manifold ${\displaystyle M}$) then the tangent vectors at each point in the manifold can be classed into three different types. A tangent vector ${\displaystyle X}$ is

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