Inmathematical physics, thecausal structure of aLorentzian manifold describes thecausal relationships between points in the manifold.

Inmodern physics (especiallygeneral relativity)spacetime is represented by aLorentzian 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 ofcurvature. Discussions of the causal structure for such manifolds must be phrased in terms ofsmoothcurves joining pairs of points. Conditions on thetangent vectors of the curves then define the causal relationships.

If${\displaystyle (M,g)}$ is aLorentzian manifold (formetric${\displaystyle g}$ onmanifold${\displaystyle M}$) then the nonzero tangent vectors at each point in the manifold can be classified into threedisjoint types.A tangent vector${\displaystyle X}$ is:

• timelike if
• null orlightlike if${\displaystyle g(X,X)=0}$
• spacelike if${\displaystyle g(X,X)>0}$

Here we use the${\displaystyle (-,+,+,+,\cdots )}$metric signature. We say that a tangent vector isnon-spacelike if it is null or timelike.

The canonical Lorentzian manifold isMinkowski spacetime, where${\displaystyle M={\mathbb{R}}^4}$ and${\displaystyle g}$ is theflatMinkowski 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${\displaystyle {\mathbb{R}}^4}$ and hence the tangent vectors may be identified with points in the space. The four-dimensional vector${\displaystyle X=(t,r)}$ is classified according to the sign of${\displaystyle g(X,X)=-c^2{t}^2+\Vert r{\Vert }^2}$, where${\displaystyle r\in {\mathbb{R}}^3}$ is aCartesian coordinate in 3-dimensional space,${\displaystyle c}$ is the constant representing the universal speed limit, and${\displaystyle t}$ is time. The classification of any vector in the space will be the same in all frames of reference that are related by aLorentz transformation (but not by a generalPoincaré transformation because the origin may then be displaced) because of the invariance of the metric.

At each point in${\displaystyle M}$ the timelike tangent vectors in the point'stangent space can be divided into two classes. To do this we first define anequivalence relation on pairs of timelike tangent vectors.

If${\displaystyle X}$ and${\displaystyle Y}$ are two timelike tangent vectors at a point we say that${\displaystyle X}$ and${\displaystyle Y}$ are equivalent (written${\displaystyle X\sim Y}$) if.

There are then twoequivalence classes which between them contain all timelike tangent vectors at the point.We can (arbitrarily) call one of these equivalence classesfuture-directed and call the otherpast-directed. Physically this designation of the two classes of future- and past-directed timelike vectors corresponds to a choice of anarrow of time at the point. The future- and past-directed designations can be extended to null vectors at a point by continuity.

ALorentzian manifold istime-orientable^[1] if a continuous designation of future-directed and past-directed for non-spacelike vectors can be made over the entire manifold.

Apath in${\displaystyle M}$ is acontinuous map${\displaystyle \mu :\mathrm{\Sigma }\to M}$ where${\displaystyle \mathrm{\Sigma }}$ is a nondegenerate interval (i.e., a connected set containing more than one point) in${\displaystyle \mathbb{R}}$. Asmooth path has${\displaystyle \mu }$ differentiable an appropriate number of times (typically${\displaystyle C^{\mathrm{\infty }}}$), and aregular path has nonvanishing derivative.

Acurve in${\displaystyle M}$ is the image of a path or, more properly, an equivalence class of path-images related by re-parametrisation, i.e.homeomorphisms ordiffeomorphisms of${\displaystyle \mathrm{\Sigma }}$. When${\displaystyle M}$ is time-orientable, the curve isoriented if the parameter change is required to bemonotonic.

Smooth regular curves (or paths) in${\displaystyle M}$ can be classified depending on their tangent vectors. Such a curve is

• chronological (ortimelike) if the tangent vector is timelike at all points in the curve. Also called aworld line.^[2]
• 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 (ornon-spacelike) if the tangent vector is timelikeor null at all points in the curve.

The requirements of regularity and nondegeneracy of${\displaystyle \mathrm{\Sigma }}$ 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${\displaystyle M}$ 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.

