Nonlinear Radiation Gauge for Near Kerr Spacetimes

Andersson, Lars; Bäckdahl, Thomas; Blue, Pieter; Ma, Siyuan

doi:10.1007/s00220-022-04461-3

Cited by 8 publications

(11 citation statements)

References 44 publications

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“…The system of linearised gravity is formulated for spin-weighted (tetrad) quantities in the Geroch-Held-Penrose (GHP) formalism and assumes that the linearised perturbations satisfy a set of "linear gauge conditions", known as (linear) outgoing radiation gauge conditions. In the follow-up paper [2], such gauge conditions are shown to arise from a choice of (nonlinear) outgoing radiation gauge for sufficiently small nonlinear perturbations of Kerr. The outgoing radiation gauge is tailored to the ingoing principal null geodesics of Kerr.…”

Section: Related Workmentioning

confidence: 97%

“…To move from the nonlinear to the linear theory, one has to implement a linearisation procedure to linearise the vacuum Einstein equations in the gauge chosen in Step 1. 2 The linearised system obtained should come with a natural notion of solutions and initial data and inherit well-posedness from the nonlinear equations.…”

Section: • Step 2: Linearisationmentioning

confidence: 99%

“…To formulate nonlinear perturbations of the Kerr exterior manifold, we start by choosing real parameters a and M , with |a| < M . We fix the manifold-with-boundary M, which we identify with the Kerr ambient manifold (2) with star-normalised, outgoing principal differentiable structure (s, s, θ A ). We introduce the fixed smooth scalar functions k a,M and k a,M of the fixed coordinates on M.…”

Section: Nonlinear Perturbationsmentioning

confidence: 99%

“…Other instances of geometric gauges which have been employed to formulate the vacuum Einstein equations are double-null gauges [13,18], constant mean curvature (and maximal) gauges [14], Bondi gauges [39] and radiation gauges [2]. Geometric gauges may be contrasted to wave gauges [43,44], for which the coordinates are solutions to a system of wave equations.…”

Section: A Geometric Frameworkmentioning

confidence: 99%

“…The outgoing radiation gauge is tailored to the ingoing principal null geodesics of Kerr. In [2], the choice of gauge is combined with a choice of null tetrads similarly tailored to the ingoing principal null geodesics of Kerr. The vacuum Einstein equations for the tetrad quantities in the outgoing radiation gauge are proven to be well-posed.…”

Section: Related Workmentioning

confidence: 99%

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A new gauge for gravitational perturbations of Kerr spacetimes I: The linearised theory

Benomio¹

2022

Preprint

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We propose a new geometric framework to address the stability of the Kerr solution to gravitational perturbations in the full sub-extremal range |a| < M . Central to our framework is a new formulation of nonlinear gravitational perturbations of Kerr, whose two novel ingredients are the choice of a geometric gauge and non-integrable null frames both tailored to the outgoing principal null geodesics of Kerr. The vacuum Einstein equations for the perturbations are formulated in our gauge as a system of equations for the connection coefficients and curvature components relative to the chosen frames. When renormalised with respect to Kerr, the null structure equations with the form of outgoing transport equations do not possess any derivatives of renormalised connection coefficients on the right hand side.In this work, we derive the linearised vacuum Einstein equations around Kerr in the new framework. The system of linearised gravity exhibits two key structural properties. The first is well-known and consists of the exact decoupling of two gauge invariant linearised quantities in the system, satisfying two decoupled spin ±2 Teukolsky equations. The second is a new, gauge dependent structure in the outgoing transport equations for the linearised connection coefficients inherited from the nonlinear system: Unlike previous works, these equations do not contain any derivatives of linearised connection coefficients on the right hand side and induce a new hierarchy of only outgoing transport equations including all gauge dependent linearised quantities.Our new framework is designed to effectively capture the stabilising properties of the red-shifted transport equations, thereby isolating one of the crucial structures of the problem. Such a feature is suggestive of future simplifications in the analysis. As an illustration, our companion work [6] employs the system of linearised gravity and its enhanced red-shifted transport equations, specialised to the |a| = 0 case, to produce a new simplified proof of linear stability of the Schwarzschild solution. The full linear stability analysis in the full sub-extremal range |a| < M is deferred to future work. As already apparent from [6], our framework will allow to combine the new structure in the transport equations with the elliptic part of the system to establish a linear orbital stability result without loss of derivatives, indicating that the framework may be well suited to address nonlinear stability in the full sub-extremal range |a| < M .

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Section: Related Workmentioning

confidence: 97%

Section: • Step 2: Linearisationmentioning

confidence: 99%

Section: Nonlinear Perturbationsmentioning

confidence: 99%

Section: A Geometric Frameworkmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

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A new gauge for gravitational perturbations of Kerr spacetimes I: The linearised theory

Benomio¹

2022

Preprint

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Geometry and Analysis in Black Hole Spacetimes

Andersson

2012

Tutorials, Schools, and Workshops in the Mathematical Sciences

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Sharp Decay for Teukolsky Equation in Kerr Spacetimes

Zhang

2023

Commun. Math. Phys.

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In this work, we derive the global sharp decay, as both a lower and an upper bounds, for the spin $$\pm {\mathfrak {s}}$$ ± s components, which are solutions to the Teukolsky equation, in the black hole exterior and on the event horizon of a slowly rotating Kerr spacetime. These estimates are generalized to any subextreme Kerr background under an integrated local energy decay estimate. Our results apply to the scalar field $$({\mathfrak {s}}=0)$$ ( s = 0 ) , the Maxwell field $$({\mathfrak {s}}=1)$$ ( s = 1 ) and the linearized gravity $$({\mathfrak {s}}=2)$$ ( s = 2 ) and confirm the Price’s law decay that is conjectured to be sharp. Our analyses rely on a novel global conservation law for the Teukolsky equation, and this new approach can be applied to derive the precise asymptotics for solutions to semilinear wave equations.

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Nonlinear Radiation Gauge for Near Kerr Spacetimes

Cited by 8 publications

References 44 publications

A new gauge for gravitational perturbations of Kerr spacetimes I: The linearised theory

A new gauge for gravitational perturbations of Kerr spacetimes I: The linearised theory

Geometry and Analysis in Black Hole Spacetimes

Sharp Decay for Teukolsky Equation in Kerr Spacetimes

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