In view of the initial value formulation for Einstein's equations, global hyperbolicity is seen to be a very natural condition in the context of general relativity, in the sense that given arbitrary initial data, there is a unique maximal globally hyperbolic solution of Einstein's equations. See more In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that … See more Global hyperbolicity, in the first form given above, was introduced by Leray in order to consider well-posedness of the Cauchy problem for the wave equation on the manifold. In 1970 Geroch proved the equivalence of definitions 1 and 2. Definition 3 under … See more There are several equivalent definitions of global hyperbolicity. Let M be a smooth connected Lorentzian manifold without boundary. We make the following preliminary definitions: • M is non-totally vicious if there is at least one point such that … See more • Causality conditions • Causal structure • Light cone See more WebMay 20, 2024 · Finally, we can define a globally hyperbolic spacetime as a spacetime for which there exists (at least one) achronal set Σ for which D ( Σ) is the entire spacetime. …
Boundary conditions and vacuum fluctuations in \({\mathrm …
WebJan 9, 2014 · Stefan Hollands, Robert M. Wald We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime. Our review emphasizes the conceptual issues arising in the formulation of the theory and presents known results in a mathematically precise way. WebDec 19, 2024 · A Duistermaat–Guillemin–Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact stationary spacetime is achieved by generalising the recent construction by Strohmaier and Zelditch (Adv Math 376:107434, 2024) to a vector bundle setting. pink rock candy sticks
Globally hyperbolic spacetimes: slicings, boundaries and ...
Weba spacetime outside its globally hyperbolic region (the boundary of the globally hyperbolic region is called the Cauchy horizon). The claim of scc is that, for most spacetimes, such extensions cannot be made. Note that in most cases one may use the constraints (3) to solve for A and d,A, given U and d ... WebDec 20, 2024 · the spacetime is globally hyperbolic with a non-compact Cauchy surface \Sigma , (3) there exists a closed trapped surface \mathscr {T}. The proof of this theorem is derived by contradiction (see, e.g., [ 47, 48 ]). It starts by assuming that the spacetime is null geodesically complete. WebAn interesting result relating spacelike geodesic completeness to global hyperbolicity was given in [18, Proposition 5.3]. The author proved that an ultra-static spacetime (M,g) is globally hyperbolic if and only if the global Cauchy surface is geodesically complete. The physical advantage steering dynamics