A time machine for free fall into the past

Abstract

Inspired by some recent works of Tippett-Tsang and Mallary-Khanna-Price, we present a new spacetime model containing closed timelike curves (CTCs). This model is obtained postulating an ad hoc Lorentzian metric on R 4 , which differs from the Minkowski metric only inside a spacetime region bounded by two concentric tori. The resulting spacetime is topologically trivial, free of curvature singularities and is both time and space orientable; besides, the inner region enclosed by the smaller torus is flat and displays geodesic CTCs. Our model shares some similarities with the time machine of Ori and Soen but it has the advantage of a higher symmetry in the metric, allowing for the explicit computation of a class of geodesics. The most remarkable feature emerging from this computation is the presence of future-oriented timelike geodesics starting from a point in the outer Minkowskian region, moving to the inner spacetime region with CTCs, and then returning to the initial spatial position at an earlier time; this means that time travel to the past can be performed by free fall across our time machine. The amount of time travelled into the past is determined quantitatively; this amount can be made arbitrarily large keeping non-large the proper duration of the travel. An important drawback of the model is the violation of the classical energy conditions, a common feature of many time machines. Other problems emerge from our computations of the required (negative) energy densities and of the tidal accelerations; these are small only if the time machine is gigantic.

Some Remarks on a New Exotic Spacetime for Time Travel by Free Fall

“…A]; see also the forthcoming Eq. Paying due attention to the usual coordinate singularity at ρ = 0, it can be checked that the line element (2.2) does indeed define a non-degenerate, symmetric bilinear form of constant signature (3, 1), i.e., a Lorentzian metric g on the whole spacetime manifold R 4 (see [10] for more details). Furthermore, on account of the presumed regularity features of X , the said metric g is granted by construction to be of class C k , with k 2; this suffices to infer that the corresponding Riemann curvature tensor Riem g is of class C k−2 (whence, at least continuous) and, in particular, free of singularities, alike all the associated curvature invariants.…”

“…The spacetime under analysis exhibits a number of symmetries; hereafter these symmetries are discussed, together with some of their most relevant implications (see [10,Sec. 4] for more details on this theme).…”

“…Consider the fundamental observers of Section 3, with 4-velocity coinciding with the timelike element E (0) of the tetrad (3). In [10] it is shown that the energy density measured by such observers at a point p ∈ T of coordinate (t, ϕ, ρ, z) can be expressed as…”

“…Several spacetime models sharing the above exotic features are known nowadays; see [22,23,24,39] for comprehensive reviews. In [10] a four-fold classification of the existing literature was suggested: First class. Exact solutions of the Einstein's equations, where closed timelike curves (CTCs) are produced by strong angular momenta or naked singularities.…”

Spinning up a Time Machine

“…Exceptions are recent ad hoc metrics such as in Ref. [9]. As a result research into quantum objects interacting with CTCs has used very simple toy models which simply model the CTCs with unusual boundary conditions [11,12].…”