Bulletin of the American Physical Society
APS April Meeting 2013
Volume 58, Number 4
Saturday–Tuesday, April 13–16, 2013; Denver, Colorado
Session R10: General Relativity: Mathematical Aspects |
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Sponsoring Units: GGR Room: Governor's Square 12 |
Monday, April 15, 2013 1:30PM - 1:42PM |
R10.00001: A positive energy theorem for the gravitational Dirichlet problem William Kelly, Donald Marolf Gravity in the presence of a Dirichlet boundary condition (fixed metric on some time-like surface) has been extensively studied in the context of the AdS/CFT correspondence. We show that this system is stable in the sense that energy is bounded below by generalizing Witten's proof of the positive energy theorem. Our proof applies to 3+1 dimensional spacetimes with vanishing cosmological constant, though we expect similar results to hold in higher dimensions and for negative cosmological constant. We also prove, under the same conditions, the familiar inequality $M \ge (Q^2 + P^2)^{1/2}$, where $M$ is the ADM mass and $Q$ and $P$ are the total electric and magnetic charge of the spacetime. [Preview Abstract] |
Monday, April 15, 2013 1:42PM - 1:54PM |
R10.00002: Linked Gravitational Radiation Amy Thompson, Joseph Swearngin, Alexander Wickes, Jan Willem Dalhuisen, Dirk Bouwmeester The electromagnetic knot is a topologically nontrivial solution to the vacuum Maxwell equations with the property that any two field lines belonging to either the electric, magnetic, or Poynting vector fields are closed and linked exactly once [1]. The relationship between the vacuum Maxwell and linearized Einstein equations, as expressed in the form of the spin-$N$ massless field equations, suggests that gravitational radiation possesses analogous topologically nontrivial field configurations. Using twistor methods we find the analogous spin-$2$ solutions of Petrov types N, D, and III. Aided by the concept of tendex and vortex lines as recently developed for the physical interpretation of solutions in general relativity [2], we investigate the physical properties of these knotted gravitational fields by characterizing the topology of their associated tendex and vortex lines.\\[4pt] [1] Ranada, A. F. and Trueba, J. L., Mod. Nonlinear Opt. III, 119, 197 (2002).\newline [2] Nichols, D. A., et al., Phys. Rev. D, 84 (2011). [Preview Abstract] |
Monday, April 15, 2013 1:54PM - 2:06PM |
R10.00003: Approximate Isometries as an Eigenvalue Problem and Angular Momentum Shawn Wilder, Chris Beetle In relativistic physics, a precise definition of a black hole's angular momentum is possible only when its horizon possesses an axial symmetry. Unfortunately most black hole horizons have no such symmetry. However, it is possible to pose an eigenvalue problem that has solutions corresponding to any manifold's ``approximate Killing fields.'' This allows one to generalize formulae requiring symmetry to cases where no symmetry is present and thus define, for example, the spin of an arbitrary black hole. This talk will discuss work using perturbation theory of a horizon to quantify the stability of quantities generalized in this way. We will present precise conditions for the stability of solutions to the eigenvalue problem, and discuss potential applications to numerical relativity. [Preview Abstract] |
Monday, April 15, 2013 2:06PM - 2:18PM |
R10.00004: History of a black hole horizon Dieter Brill The horizon of a general (non-eternal) black hole is initially much more complicated than in the well-known case of a spherically symmetric spacetime; but finally it becomes simple and acquires an asymptotic ``no hair'' state. It is therefore simplest to evolve the horizon backwards in time from the final condition, by following its congruence of null geodesic generators. During the evolution, significant events occur when generators cross and exit the horizon, leaving behind a spacelike crease. This happens typically when matter crosses the horizon, but the crossing is not causally related to the crease. The crease set (if it were a priori known) can be an initial condition for horizon development forward in time. Examples will be given in 3- and 4-dimensional spacetimes. [Preview Abstract] |
Monday, April 15, 2013 2:18PM - 2:30PM |
