{{Short description|Causal relationships between points in a manifold}}
In [[mathematical physics]], the '''causal structure''' of a [[Lorentzian manifold]] describes the possible [[Causality (physics)|causal relationships]] between points in the manifold. Lorentzian manifolds can be classified according to the types of causal structures they admit (''[[causality conditions]]'').
== Introduction == In [[modern physics]] (especially [[general relativity]]) [[spacetime]] 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.
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 function|smooth]] [[curve]]s joining pairs of points. Conditions on the [[tangent vectors]] of the curves then define the causal relationships.
=== Tangent vectors === [[File:World line.svg|250px|thumb|right|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.]]
If <math>\,(M,g)</math> is a [[Lorentzian manifold]] (for [[metric tensor|metric]] <math>g</math> on [[manifold]] <math>M</math>) then the nonzero tangent vectors at each point in the manifold can be classified into three [[Disjoint sets|disjoint]] types. A tangent vector <math>X</math> is: * '''timelike''' if <math>\,g(X,X) < 0</math> * '''null''' or '''lightlike''' if <math>\,g(X,X) = 0</math> * '''spacelike''' if <math>\,g(X,X) > 0</math> Here we use the <math>(-,+,+,+,\cdots)</math> [[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 <math>M=\mathbb{R}^4</math> and <math>g</math> is the [[Curvature of space|flat]] [[Minkowski 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 <math>\mathbb{R}^4</math> and hence the tangent vectors may be identified with points in the space. The four-dimensional vector <math>X = (t,r)</math> is classified according to the sign of <math>g(X,X) = - c^2 t^2 + \|r\|^2</math>, where <math>r \in \mathbb{R}^3</math> is a [[Cartesian coordinate system|Cartesian]] coordinate in 3-dimensional space, <math>c</math> is the constant representing the universal speed limit, and <math>t</math> 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.
=== Time-orientability === At each point in <math>M</math> 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 <math>X</math> and <math>Y</math> are two timelike tangent vectors at a point we say that <math>X</math> and <math>Y</math> are equivalent (written <math>X \sim Y</math>) if <math>\,g(X,Y) < 0</math>.
There are then two [[equivalence class]]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]] 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'''<ref>{{harvnb|Hawking|Israel|1979|p=255}}</ref> 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 <math>M</math> is a [[Smooth function#Parametric continuity|continuous]] map <math>\mu : \Sigma \to M</math> where <math>\Sigma</math> is a nondegenerate interval (i.e., a connected set containing more than one point) in <math>\mathbb{R}</math>. A '''smooth''' path has <math>\mu</math> differentiable an appropriate number of times (typically <math>C^\infty</math>), and a '''regular''' path has nonvanishing derivative.
A '''curve''' in <math>M</math> is the image of a path or, more properly, an equivalence class of path-images related by re-parametrisation, i.e. [[homeomorphism]]s or [[diffeomorphism]]s of <math>\Sigma</math>. When <math>M</math> is time-orientable, the curve is '''oriented''' if the parameter change is required to be [[monotonic function|monotonic]].
Smooth regular curves (or paths) in <math>M</math> 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]].<ref>{{cite web |last1=Galloway |first1=Gregory J. |title=Notes on Lorentzian causality |url=https://www.math.miami.edu/~galloway/vienna-course-notes.pdf |website=ESI-EMS-IAMP Summer School on Mathematical Relativity |publisher=University of Miami |access-date=2 July 2021|page=4}}</ref> * '''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 <math>\Sigma</math> 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 <math>M</math> 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'''.
=== Causal relations ===
There are several causal [[relation (mathematics)|relations]] between points <math>x</math> and <math>y</math> in the manifold <math>M</math>.
