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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 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 \,(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 g on manifold 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 space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
, 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 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 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 M is a continuous 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 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 \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 ord ...
. 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 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 space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
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 of M we define :I^\pm = \bigcup_ I^\pm(x) :J^\pm = \bigcup_ J^\pm(x) For S, T two subsets 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. * 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. 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). 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 In mathematics and physics, ''n''-dimensional anti-de Sitter space (AdS''n'') is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872� ...
, 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 holes, the future timelike boundary ends on a singularity in some places. * For the Big Bang, 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 * Causal sets *
Cauchy surface In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as ...
* Closed timelike curve *
Cosmic censorship hypothesis The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity. Singularities that arise in the solutions of Einstein's equations are typically ...
* Globally hyperbolic manifold * 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) *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) * D. Malament;
The class of continuous timelike curves determines the topology of spacetime
'; J. Math. Phys. 18 7:1399-1404 (1977); (Geometry, Causal Structure) * A.A. Robb ;
A theory of time and space
'; Cambridge University Press, 1914; (Geometry, Causal Structure) * A.A. Robb ;
The absolute relations of time and space
'; Cambridge University Press, 1921; (Geometry, Causal Structure) * A.A. Robb ;
Geometry of Time and Space
'; Cambridge University Press, 1936; (Geometry, Causal Structure) * 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)


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