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

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) is a Lorentzian manifold (for metric g on manifold M) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types. A tangent vector X is: 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 g 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 X = (t,r) is classified according to the sign of, where is a Cartesian coordinate in 3-dimensional space, c is the constant representing the universal speed limit, and t 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.

Time-orientability

At each point in 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 and Y are two timelike tangent vectors at a point we say that X and Y are equivalent (written X \sim Y) if. There are then two equivalence classes which between them contain all timelike tangent vectors at the point. We can (arbitrarily) call one of these equivalence classes future-directed and call the other past-directed. Physically this designation of the two classes of future- and past-directed timelike vectors corresponds to a choice of an arrow of time at the point. The future- and past-directed designations can be extended to null vectors at a point by continuity. A Lorentzian manifold is time-orientable if a continuous designation of future-directed and past-directed for non-spacelike vectors can be made over the entire manifold.

Curves

A path in M is a continuous map where \Sigma is a nondegenerate interval (i.e., a connected set containing more than one point) in \mathbb{R}. A smooth path has \mu differentiable an appropriate number of times (typically C^\infty), and a regular path has nonvanishing derivative. A curve in M is the image of a path or, more properly, an equivalence class of path-images related by re-parametrisation, i.e. homeomorphisms or diffeomorphisms of \Sigma. When M is time-orientable, the curve is oriented if the parameter change is required to be monotonic. Smooth regular curves (or paths) in M can be classified depending on their tangent vectors. Such a curve is The requirements of regularity and nondegeneracy of \Sigma ensure that closed causal curves (such as those consisting of a single point) are not automatically admitted by all spacetimes. If the manifold is time-orientable then the non-spacelike curves can further be classified depending on their orientation with respect to time. A chronological, null or causal curve in M is These definitions only apply to causal (chronological or null) curves because only timelike or null tangent vectors can be assigned an orientation with respect to time.

Causal relations

There are several causal relations between points x and y in the manifold M. These relations satisfy the following properties: For a point x in the manifold M we define We similarly define Points contained in , I^+(x), for example, can be reached from x by a future-directed timelike curve. The point x can be reached, for example, from points contained in ,J^-(x) by a future-directed non-spacelike curve. In Minkowski spacetime the set ,I^+(x) is the interior of the future light cone at x. The set ,J^+(x) is the full future light cone at x, including the cone itself. These sets defined for all x in M, are collectively called the causal structure of M. For S a subset of M we define For S, T two subsets of M we define

Properties

See Penrose (1972), p13. Topological properties:

Conformal geometry

Two metrics ,g and \hat{g} are conformally related if for some real function \Omega called the conformal factor. (See conformal map). Looking at the definitions of which tangent vectors are timelike, null and spacelike we see they remain unchanged if we use ,g or \hat{g}. As an example suppose X is a timelike tangent vector with respect to the ,g metric. This means that. We then have that so X is a timelike tangent vector with respect to the \hat{g} too. It follows from this that the causal structure of a Lorentzian manifold is unaffected by a conformal transformation. A null geodesic remains a null geodesic under a conformal rescaling.

Conformal infinity

An infinite metric admits geodesics of infinite length/proper time. However, we can sometimes make a conformal rescaling of the metric with a conformal factor which falls off sufficiently fast to 0 as we approach infinity to get the conformal boundary of the manifold. The topological structure of the conformal boundary depends upon the causal structure. In various spaces:

Gravitational singularity

If a geodesic terminates after a finite affine parameter, and it is not possible to extend the manifold to extend the geodesic, then we have a singularity. The absolute event horizon is the past null cone of the future timelike infinity. It is generated by null geodesics which obey the Raychaudhuri optical equation.