causal future

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:
* timelike if $\,g(X,X) < 0$
* null or lightlike if $\,g(X,X) = 0$
* spacelike if $\,g(X,X) > 0$
Here we use the $(-,+,+,+,\cdots)$ 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 $M=\mathbb^4$ 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 $\mathbb^4$ 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 $g(X,X) = - c^2 t^2 + \, r\, ^2$, where $r \in \mathbb^3$ 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 $\,g(X,Y) < 0$.

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 $\mu : \Sigma \to M$ where $\Sigma$ is a nondegenerate interval (i.e., a connected set containing more than one point) in $\mathbb$. 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
* chronological (or timelike) if the tangent vector is timelike at all points in the curve. Also called a world line.
* null if the tangent vector is null at all points in the curve.
* spacelike if the tangent vector is spacelike at all points in the curve.
* causal (or non-spacelike) if the tangent vector is timelike ''or'' null at all points in the curve.
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
* future-directed if, for every point in the curve, the tangent vector is future-directed.
* past-directed if, for every point in the curve, the tangent vector is past-directed.
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.

* A closed timelike curve is a closed curve which is everywhere future-directed timelike (or everywhere past-directed timelike).
* A closed null curve is a closed curve which is everywhere future-directed null (or everywhere past-directed null).
* The holonomy of the ratio of the rate of change of the affine parameter around a closed null geodesic is the redshift factor.

Causal relations

There are several causal relations between points $x$ and $y$ in the manifold $M$.

* $x$ chronologicall