• Aclosed timelike curve is a closed curve which is everywhere future-directed timelike (or everywhere past-directed timelike).
• Aclosed null curve is a closed curve which is everywhere future-directed null (or everywhere past-directed null).
• Theholonomy of the ratio of the rate of change of the affine parameter around a closed null geodesic is theredshift factor.

There are several causalrelations between points${\displaystyle x}$ and${\displaystyle y}$ in the manifold${\displaystyle M}$.

• ${\displaystyle x}$chronologically precedes${\displaystyle y}$ (often denoted${\displaystyle x\ll y}$) if there exists a future-directed chronological (timelike) curve from${\displaystyle x}$ to${\displaystyle y}$.
• ${\displaystyle x}$strictly causally precedes${\displaystyle y}$ (often denoted) if there exists a future-directed causal (non-spacelike) curve from${\displaystyle x}$ to${\displaystyle y}$.
• ${\displaystyle x}$causally precedes${\displaystyle y}$ (often denoted${\displaystyle x\prec y}$ or${\displaystyle x\le y}$) if${\displaystyle x}$ strictly causally precedes${\displaystyle y}$ or${\displaystyle x=y}$.
• ${\displaystyle x}$horismos${\displaystyle y}$^[3] (often denoted${\displaystyle x\to y}$ or${\displaystyle x\nearrow y}$) if${\displaystyle x=y}$ or there exists a future-directed null curve from${\displaystyle x}$ to${\displaystyle y}$^[4] (or equivalently,${\displaystyle x\prec y}$ and${\displaystyle x\ll ̸y}$ implies${\displaystyle x\prec y}$ (this follows trivially from the definition)^[5]
• ${\displaystyle x\ll y}$,${\displaystyle y\prec z}$ implies${\displaystyle x\ll z}$^[5]
• ${\displaystyle x\prec y}$,${\displaystyle y\ll z}$ implies${\displaystyle x\ll z}$^[5]
• ${\displaystyle \ll }$,,${\displaystyle \prec }$ aretransitive.^[5]${\displaystyle \to }$ is not transitive.^[6]
• ${\displaystyle \prec }$,${\displaystyle \to }$ arereflexive^[4]

For a point${\displaystyle x}$ in the manifold${\displaystyle M}$ we define^[5]

• Thechronological future of${\displaystyle x}$, denoted${\displaystyle I^{+}(x)}$, as the set of all points${\displaystyle y}$ in${\displaystyle M}$ such that${\displaystyle x}$ chronologically precedes${\displaystyle y}$:

${\displaystyle I^{+}(x)=\{y\in M|x\ll y\}}$

• Thechronological past of${\displaystyle x}$, denoted${\displaystyle I^{-}(x)}$, as the set of all points${\displaystyle y}$ in${\displaystyle M}$ such that${\displaystyle y}$ chronologically precedes${\displaystyle x}$:

${\displaystyle I^{-}(x)=\{y\in M|y\ll x\}}$

We similarly define

• Thecausal future (also called theabsolute future) of${\displaystyle x}$, denoted${\displaystyle J^{+}(x)}$, as the set of all points${\displaystyle y}$ in${\displaystyle M}$ such that${\displaystyle x}$ causally precedes${\displaystyle y}$:

${\displaystyle J^{+}(x)=\{y\in M|x\prec y\}}$

• Thecausal past (also called theabsolute past) of${\displaystyle x}$, denoted${\displaystyle J^{-}(x)}$, as the set of all points${\displaystyle y}$ in${\displaystyle M}$ such that${\displaystyle y}$ causally precedes${\displaystyle x}$:

${\displaystyle J^{-}(x)=\{y\in M|y\prec x\}}$

• Thefuture null cone of${\displaystyle x}$ as the set of all points${\displaystyle y}$ in${\displaystyle M}$ such that${\displaystyle x\to y}$.
• Thepast null cone of${\displaystyle x}$ as the set of all points${\displaystyle y}$ in${\displaystyle M}$ such that${\displaystyle y\to x}$.
• Thelight cone of${\displaystyle x}$ as the future and past null cones of${\displaystyle x}$ together.^[7]
• elsewhere as points not in the light cone, causal future, or causal past.^[7]

Points contained in${\displaystyle I^{+}(x)}$, for example, can be reached from${\displaystyle x}$ by a future-directed timelike curve.The point${\displaystyle x}$ can be reached, for example, from points contained in${\displaystyle J^{-}(x)}$ by a future-directed non-spacelike curve.

