Optimal Time Travel in the Gödel Universe

Abstract

Using the theory of optimal rocket trajectories in general relativity, recently developed in [HN11], we present a candidate for the minimum total integrated acceleration closed timelike curve in the Gödel universe, and give evidence for its minimality. The total integrated acceleration of this curve is lower than Malament's conjectured value [Mal84], as was already implicit in the work of Manchak [Man11]; however, Malament's conjecture does seem to hold for periodic closed timelike curves.Partially supported by FCT (Portugal). 1 We use the conventions of [MTW73], including geometrized units (c = G = 1).

The Rocket Problem in General Relativity

“…. , u m , τ ), 2 We take piecewise smooth functions on [τ 0 , τ 1 ] to be given by restrictions of smooth functions defined on R to the subintervals of a partition of [τ 0 , τ 1 ]; in particular, piecewise smooth functions and all their derivatives have one-sided limits at all points. 3 It is possible to formulate the Mayer problem with much lower regularity; here, for simplicity, we consider only (piecewise) smooth functions, as is usual in differential geometry.…”

“…), it turns out that the geometric perspective of general relativity brings fresh insights into this classical application of control theory -for instance, the relation of the primer equation with the Jacobi equation (Theorem 4.1), or the fact that ignorable coordinates restrict variations to geodesics with the same Killing conserved quantities (Theorem 8.1). Moreover, this theory can be used to address theoretical issues in general relativity [2].…”

Essay Review:Topics in the Foundations of General Relativity and Newtonian Gravitation Theory

“…In fact, there is a lower bound on the total acceleration needed to complete such a journey (Malament 1985). Accordingly, one recent area of study concerns optimal time travel in the space-time (Manchak 2011;Natário 2012).…”