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

Contents

IntroductionEdit

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.

Tangent vectorsEdit

If   is a Lorentzian manifold (for metric   on manifold  ) then the tangent vectors at each point in the manifold can be classed into three different types. A tangent vector   is

  • timelike if  
  • null or lightlike if  
  • spacelike if  

(Here we use the   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).

Time-orientabilityEdit

At each point in   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   and   are two timelike tangent vectors at a point we say that   and   are equivalent (written  ) 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[1] if a continuous designation of future-directed and past-directed for non-spacelike vectors can be made over the entire manifold.

CurvesEdit

A path in   is a continuous map   where   is a nondegenerate interval (i.e., a connected set containing more than one point) in  . A smooth path has   differentiable an appropriate number of times (typically  ), and a regular path has nonvanishing derivative.

A curve in   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  . When   is time-orientable, the curve is oriented if the parameter change is required to be monotonic.

Smooth regular curves (or paths) in   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.
  • 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   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   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 relationsEdit

There are two types of causal relations between points   and   in the manifold  .

  •   chronologically precedes   (often denoted  ) if there exists a future-directed chronological (timelike) curve from   to  .
  •   strictly causally precedes   (often denoted  ) if there exists a future-directed causal (non-spacelike) curve from   to  .
  •   causally precedes   (often denoted   or  ) if   strictly causally precedes   or  .
  •   horismos (light cone)  [2] (often denoted   or  ) if   and  .

These relations are transitive:[3]

  •  ,   implies  
  •  ,   implies  

and satisfy[3]

  •   implies   (this follows trivially from the definition)
  •  ,   implies  
  •  ,   implies  

For a point   in the manifold   we define[3]

  • The chronological future of  , denoted  , as the set of all points   in   such that   chronologically precedes  :
 
  • The chronological past of  , denoted  , as the set of all points   in   such that   chronologically precedes  :
 

We similarly define

  • The causal future (also called the absolute future) of  , denoted  , as the set of all points   in   such that   causally precedes  :
 
  • The causal past (also called the absolute past) of  , denoted  , as the set of all points   in   such that   causally precedes  :
 

Points contained in  , for example, can be reached from   by a future-directed timelike curve. The point   can be reached, for example, from points contained in   by a future-directed non-spacelike curve.

As a simple example, in Minkowski spacetime the set   is the interior of the future light cone at  . The set   is the full future light cone at  , including the cone itself.

These sets   defined for all   in  , are collectively called the causal structure of  .

For   a subset of   we define[3]

 
 

For   two subsets of   we define

  • The chronological future of   relative to  ,  , is the chronological future of   considered as a submanifold of  . Note that this is quite a different concept from   which gives the set of points in   which can be reached by future-directed timelike curves starting from  . In the first case the curves must lie in   in the second case they do not. See Hawking and Ellis.
  • The causal future of   relative to  ,  , is the causal future of   considered as a submanifold of  . Note that this is quite a different concept from   which gives the set of points in   which can be reached by future-directed causal curves starting from  . In the first case the curves must lie in   in the second case they do not. See Hawking and Ellis.
  • A future set is a set closed under chronological future.
  • A past set is a set closed under chronological past.
  • An indecomposable past set is a past set which isn't the union of two different open past proper subsets.
  •   is a proper indecomposable past set (PIP).
  • A terminal indecomposable past set (TIP) is an IP which isn't a PIP.
  • The future Cauchy development of  ,   is the set of all points   for which every past directed inextendible causal curve through   intersects   at least once. Similarly for the past Cauchy development. The Cauchy development is the union of the future and past Cauchy developments. Cauchy developments are important for the study of determinism.
  • A subset   is achronal if there do not exist   such that  , or equivalently, if   is disjoint from  .
  • A Cauchy surface is a closed achronal set whose Cauchy development is  .
  • A metric is globally hyperbolic if it can be foliated by Cauchy surfaces.
  • The chronology violating set is the set of points through which closed timelike curves pass.
  • The causality violating set is the set of points through which closed causal curves pass.
  • For a causal curve  , the causal diamond is   (here we are using the looser definition of 'curve' whereon it is just a set of points). In words: the causal diamond of a particle's world-line   is the set of all events that lie in both the past of some point in   and the future of some point in  .

PropertiesEdit

See Penrose, p13.

  • A point   is in   if and only if   is in  .
  •  
  •  
  •  
  •  
  • The horismos is generated by null geodesic congruences.

Topological properties:

  •   is open for all points   in  .
  •   is open for all subsets  .
  •   for all subsets  . Here   is the closure of a subset  .
  •  

Conformal geometryEdit

Two metrics   and   are conformally related[4] if   for some real function   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   or   As an example suppose   is a timelike tangent vector with respect to the   metric. This means that  . We then have that   so   is a timelike tangent vector with respect to the   too.

It follows from this that the causal structure of a Lorentzian manifold is unaffected by a conformal transformation.

See alsoEdit

NotesEdit

ReferencesEdit

  • Hawking, S.W.; Ellis, G.F.R. (1973), The Large Scale Structure of Space-Time, Cambridge: Cambridge University Press, ISBN 0-521-20016-4
  • Hawking, S.W.; Israel, W. (1979), General Relativity, an Einstein Centenary Survey, Cambridge University Press, ISBN 0-521-22285-0
  • Penrose, R. (1972), Techniques of Differential Topology in Relativity, SIAM, ISBN 0898710057

Further readingEdit

External linksEdit