Charges for linearized gravity

Aksteiner, Steffen; Andersson, Lars

doi:10.1088/0264-9381/30/15/155016

Cited by 12 publications

(17 citation statements)

References 48 publications

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“…for the Mathematica based symbolic differential geometry suite xAct [31], exploits in a systematic way the above mentioned decompositions and allows one to carry out investigations which are not feasible to do by hand. The related SpinFrames package [32] developed by two of the authors implements computations in tetrad components using the Newman-Penrose (NP) and Geroch-Held-Penrose (GHP) [10] formalisms.…”

Section: A 2-spinors and Irreducible Decompositionsmentioning

confidence: 99%

New identities for linearized gravity on the Kerr spacetime

2019

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In this paper we derive a differential identity for linearized gravity on the Kerr spacetime and more generally on vacuum spacetimes of Petrov type D. We show that a linear combination of second derivatives of the linearized Weyl tensor can be formed into a complex symmetric 2-tensor M ab which solves the linearized Einstein equations. The identity makes this manifest by relating M ab to two terms solving the linearized Einstein equations by construction. The self-dual Weyl curvature of M ab gives a covariant version of the Teukolsky-Starobinsky identities for linearized gravity which, in addition to the two classical identities for linearized Weyl scalars with extreme spin weights, includes three additional equations. In particular, they are not consequences of the classical Teukolsky-Starobinsky identities, but are additional integrability conditions for linearized gravity. The result has direct application in the construction of symmetry operators and also yields a set of non-trivial gauge invariants for linearized gravity. * steffen.aksteiner@aei.mpg.de †

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Section: A 2-spinors and Irreducible Decompositionsmentioning

confidence: 99%

New identities for linearized gravity on the Kerr spacetime

2019

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“…The DHR variables Ψ ab and Ψ DHR satisfy, respectively, a tensor and spin-weighted Regge-Wheeler equation [1] 21 Note thatψ 4 m a m b corresponds (up to numerical factors) to the quantity (1) α ab in [1]. 22 For Petrov type-D spacetimes, the rescaling factor fromψ 4 to ψ is given by the coefficient of the unique Killing spinor in the principal null frame [24,36]. Up to constant factors this is equivalent to rescaling by (−Ψ 2 where the potential is given by…”

Section: Canonical Energy Of An Associated Perturbationmentioning

confidence: 99%

Canonical energy and Hertz potentials for perturbations of Schwarzschild spacetime

Prabhu

Wald

2018

Class. Quantum Grav.

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Canonical energy is a valuable tool for analyzing the linear stability of black hole spacetimes; positivity of canonical energy for all perturbations implies mode stability, whereas the failure of positivity for any perturbation implies instability. Nevertheless, even in the case of 4-dimensional Schwarzschild spacetime -which is known to be stable -manifest positivity of the canonical energy is difficult to establish, due to the presence of constraints on the initial data as well as the gauge dependence of the canonical energy integrand. Consideration of perturbations generated by a Hertz potential would appear to be a promising way to improve this situation, since the constraints and gauge dependence are eliminated when the canonical energy is expressed in terms of the Hertz potential. We prove that the canonical energy of a metric perturbation of Schwarzschild that is generated by a Hertz potential is positive. We relate the energy quantity arising in the linear stability proof of Dafermos, Holzegel and Rodnianski (DHR) to the canonical energy of an associated metric perturbation generated by a Hertz potential. We also relate the Regge-Wheeler variable of DHR to the ordinary Regge-Wheeler and twist potential variables of the associated perturbation. Since the Hertz potential formalism can be generalized to a Kerr black hole, our results may be useful for the analysis of the linear stability of Kerr. * kartikprabhu@cornell.edu † rmwa@uchicago.edu not been able to directly establish the positivity of the potential energy U , Eq. 1.3, even for the Schwarzschild background. 3 The two main obstacles to showing positivity of U appear to be the following: (1) The variable q ab is not free data but is subject to the linearized Hamiltonian constraint Eq. 1.4b.(2) Even though U is gauge-invariant, the integrand in Eq. 1.3 is not. Thus, it would seem that to show positivity of U one would need to utilize the constraint equations effectively and make a suitable choice of gauge (as was done for the proof of positivity of K in [8]). However, we have not, as yet, found a way to do this. A promising strategy to prove positivity would be rewrite the canonical energy in terms of unconstrained variables. In 4-dimensions, for the case of algebraically special spacetimes such as Schwarzschild and Kerr, a possible choice of such variables are the Hertz-Bromowich-Debye-Whittaker-Penrose potentials (henceforth Hertz potentials) [11][12][13][14]. The Hertz potentials, which solve the Teukolsky equation [15], can be used to generate (complex) metric perturbations that solve the Einstein equation. The initial data corresponding to the Hertz potentials is unconstrained, and one could attempt to prove positivity of canonical energy for perturbations generated by a Hertz potential. 4 The main purpose of this paper is to show that for perturbations of Schwarzschild generated by a Hertz potential, the canonical energy is indeed positive. Since the Hertz potential formalism can be straightforwardly extended to Kerr spacetime, it is possible t...

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“…We empasize that symbolic computations have played an essential role in the results on fields with non-zero spin presented here, in particular in sections 4.2, 4.3. The packages Sym-Manipulator [8] and SpinFrames [2], developed for the Mathematica based symbolic differential geometry suite xAct [23] makes it possible to systematically exploit decompositions in terms of irreducible representations of the spin group Spin (1,3), and allows one to carry out investigations that are not feasible by hand.…”

Section: Spin Geometry In 3 + 1 Dimensionsmentioning

confidence: 99%

Wave and Dirac equations on manifolds

Andersson¹,

Bär²

2018

Space – Time – Matter

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We review some recent results on geometric equations on Lorentzian manifolds such as the wave and Dirac equations. This includes well-posedness and stability for various initial value problems, as well as results on the structure of these equations on black-hole spacetimes (in particular, on the Kerr solution), the index theorem for hyperbolic Dirac operators and properties of the class of Green-hyperbolic operators. Mathematical Subject Classification: 53C27, 53C50, 83C57, 83C60, 58J45

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Charges for linearized gravity

Cited by 12 publications

References 48 publications

New identities for linearized gravity on the Kerr spacetime

New identities for linearized gravity on the Kerr spacetime

Canonical energy and Hertz potentials for perturbations of Schwarzschild spacetime

Wave and Dirac equations on manifolds

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