Modeling the dynamics of black holes through balanced equations of motion in the null gauge

Abstract

We develop further the general framework for modeling the dynamics of the motion of black holes, presented in [E. Gallo and O. M. Moreschi, Modeling the dynamics of black holes through balanced equations of motion, Int. J. Geom. Meth. Mod. Phys. 16(3) (2019) 1950034], by employing a convenient null gauge, in general relativity, for the construction of the balanced equations of motion. This null gauge has the property that the asymptotic structure is intimately related to the interior one; in particular there is a strong connection between the field equations and the balanced equations of motion. Our work is very related to what we have called "Robinson-Trautman (RT) geometries" [S. Dain, O. M. Moreschi and R. J. Gleiser, Photon rockets and the Robinson-Trautman geometries, Class. Quantum Grav. 13(5) (1996) 1155-1160] in the past. These geometries are used in the sense of the general framework, we have presented in [E. Gallo and O. M. Moreschi, Modeling the dynamics of black holes through balanced equations of motion, Int. J. Geom. Meth. Mod. Phys. 16(3) (2019) 1950034]. We present the balanced equations of motion in second order of the acceleration. We solve the required components of the field equation at their respective required orders, G2 and G3, in terms of the gravitational constant. We indicate how this approach can be extended to higher orders.