Causal structure

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

Lorentzian manifolds can be classified according to the types of causal structures they admit (causality conditions).

Introduction

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.

Tangent vectors

If

(

M

,

g

)

{\displaystyle \,(M,g)}

is a Lorentzian manifold (for metric

g

{\displaystyle g}

on manifold

M

{\displaystyle M}

) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types.

A tangent vector

X

{\displaystyle X}

is:

timelike if

g

(

X

,

X

)

<

0

{\displaystyle \,g(X,X)<0}

null or lightlike if

g

(

X

,

X

)

=

0

{\displaystyle \,g(X,X)=0}

spacelike if

g

(

X

,

X

)

>

0

{\displaystyle \,g(X,X)>0}

Here we use the

(

−

,

+

,

+

,

+

,

⋯

)

{\displaystyle (-,+,+,+,\cdots )}

metric signature. We say that a tangent vector is non-spacelike if it is null or timelike.