Recent results on the maximization of the charged-particle action I(x_0,x_1) in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of I(x_0,x_1) over a given causal homotopy class C of curves connecting two causally related events x_0 ? x_1. Action I(x_0,x_1) is proved to admit a maximum on C, and also one in the adherence of each timelike homotopy class C. Moreover, the maximum ? on C is timelike if C contains a timelike curve (and the degree of differentiability of all the elements is at least C^2). In particular, this last result yields a completeAvez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x_0 << x_1 can be connected by means of a timelike solution of the Lorentz force equation corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m different from 0) in the non-exact case. As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied.
Connecting solutions of the Lorentz force equation do exist
MINGUZZI, ETTORE;
2006-01-01
Abstract
Recent results on the maximization of the charged-particle action I(x_0,x_1) in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of I(x_0,x_1) over a given causal homotopy class C of curves connecting two causally related events x_0 ? x_1. Action I(x_0,x_1) is proved to admit a maximum on C, and also one in the adherence of each timelike homotopy class C. Moreover, the maximum ? on C is timelike if C contains a timelike curve (and the degree of differentiability of all the elements is at least C^2). In particular, this last result yields a completeAvez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x_0 << x_1 can be connected by means of a timelike solution of the Lorentz force equation corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m different from 0) in the non-exact case. As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
