Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
Introduction
In modern physics 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 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 is a Lorentzian manifold then the tangent vectors at each point in the manifold can be classified into three different types.A tangent vector is
- timelike if
- null or lightlike if
- spacelike if
These names come from the simpler case of Minkowski spacetime.
Time-orientability
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 if.
There are then two equivalence classes which between them contain all timelike tangent vectors at the point.
We can 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 is a continuous map where is a nondegenerate interval in. A smooth path has differentiable an appropriate number of times, 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 in can be classified depending on their tangent vectors. Such a curve is
- chronological 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 if the tangent vector is timelike or null at all points in the curve.
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.
- A closed timelike curve is a closed curve which is everywhere future-directed timelike.
- A closed null curve is a closed curve which is everywhere future-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
- chronologically precedes if there exists a future-directed chronological curve from to.
- strictly causally precedes if there exists a future-directed causal curve from to.
- causally precedes if strictly causally precedes or.
- horismos if and.
- , implies
- , implies
- implies
- , implies
- , implies
- 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 :
- The causal future of, denoted, as the set of all points in such that causally precedes :
- The causal past of, denoted, as the set of all points in such that causally precedes :
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
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.
- A terminal indecomposable past set 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 . 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.
Properties
- A point is in if and only if is in.
- The horismos is generated by null geodesic congruences.
- is open for all points in.
- is open for all subsets.
- for all subsets. Here is the closure of a subset.
Conformal geometry
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.