Causal structure

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The causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.

1 Introduction
2 Causal structure
- 2.1 Properties
3 Conformal geometry
4 See also
5 References
6 External links

[edit] 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.

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.

[edit] Tangent vectors

If $\,(M,g)$ is a Lorentzian manifold (so $g$ is the metric on the manifold $M$ ) then the tangent vectors at each point in the manifold can be classed into three different types. A tangent vector $X$ is

timelike if $\,g(X,X) > 0$
null if $\,g(X,X) = 0$
spacelike if $\,g(X,X) < 0$

(Here we use the $(+,-,-,-,\cdots)$ 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).

[edit] 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.

[edit] Curves

Smooth curves (which possess tangent vectors) in $M$ can be classified depending on their tangent vectors. A smooth curve is

chronological (or timelike) if the tangent vector is timelike at all points in the curve.
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.

Note that these definitions mean that the tangent vector of a chronological curve cannot vanish (i.e. be equal to the zero vector). The tangent vector of a causal curve may vanish.

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 the manifold $M$ 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 chronological, null and causal curves because only timelike or null tangent vectors can be assigned an orientation with respect to time.

[edit] Causal relations

There are two types of causal relations between points $x$ and $y$ in the manifold $M$ .

$x$ chronologically precedes $y$ (often denoted $\,x \ll y$ ) if there exists a future-directed chronological (timelike) curve from $x$ to $y$ .

$x$ causally precedes $y$ (often denoted $x \prec y$ ) if there exists a future-directed causal (non-spacelike) curve from $x$ to $y$ .

These relations are transitive:

$x \ll y$ , $y \ll z$ implies $x \ll z$
$\,x \prec y$ , $\,y \prec z$ implies $\,x \prec z$

and satisfy

$x \ll y$ implies $x \prec y$ (this follows trivially from the definition)
$x \ll y$ , $y \prec z$ implies $x \ll z$
$x \prec y$ , $y \ll z$ implies $x \ll z$

[edit] Causal structure

For a point $x$ in the manifold $M$ we define

The chronological future of $x$ , denoted $\,I^+(x)$ , as the set of all points $y$ in $M$ such that $x$ chronologically precedes $y$ :

$\,I^+(x) = \{ y \in M | x \ll y\}$

The chronological past of $x$ , denoted $\,I^-(x)$ , as the set of all points $y$ in $M$ such that $y$ chronologically precedes $x$ :

$\,I^-(x) = \{ y \in M | y \ll x\}$

We similarly define

The causal future (also called the absolute future) of $x$ , denoted $\,J^+(x)$ , as the set of all points $y$ in $M$ such that $x$ causally precedes $y$ :

$\,J^+(x) = \{ y \in M | x \prec y\}$

The causal past (also called the absolute past) of $x$ , denoted $\,J^-(x)$ , as the set of all points $y$ in $M$ such that $y$ causally precedes $x$ :

$\,J^-(x) = \{ y \in M | y \prec x\}$

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.

As a simple example, 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 $\,I^+(x) ,I^-(x), J^+(x), J^-(x)$ defined for all $x$ in $M$ , are collectively called the causal structure of $M$ .

For $S$ a subset of $M$ we define

$I^\pm(S) = \bigcup_{x \in S} I^\pm(x) \qquad J^\pm(S) = \bigcup_{x \in S} J^\pm(x)$

For $S, T$ two subsets of $M$ we define

The chronological future of $S$ relative to $T$ :

$I^+(S,T) = I^+(S) \cap T$

The causal future of $S$ relative to $T$ :

$J^+(S,T) = J^+(S) \cap T$

[edit] Properties

A point $x$ is in $\,I^-(y)$ if and only if $y$ is in $\,I^+(x)$ .
$x \prec y \implies I^-(x) \subset I^-(y)$
$x \prec y \implies I^+(y) \subset I^+(x)$
$I^+[S] = I^+[I^+[S]] \subset J^+[S] = J^+[J^+[S]]$
$I^-[S] = I^-[I^-[S]] \subset J^-[S] = J^-[J^-[S]]$

Topological properties:

$I^\pm(x)$ is open for all points $x$ in $M$ .
$I^\pm[S]$ is open for all subsets $S \subset M$ .
$I^\pm[S] = I^\pm[\bar{S}]$ for all subsets $S \subset M$ . Here $\bar{S}$ is the closure of a subset $S$ .
$J^\pm[S] \subset \overline{I^\pm[S]}$

[edit] Conformal geometry

Two metrics $\,g$ and $\hat{g}$ are conformally related if $\hat{g} = \Omega^2 g$ for some real function $Ω$ 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 $\,g(X,X) > 0$ . We then have that $\hat{g}(X,X) = \Omega^2 g(X,X) > 0$ so $X$ is a timelike tangent vector with respect to the $\hat{g}$ too.