Small black holes: geometrical preliminaries

Abstract

To model a small black hole moving in an external field we follow the approach of Futamase, Hogan, and Itoh. The space‐time model of the external field provides us with a background space‐time. If the external field is a vacuum gravitational field, due to a large black hole for example, then the background space‐time is a non‐flat solution of Einstein's vacuum field equations. If we wish to model a small charged black hole moving in external gravitational and electromagnetic fields then the background space‐time is a solution of the vacuum Einstein‐Maxwell field equations. This chapter includes the charged case because it provides a relatively easily manageable example to illustrate the method of derivation of the equations of motion including radiation reaction. The world‐line or history of the black hole will be a non‐singular time‐like world‐line in the background space‐time. The field of the small black hole is introduced as a small perturbation of the background space‐time which, inter alia, is singular on this world‐line. The equations of motion of the black hole will be differential equations for this world‐line, obtained using the field equations which determine the small perturbation. The perturbed space‐time will be an approximate solution of the vacuum Einstein or Einstein‐Maxwell field equations as appropriate. To lay the foundations for this approach to equations of motion the geometry of background space-time must be studied.