In
mathematical physics
Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developme ...
, the causal structure of a
Lorentzian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
describes the
causal relationships
Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...
between points in the manifold.
Introduction
In
modern physics (especially
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
)
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
is represented by a
Lorentzian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
. The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events.
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
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s joining pairs of points. Conditions on the
tangent vectors
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
of the curves then define the causal relationships.
Tangent vectors
If
is a
Lorentzian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
(for
metric on
manifold ) then the nonzero tangent vectors at each point in the manifold can be classified into three
disjoint types.
A tangent vector
is:
* timelike if
* null or lightlike if
* spacelike if
Here we use the
metric signature
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and ...
. We say that a tangent vector is non-spacelike if it is null or timelike.
The canonical Lorentzian manifold is
Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
, where
and
is the
flat Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
. 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
In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
(but not by a general
Poincaré transformation because the origin may then be displaced) because of the invariance of the metric.
Time-orientability
At each point in
the timelike tangent vectors in the point's
tangent space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
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
The arrow of time, also called time's arrow, is the concept positing the "one-way direction" or "asymmetry" of time. It was developed in 1927 by the British astrophysicist Arthur Eddington, and is an unsolved general physics question. This ...
at the point. The future- and past-directed designations can be extended to null vectors at a point by continuity.
A
Lorentzian manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
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 (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.
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...
s or
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two ...
s of
. When
is time-orientable, the curve is oriented if the parameter change is required to be
monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
.
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
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics.
The concept of a "world line" is distinguished from c ...
.
* 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
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
of the ratio of the rate of change of the affine parameter around a closed null geodesic is the redshift factor.
Causal relations
There are several 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
).
These relations satisfy the following properties:
*
implies
(this follows trivially from the definition)
*
,
implies
*
,
implies
*
,
,
are
transitive.
is not transitive.
*
,
are
reflexive[
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 :
:
* The future null cone of as the set of all points in such that .
* The past null cone of as the set of all points in such that .
* The light cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
of as the future and past null cones of together.
* elsewhere as points not in the light cone, causal future, or causal past.[
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.
In ]Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
the set is the interior of the future light cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
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 ,