R10.00005: Exact example of backreaction of small scale inhomogeneities in cosmology Stephen Green, Robert Wald We construct a one-parameter family of polarized vacuum Gowdy spacetimes on a torus. In the limit as the parameter $N$ goes to infinity, the metric uniformly approaches a smooth ``background metric.'' However, spacetime derivatives of the metric do not approach a limit. As a result, we find that the background metric itself is not a solution of the vacuum Einstein equation. Rather, it is a solution of the Einstein equation with an ``effective stress-energy tensor,'' which is traceless and satisfies the weak energy condition. This is an explicit example of backreaction due to small scale inhomogeneities. We comment on the non-vacuum case, where we have proven in previous work that, provided the matter stress-energy tensor satisfies the weak energy condition, no additional backreaction is possible. [Preview Abstract] |
Monday, April 15, 2013 2:30PM - 2:42PM |
R10.00006: Gauge Conditions and Black hole Stability Kartik Prabhu Hollands and Wald showed that dynamic stability of a black hole is equivalent to the positivity of canonical energy on a space of linearised perturbations satisfying certain boundary conditions and gauge conditions. The boundary/gauge conditions are naturally formulated on the space of initial data for the perturbations in terms of orthogonality to gauge transformations. These perturbations can be uniquely specified in terms of transverse-traceless tensors. Using these transverse-traceless data, positivity of kinetic energy for perturbations can be proven. [Preview Abstract] |
Monday, April 15, 2013 2:42PM - 2:54PM |
R10.00007: Dynamical and Thermodynamic Stability of Perfect Fluid Stars Joshua Schiffrin, Stephen Green, Robert Wald We explore the stability of stationary axisymmetric perfect fluid configurations to axisymmetric perturbations in general relativity. We consider the class of perturbations which keep the particle number, entropy, and angular momentum of each fluid element fixed. We show that the condition for dynamical stability with respect to such perturbations is equivalent to positivity of the canonical energy. Additionally we show that, with respect to this class of perturbations, dynamical stability is equivalent to thermodynamic stability. [Preview Abstract] |
Monday, April 15, 2013 2:54PM - 3:06PM |
R10.00008: Comments on the BKL conjecture for spatially inhomogeneous cosmologies Beverly K. Berger Long ago, Belinskii, Khalatnikov, and Lifshitz (BKL) argued that the approach to the singularity in generic gravitational collapse behaved locally as a spatially homogeneous cosmology that was either velocity dominated (Kasner-like) or oscillatory (Mixmaster-like). While mathematical proofs of the BKL conjecture for apparently Mixmaster-like, spatially inhomogeneous models do not yet exist, several frameworks have been proposed to make the conjecture precise and to offer a roadmap for mathematical results. However, detailed examination of models with $G_2$ spatial symmetry have found recurrent spiky solutions that may be non-BKL-like or indicative of a broader BKL-like phenomenology (W.C. Lim et al, Phys. Rev. D {\bf 79}, 123526 (2009)). These issues will be explored with the BKL Simulator, a realization of the BKL conjecture. Spatial dependence is simulated by evolving spatially homogeneous Mixmaster models with slowly varying initial data. Numerical and analytic results in one spatial dimension will be presented along with comparison to genuine spatially inhomogeneous simulations. [Preview Abstract] |
Monday, April 15, 2013 3:06PM - 3:18PM |
R10.00009: On the consequences of the weak field approximation John Laubenstein General Relativity reduces to Newtonian gravity within the appropriate limit. But, what is that limit? The conventional response is that of the weak field approximation in which the gravitating source is weak and velocities are low. But, this is a far cry from a quantitative statement. In that regard, the weak field may be defined more quantitatively as one in which any error introduced is far beyond the level of precision required. Since the field can always be made incrementally weaker there is no limit as to the degree of precision that can be achieved. In this regard, GR reduces exactly to Newtonian gravity at the limit where velocity goes to zero. It is only out of convenience that we extend this to include those conditions where v $<<$ c with the argument that any error is arbitrarily small. However, in practice GR can be shown to reduce to an exact Newtonian expression at v $>$ 0. How can this observation fit with the quantitative definition of the weak field? This paper explores the consequences of the weak field approximation and the fact that GR reduces directly to Newtonian gravity within the weak field as opposed to the more specific condition where v $=$ zero. [Preview Abstract] |
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