* <math>x</math> '''chronologically precedes''' <math>y</math> (often denoted <math>\,x \ll y</math>) if there exists a future-directed chronological (timelike) curve from <math>x</math> to {{nowrap|<math>y</math>.}} * <math>x</math> '''strictly causally precedes''' <math>y</math> (often denoted <math>x < y</math>) if there exists a future-directed causal (non-spacelike) curve from <math>x</math> to <math>y</math>. * <math>x</math> '''causally precedes''' <math>y</math> (often denoted <math>x \prec y</math> or <math>x \le y</math>) if <math>x</math> strictly causally precedes <math>y</math> or <math>x=y</math>. * <math>x</math> '''horismos''' <math>y</math><ref>{{harvnb|Penrose|1972|p=15}}</ref> (often denoted <math>x \to y</math> or <math> x \nearrow y </math>) if <math>x=y</math> or there exists a future-directed null curve from <math>x</math> to <math>y</math><ref name=Papadopoulos>{{cite journal |last1=Papadopoulos |first1=Kyriakos |last2=Acharjee |first2=Santanu |last3=Papadopoulos |first3=Basil K. |title=The order on the light cone and its induced topology |journal=International Journal of Geometric Methods in Modern Physics |date=May 2018 |volume=15 |issue=5 |pages=1850069–1851572 |doi=10.1142/S021988781850069X |arxiv=1710.05177 |bibcode=2018IJGMM..1550069P |s2cid=119120311 }}</ref> (or equivalently, <math>x \prec y</math> and <math>x \not\ll y</math> implies <math>x \prec y</math> (this follows trivially from the definition))<ref name="Penrose12" /> * <math>x \ll y</math>, <math>y \prec z</math> implies <math>x \ll z</math><ref name="Penrose12">{{harvnb|Penrose|1972|p=12}}</ref> * <math>x \prec y</math>, <math>y \ll z</math> implies <math>x \ll z</math><ref name="Penrose12" /> * <math>\ll</math>, <math><</math>, <math>\prec</math> are [[transitive relation|transitive]].<ref name="Penrose12" /> <math>\to</math> is not transitive. The causal relation <math>\prec</math> is the smallest transitive extension of the horismos relation <math>\to</math>.<ref>{{cite journal |last1=Stoica |first1=O. C. |title=Spacetime Causal Structure and Dimension from Horismotic Relation |journal=Journal of Gravity |date=25 May 2016 |volume=2016 |pages=1–6 |doi=10.1155/2016/6151726|doi-access=free|arxiv=1504.03265 }}</ref> * <math>\prec</math>, <math>\to</math> are [[reflexive relation|reflexive]]<ref name="Papadopoulos" />
For a point <math>x</math> in the manifold <math>M</math> we define<ref name="Penrose12">{{harvnb|Penrose|1972|p=12}}</ref> * The '''chronological future''' of <math>x</math>, denoted <math>\,I^+(x)</math>, as the set of all points <math>y</math> in <math>M</math> such that <math>x</math> chronologically precedes <math>y</math>:
:<math>\,I^+(x) = \{ y \in M | x \ll y\}</math> * The '''chronological past''' of <math>x</math>, denoted <math>\,I^-(x)</math>, as the set of all points <math>y</math> in <math>M</math> such that <math>y</math> chronologically precedes <math>x</math>:
:<math>\,I^-(x) = \{ y \in M | y \ll x\}</math>
We similarly define * The '''causal future''' (also called the '''absolute future''') of <math>x</math>, denoted <math>\,J^+(x)</math>, as the set of all points <math>y</math> in <math>M</math> such that <math>x</math> causally precedes <math>y</math>:
:<math>\,J^+(x) = \{ y \in M | x \prec y\}</math> * The '''causal past''' (also called the '''absolute past''') of <math>x</math>, denoted <math>\,J^-(x)</math>, as the set of all points <math>y</math> in <math>M</math> such that <math>y</math> causally precedes <math>x</math>:
:<math>\,J^-(x) = \{ y \in M | y \prec x\}</math>
* The '''future null cone''' of <math>x</math> as the set of all points <math>y</math> in <math>M</math> such that <math>x \to y</math>. * The '''past null cone''' of <math>x</math> as the set of all points <math>y</math> in <math>M</math> such that <math>y \to x</math>. * The [[light cone]] of <math>x</math> as the future and past null cones of <math>x</math> together.<ref name=Sard78>{{harvnb|Sard|1970|p=78}}</ref> * '''elsewhere''' as points not in the light cone, causal future, or causal past.<ref name=Sard78/>
Points contained in <math>\, I^+(x)</math>, for example, can be reached from <math>x</math> by a future-directed timelike curve. The point <math>x</math> can be reached, for example, from points contained in <math>\,J^-(x)</math> by a future-directed non-spacelike curve.
