Causal structure

In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold. Lorentzian manifolds can be classified according to the types of causal structures they admit (causality conditions).

In mathematical physics, the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold. Lorentzian manifolds can be classified according to the types of causal structures they admit (causality conditions). null or lightlike if \,g(X,X) = 0 spacelike if \,g(X,X) > 0 Here we use the (-,+,+,+,\cdots) 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 M=\mathbb{R}^4 and g is the 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 \mathbb{R}^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{R}^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 (but not by a general Poincaré transformation because the origin may then be displaced) because of the invariance of the metric. 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 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. The causal relation \prec is the smallest transitive extension of the horismos relation \to . \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) = \{ y \in M | x \ll y\} 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) = \{ y \in M | y \ll 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) = \{ y \in M | x \prec y\} 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) = \{ y \in M | y \prec 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 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 the set \,I^+(x) is the interior of the future light cone 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[S] = \bigcup_{x \in S} I^\pm(x) J^\pm[S] = \bigcup_{x \in S} J^\pm(x) For S, T two subsets of M we define The chronological future of S relative to T , I^+[S;T] , is the chronological future of S considered as a submanifold of T . Note that this is quite a different concept from I^+[S] \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^+[S;T] , is the causal future of S considered as a submanifold of T . Note that this is quite a different concept from J^+[S] \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^{+}[S] . 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(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 \gamma(t_1) in the causal past of \gamma(t_2) . 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 . In the discrete version, the causal diamond is the set of all the causal paths that connect \gamma(t_2) from \gamma(t_1) . I^+[S] = I^+[I^+[S]] \subset J^+[S] = J^+[J^+[S]] I^-[S] = I^-[I^-[S]] \subset J^-[S] = J^-[J^-[S]] The horismos is generated by null geodesic congruences. Topological properties: I^\pm(x) is open for all points x in M . I^\pm[S] is open for all subsets S \subset M . I^\pm[S] = I^\pm[\overline{S}] for all subsets S \subset M . Here \overline{S} is the closure of a subset S . I^\pm[S] \subset \overline{J^\pm[S]} Two metrics \,g and \hat{g} are conformally related if \hat{g} = \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{g} . As an example suppose X is a timelike tangent vector with respect to the \,g metric. This means that \,g(X,X) . We then have that \hat{g}(X,X) = \Omega^2 g(X,X) so X is a timelike tangent vector with respect to the \hat{g} 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.

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; ; 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; ; 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)

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