Time machines with the compactly determined Cauchy horizon

TL;DR

This paper explores spacetimes with compactly determined Cauchy horizons, revealing that such regions cannot be maximally extended and may produce observable high-energy photon phenomena near the horizon.

Contribution

It introduces the concept of compactly determined Cauchy horizons and demonstrates their implications for spacetime extension and potential observable effects.

Findings

No globally hyperbolic maximal extension exists for these spacetimes.

Null geodesics near the horizon can lead to infinite energy accumulation.

Potential observable consequences of high-energy photon flux are suggested.

Abstract

The building of a time machine, if possible at all, requires the relevant regions of spacetime to be compact (that is, physically speaking, free from sources of unpredictability such as infinities and singularities). Motivated by this argument we consider the spacetimes with the compactly determined Cauchy horizons (CDCHs), the defining property of which is the compactness of $\overline{J^-(\EuScript U)}\cap J^+(\EuScript S_0)$, where $\EuScript U$ is an open subset of the Cauchy horizon and $\EuScript S_0$ is a Cauchy surface of the initial globally hyperbolic region $\ingh$. The following two facts are established: 1) $\ingh$ has no globally hyperbolic maximal extension. This means that by shaping appropriately a precompact portion of a globally hyperbolic region one can \emph{force} the Universe to produce either a closed causal curve, or a quasiregular singularity, whichever it abhors less; 2) Before a CDCH is formed a null geodesic appears which infinitely approaches the horizon returning again and again in the same --- arbitrarily small --- region. The energy of the photon moving on such a geodesic increases with each passage, or at least falls insufficiently fast. As a result, an observer located in the mentioned region would see a bunch of photons passing through his laboratory with the arbitrarily large total energy. We speculate that this phenomenon may have observable consequences.