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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 developm ...
, 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 between points in the manifold.


Introduction

In
modern physics Modern physics is a branch of physics that developed in the early 20th century and onward or branches greatly influenced by early 20th century physics. Notable branches of modern physics include quantum mechanics, special relativity and general ...
(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 In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
. Discussions of the causal structure for such manifolds must be phrased in terms of smooth
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 of the curves then define the causal relationships.


Tangent vectors

If \,(M,g) 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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
g on
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
M) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types. A tangent vector X is: * timelike if \,g(X,X) < 0 * null or lightlike if \,g(X,X) = 0 * spacelike if \,g(X,X) > 0 Here we use the (-,+,+,+,\cdots)
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 Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
, where M=\mathbb^4 and g 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 \mathbb^4 and hence the tangent vectors may be identified with points in the space. The four-dimensional vector X = (t,r) is classified according to the sign of g(X,X) = - c^2 t^2 + \, r\, ^2, where r \in \mathbb^3 is a Cartesian coordinate in 3-dimensional space, c is the constant representing the universal speed limit, and t 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 M 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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
on pairs of timelike tangent vectors. If X and Y are two timelike tangent vectors at a point we say that X and Y are equivalent (written X \sim Y) if \,g(X,Y) < 0. There are then two
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements 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 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 M is a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...
map \mu : \Sigma \to M where \Sigma is a nondegenerate interval (i.e., a connected set containing more than one point) in \mathbb. A smooth path has \mu differentiable an appropriate number of times (typically C^\infty), and a regular path has nonvanishing derivative. A curve in M 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 isom ...
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 \Sigma. When M 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 order ...
. Smooth regular curves (or paths) in M 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 \Sigma 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 M 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 In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van ...
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 x and y in the manifold M. * x chronologically precedes y (often denoted \,x \ll y) if there exists a future-directed chronological (timelike) curve from x to * x strictly causally precedes y (often denoted x < y) if there exists a future-directed causal (non-spacelike) curve from x to y. * x causally precedes y (often denoted x \prec y or x \le y) if x strictly causally precedes y or x=y. * x horismos y (often denoted x \to y or x \nearrow y ) if x=y or there exists a future-directed null curve from x to y (or equivalently, x \prec y and x \not\ll y). These relations satisfy the following properties: * x \ll y implies x \prec y (this follows trivially from the definition) * x \ll y, y \prec z implies x \ll z * x \prec y, y \ll z implies x \ll z * \ll, <, \prec are transitive. \to is not transitive. * \prec, \to are reflexive For a point x in the manifold M we define * The chronological future of x, denoted \,I^+(x), as the set of all points y in M such that x chronologically precedes y: :\,I^+(x) = \ * The chronological past of x, denoted \,I^-(x), as the set of all points y in M such that y chronologically precedes x: :\,I^-(x) = \ We similarly define * The causal future (also called the absolute future) of x, denoted \,J^+(x), as the set of all points y in M such that x causally precedes y: :\,J^+(x) = \ * The causal past (also called the absolute past) of x, denoted \,J^-(x), as the set of all points y in M such that y causally precedes x: :\,J^-(x) = \ * The future null cone of x as the set of all points y in M such that x \to y. * The past null cone of x as the set of all points y in M such that y \to x. * 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 x as the future and past null cones of x together. * elsewhere as points not in the light cone, causal future, or causal past. Points contained in \, I^+(x), for example, can be reached from x by a future-directed timelike curve. The point x can be reached, for example, from points contained in \,J^-(x) by a future-directed non-spacelike curve. In
Minkowski spacetime 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 ...
the set \,I^+(x) 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 x. The set \,J^+(x) is the full future light cone at x, including the cone itself. These sets \,I^+(x) ,I^-(x), J^+(x), J^-(x) defined for all x in M, are collectively called the causal structure of M. For S a
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of M we define :I^\pm = \bigcup_ I^\pm(x) :J^\pm = \bigcup_ J^\pm(x) For S, T two
subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
s of M we define * The chronological future of S relative to T, I^+ ;T/math>, is the chronological future of S considered as a submanifold of T. Note that this is quite a different concept from I^+ \cap T which gives the set of points in T which can be reached by future-directed timelike curves starting from S. In the first case the curves must lie in T in the second case they do not. See Hawking and Ellis. * The causal future of S relative to T, J^+ ;T/math>, is the causal future of S considered as a submanifold of T. Note that this is quite a different concept from J^+ \cap T which gives the set of points in T which can be reached by future-directed causal curves starting from S. In the first case the curves must lie in T 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. * An IP which does not coincide with the past of any point in M is called a terminal indecomposable past set (TIP). * A proper indecomposable past set (PIP) is an IP which isn't a TIP. I^-(x) is a proper indecomposable past set (PIP). * The future Cauchy development of S, D^+ (S) is the set of all points x for which every past directed inextendible causal curve through x intersects S 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 Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
. * A subset S \subset M is achronal if there do not exist q,r \in S such that r \in I^(q), or equivalently, if S is disjoint from I^ /math>. Causal diamond * A Cauchy surface is a closed achronal set whose Cauchy development is M. * 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. * The boundary of the causality violating set is a
Cauchy horizon In physics, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem (a particular boundary value problem of the theory of partial differential equations). One side of the horizon contains closed space-like geodes ...
. If the Cauchy horizon is generated by closed null geodesics, then there's a redshift factor associated with each of them. * For a causal curve \gamma, the causal diamond is J^+(\gamma) \cap J^-(\gamma) (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 \gamma is the set of all events that lie in both the past of some point in \gamma and the future of some point in \gamma.


