Exact solution for the metric and the motion of two bodies in (1+1)-dimensional gravity

Abstract

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.1

Canonical reduction for dilatonic gravity in3+1dimensions

“…However, the problematic issue for General Relativity (GRT) is that the Einstein tensor is topologically trivial in 1 + 1 dimensions and cannot yield the correct Newtonian limit. Through the addition of an auxiliary field corresponding to a particle known as a dilaton, this problem can be addressed and yields a successful manybody gravity theory [1][2][3].…”

“…Two of the oldest and most notoriously vexing problems in gravitational theory (which are possibly related to each other) are (i) obtaining a quantum gravity theory which is renormalizable and therefore amenable to meaningful physical predictions, and (ii) determining the (selfconsistent) motion of N bodies and the resultant metric they collectively produce under their mutual gravita-tional influence [1]. In the latter case, lower-dimensional theories such as 1 + 1 dimensional gravity, (meaning one spatial dimension and one time dimension) have been examined in large part because problems in quantum gravity become much more mathematically tractable in this context.…”

Nordström gravity coupled to point particles in (1+1) dimensions

“…We show that the theory is formally equivalent to scalar field theory in flat spacetime, and we use this equivalence to solve the initial value problem for the system. The dynamics of systems of point particles is of considerable interest in general relativity [13][14][15][16], and has also been studied in the context of (1 þ 1)-dimensional Nordström gravity [4]. In (1 þ 1)-dimensional Nordström gravity there is no gravitational radiation, but the gravitational interaction of the particles still gives rise to nontrivial particle dynamics.…”

“…This theory was proposed as a model of general relativity in the 1980s, and has been investigated by a number of authors [2][3][4][5]. Features of the model that have been studied include black hole solutions [6], gravitational collapse [7], and chaotic particle dynamics [8,9].…”

Wien peaks and the Lambert W function

“…4 Secondly, while many examples from physics where the Lambert W function arises have now been found (see, e.g., Refs. [18,[20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]), the problem of determining closed-form expressions for the Wien peaks provides what is undoubtedly the simplest illustration of the use of this function in physics.…”