The
causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
In modern physics (especially
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
)
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
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 structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
Introduction
In modern physics (especially
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
)
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
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
flatIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
. 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
curvatureIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
. Discussions of the causal structure for such manifolds must be phrased in terms of
smoothIn mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.Most of...
curveIn mathematics, a curve consists of the points through which a continuously moving point passes. This notion captures the intuitive idea of a geometrical one-dimensional object, which furthermore is connected in the sense of having no discontinuities or gaps. Simple examples include the sine wave...
s joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships.
Tangent vectors
If is a Lorentzian manifold (so is the
metricIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
on the
manifoldIn mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the 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 if
- spacelike if
(Here we use the
metric signatureThe signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
) 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-orientability
At each point in the timelike tangent vectors in the point's
tangent spaceIn mathematics, the tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.-Informal description:In...
can be divided into two classes. To do this we first define an
equivalence relationIn mathematics, an equivalence relation is, loosely, a binary relation on a set that specifies how to split up the set into subsets such that every element of the larger set is in exactly one of the subsets...
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 classIn mathematics, given a set X and an equivalence relation ~ on X, the equivalence class of an element a in X is the subset of all elements in X which are equivalent to a:...
es 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 timeIn the natural sciences, arrow of time, or time’s arrow, is a term coined in 1927 by British astronomer Arthur Eddington used to distinguish a direction of time on a four-dimensional relativistic map of the world, which, according to Eddington, can be determined by a study of organizations of...
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
Smooth regular curves in 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.
The assumption of regularity means that the tangent vector never vanishes (it is necessary to make this assumption otherwise every spacetime would admit closed causal curves, for instance curves whose image is a single point).
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 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.
Causal relations
There are two types of causal
relationsIn mathematics , a relation is a property that assigns truth values to combinations of k individuals. Typically, the property describes a possible connection between the components of a k-tuple...
between points and in the manifold .
- chronologically precedes (often denoted ) if there exists a future-directed chronological (timelike) curve from to .
- causally precedes (often denoted or ) if there exists a future-directed causal (non-spacelike) curve from to or .
- stricly causally precedes (often denoted ) if there exists a future-directed causal (non-spacelike) curve from to .
- horismos (often denoted or ) if and .
These relations are
transitiveIn mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
:
and satisfy
- implies (this follows trivially from the definition)
- , implies
- , implies
Causal structure
For a point in the manifold we define
- 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
interiorIn mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S....
of the future
light coneA Light cone is the path that a flash of light would take through spacetime. As time progresses, the light from the flash spreads out in a circle, and the result is a 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
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
of we define
For two
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. Notice that A and B may coincide...
s of we define
- The chronological future of relative to :
- The causal future of relative to :
Properties
See Penrose, p13.
- A point is in if and only if is in .
TopologicalTopology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example deformations that involve stretching, but no tearing or gluing...
properties:
- is open for all points in .
- is open for all subsets .
- for all subsets . Here is the closure
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a member of the set...
of a subset .
Conformal geometry
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.
It follows from this that the causal structure of a Lorentzian manifold is unaffected by a conformal transformation.
External links
- Turing Machine Causal Networks by Enrique Zeleny, the Wolfram Demonstrations Project
The Wolfram Demonstrations Project is developed by Wolfram Research, whose stated goal is to bring computational exploration to the widest possible audience. It consists of an organized, open-source collection of small interactive programs called Demonstrations, which are meant to visually and...