In [[Minkowski spacetime]] the set <math>\,I^+(x)</math> is the [[interior (topology)|interior]] of the future [[light cone]] at <math>x</math>. The set <math>\,J^+(x)</math> is the full future light cone at <math>x</math>, including the cone itself.
These sets <math>\,I^+(x) ,I^-(x), J^+(x), J^-(x)</math> defined for all <math>x</math> in <math>M</math>, are collectively called the '''causal structure''' of <math>M</math>.
For <math>S</math> a [[subset]] of <math>M</math> we define<ref name = "Penrose12"/>
:<math>I^\pm[S] = \bigcup_{x \in S} I^\pm(x) </math> :<math>J^\pm[S] = \bigcup_{x \in S} J^\pm(x) </math>
For <math>S, T</math> two [[subset]]s of <math>M</math> we define
* The '''chronological future of <math>S</math> relative to <math>T</math>''', <math>I^+[S;T]</math>, is the chronological future of <math>S</math> considered as a [[submanifold]] of <math>T</math>. Note that this is quite a different concept from <math>I^+[S] \cap T</math> which gives the set of points in <math>T</math> which can be reached by future-directed timelike curves starting from <math>S</math>. In the first case the curves must lie in <math>T</math> in the second case they do not. See Hawking and Ellis. * The '''causal future of <math>S</math> relative to <math>T</math>''', <math>J^+[S;T]</math>, is the causal future of <math>S</math> considered as a submanifold of <math>T</math>. Note that this is quite a different concept from <math>J^+[S] \cap T</math> which gives the set of points in <math>T</math> which can be reached by future-directed causal curves starting from <math>S</math>. In the first case the curves must lie in <math>T</math> 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 <math>M</math> is called a '''terminal indecomposable past set''' (TIP). * A '''proper indecomposable past set''' (PIP) is an IP which isn't a TIP. <math>I^-(x)</math> is a proper indecomposable past set (PIP). * The future '''[[Cauchy development]]''' of <math>S</math>, <math>D^+ (S)</math> is the set of all points <math>x</math> for which every past directed inextendible causal curve through <math>x</math> intersects <math>S</math> 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 <math>S \subset M</math> is '''achronal''' if there do not exist <math>q,r \in S</math> such that <math>r \in I^{+}(q)</math>, or equivalently, if <math>S</math> is disjoint from <math>I^{+}[S]</math>. {{anchor|diamond}}[[file:PhysRevD.99.086006 Fig2 CausalDiamond.png|thumb|Causal diamond]] * A '''[[Cauchy surface]]''' is a closed achronal set whose Cauchy development is <math>M</math>. * 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 <math>\gamma</math>, the '''causal diamond''' is <math>J^+(\gamma(t_1)) \cap J^-(\gamma(t_2))</math> (here we are using the looser definition of 'curve' whereon it is just a set of points), being the point <math>\gamma(t_1)</math> in the causal past of <math>\gamma(t_2)</math>. In words: the causal diamond of a particle's world-line <math>\gamma</math> is the set of all events that lie in both the past of some point in <math>\gamma</math> and the future of some point in <math>\gamma</math>. In the discrete version, the causal diamond is the set of all the causal paths that connect <math>\gamma(t_2)</math> from <math>\gamma(t_1)</math>.
=== Properties === See Penrose (1972), p13.
* A point <math>x</math> is in <math>\,I^-(y)</math> if and only if <math>y</math> is in <math>\,I^+(x)</math>. * <math>x \prec y \implies I^-(x) \subset I^-(y)</math> * <math>x \prec y \implies I^+(y) \subset I^+(x)</math> * <math>I^+[S] = I^+[I^+[S]] \subset J^+[S] = J^+[J^+[S]]</math> * <math>I^-[S] = I^-[I^-[S]] \subset J^-[S] = J^-[J^-[S]]</math> * The horismos is generated by null geodesic congruences.
