The Initial Value Problem in General Relativity

Null infinity as an open Hamiltonian system

“…In fact, the Rachaudhuri equation turns into a constraint (3.51) on the kinematical phase space, and it selects a unique value of λ on every such gauge orbit for given initial conditions on a cross-section C o of N (the initial values are λ o = λ| Co andλ o = L λ| Co ). The construction is reminiscent of conformal methods on a spacelike hypersurface, where the orbits of three-dimensional conformal transformations are used often to determine a local gauge-fixing for the Hamiltonian constraint, see [33][34][35].…”

Counterfactuals in the Initial Value Formulation of General Relativity

“…8 Nothing substantial for our discussion hinges on this assumption, which, on the other hand, actually amounts to a gauge choice in the slicing of the 3 + 1 decomposition; it allows to ease the discussion of the partial differential equation we end up with. Moreover, several very interesting results have been proven using such constant mean curvature data, in particular on closed manifolds (using Yamabe theorem, which ensures that any Riemannian metric on a closed manifold with dimension no less than three can be conformally transformed into a metric with constant scalar curvature); for non-constant mean curvature data, much less is known about existence and uniqueness of solutions (see (Isenberg [2014], §16.4) and references therein). As a consequence, dropping the constant mean curvature assumption actually further strengthens the difficulties for evaluating counterfactuals, since then one does not even know whether initial data corresponding to the antecedent of the counterfactual to be evaluated exist (that is, whether appropriate solutions to the constraints exist).…”

“…5). For instance, the Corvino-Schoen asymptotic exterior gluing technique 'allows one to smoothly glue any interior region of an asymptotically Euclidean solution to an exterior region of a slice of a Kerr solution' (Isenberg [2014], p. 317). 28 Moreover, by combining various gluing techniques, Chruściel et al ([2011]) were able to show that 'for any chosen set of N asymptotically Euclidean solutions of the constrains representing black holes, stars, or other astrophysical objects of interest, one can construct a new asymptotically Euclidean solution that includes interior regions of these N chosen solutions, placed as desired (so long as the distances between the bodies are sufficiently large) and with the desired relative momenta' (Isenberg [2014], p. 318).…”

“…The initial value formulation can be extended to certain non-globally hyperbolic situations(Friedman [2004]). Here we aim to consider issues arising for the evaluation of counterfactuals already in the friendliest (that is, globally hyperbolic) GR spacetimes in a Cauchy initial data setting.6 For the sake of readability, and in order to focus on the conceptual aspects related to counterfactuals, we privilege in this paper a qualitative discussion over technical details; for the latter, we refer the interested reader to(Isenberg [2014])-which we broadly follow for the technical aspects in this article-as well as references therein.5Downloaded from https://academic.oup.com/bjps/advance-article-abstract/doi/10.1093/bjps/axy066/5107653 by University of Queensland user on 09 April 2019…”

Geodesic Incompleteness and Partially Covariant Gravity

“…Due to the close relation of the Hořava-Lifshitz equations to a generalized form of the Ricci flow equation, there are new possibilities for the construction of topological quantum gravity theories by utilizing the special mathematical properties of the latter [11]. Secondly, the Ricci flow, as well as other non-relativistic geometric flows, such as the mean-curvature flow, has long been known for its interesting properties, especially with regard to the formation of singularities [12][13][14][15][16], in association with corresponding properties known for the Einstein flow [17][18][19], although a clear relation among the nature of singularities between different non-relativistic flows and general relativity is presently elusive. Thirdly, the importance of alternative geometries such as the Newton-Cartan (NC) geometry in gravitation through a non-relativistic expansion is useful in delineating more precisely the relations of non-relativistic gravity and general relativity [20][21][22].…”