Historical research guide · 0408 · 1997
Superluminal subway: The Krasnikov tube
Allen E. Everett · Thomas A. Roman
In one minute
Miguel Alcubierre’s 1994 warp metric has a catch that Sergei Krasnikov spotted: the crew at the centre of the bubble is causally cut off from its leading wall, so they can neither raise a bubble on demand nor steer one that exists. Allen Everett and Thomas Roman, at Tufts, take up Krasnikov’s alternative — a geometry the ship lays down behind itself as it flies — and carry it from two dimensions up to four. What comes out is a tube along the outbound path, flat inside but with the light cones tipped open in one direction, so that the return leg can be made as quick as you like by Earth’s clocks while the outbound leg keeps to the usual rules. Then they price the structure. Holding the tube open calls for a shell of energy below the vacuum level only a few thousand Planck lengths thick, and the totals they estimate are enormous. Two separated tubes, they show, make a time machine.
Connection
Chapter 4 is metric engineering, and this is the paper that answered the causal objection to warp bubbles and then named the bill for the alternative. It reaches chapter 2 because that bill is written entirely in the currency of the vacuum: how thin a region of energy below the ambient vacuum level quantum field theory will tolerate, and for how long.
From the source
A photon sent forward from a ship at the centre of an Alcubierre bubble reaches a point inside the bubble wall where it stops moving relative to the bubble and is simply carried along with it, so the outer edge of the wall lies permanently outside the ship’s forward light cone — meaning the crew can neither create the bubble on demand nor control one once it exists, and someone else must have warped the region beforehand, like a trolley car laid on in advance.Section 2, the photon argument
Krasnikov’s geometry avoids that objection because every modification of the metric happens in the causal future of the launch point, along the ship’s own outbound world line. The consequence is the paper’s headline property: the one-way trip to a distant star is bounded by the usual restrictions of special relativity, while the round-trip time measured by clocks on Earth can be made arbitrarily short.Abstract; Section 2, discussion following Equation 6; Section 7
Extending Krasnikov’s two-dimensional metric to four dimensions produces a tube of finite radius centred on the ship’s path, connecting Earth and the star, static once it has been created, flat inside, with the light cones opened out so that travel in one direction along it is superluminal as seen by observers outside.Section 3, Equations 12 and 13
A single Krasnikov tube contains no closed timelike curves, but a system of two non-overlapping tubes can be used to construct a time machine — a feature it shares with two-wormhole and two-warp-bubble systems, and one that would already pose a causality problem for tubes of merely laboratory size.Section 4, A Superluminal Subway and Closed Timelike Curves
Applying the Ford and Roman quantum inequality in the short-sampling-time limit, maintaining the tube long after its formation requires a band of energy density below the vacuum level no thicker than a few thousand Planck lengths, and the rough total for a tube one metre long and one metre wide comes out around ten to the sixty-third grams — the same order of difficulty already found for Alcubierre bubbles and for traversable wormholes.Section 6, Equations 44, 51 and 52
The authors show their own escape route and mark its cost. If the light cones are opened only slightly, the negative energy densities in the tube wall become very small, the quantum inequality is satisfied with a wall a centimetre thick, and a region within which superluminal travel is in principle allowed could be established — but at the parameter value that permits it, the speed of a light ray travelling back along the tube would exceed the usual one by about one part in ten to the sixty-six.Section 6, final paragraph
The way in
https://doi.org/10.1103/PhysRevD.56.2100How to cite it
Allen E. Everett, Thomas A. Roman (1997) Superluminal subway: The Krasnikov tube doi:10.1103/PhysRevD.56.2100