Exact charged two-body motion and the static balance condition in lineal gravity

Mann, Robert B.; Ohta, T.

doi:10.1088/0264-9381/17/19/307

Cited by 10 publications

(28 citation statements)

References 23 publications

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“…The relativistic Hamiltonian, determined by Eq. (12), is invariant under the symmetry ͑p i , ⑀͒ → ͑−p i ,−⑀͒. Contrary to the Newtonian case, this relativistic symmetry is not manifest in our surface of section.…”

Section: Global Structure Of Phase Spacementioning

confidence: 74%

“…Since we were unable to find a closed form solution to either the relativistic determining equation (12) or the equations of motion (16) and (17), it was necessary to employ numerical techniques to study the motion. Using a MATLAB integration routine (ode15s) we were able to solve the equations of motion in the N, pN, and R systems.…”

Section: Methods For Solving the Equations Of Motionmentioning

confidence: 99%

“…Expressions for Eqs. (12), (20), and (24) in terms of the new coordinates are given in Ref. [17] for the case when m 1 = m 2 = m 3 .…”

Section: ͑28͒mentioning

confidence: 99%

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Three-body dynamics in a(1+1)-dimensional relativistic self-gravitating system

Malecki

Mann

2004

Phys. Rev. E

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The results of our study of the motion of a three particle, self-gravitating system in general relativistic lineal gravity is presented for an arbitrary ratio of the particle masses. We derive a canonical expression for the Hamiltonian of the system and discuss the numerical solution of the resulting equations of motion. This solution is compared to the corresponding nonrelativistic and post-Newtonian approximation solutions so that the dynamics of the fully relativistic system can be interpreted as a correction to the one-dimensional Newtonian self-gravitating system. We find that the structure of the phase space of each of these systems yields a large variety of interesting dynamics that can be divided into three distinct regions: annulus, pretzel, and chaotic; the first two being regions of quasiperiodicity while the latter is a region of chaos. By changing the relative masses of the three particles we find that the relative sizes of these three phase space regions changes, and that this deformation can be interpreted physically in terms of the gravitational interactions of the particles. Furthermore, we find that many of the interesting characteristics found in the case where all of the particles share the same mass also appear in our more general study. We find that there are additional regions of chaos in the unequal mass system which are not present in the equal mass case. We compare these results to those found in similar systems.

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Section: Global Structure Of Phase Spacementioning

confidence: 74%

Section: Methods For Solving the Equations Of Motionmentioning

confidence: 99%

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Three-body dynamics in a(1+1)-dimensional relativistic self-gravitating system

Malecki

Mann

2004

Phys. Rev. E

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“…An obvious thing to try is to incorporate additional couplings to electromagnetism and a cosmological constant. A study of the quantization of the Hamiltonian (20) should also afford interesting insight into the nature of (2+1) quantum gravity coupled to matter.…”

Section: Resultsmentioning

confidence: 99%

“…For example, the general form of the solution has been obtained for lineal gravity [15] for arbitrary N , after which a variety of exact solutions for N = 2 were obtained in various contexts that include both charge and cosmological expansion/contraction [16,17,18,19], by investigating the Hamiltonian of such a system through canonical reduction. Several interesting exact solutions to the N -body equilibrium problem [20,21,22,23] in (1+1) dimensions have also been obtained.In this paper, we follow a similar approach in (2+1) gravity to obtain canonical equations of motion to analyze a two-body system. The analysis of the N -body problem in (2+1) dimensions also has an interesting history, beginning with construction of a spinning point-particle solution [4] and then a consideration of the quantum scattering problem [24].…”

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confidence: 98%

Analysis of two-particle systems in 2 + 1 gravity through Hamiltonian dynamics

Yale

Mann

Ohta

2010

Class. Quantum Grav.

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We study the dynamics of particles coupled to gravity in (2 + 1) dimensions. Using the ADM formalism, we derive the general Hamiltonian for an N -body system and analyze the dynamics of a two-particle system. Nonlinear terms are found up to second order in κ in the general case, and to every order in the quasi-static limit. IntroductionThe study of 2+1 dimensional gravity [1,2,3,4,5,6,7,8,9] has been around for several decades. Its attraction was rooted in its mathematical simplicity, which afforded some insight into the dynamics and behaviour of general relativistic gravity. Indeed, in this framework, the Einstein tensor can be expressed in terms of the curvature tensor, such that vacuum must be locally flat. This framework became even more popular after the discovery of the BTZ black hole [10,11] and is now a standard tool practitioners of quantum gravity employ in understanding their subject [12,13,14].We are concerned here with the problem of N -body dynamics, which is a long-standing one in relativity due to its notorious difficulty. In lower dimensional settings, this problem becomes much simpler. For example, the general form of the solution has been obtained for lineal gravity [15] for arbitrary N , after which a variety of exact solutions for N = 2 were obtained in various contexts that include both charge and cosmological expansion/contraction [16,17,18,19], by investigating the Hamiltonian of such a system through canonical reduction. Several interesting exact solutions to the N -body equilibrium problem [20,21,22,23] in (1+1) dimensions have also been obtained.In this paper, we follow a similar approach in (2+1) gravity to obtain canonical equations of motion to analyze a two-body system. The analysis of the N -body problem in (2+1) dimensions also has an interesting history, beginning with construction of a spinning point-particle solution [4] and then a consideration of the quantum scattering problem [24]. Further developments came upon realizing that the problem could be analyzed from a topological perspective [25], and an implicit solution for the metric and the motion of N interacting particles was obtained [26]. Based on a mapping from multivalued Minkowskian coordinates to single-valued ones, it becomes explicit for two particles with any speed and for any number of particles with small speed. It is possible to show that the collision of point particles in 2+1 AdS spacetime can result in the formation of a black hole [27].The connection between these approaches and more traditional canonical methods as employed in (1+1) [15] and (3+1) dimensions [28] has not been explicated. In this paper, we address this issue. We begin by finding the total action corresponding to the system, which consists of the Einstein-Hilbert action for the field as well as a term corresponding to coupling gravity to matter. A variational approach will then lead to coordinate conditions and constraints, which will in turn produce the total Hamiltonian as an expansion in powers of the gravitational coupling κ. The Hamilton...

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Particles on a circle in canonical lineal gravity

Mann¹

2001

Class. Quantum Grav.

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A description of the canonical formulation of lineal gravity minimally coupled to N point particles in a circular topology is given. The Hamiltonian is found to be equal to the time-rate of change of the extrinsic curvature multiplied by the proper circumference of the circle. Exact solutions for pure gravity and for gravity coupled to a single particle are obtained. The presence of a single particle significantly modifies the spacetime evolution by either slowing down or reversing the cosmological expansion of the circle, depending upon the choice of parameters.

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Exact charged two-body motion and the static balance condition in lineal gravity

Abstract: We find an exact solution to the charged 2-body problem in (1 + 1) dimensional lineal gravity which provides the first example of a relativistic system that generalizes the Majumdar-Papapetrou condition for static balance.1

Cited by 10 publications

References 23 publications

Three-body dynamics in a(1+1)-dimensional relativistic self-gravitating system

Three-body dynamics in a(1+1)-dimensional relativistic self-gravitating system

Analysis of two-particle systems in 2 + 1 gravity through Hamiltonian dynamics

Particles on a circle in canonical lineal gravity

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