STM-D-0671Paper1998Published and peer-reviewed

Superluminal Travel Requires Negative Energies

Ken D. Olum

Abstract and summary · read the original at the source

In one page

Ken Olum, at the Institute of Cosmology at Tufts, asks a sharp question and answers it with a theorem. First he has to settle what faster-than-light travel even means, because a metric can look like a shortcut and turn out to be flat space wearing odd coordinates — he writes one down in which the distant star is simply being carried closer. His replacement definition is about competition between paths: the route counts as superluminal only if it reaches the destination surface earlier than every neighbouring route. With that definition, and with ordinary matter somewhere along the path, Olum proves that such a route demands a violation of the weak energy condition — somewhere along it the energy density must sit below the ambient vacuum level. Then he answers his own headline. Does that make superluminal travel impossible? No: quantum fields do not obey the weak energy condition, and he shows that the Casimir effect, the most measured example there is, already satisfies his own condition.

Why it matters hereChapter 4 is metric engineering, and this is the theorem that tells every warp and tube design exactly what ingredient it must supply: a region of the vacuum held below its ambient level, on the path itself. That makes it a chapter 2 paper as much as a chapter 4 one, because Olum closes by pointing at the Casimir effect — a laboratory system that already produces the ingredient and, he shows, already satisfies his definition of a shortcut.

What it claims

01A metric can appear to permit superluminal travel and be nothing of the kind. Olum writes down a two-dimensional metric in which a null ray from the origin reaches arbitrarily large distance before a fixed time, then removes the illusion with a single change of coordinate: the spacetime is flat, and the star that looked reachable was travelling toward the traveller. Reading a metric is therefore not enough — one needs a way to tell bringing a place closer apart from arranging a genuine shortcut.Equations 1 and 2, and Figure 1
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02Olum proposes the definition the field has used since. A causal path is superluminal from A to B only if there are small spacelike 2-surfaces around A and around B, each woven from spacelike geodesics through its own point, such that no point of the destination surface lies in the causal future of any point of the origin surface except the endpoint pair itself. In plain terms: the path works better than every path beside it.Condition 1, with Figure 3
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03The theorem. Given a path satisfying that condition, and given the generic condition — which holds wherever there is any normal matter or any transverse tidal force along the route — the weak energy condition must be violated at some point of the path. Olum gets there by showing the path must be a null geodesic with no conjugate points, running the Raychaudhuri equation forward to force the expansion negative at the destination, and then proving directly that the same expansion cannot be negative there.Equations 3 to 8, and the paragraph beginning ‘Thus we see that any spacetime that admits superluminal travel’
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04For the simpler case where flat space is modified only inside a compact region — the finite-duration Alcubierre bubble and the finite-length Krasnikov tube are both of this type — the same conclusion follows from the older closed-timelike-curve theorems of Tipler and Hawking, by identifying the two faces of the modified slab and reading off the Cauchy horizon that appears.The paragraph beginning ‘In this simple case we can show that WEC must be violated’
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05Olum marks what his result adds to the earlier theorems and what it does not. Tipler and Hawking rule out the construction of closed timelike curves from a compact region unless there is a singularity or an energy-condition violation on the boundary; Olum’s theorem rules out the existence of the shortcut, a spacetime singularity is no substitute, and the violation has to occur along the very path to be travelled. Extending it to catch more time machines, he notes, is not easily accomplished.The two paragraphs beginning ‘One can compare this theorem with those of Tipler and Hawking’
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06The paper’s own answer to its title. Does the theorem mean superluminal travel is impossible? Olum writes no, because the weak energy condition is not obeyed by systems of quantum fields — and the Casimir effect provides an example that satisfies his Condition 1. Between two circular conducting plates the stress-energy is the familiar Casimir form, the Raychaudhuri equation runs the other way, the null geodesics are defocused, and a ray through the centre arrives ahead of its neighbours. What he leaves open is the accounting: the mass of the plates and of the structure holding them apart is not yet in the calculation, so whether a path avoiding the apparatus entirely arrives earlier still is the measurement to make.Figure 5, Equations 9 to 11, and the closing paragraph
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Read it · abstract

Abstract

I investigate the relationship between faster-than-light travel and weak-energy-condition violation, i.e., negative energy densities. In a general spacetime it is difficult to define faster-than-light travel, and I give an example of a metric which appears to allow superluminal travel, but in fact is just flat space. To avoid such difficulties, I propose a definition of superluminal travel which requires that the path to be traveled reach a destination surface at an earlier time than any neighboring path. With this definition (and assuming the generic condition) I prove that superluminal travel requires weak-energy-condition violation.

The way in

https://doi.org/10.1103/PhysRevLett.81.3567Published as Physical Review Letters 81, 3567 (1998) under the APS default licence. The preprint is on arXiv as gr-qc/9805003, version 2 dated 14 October 1998, carried under the arXiv assumed-1991-2003 licence, which grants arXiv the right to distribute 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 itself carries no licence statement either. So this page carries the summary, the claims and Olum’s own abstract and sends the reader to the source; the claims below are read against that preprint. Olum was at the Institute of Cosmology, Department of Physics and Astronomy, Tufts University; the work was supported in part by the National Science Foundation. Two companion sheets in this library sit on either side of this one: Krasnikov’s Hyperfast travel in general relativity, Physical Review D 57, 4760 (1998), and Everett and Roman’s Superluminal subway: The Krasnikov tube, Physical Review D 56, 2100 (1997).

How to cite it

Ken D. Olum (1998) Superluminal Travel Requires Negative Energies. doi:10.1103/PhysRevLett.81.3567

Where it sits in the curriculum

The metric, warp drives and wormholes What the vacuum is