Van Vleck determinants: Traversable wormhole spacetimes.

Abstract

Prompted by various questions regarding the putative existence, stability, and chronological properties of traversable wormholes, a number of authors have presented calculations of the renormalized stress-energy tensor in wormhole spacetimes. In particular, the use of point-splitting techniques leads to expressions that contain the van Vleck determinant as a common prefactor. Recent technical advances permit one to undertake extensive computations of the van Vleck determinant in traversable wormhole spacetimes, at least in the short-throat flat-space approximation. This paper presents several such computations for various model spacetimes. Implications with regard to Hawking's chronology protection conjecture are discussed. In particular, any attempt to transform a single isolated wormhole into a time machine results in large vacuum polarization effects. These vacuum polarization effects are sufficient to disrupt the internal structure of the wormhole long before the onset of Planck scale physics, and before the onset of time travel. Thus for isolated wormholes, vacuum polarization effects are sufficient to enforce Hawking's chronology protection conjecture. On the other hand, it is possible to conceive of a putative time machine built out of two or more wormholes, each of which taken in isolation is not itself a time machine. Such ``Roman configurations'' are much more subtle to analyze. For ``reasonable'' configurations (traversable by humans) the vacuum polarization effects in such multiple wormhole putative time machines become large long before the onset of Planck scale physics. The disruption scale for would-be ``traversable time machines'' is well above the Planck length. On the other hand, for some particularly bizarre configurations (not traversable by humans) the vacuum polarization effects can be arranged to be arbitrarily small at the onset of Planck scale physics. This indicates that the disruption scale has been pushed down into the Planck slope. This is mildly disturbing. Ultimately, for these bizarre configurations, questions regarding the truth or falsity of Hawking's chronology protection can only be addressed by entering the uncharted wastelands of full fledged quantum gravity.