STM-D-0676Paper1998Published and peer-reviewed

Hyperfast travel in general relativity

S. V. Krasnikov

Abstract and summary · read the original at the source

In one page

Sergei Krasnikov, at the Pulkovo observatory in St Petersburg, asks the traveller’s question rather than the theorist’s one. Not how fast can a ship go, but: can a ship that is allowed to reshape the geometry it passes through reach a distant star and come home sooner than a photon sent on the same errand? He splits the answer in two, and the split is the paper. For the outbound leg he proves a hard result: in a globally hyperbolic spacetime, a pilot who starts from the same event as the light cannot beat it to the destination, however cleverly the metric is worked on the way. For the round trip the answer flips, because the return leg lies in the future of the departure and can be prepared. He writes down a metric — later named the Krasnikov tube — in which the light cones are tipped open along the outbound track, and an astronaut who leaves in 2000 and arrives at Deneb in 3600 is back on Earth in 2002.

Why it matters hereThis is one of the founding constructions of chapter 4: the ship does not travel through space faster, it lays down a modified region behind itself and the distance is what is engineered. It reaches chapter 2 in the discussion, where Krasnikov prices the structure in vacuum terms and then names the loophole he thinks worth hunting — the exact place where the quantum inequality that limits energy below the ambient vacuum level stops applying.

What it claims

01The question is posed as a race with a fair start. A beam of test particles is emitted from Earth to Deneb at an event S, reflects and returns; then a spaceship — something that does act on the surrounding geometry — is launched at the same event. Krasnikov asks whether the ship can come back sooner than the first particle, using only causal means, and calls the constraint he imposes on the ship utter causality: no tachyons, and no way of changing the metric that would breach it.Section 1, Introduction; Section 2, Causal changes
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02The one-way result. Proposition 1 states that for globally hyperbolic spacetimes diverging by the launch event, the ship’s arrival at the destination cannot precede the arrival of the test particles. Krasnikov calls the proposition seemingly self-evident and notes that its proof turned out to be quite tedious; it occupies the Appendix. In plainer terms: nothing the pilot does after departure gets the pilot to Deneb early.Section 3, Proposition 1, proved in the Appendix
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03Applied to the Alcubierre metric, the same reasoning names the price precisely. The curve separating the flat exterior from the curved bubble region is spacelike exactly when the bubble speed exceeds one, and Alcubierre’s own field equation puts matter immediately inside that curve — so the leading edge of the bubble is the world line of that matter, and shortening the outbound trip this way would require tachyonic matter. The alternative Krasnikov names is prosaic and buildable: place devices along the route in advance and programme them to act at set moments. That can establish a regular service; it does not help the first flight.Section 3, Example 4, with equation 4 and the discussion of the curve λ+
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04The round trip is a different problem, because the segment between arrival and return lies in the causal future of the start and can be prepared. Krasnikov’s Example 5, labelled the warp drive, is a two-dimensional metric with three regions: a flat outside, a narrow curved transition strip, and a flat inside region where the light cones are opened out. A photon heading back along the inside region reaches the launch point almost at the launch event — so an arbitrarily distant journey can be made in an arbitrarily short time as measured on Earth. His illustration: the astronaut departs in 2000, reaches Deneb in 3600 by Deneb’s clocks, and returns to Earth in 2002.Section 4, Example 5, with Figure 3
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05Example 6 reaches the same result with a wormhole and no exotic geometry along the way: keep both mouths near Earth, carry one of them on the outbound trip at near light speed, and return through the throat. Because the mouth only ever moves away from Earth, causality is preserved and none of the difficulties of a wormhole time machine arise. Krasnikov’s own name for devices of this class is a space machine — in his phrase, the square root of a time machine, since applying the same trick twice would close a timelike curve.Section 4, Example 6; Section 5, opening paragraph
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06The discussion sorts space machines into two types and marks the open question in each. Compactly generated ones, Examples 4 to 6, need energy below the ambient vacuum level, and the quantum inequality applied to a four-dimensional version of Example 5 returned a requirement of order ten to the thirty-second galaxy masses. Krasnikov’s response is not to shrink the geometry but to attack the inequality’s own premise: it was derived assuming that small regions look approximately Minkowski, and near the Cauchy horizon of a wormhole about to become a time machine that assumption fails outright — in Misner space with a massless scalar field the sampled energy runs to minus infinity. His conclusion is a research instruction: one need not create a time machine to break the bound, only almost create one. Noncompact space machines need no violation at all, but the evolution of spacetimes that are not globally hyperbolic is not yet understood well enough to drive one.Section 5, Discussion, types 1 and 2
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Read it · abstract

Abstract

The problem is discussed of whether a traveller can reach a remote object and return back sooner than a photon would when taken into account that the traveller can partly control the geometry of his world. It is argued that under some reasonable assumptions in globally hyperbolic spacetimes the traveller cannot hasten reaching the destination. Nevertheless, it is perhaps possible for him to make an arbitrarily long round-trip within an arbitrarily short (from the point of view of a terrestrial observer) time.

The way in

https://doi.org/10.1103/PhysRevD.57.4760Published as Physical Review D 57, 4760 (1998) under the APS default licence. The preprint is on arXiv as gr-qc/9511068, version 6 dated 9 March 1998, carried under the arXiv assumed-1991-2003 licence, which grants arXiv distribution rights and nothing further — checked on the arXiv record for this paper on 2026-09-08, where no Creative Commons statement appears, and the preprint text carries none either. So this page holds the summary, the claims and Krasnikov’s own abstract and sends the reader to the source; the claims are read against that preprint. Krasnikov was at the Central Astronomical Observatory at Pulkovo, St Petersburg; the work was partially supported by RFFI grant 96-02-19528. This is the paper whose Example 5 the field renamed the Krasnikov tube. Three companion sheets in this library carry the rest of that conversation: Everett and Roman’s four-dimensional extension and costing, Superluminal subway: The Krasnikov tube, Physical Review D 56, 2100 (1997); Olum’s theorem, Superluminal Travel Requires Negative Energies, Physical Review Letters 81, 3567 (1998); and Krasnikov’s own later reply, Quantum inequalities do not forbid spacetime shortcuts, Physical Review D 67, 104013 (2003).

How to cite it

S. V. Krasnikov (1998) Hyperfast travel in general relativity. doi:10.1103/PhysRevD.57.4760

Where it sits in the curriculum

The metric, warp drives and wormholes What the vacuum is