Gauge-invariant Hamiltonian formalism for spherically symmetric gravitating shells

Hájı́ček, P.; Bičák, Jiřı́

doi:10.1103/physrevd.56.4706

Cited by 32 publications

(51 citation statements)

References 19 publications

(37 reference statements)

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“…Attempts to derive a Hamiltonian formalism for such shells are for example Refs. [9], [6], [10], [11], [5] and [12]. The Hamiltonian (or super-Hamiltonian) is either guessed directly from equations of motion (Refs.…”

Section: Introductionmentioning

confidence: 99%

Spherically symmetric gravitating shell as a reparametrization-invariant system

Hájı́ček

1998

Phys. Rev. D

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The subject of this paper is spherically symmetric thin shells made of a baryotropic ideal fluid which moves under the influence of its own gravitational field as well as that of a central black hole; the cosmological constant is assumed to be zero. The general super-Hamiltonian derived in a previous paper is rewritten for this spherically symmetric special case. The dependence of the resulting action on the gravitational variables is trivialized by a transformation due to Kuchař. The resulting variational principle depends only on shell variables, is reparametrization invariant, and includes both first-and second-class constraints. Several equivalent forms of the constrained system are written down. The exclusion of the second-class constraints leads to a super-Hamiltonian which appears to overlap with that by Ansoldi et al. in a quarter of the phase space. As the Kuchař variables are singular at the horizons of both Schwarzschild spacetimes inside and outside the shell, the dynamics is first well defined only inside of 16 disjoint sectors. The 16 sectors are, however, shown to be contained in a single, connected symplectic manifold and the constraints are extended to this manifold by continuity. Poisson brackets between no two independent spacetime coordinates of the shell vanish at any intersection of two horizons.

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Section: Introductionmentioning

confidence: 99%

Spherically symmetric gravitating shell as a reparametrization-invariant system

Hájı́ček

1998

Phys. Rev. D

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“…Thus the question of how to perform and study the canonical quantization remains open. An important step in this direction is provided by the work of Kuchař and Hájíček [3,7]. In this paper the authors obtain the classical equations of motion of the shell as the exact finite dimensional reduction of the full relativistic Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

“…In order to make our presentation as self contained as possible, as well as to fix notation and conventions, we recall briefly some of the details of the Kuchař-Hájíček [3,7] construction of the canonical dynamics of gravitational shells. The geometry of the shell is described by…”

Section: Formulation Of the Problemmentioning

confidence: 99%

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Quantum collapse of a small dust shell

et al. 2002

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The full quantum mechanical collapse of a small relativistic dust shell is studied analytically, asymptotically and numerically starting from the exact finite dimensional classical reduced Hamiltonian recently derived by Hájíček and Kuchař. The formulation of the quantum mechanics encounters two problems. The first is the multivalued nature of the Hamiltonian and the second is the construction of an appropriate self adjoint momentum operator in the space of the shell motion which is confined to a half line. The first problem is solved by identifying and neglecting orbits of small action in order to obtain a single valued Hamiltonian. The second problem is solved by introducing an appropriate lapse function. The resulting quantum mechanics is then studied by means of analytical and numerical techniques. We find that the region of total collapse has very small probability. We also find that the solution concentrates around the classical Schwarzschild radius. The present work obtains from first principles a quantum mechanics for the shell and provides numerical solutions, whose behavior is explained by a detailed WKB analysis for a wide class of collapsing shells.

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“…The super-Hamiltonian of Ref. [8] is simple and amenable to the existing gauge invariant quantization methods (like e.g., Refs. [17] and [18]).…”

Section: Introductionmentioning

confidence: 99%

Black hole interacting with matter as a simple dynamical system

Hájı́ček

1999

Journal of Mathematical Physics

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Recently, a variational principle has been derived from Einstein-Hilbert and a matter Lagrangian for the spherically symmetric system of a dust shell and a black hole. The so-called physical region of the phase space, which contains all physically meaningful states of the system defined by the variational principle, is specified; it has a complicated boundary. The principle is then transformed to new variables that remove some problems of the original formalism: the whole phase space is covered (in particular, the variables are regular at all horizons), the constraint has a polynomial form, and the constraint equation is uniquely solvable for two of the three conserved momenta. The solutions for the momenta are written down explicitly. The symmetry group of the system is studied. The equations of motion are derived from the transformed principle and are shown to be equivalent to the previous ones. Some lowerdimensional systems are constructed by exclusion of cyclic variables, and some of their properties are found.

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Gauge-invariant Hamiltonian formalism for spherically symmetric gravitating shells

Cited by 32 publications

References 19 publications

Spherically symmetric gravitating shell as a reparametrization-invariant system

Spherically symmetric gravitating shell as a reparametrization-invariant system

Quantum collapse of a small dust shell

Black hole interacting with matter as a simple dynamical system

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