Properties

See Penrose (1972), p13. * A point x is in \,I^-(y) if and only if y is in \,I^+(x). * x \prec y \implies I^-(x) \subset I^-(y) * x \prec y \implies I^+(y) \subset I^+(x) * I^+ = I^+ ^+[S_\subset_J^+_=_J^+[J^+[S.html" ;"title=".html" ;"title="^+[S">^+[S \subset J^+ = J^+[J^+[S">.html" ;"title="^+[S">^+[S \subset J^+ = J^+[J^+[S * I^- = I^-[I^-[S \subset J^- = J^-[J^-[S * The horismos is generated by null geodesic congruences. Topology, Topological properties: * I^\pm(x) is open for all points x in M. * I^\pm /math> is open for all subsets S \subset M. * I^\pm = I^\pm
overline An overline, overscore, or overbar, is a typographical feature of a horizontal line drawn immediately above the text. In old mathematical notation, an overline was called a '' vinculum'', a notation for grouping symbols which is expressed in m ...
/math> for all subsets S \subset M. Here \overline is the closure of a subset S. * I^\pm \subset \overline


Conformal geometry

Two metrics \,g and \hat are conformally related if \hat = \Omega^2 g for some real function \Omega called the conformal factor. (See
conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
). Looking at the definitions of which tangent vectors are timelike, null and spacelike we see they remain unchanged if we use \,g or \hat. As an example suppose X is a timelike tangent vector with respect to the \,g metric. This means that \,g(X,X) < 0. We then have that \hat(X,X) = \Omega^2 g(X,X) < 0 so X is a timelike tangent vector with respect to the \hat too. It follows from this that the causal structure of a Lorentzian manifold is unaffected by a
conformal transformation In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\in ...
. A null geodesic remains a null geodesic under a conformal rescaling.


Conformal infinity In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity. It is an ext ...

An infinite metric admits geodesics of infinite length/proper time. However, we can sometimes make a conformal rescaling of the metric with a conformal factor which falls off sufficiently fast to 0 as we approach infinity to get the conformal boundary of the manifold. The topological structure of the conformal boundary depends upon the causal structure. * Future-directed timelike geodesics end up on i^+, the future timelike infinity. * Past-directed timelike geodesics end up on i^-, the past timelike infinity. * Future-directed null geodesics end up on ℐ+, the future null infinity. * Past-directed null geodesics end up on ℐ, the past null infinity. * Spacelike geodesics end up on spacelike infinity. * For
Minkowski space 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 ...
, i^\pm are points, ℐ± are null sheets, and spacelike infinity has codimension 2. * For anti-de Sitter space, there's no timelike or null infinity, and spacelike infinity has codimension 1. * For
de Sitter space In mathematical physics, ''n''-dimensional de Sitter space (often abbreviated to dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canoni ...
, the future and past timelike infinity has codimension 1.


Gravitational singularity A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by definition no longer part of the regular sp ...

If a geodesic terminates after a finite affine parameter, and it is not possible to extend the manifold to extend the geodesic, then we have a singularity. * For
black hole A black hole is a region of spacetime where gravity is so strong that nothing, including light or other electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can def ...
s, the future timelike boundary ends on a singularity in some places. * For the
Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
, the past timelike boundary is also a singularity. The absolute event horizon is the past null cone of the future timelike infinity. It is generated by null geodesics which obey the Raychaudhuri optical equation.


See also

* Causal dynamical triangulation (CDT) *
Causality conditions In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 197 ...
* Causal sets * Cauchy surface *
Closed timelike curve In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van ...
* Cosmic censorship hypothesis *
Globally hyperbolic manifold In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifo ...
* Malament–Hogarth spacetime *
Penrose diagram In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity. It is an ext ...
* Penrose–Hawking singularity theorems *
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 ...


Notes


References

* * * *


Further reading

* G. W. Gibbons, S. N. Solodukhin; ''The Geometry of Small Causal Diamonds'' arXiv:hep-th/0703098 (Causal intervals) * S.W. Hawking, A.R. King, P.J. McCarthy;
A new topology for curved space–time which incorporates the causal, differential, and conformal structures
'; J. Math. Phys. 17 2:174-181 (1976); (Geometry,
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 (especially general relativity) spacetime is represented by a Lorentzian ma ...
) *A.V. Levichev; ''Prescribing the conformal geometry of a lorentz manifold by means of its causal structure''; Soviet Math. Dokl. 35:452-455, (1987); (Geometry,
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 (especially general relativity) spacetime is represented by a Lorentzian ma ...
) * D. Malament;
The class of continuous timelike curves determines the topology of spacetime
'; J. Math. Phys. 18 7:1399-1404 (1977); (Geometry,
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 (especially general relativity) spacetime is represented by a Lorentzian ma ...
) * A.A. Robb ;
A theory of time and space
'; Cambridge University Press, 1914; (Geometry,
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 (especially general relativity) spacetime is represented by a Lorentzian ma ...
) * A.A. Robb ;
The absolute relations of time and space
'; Cambridge University Press, 1921; (Geometry,
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 (especially general relativity) spacetime is represented by a Lorentzian ma ...
) * A.A. Robb ;
Geometry of Time and Space
'; Cambridge University Press, 1936; (Geometry,
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 (especially general relativity) spacetime is represented by a Lorentzian ma ...
) * R.D. Sorkin, E. Woolgar; ''A Causal Order for Spacetimes with C^0 Lorentzian Metrics: Proof of Compactness of the Space of Causal Curves''; Classical & Quantum Gravity 13: 1971-1994 (1996); arXiv:gr-qc/9508018 (
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 (especially general relativity) spacetime is represented by a Lorentzian ma ...
)


External links


Turing Machine Causal Networks
by Enrique Zeleny, the
Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
* {{MathWorld , title=Causal Network , urlname=CausalNetwork Lorentzian manifolds Theory of relativity General relativity Theoretical physics