[[Topology|Topological]] properties: * <math>I^\pm(x)</math> is open for all points <math>x</math> in <math>M</math>. * <math>I^\pm[S]</math> is open for all subsets <math>S \subset M</math>. * <math>I^\pm[S] = I^\pm[\overline{S}]</math> for all subsets <math>S \subset M</math>. Here <math>\overline{S}</math> is the [[Closure (mathematics)|closure]] of a subset <math>S</math>. * <math>I^\pm[S] \subset \overline{J^\pm[S]}</math>
== Conformal geometry ==
Two metrics <math>\,g</math> and <math>\hat{g}</math> are '''conformally related'''<ref>{{harvnb|Hawking|Ellis|1973|p=42}}</ref> if <math>\hat{g} = \Omega^2 g</math> for some real function <math>\Omega</math> called the '''conformal factor'''. (See [[conformal map#Riemannian geometry|conformal map]]).
Looking at the definitions of which tangent vectors are timelike, null and spacelike we see they remain unchanged if we use <math>\,g</math> or <math>\hat{g}</math>. As an example suppose <math>X</math> is a timelike tangent vector with respect to the <math>\,g</math> metric. This means that <math>\,g(X,X) < 0</math>. We then have that <math>\hat{g}(X,X) = \Omega^2 g(X,X) < 0</math> so <math>X</math> is a timelike tangent vector with respect to the <math>\hat{g}</math> too.
It follows from this that the causal structure of a Lorentzian manifold is unaffected by a [[conformal transformation]].
A null geodesic remains a null geodesic under a conformal rescaling.
== Conformal infinity == {{Main|Conformal infinity}} 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 <math>i^+</math>, the '''future timelike infinity'''. * Past-directed timelike geodesics end up on <math>i^-</math>, the '''past timelike infinity'''. * Future-directed null geodesics end up on ℐ<sup>+</sup>, the '''future [[null infinity]]'''. * Past-directed null geodesics end up on ℐ<sup>−</sup>, the '''past null infinity'''. * Spacelike geodesics end up on '''spacelike infinity'''.
In various spaces: * [[Minkowski space]]: <math>i^\pm</math> are points, ℐ<sup>±</sup> are null sheets, and spacelike infinity has [[codimension]] 2. * [[Anti-de Sitter space]]: there's no timelike or null infinity, and spacelike infinity has codimension 1. * [[de Sitter space]]: the future and past timelike infinity has codimension 1.
== Gravitational singularity == {{Main|Gravitational singularity}}
A geodesic is called ''extendible'' if there exists a point <math>p</math> such that, for every neighborhood <math>O</math> of <math>p</math>, there exists a value <math>t_0</math> such that <math>\gamma(t) \in O</math> for all <math>t > t_0</math>. Otherwise, the geodesic is ''inextendible''. A geodesic is said to be ''complete'' if its affine parameter can be extended to both <math>+\infty</math> and <math>-\infty</math>.<ref name="Reall">{{cite web |last1=Reall |first1=Harvey |title=Black Holes |url=https://www.damtp.cam.ac.uk/user/hsr1000/black_holes_lectures_2014.pdf |website=www.damtp.cam.ac.uk |access-date=23 June 2025}}</ref>
A spacetime manifold is geodesically complete if every inextendible causal geodesic is complete. If at least one inextendible causal geodesic is incomplete, then the spacetime is said to be geodesically incomplete. If the spacetime manifold itself can be extended (i.e., it is extendible as a differentiable manifold), it must also be geodesically incomplete. The manifold is said to have a ''singularity'' if the spacetime is both geodesically incomplete and inextendible as a manifold.<ref name="Reall"/><ref>{{cite book |last1=Ferrari |first1=Valeria |last2=Gualtieri |first2=Leonardo |last3=Pani |first3=Paolo |title=General relativity and its applications: black holes, compact stars and gravitational waves |date=2021 |publisher=CRC Press |location=Boca Raton London New York |isbn=978-1138589773 |edition=First|url=https://www.roma1.infn.it/teongrav/onde19_20/addendum3.pdf}}</ref>
* For [[black hole]]s, the future timelike boundary ends on a [[gravitational singularity]] in some places. * For the [[Big Bang]], the past timelike boundary is also a singularity.
