Causal structure

In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. 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 ( M , g ) {displaystyle ,(M,g)} is a Lorentzian manifold (for metric g {displaystyle g} on manifold M {displaystyle M} ) then the tangent vectors at each point in the manifold can be classified into three different types.A tangent vector X {displaystyle X} is (Here we use the ( − , + , + , + , ⋯ ) {displaystyle (-,+,+,+,cdots )} metric signature). A tangent vector is called 'non-spacelike' if it is null or timelike. These names come from the simpler case of Minkowski spacetime (see Causal structure of Minkowski spacetime). At each point in M {displaystyle M} 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. If X {displaystyle X} and Y {displaystyle Y} are two timelike tangent vectors at a point we say that X {displaystyle X} and Y {displaystyle Y} are equivalent (written X ∼ Y {displaystyle Xsim Y} ) if g ( X , Y ) < 0 {displaystyle ,g(X,Y)<0} .