InMinkowski spacetime the set${\displaystyle I^{+}(x)}$ is theinterior of the futurelight cone at${\displaystyle x}$. The set${\displaystyle J^{+}(x)}$ is the full future light cone at${\displaystyle x}$, including the cone itself.

These sets${\displaystyle I^{+}(x),I^{-}(x),J^{+}(x),J^{-}(x)}$defined for all${\displaystyle x}$ in${\displaystyle M}$, are collectively called thecausal structure of${\displaystyle M}$.

For${\displaystyle S}$ asubset of${\displaystyle M}$ we define^[5]

${\displaystyle I^{\pm }[S]=\bigcup _{x\in S}{I}^{\pm }(x)}$

${\displaystyle J^{\pm }[S]=\bigcup _{x\in S}{J}^{\pm }(x)}$

For${\displaystyle S,T}$ twosubsets of${\displaystyle M}$ we define

• Thechronological future of${\displaystyle S}$ relative to${\displaystyle T}$,${\displaystyle I^{+}[S;T]}$, is the chronological future of${\displaystyle S}$ considered as asubmanifold of${\displaystyle T}$. Note that this is quite a different concept from${\displaystyle I^{+}[S]\cap T}$ which gives the set of points in${\displaystyle T}$ which can be reached by future-directed timelike curves starting from${\displaystyle S}$. In the first case the curves must lie in${\displaystyle T}$ in the second case they do not. See Hawking and Ellis.
• Thecausal future of${\displaystyle S}$ relative to${\displaystyle T}$,${\displaystyle J^{+}[S;T]}$, is the causal future of${\displaystyle S}$ considered as a submanifold of${\displaystyle T}$. Note that this is quite a different concept from${\displaystyle J^{+}[S]\cap T}$ which gives the set of points in${\displaystyle T}$ which can be reached by future-directed causal curves starting from${\displaystyle S}$. In the first case the curves must lie in${\displaystyle T}$ in the second case they do not. See Hawking and Ellis.
• Afuture set is a set closed under chronological future.
• Apast set is a set closed under chronological past.
• Anindecomposable 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${\displaystyle M}$ is called aterminal indecomposable past set (TIP).
• Aproper indecomposable past set (PIP) is an IP which isn't a TIP.${\displaystyle I^{-}(x)}$ is a proper indecomposable past set (PIP).
• The futureCauchy development of${\displaystyle S}$,${\displaystyle D^{+}(S)}$ is the set of all points${\displaystyle x}$ for which every past directed inextendible causal curve through${\displaystyle x}$ intersects${\displaystyle S}$ 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 ofdeterminism.
• A subset${\displaystyle S\subset M}$ isachronal if there do not exist${\displaystyle q,r\in S}$ such that${\displaystyle r\in I^{+}(q)}$, or equivalently, if${\displaystyle S}$ is disjoint from${\displaystyle I^{+}[S]}$.

• ACauchy surface is a closed achronal set whose Cauchy development is${\displaystyle M}$.
• A metric isglobally hyperbolic if it can be foliated by Cauchy surfaces.
• Thechronology violating set is the set of points through which closed timelike curves pass.
• Thecausality violating set is the set of points through which closed causal curves pass.
• The boundary of the causality violating set is aCauchy horizon. If the Cauchy horizon is generated by closed null geodesics, then there's aredshift factor associated with each of them.
• For a causal curve${\displaystyle \gamma }$, thecausal diamond is${\displaystyle J^{+}(\gamma (t_1))\cap J^{-}(\gamma (t_2))}$ (here we are using the looser definition of 'curve' whereon it is just a set of points), being the point${\displaystyle \gamma (t_1)}$ in the causal past of${\displaystyle \gamma (t_2)}$. In words: the causal diamond of a particle's world-line${\displaystyle \gamma }$ is the set of all events that lie in both the past of some point in${\displaystyle \gamma }$ and the future of some point in${\displaystyle \gamma }$. In the discrete version, the causal diamond is the set of all the causal paths that connect${\displaystyle \gamma (t_2)}$ from${\displaystyle \gamma (t_1)}$.

