Gravitational Waves and Causality

Ehrlich, Paul E.; Emch, Gérard G.

doi:10.1142/s0129055x92000066

Cited by 25 publications

(39 citation statements)

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“…In order to obtain a detailed study on existence and multiplicity of connecting geodesics in exact gravitational waves, Ehrlich and Emch introduced in [14] the concept of "first astigmatic conjugate pairs" for the coordinate u (defined for an ODE system in [17,Definition 3.12]). In that reference they concluded: (a) the existence of a unique connecting geodesic (which is causal for causally related points) whenever u 1 appears before the first astigmatic conjugate point of u 0 , and (b) the non-geodesic connectedness when u 1 is the first astigmatic conjugate point.…”

Section: Conjugate Pointsmentioning

confidence: 99%

Causality and conjugate points in general plane waves

Flores

Sánchez

2003

Class. Quantum Grav.

125

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Abstract.Let M = M0 × R 2 be a pp-wave type spacetime endowed with the metric ·, · z = ·, · x + 2 du dv + H(x, u) du 2 , where (M0, ·, · x) is any Riemannian manifold and H(x, u) an arbitrary function. We show that the behaviour of H(x, u) at spatial infinity determines the causality of M, say: (a) if −H(x, u) behaves subquadratically (i.e, essentially −H(x, u) ≤ R1(u)|x| 2−ǫ for some ǫ > 0 and large distance |x| to a fixed point) and the spatial part (M0, ·, · x) is complete, then the spacetime M is globally hyperbolic, (b) if −H(x, u) grows at most quadratically (i.e, −H(x, u) ≤ R1(u)|x| 2 for large |x|) then it is strongly causal and (c) M is always causal, but there are non-distinguishing examples (and thus, non-strongly causal), even when −H(x, u) ≤ R1(u)|x| 2+ǫ , for small ǫ > 0.Therefore, the classical model M0 = R 2 , H(x, u) = i,j hij (u)xixj( ≡ 0), which is known to be strongly causal but not globally hyperbolic, lies in the critical quadratic situation with complete M0. This must be taken into account for realistic applications. In fact, we argue that −H will be subquadratic (and the spacetime globally hyperbolic) if M is asymptotically flat. The relation of these results with the notion of astigmatic conjugacy and the existence of conjugate points is also discussed.

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Section: Conjugate Pointsmentioning

confidence: 99%

Causality and conjugate points in general plane waves

Flores

Sánchez

2003

Class. Quantum Grav.

125

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“…As a matter of fact similar examples already appeared in the literature. In fact, after the discovery by Penrose [42] that pp-waves are not globally hyperbolic Ehrlich and Emch [14] showed that they are not even causally simple (for them a ab = δ ab , a, b = 1, 2, b t = 0 and U is a suitable time dependent quadratic form in q 1 and q 2 ). In their study of the general plane waves [17,18] Flores and Sánchez prove (using only differential geometric tools) that if a t is independent of time, (S, a) is a complete Riemannian manifold, b t = 0, and U has a subquadratic behavior at spatial infinity then, (a) the spacetime (M, g) is globally hyperbolic [17,Theor.…”

Section: The Bolza Problemmentioning

confidence: 99%

Eisenhart's theorem and the causal simplicity of Eisenhart's spacetime

Minguzzi¹

2007

Class. Quantum Grav.

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Abstract. We give a causal version of Eisenhart's geodesic characterization of classical mechanics. We emphasize the geometric, coordinate independent properties needed to express Eisenhart's theorem in light of modern studies on the Bargmann structures (lightlike dimensional reduction, pp-waves). The construction of the space metric, Coriolis 1-form and scalar potential through which the theorem is formulated is shown in detail, and in particular it is proved a one-to-one correspondence between Newtonian frames and Abelian connections on suitable lightlike principal bundles. The relation of Eisenhart's theorem in the lightlike case with a Fermat type principle is pointed out. The operation of lightlike lift is introduced and the existence of minimizers for the classical action is related to the causal simplicity of Eisenhart's spacetime.

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“…Notice that every metric tensor of the form (21) admits the homothetic vector field ξ = 2u∂ u + x∂ x + y∂ y . A metric tensor which have the form (21) with additionally P ≡ 0 and h ≡ 0 is called a plane wave [6,17]. These plane waves are a subclass of the more general class of pp-waves which have the form ds 2 = −2H (v, x, y) dv 2 − 2 du dv + dx 2 + dy 2 (22) with an arbitrary real function H , which does not depend on u (notice that sometimes in the literature the names of the first two coordinates are changed).…”

Section: With a Matrix P A Function H And A Symmetric Matrix Q Whicmentioning

confidence: 99%

Conformal vector fields on spacetimes

Steller

2006

Ann Glob Anal Geom

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We study conformal vector fields and their zeros on spacetimes which are nonconformally-flat. Depending on the Petrov type, we classify all conformal vector fields with zeros. The problems of reducing a conformal vector field to a homothetic vector field are considered. We show that a spacetime admitting a proper homothetic vector field is (locally) a plane wave. This precises a well-known theorem of Alekseevski, where all these spacetimes are determined in a more general form. (2000): Primary 53A30, Secondary 53C50 Mathematics Subject Classifications

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Gravitational Waves and Causality

Cited by 25 publications

References 0 publications

Causality and conjugate points in general plane waves

Causality and conjugate points in general plane waves

Eisenhart's theorem and the causal simplicity of Eisenhart's spacetime

Conformal vector fields on spacetimes

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