Canonical reduction of two-dimensional gravity for particle dynamics

Ohta, T.; Mann, Robert B.

doi:10.1088/0264-9381/13/9/022

Cited by 35 publications

(101 citation statements)

References 24 publications

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“…We can solve the constraint equations (11) and (12) in terms of the quantities (Ψ ′ / √ γ) ′ and π ′ , since they are the only linear terms present. The generator obtained from the end point variation can then be transformed to fix the coordinate conditions.…”

Section: Canonical Reduction Of the N-body Problem In Lineal Gravitymentioning

confidence: 99%

Chaos in a three-body self-gravitating cosmological spacetime

2007

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We investigate the equal-mass 3-body system in general relativistic lineal gravity in the presence of a cosmological constant Λ. The cosmological vacuum energy introduces features that do not have a non-relativistic counterpart, inducing a competing expansion/contraction of spacetime that competes with the gravitational self-attraction of the bodies. We derive a canonical expression for the Hamiltonian of the system and discuss the numerical solution of the resulting equations of motion. As for the system with Λ = 0, we find that the structure of the phase space yields a rich variety of interesting dynamics that can be divided into three distinct regions: annulus, pretzel, and chaotic; the first two being regions of quasi-periodicity while the latter is a region of chaos. However unlike the Λ = 0 case, we find that a negative cosmological constant considerably diminishes the amount of chaos in the system, even beyond that of the Λ = 0 non-relativistic system. By contrast, a positive cosmological constant considerably enhances the amount of chaos, typically leading to KAM breakdown.

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Section: Canonical Reduction Of the N-body Problem In Lineal Gravitymentioning

confidence: 99%

Chaos in a three-body self-gravitating cosmological spacetime

2007

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“…It also reduces to the Newtonian N -body gravitational action in the nonrelativistic limit [5][6][7]. The action for the gravitational scalar-tensor formulation in 1 + 1 dimensions [1,3,4,8] coupled to N particles is, in the presence of a cosmological constant Λ …”

Section: Introductionmentioning

confidence: 99%

Canonical reduction for dilatonic gravity in3+1dimensions

et al. 2016

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We generalize the 1 + 1-dimensional gravity formalism of Ohta and Mann to 3 + 1 dimensions by developing the canonical reduction of a proposed formalism applied to a system coupled with a set of point particles. This is done via the Arnowitt-Deser-Misner method and by eliminating the resulting constraints and imposing coordinate conditions. The reduced Hamiltonian is completely determined in terms of the particles' canonical variables (coordinates, dilaton field and momenta). It is found that the equation governing the dilaton field under suitable gauge and coordinate conditions, including the absence of transverse-traceless metric components, is a logarithmic Schrödinger equation. Thus, although different, the 3 + 1 formalism retains some essential features of the earlier 1 + 1 formalism, in particular the means of obtaining a quantum theory for dilatonic gravity.

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“…Exact solution for 2 point sources on a line, 3 point sources on a line and N point sources on a circle have been found. For 3 point sources the system is chaotic and is a simple model where to study relativistic chaos [20] besides the Kasner-Misner mixmaster chaotic cosmological models. Before doing so, we shall model the mass distribution by a smeared delta function ρ [22], by starting with the following field equations associated with the signature (+, −, −, −)…”

Section: 1mentioning

confidence: 99%

The Euclidean gravitational action as black hole entropy, singularities, and spacetime voids

Castro

2008

Journal of Mathematical Physics

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We argue why the static spherically symmetric (SSS) vacuum solutions of Einstein's equations described by the textbook Hilbert metric g µν (r) is not diffeomorphic to the metric g µν (|r|) corresponding to the gravitational field of a point mass delta function source at r = 0. By choosing a judicious radial function R(r) = r + 2G|M |Θ(r) involving the Heaviside step function, one has the correct boundary condition R(r = 0) = 0 , while displacing the horizon from r = 2G|M | to a location arbitrarily close to r = 0 as one desires, r h → 0, where stringy geometry and quantum gravitational effects begin to take place. We solve the field equations due to a delta function point mass source at r = 0, and show that the Euclidean gravitational action (inh units) is precisely equal to the black hole entropy (in Planck area units). This result holds in any dimensions D ≥ 3 . In the Reissner-Nordsrom (massive-charged) and Kerr-Newman black hole case (massive-rotating-charged) we show that the Euclidean action in a bulk domain bounded by the inner and outer horizons is the same as the black hole entropy. When one smears out the point-mass and point-charge delta function distributions by a Gaussian distribution, the areaentropy relation is modified. We postulate why these modifications should furnish the logarithmic corrections (and higher inverse powers of the area) to the entropy of these smeared Black Holes. To finalize, we analyse the Bars-Witten stringy black hole in 1 + 1 dim and its relation to the maximal acceleration principle in phase spaces and Finsler geometries.

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Canonical reduction of two-dimensional gravity for particle dynamics

Cited by 35 publications

References 24 publications

Chaos in a three-body self-gravitating cosmological spacetime

Chaos in a three-body self-gravitating cosmological spacetime

Canonical reduction for dilatonic gravity in3+1dimensions

The Euclidean gravitational action as black hole entropy, singularities, and spacetime voids

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