Quantum inequalities do not forbid spacetime shortcuts

What it claims

1. 01Krasnikov gives the class a precise definition. Take Minkowski space, remove a timelike cylinder, and glue in something else; the result is a shortcut if some pair of points that were spacelike separated in the original are causally connected in the new spacetime, with the spacetime required to stay globally hyperbolic. The class defined this way comprises the Alcubierre bubble, the Krasnikov tube and a certain type of wormhole, and it makes precise the sense in which travel is faster than light — the ship never outruns a passing photon locally; the comparison is between two different spacetimes.Section I, Introduction, the Definition and the Examples

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2. 02The standard argument runs as follows. The weak energy condition must break down somewhere in each of these spacetimes, which classically is forbidden, so the shortfall has to come from quantum fields. Their energy density, averaged over a sampling interval, is bounded below by a constant divided by the fourth power of that interval, and tying the sampling interval to the local curvature radius closes the loop with the Einstein equations to give a density of order the Planck density. Summing that over a domain wall around a region at least a metre across — the size needed to pass a human — yields a total amount of energy below the vacuum level of around ten to the thirty-second solar masses, which is the figure usually taken as showing that shortcuts are unphysical.Section II B, equations 4 to 9

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3. 03The worked example that sets up the whole paper is the Morris-Thorne wormhole known as absurdly benign, flat everywhere except in a thin spherical layer. Its total comes out near one thousandth of a solar mass for a throat a metre across, which looks far more attractive than the standard figure — until the quantum inequality is imposed, at which point the thickness of the curved layer must be of order ten to the minus thirty-five Planck lengths. Krasnikov’s comment is that this makes the wormhole much more absurd than benign, and the frugality was owed entirely to that thickness.Section II B, the Example, equations 10 to 12

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4. 04First way out: the chain that produces the Planck density assumes the components of the Riemann and Einstein tensors are of roughly the same order, which holds only when the maximum Weyl curvature is no larger than the maximum Ricci curvature. That condition breaks down more often than not — in any curved and empty region, such as the neighbourhood of any star, the ratio is infinite — and relaxing it removes the Planck-density restriction and invalidates the total. Krasnikov adds two further openings: for non-Minkowskian spacetimes the quantum inequality has never been proved, the two-dimensional conformally trivial case excepted, and Olum and Graham showed that a system of two interacting scalar fields can violate it, near a domain wall at that.Section III, opening paragraphs, and Section III A, equations 13 and 14

   What to watch

5. 05Second way out: a large total can be a meaningless number. In classical electrostatics the field energy outside a small ball around a point charge can be made to come out at ten to the thirty-second solar masses, and nobody concludes from that figure that the point charge is unphysical — the number is large only because the dominant contribution comes from a trans-Planckian region where the model itself no longer applies. Krasnikov argues that the totals quoted against shortcuts are of the same character. He also notes that the quantum inequality already fails in flat spacetime warped into a cylinder, where the Casimir energy density is negative and constant, which is why the rule has to be supplemented by a condition on the sampling interval in the first place.Section III B, equation 17; Section III A, equations 15 and 16

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6. 06The constructive result is a wormhole Krasnikov calls a portal: static, globally hyperbolic, and flat everywhere outside a compact region, whose spatial section contains two hoops that are in fact a single hoop, so that a traveller passing through one emerges from the other with the spacetime around them empty and flat throughout the journey. Abandoning spherical symmetry shrinks the volume of the region where the weak energy condition fails by about thirty-five orders of magnitude, so a human-sized portal needs about one hundredth of a solar mass — roughly the energy of a supernova — and the quantum inequality, if it holds, only requires that the hoops be thin, not absurdly thin. Combining the portal with Van Den Broeck’s trick, a capsule that is Planck-sized from outside and arbitrarily roomy within, brings the bill to about one milligram to sustain the capsule and a tenth of that for the portal.Section III C 1, the Portal, and Section III C 2, Van Den Broeck’s trick, equation 20

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The way in

https://doi.org/10.1103/PhysRevD.67.104013Published as Physical Review D 67, 104013 (2003) under the APS default licence. The preprint is on arXiv as gr-qc/0207057, version 3 dated 19 May 2003, carried under the arXiv assumed-1991-2003 licence, which grants arXiv distribution rights and nothing further — checked on the arXiv record on 2026-09-08, where no Creative Commons statement appears, and the preprint carries none. So this sheet holds the summary, the claims and Krasnikov’s own abstract and sends the reader to the source; the claims are read against that preprint, whose section numbering is used in the locators. Krasnikov writes from the Central Astronomical Observatory at Pulkovo, St Petersburg, and thanks L. H. Ford and M. J. Pfenning for critical comments and C. J. Fewster for remarks on the difference quantum inequality. This paper is his reply in a long exchange, and two of its other halves are in this library: his own Hyperfast travel in general relativity, Physical Review D 57, 4760 (1998), the paper whose Example 5 the field renamed the Krasnikov tube; and Everett and Roman’s Superluminal subway: The Krasnikov tube, Physical Review D 56, 2100 (1997), which carried that tube to four dimensions and priced it with the quantum inequality this paper reopens. Each position is stated in its own authors’ terms. The skeleton listed ch04 alone; ch02 is added, because the argument is conducted almost entirely in the currency of the vacuum.

How to cite it

S. Krasnikov (2003) Quantum inequalities do not forbid spacetime shortcuts. doi:10.1103/PhysRevD.67.104013

Where it sits in the curriculum

The metric, warp drives and wormholesWhat the vacuum is