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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 smoothcurves joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships.
Subdivision of Minkowski spacetime with respect to a point in four disjoint sets. The light cone, the causal future, the causal past, and elsewhere. The terminology is defined in this article.
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 is the flatMinkowski 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 is classified according to the sign of , where is a Cartesian coordinate in 3-dimensional space, is the constant representing the universal speed limit, and 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.
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 if a continuous designation of future-directed and past-directed for non-spacelike vectors can be made over the entire manifold.
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. 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 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.
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 (often denoted or ) if or there exists a future-directed null curve from to  (or equivalently, and ).
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 (IP) 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 .
See Penrose (1972), p13.
A point is in if and only if is in .
The horismos is generated by null geodesic congruences.
for all subsets . Here is the closure of a subset .
Two metrics and are conformally related 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.