The [[Absolute horizon|absolute event horizon]] is the past null cone of the future timelike infinity. It is generated by null geodesics which obey the [[Raychaudhuri equation|Raychaudhuri optical equation]].
== See also ==
* [[Causal dynamical triangulation]] (CDT) * [[Causality conditions]] * [[Causal sets]] * [[Cauchy surface]] * [[Closed timelike curve]] * [[Cosmic censorship hypothesis]] * [[Globally hyperbolic manifold]] * [[Malament–Hogarth spacetime]] * [[Null infinity]] * [[Penrose diagram]] * [[Penrose–Hawking singularity theorems]] * [[Spacetime]]
== Notes == <references/>
==References==
*{{citation | authorlink1=Stephen Hawking|first1=S.W.|last1=Hawking|authorlink2=George Francis Rayner Ellis|first2=G.F.R.|last2=Ellis| title=[[The Large Scale Structure of Space-Time]] | location=Cambridge | publisher=Cambridge University Press | year=1973 | isbn=0-521-20016-4}} *{{citation | authorlink1=Stephen Hawking|first1=S.W.|last1=Hawking|authorlink2=Werner Israel|first2=W.|last2=Israel| title = General Relativity, an Einstein Centenary Survey| publisher = Cambridge University Press | year=1979 | isbn=0-521-22285-0}} *{{citation | authorlink = Roger Penrose|first=R.|last=Penrose| title = Techniques of Differential Topology in Relativity| publisher = SIAM | year = 1972 | isbn=0898710057}} *{{cite book|last=Sard|first=R. D.|title=Relativistic Mechanics – Special Relativity and Classical Particle Dynamics|url=https://archive.org/details/relativisticmech0000sard|url-access=registration|year=1970|publisher=W. A. Benjamin|location=New York|isbn=978-0805384918}}
==Further reading== {{refbegin}} *[[Gary Gibbons|G. W. Gibbons]], S. N. Solodukhin; ''The Geometry of Small Causal Diamonds'' [[arXiv:hep-th/0703098]] (Causal intervals) *[[Stephen Hawking|S.W. Hawking]], A.R. King, P.J. McCarthy; ''[https://archive.today/20130113043425/http://link.aip.org/link/?JMAPAQ/17/174/1 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]]) *[[David Malament|D. Malament]]; ''[https://archive.today/20130112104622/http://link.aip.org/link/?JMAPAQ/18/1399/1 The class of continuous timelike curves determines the topology of spacetime]''; J. Math. Phys. 18 7:1399–1404 (1977); (Geometry, [[Causal Structure]]) *[[Alfred Robb|A.A. Robb]]; ''[https://archive.org/details/theoryoftimespac00robbrich A theory of time and space]''; Cambridge University Press, 1914; (Geometry, [[Causal Structure]]) *[[Alfred Robb|A.A. Robb]]; ''[https://archive.org/details/relationsoftime00robbuoft The absolute relations of time and space]''; Cambridge University Press, 1921; (Geometry, [[Causal Structure]]) *[[Alfred Robb|A.A. Robb]]; ''[https://archive.org/details/geometryoftimean032218mbp Geometry of Time and Space]''; Cambridge University Press, 1936; (Geometry, [[Causal Structure]]) *[[Rafael Sorkin|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]]) {{refend}}
== External links == * [http://demonstrations.wolfram.com/TuringMachineCausalNetworks/ Turing Machine Causal Networks] by Enrique Zeleny, the [[Wolfram Demonstrations Project]] * {{MathWorld |title=Causal Network |urlname=CausalNetwork}}
[[Category:Lorentzian manifolds]] [[Category:Theory of relativity]] [[Category:Mathematics of general relativity]]