New spinorial approach to mass inequalities for black holes in general relativity

Kopiński, Jarosław; Kroon, Juan A. Valiente

doi:10.1103/physrevd.103.024057

“…However, the use of the skew symmetric tensor J i j in the form of the perturbation does not seem strictly necessary and could likely be removed. Another possibility would be to investigate further a spin geometry approach to proving the Penrose inequality, although this would also involve sophisticated estimates from PDE theory [19][20][21].…”

Section: Discussionmentioning

confidence: 99%

A PDE proof of the Penrose inequality for perturbations of Schwarzschild initial data

Williams

¹

2022

Class. Quantum Grav.

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The classical Penrose inequality relates the mass of an asymptotically flat spacetime to the total area of its black holes. We present a PDE-based proof of the Penrose inequality for a special class of perturbations of Schwarzschild initial data. The proof is based around linearising solutions to a version of the Jang equation which is modified to deal with warped product metrics. We discuss the possibility of generalising the argument to a wider class of perturbations.

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“…The subsequent analysis will further refinement of a 1 + 1 + 2 spinorial split. This is analogous to the 1 + 3 split when an additional direction ρ a is singled out-see [15,21,23,24,30] for further discussion on the space-spinor formalism, the 1 + 3 split, the 1 + 1 + 2 split and applications. The basic elements of this formalism will be discussed on a general manifold with metric (M, g) and towards the end of this section, some results specific for Friedrich's representation of the spatial infinity of the Minkowski spacetime (M, η) will be given.…”

Section: The Space-spinor Formalismmentioning

confidence: 91%

“…Further discussion on the spinorial 1 + 1 + 2 split can be found in [15,23,24]. Particularising the previous general discussion to Friedrich's representation of the spatial infinity of the Minkowski spacetime and using the lift to the fibre space of the null frame (9a)-(9d) and exploiting (23) one can directly read…”

Section: The Space-spinor Decomposition Of the Connectionmentioning

confidence: 96%

Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

Gasperin¹,

Kroon²

2021

Class. Quantum Grav.

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Linear zero-rest-mass fields generically develop logarithmic singularities at the critical sets where spatial infinity meets null infinity. Friedrich's representation of spatial infinity is ideally suited to study this phenomenon. These logarithmic singularities are an obstruction to the smoothness of the zero-rest-mass field at null infinity and, in particular, to peeling. In the case of the spin-2 field it has been shown that these logarithmic singularities can be precluded if the initial data for the field satisfies a certain regularity condition involving the vanishing, at spatial infinity, of a certain spinor (the linearised Cotton spinor) and its totally symmetrised derivatives. In this article we investigate the relation between this regularity condition and the staticity of the spin-2 field. It is shown that while any static spin-2 field satisfies the regularity condition, not every solution satisfying the regularity condition is static. This result is in contrast with what happens in the case of General Relativity where staticity in a neighbourhood of spatial infinity and the smoothness of the field at future and past null infinities are much more closely related.

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“…The subsequent analysis will further refinement of a 1 + 1 + 2 spinorial split. This is analogous to the 1 + 3 split when an additional direction ρ a is singled out -see [32,21,24,23,16] for further discussion on the space-spinor formalism, the 1 + 3 split, the 1 + 1 + 2 split and applications. The basic elements of this formalism will be discussed on a general manifold with metric (M, g) and towards the end of this section, some results specific for Friedrich's representation of the spatial infinity of the Minkowski spacetime (M, η) will be given.…”

Section: The Space-spinor Formalismmentioning

confidence: 98%

Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

Gasperin,

Kroon

2021

Preprint

Self Cite

0

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Linear zero-rest-mass fields generically develop logarithmic singularities at the critical sets where spatial infinity meets null infinity. Friedrich's representation of spatial infinity is ideally suited to study this phenomenon. These logarithmic singularities are an obstruction to the smoothness of the zero-rest-mass field at null infinity and, in particular, to peeling. In the case of the spin-2 field it has been shown that these logarithmic singularities can be precluded if the initial data for the field satisfies a certain regularity condition involving the vanishing, at spatial infinity, of a certain spinor (the linearised Cotton spinor) and its totally symmetrised derivatives. In this article we investigate the relation between this regularity condition and the staticity of the spin-2 field. It is shown that while any static spin-2 field satisfies the regularity condition, not every solution satisfying the regularity condition is static. This result is in contrast with what happens in the case of General Relativity where staticity in a neighbourhood of spatial infinity and the smoothness of the field at future and past null infinities are much more closely related.

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New spinorial approach to mass inequalities for black holes in general relativity

Cited by 4 publications

References 32 publications

A PDE proof of the Penrose inequality for perturbations of Schwarzschild initial data

A PDE proof of the Penrose inequality for perturbations of Schwarzschild initial data

Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

Staticity and regularity for zero rest-mass fields near spatial infinity on flat spacetime

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