Causal structure

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. 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 is a Lorentzian manifold (for metric on manifold ) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types. A tangent vector is: timelike if null or lightlike if spacelike if 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. The causal relationships between points in Minkowski spacetime take a particularly simple form because the tangent space is also and hence the tangent vectors may be identified with points in the space. The four-dimensional vector is classified according to the sign of , where is a Cartesian coordinate in 3-dimensional space, is the constant representing the universal speed limit, and is time. The classification of any vector in the space will be the same in all frames of reference that are related by a Lorentz transformation (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the metric. At each point in the timelike tangent vectors in the point's tangent space can be divided into two classes. To do this we first define an equivalence relation on pairs of timelike tangent vectors.