Exact Solution for Relativistic Two-Body Motion in Dilaton Gravity

We present an exact solution to the problem of the relativistic motion of 2 point masses in $(1+1)$ dimensional dilaton gravity. The motion of the bodies is governed entirely by their mutual gravitational influence, and the spacetime metric is likewise fully determined by their stress-energy. A Newtonian limit exists, and there is a static gravitational potential. Our solution gives the exact Hamiltonian to infinite order in the gravitational coupling constant.Comment: 6 pages, latex, 3 figure

Exact Charged 2-Body Motion and the Static Balance Condition in Lineal Gravity

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.Comment: latex,7 pages, 2 figure

Self-replication and splitting of domain patterns in reaction-diffusion systems with fast inhibitor

An asymptotic equation of motion for the pattern interface in the domain-forming reaction-diffusion systems is derived. The free boundary problem is reduced to the universal equation of non-local contour dynamics in two dimensions in the parameter region where a pattern is not far from the points of the transverse instabilities of its walls. The contour dynamics is studied numerically for the reaction-diffusion system of the FitzHugh-Nagumo type. It is shown that in the asymptotic limit the transverse instability of the localized domains leads to their splitting and formation of the multidomain pattern rather than fingering and formation of the labyrinthine pattern.Comment: 9 pages (ReVTeX), 5 figures (postscript). To be published in Phys. Rev.

Exact Solutions of Relativistic Two-Body Motion in Lineal Gravity

We develop the canonical formalism for a system of $N$ bodies in lineal gravity and obtain exact solutions to the equations of motion for N=2. The determining equation of the Hamiltonian is derived in the form of a transcendental equation, which leads to the exact Hamiltonian to infinite order of the gravitational coupling constant. In the equal mass case explicit expressions of the trajectories of the particles are given as the functions of the proper time, which show characteristic features of the motion depending on the strength of gravity (mass) and the magnitude and sign of the cosmological constant. As expected, we find that a positive cosmological constant has a repulsive effect on the motion, while a negative one has an attractive effect. However, some surprising features emerge that are absent for vanishing cosmological constant. For a certain range of the negative cosmological constant the motion shows a double maximum behavior as a combined result of an induced momentum-dependent cosmological potential and the gravitational attraction between the particles. For a positive cosmological constant, not only bounded motions but also unbounded ones are realized. The change of the metric along the movement of the particles is also exactly derived.Comment: 37 pages, Latex, 24 figure

Exact Solution for the Metric and the Motion of Two Bodies in (1+1) Dimensional Gravity

We present the exact solution of two-body motion in (1+1) dimensional dilaton gravity by solving the constraint equations in the canonical formalism. The determining equation of the Hamiltonian is derived in a transcendental form and the Hamiltonian is expressed for the system of two identical particles in terms of the Lambert $W$ function. The $W$ function has two real branches which join smoothly onto each other and the Hamiltonian on the principal branch reduces to the Newtonian limit for small coupling constant. On the other branch the Hamiltonian yields a new set of motions which can not be understood as relativistically correcting the Newtonian motion. The explicit trajectory in the phase space $(r, p)$ is illustrated for various values of the energy. The analysis is extended to the case of unequal masses. The full expression of metric tensor is given and the consistency between the solution of the metric and the equations of motion is rigorously proved.Comment: 34 pages, LaTeX, 16 figure

Chaos in an Exact Relativistic 3-body Self-Gravitating System

We consider the problem of three body motion for a relativistic one-dimensional self-gravitating system. After describing the canonical decomposition of the action, we find an exact expression for the 3-body Hamiltonian, implicitly determined in terms of the four coordinate and momentum degrees of freedom in the system. Non-relativistically these degrees of freedom can be rewritten in terms of a single particle moving in a two-dimensional hexagonal well. We find the exact relativistic generalization of this potential, along with its post-Newtonian approximation. We then specialize to the equal mass case and numerically solve the equations of motion that follow from the Hamiltonian. Working in hexagonal-well coordinates, we obtaining orbits in both the hexagonal and 3-body representations of the system, and plot the Poincare sections as a function of the relativistic energy parameter $\eta$. We find two broad categories of periodic and quasi-periodic motions that we refer to as the annulus and pretzel patterns, as well as a set of chaotic motions that appear in the region of phase-space between these two types. Despite the high degree of non-linearity in the relativistic system, we find that the the global structure of its phase space remains qualitatively the same as its non-relativisitic counterpart for all values of $\eta$ that we could study. However the relativistic system has a weaker symmetry and so its Poincare section develops an asymmetric distortion that increases with increasing $\eta$. For the post-Newtonian system we find that it experiences a KAM breakdown for $\eta \simeq 0.26$: above which the near integrable regions degenerate into chaos.Comment: latex, 65 pages, 36 figures, high-resolution figures available upon reques

Chaos in a Relativistic 3-body Self-Gravitating System

We consider the 3-body problem in relativistic lineal gravity and obtain an exact expression for its Hamiltonian and equations of motion. While general-relativistic effects yield more tightly-bound orbits of higher frequency compared to their non-relativistic counterparts, as energy increases we find in the equal-mass case no evidence for either global chaos or a breakdown from regular to chaotic motion, despite the high degree of non-linearity in the system. We find numerical evidence for a countably infinite class of non-chaotic orbits, yielding a fractal structure in the outer regions of the Poincare plot.Comment: 9 pages, LaTex, 3 figures, final version to appear in Phys. Rev. Let

Exact Relativistic Two-Body Motion in Lineal Gravity

We consider the N-body problem in (1+1) dimensional lineal gravity. For 2 point masses (N=2) we obtain an exact solution for the relativistic motion. In the equal mass case we obtain an explicit expression for their proper separation as a function of their mutual proper time. Our solution gives the exact Hamiltonian to infinite order in the gravitational coupling constant.Comment: latex, 11 pages, 2 figures, final version to appear in Phys. Rev. Let