Optimal time travel in the Gödel universe

Abstract

Using the theory of optimal rocket trajectories in general relativity, recently developed in Henriques and Natário (2011), 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 (Malament 1984), as was already implicit in the work of Manchak (Gen. Relativ. Gravit. 51-60, 2011); however, Malament's conjecture does seem to hold for periodic closed timelike curves.

Causality violation without time-travel: closed lightlike paths in Gödel’s universe

“…But there is a heavy fuel cost (i.e. a large contribution to the total integrated acceleration) at each junction of successive segments: see for example Natario's calculation [10], which shows a contribution ∆T A 3.6158 to the total integrated acceleration to provide the boost required at the end point of his closed timelike curve to make the CTC periodic: i.e. to make the tangent continuous at the initial and final points of the curve.…”

“…Manchak [9] subsequently showed that Malament's conjectured lower bound can be violated. Natario [10] presented a candidate for the optimal CTC -i.e. that with the least T A: this has T A 24.9947.…”

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.…”

“…), 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].…”

Null geodesics and wave front singularities in the Gödel space–time

“…Németi et al (2008) [9] provides excellent visualizations of the light cones, closed time-like geodesics and other features in Gödel type rotating universes. Other authors, for example Gleiser et al (2006) [10] and Natário (2012) [11], have considered properties of the closed time-like geodesics in these spacetimes, and Slobodov (2008) has shown how changing the topology can be used to remove them [12].…”