See Penrose (1972), p13.

• A point${\displaystyle x}$ is in${\displaystyle I^{-}(y)}$ if and only if${\displaystyle y}$ is in${\displaystyle I^{+}(x)}$.
• ${\displaystyle x\prec y⟹I^{-}(x)\subset I^{-}(y)}$
• ${\displaystyle x\prec y⟹I^{+}(y)\subset I^{+}(x)}$
• ${\displaystyle I^{+}[S]=I^{+}[I^{+}[S]]\subset J^{+}[S]=J^{+}[J^{+}[S]]}$
• ${\displaystyle I^{-}[S]=I^{-}[I^{-}[S]]\subset J^{-}[S]=J^{-}[J^{-}[S]]}$
• The horismos is generated by null geodesic congruences.

Topological properties:

• ${\displaystyle I^{\pm }(x)}$ is open for all points${\displaystyle x}$ in${\displaystyle M}$.
• ${\displaystyle I^{\pm }[S]}$ is open for all subsets${\displaystyle S\subset M}$.
• ${\displaystyle I^{\pm }[S]=I^{\pm }[\overline{S}]}$ for all subsets${\displaystyle S\subset M}$. Here${\displaystyle \overline{S}}$ is theclosure of a subset${\displaystyle S}$.
• ${\displaystyle I^{\pm }[S]\subset \overline{J^{\pm }[S]}}$

Two metrics${\displaystyle g}$ and${\displaystyle \hat{g}}$ areconformally related^[8] if${\displaystyle \hat{g}={\mathrm{\Omega }}^{2}g}$ for some real function${\displaystyle \mathrm{\Omega }}$ called theconformal factor. (Seeconformal map).

Looking at the definitions of which tangent vectors are timelike, null and spacelike we see they remain unchanged if we use${\displaystyle g}$ or${\displaystyle \hat{g}}$. As an example suppose${\displaystyle X}$ is a timelike tangent vector with respect to the${\displaystyle g}$ metric. This means that. We then have that so${\displaystyle X}$ is a timelike tangent vector with respect to the${\displaystyle \hat{g}}$ too.

It follows from this that the causal structure of a Lorentzian manifold is unaffected by aconformal transformation.

A null geodesic remains a null geodesic under a conformal rescaling.

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 theconformal boundary of the manifold. The topological structure of the conformal boundary depends upon the causal structure.

• Future-directed timelike geodesics end up on${\displaystyle i^{+}}$, thefuture timelike infinity.
• Past-directed timelike geodesics end up on${\displaystyle i^{-}}$, thepast timelike infinity.
• Future-directed null geodesics end up on ℐ⁺, thefuturenull infinity.
• Past-directed null geodesics end up on ℐ⁻, thepast null infinity.
• Spacelike geodesics end up onspacelike infinity.

In various spaces:

• Minkowski space:${\displaystyle i^{\pm }}$ are points, ℐ^± are null sheets, and spacelike infinity hascodimension 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.

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 asingularity.

• Forblack holes, the future timelike boundary ends on asingularity in some places.
• For theBig Bang, the past timelike boundary is also a singularity.

Theabsolute event horizon is the past null cone of the future timelike infinity. It is generated by null geodesics which obey theRaychaudhuri optical equation.

• 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

• Hawking, S.W.;Ellis, G.F.R. (1973),The Large Scale Structure of Space-Time, Cambridge: Cambridge University Press,ISBN 0-521-20016-4
• Hawking, S.W.;Israel, W. (1979),General Relativity, an Einstein Centenary Survey, Cambridge University Press,ISBN 0-521-22285-0
• Penrose, R. (1972),Techniques of Differential Topology in Relativity, SIAM,ISBN 0898710057
• Sard, R. D. (1970).Relativistic Mechanics - Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin.ISBN 978-0805384918.

• Turing Machine Causal Networks by Enrique Zeleny, theWolfram Demonstrations Project
• Weisstein, Eric W."Causal Network".MathWorld.

• Lorentzian manifolds
• Theory of relativity
• General relativity
• Theoretical physics

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