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. 1email: [email protected] 2email: [email protected] The problem of motion is a notoriously difficult one in gravitational theory. Although approximation techniques exist [1], in general there is no exact solution to the problem of the motion of N bodies each interacting under their mutual gravitational influence, except in the case N = 2 for Newtonian gravity, or in (2 + 1) dimensions, where the absence of a static gravitational potential allows one to generalize the static 2-body metric to that of two bodies moving with any speed [2]. We present here an exact solution to problem of the relativistic motion of 2 point masses under gravity in (1 + 1) dimensions. The dimensionality necessitates that the gravitational theory we choose is a dilaton theory of gravity; however our choice of dilatonic gravity is such that the dilaton decouples from the classical equations of motion [3]. Consequently 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 [4]. Unlike the (2 +1) dimensional case, 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. We can thus view the whole structure of the theory from the weak field to the strong field limits. We begin with the N -body gravitational action I = ∫ dx [ 1 2κ √ −g { ΨR+ 1 2 g∇μΨ∇νΨ }