I have written a number of columns in this magazine about wormholes, warp drives, and other constructs of Einstein's general relativity (GR) that appear to offer a good physics foundation for faster-than-light travel and even for travel back in time.  All of these GR constructs come from a particular non-standard way of using Einstein's theory, an approach that might be described as "metric engineering".  Instead of considering a particular arrangement of mass and energy and asking how space would be warped and what effects would be produced by such an arrangement, in metric engineering we specify how we want space to be warped in order to produce wormholes, warp drive, etc., and then ask what arrangement of mass and energy would be required to accomplish this.  The usual outcome of this kind of GR calculation is that a certain quantity of negative mass-energy would be needed.  For example, to stabilize a wormhole, a significant quantity of negative mass-energy is needed near the wormhole's throat.

While there are no well-established physical laws that prevent the existence of negative mass or energy, in looking around in our corner of the universe we have not seen any significant amount of either.  I was co-author of a physics paper (see ) suggesting how star-scale negative mass objects (actually, wormhole mouths) might be searched for in astronomical measurements, but so far no objects with the signature our paper suggested have been observed.

Our best present route to negative energy lies in the space between two closely-spaced conducting parallel plates.  There, the Casimir Effect (see AV43: "FTL Photons", Analog, mid-December-1990) requires a net negative energy in the gap.  However, the overall energy of a Casimir plate system must be positive, so the local negative energy is bought at the expense of positive energy elsewhere.  Further, the quantity of negative energy is tiny, so that any effects for which it is responsible are very difficult to observe.  In a previous AV column, (AV53: "Squeezing the Vacuum", Analog, July-1992) we also discussed the observation that the "squeezing" of space in a strong and changing gravitational field (e.g., near the event horizon of a back hole) creates a negative energy region.  These phenomena represents what mathematicians call an "existence theorem", demonstrating that negative energy can and does exist, but not whether it might be useful for metric engineering.   They also represent intrusions of quantum mechanics into the "classical" physics of Newton and Maxwell, as extended by Einstein.

Many of the theoretical physicists who work with general relativity have fundamental objections to the very idea of wormholes and warp drives, which they consider to be"unphysical".  Some of them have decided that one should place a "picket fence" around these solutions of Einstein's equations that are considerd to be physically reasonable, and to place exotica like stable transversable wormholes, faster-than-light warp drives, and time machines in the forbidden area outside the fence, excluded because it is presumed that Nature does not allow such objects.  They are, in effect, attempting to discover new laws of physics that would place restrictions on GR solutions.

Their first attempt at building such a fence was called the Weak Energy Condition (WEC).  In essence, the WEC assumes that negative energy is the source of "problems" with GR and requires that for all observers the local energy in all space-time locations must be greater than or equal to zero.  In other words, if any possible observer would see a negative energy, that solution of Einstein's equations is excluded by the WEC.  An less restrictive variant of the WEC is the Average Weak energy Condition (AWEC), which requires that when time-averaged along some arbitrary world-line through all time, the net energy must be greater than or equal to zero, so that any time period when the energy is negative must be compensated by a period of positive energy.

The WEC, AWEC and the other similar energy rules are "made-up" Laws of Nature and are not derivable from general relativity.  They appear to be obeyed for observations of all known forms of matter and energy that do not fall within the domain of quantum mechanics.  However, even for simple situations involving quantum phenomena, for example the Casimir Effect, squeezed vacuum, or the evaporation of black holes, the WEC and AWEC are violated. 

For fence-building theorists, the dismaying failure of these energy conditions raises the question of why nature seems to need negative energy in certain circumstances.  It is clear that it has something to do with quantum mechanics.  For example, the time-energy version of Heisenberg's uncertainty principle requires that if a time interval is made sufficiently short, the fluctuations in energy must become very large.  If the energy is not allowed to fluctuate to negative values, the Heisenberg uncertainty relation doesn't work.  Similarly, the Hawking evaporation of black holes, which is a quantum-mechanics based process, involves a black hole "eating" a negative-energy photon (or other particle) while its positive-energy twin escapes into the space outside the black hole.  This process would not work, and its connection to thermodynamics would fail, if negative energy were forbidden.  These clues, suggesting a connection between GR and quantum mechanics, have attracted considerable theoretical interest because there is presently no physics formalism that connects the two theories.  Up to now, all attempts to construct such a theory of quantum gravity have either failed completely or are in such a primitive state (c.f., string theory) that they cannot make useful predictions of physical observables.

Therefore, the fence-building theorists have turned to quantum field theory, the standard model of relativistic quantum mechanics, to search for rules governing the existence on negative energy.  This work, pioneered in 1978 by Laurence H. Ford, has led to what are called "quantum inequalities" (QI).  Basically, one chooses a "sampling function", some bell-shaped curve with unit area and width T, that specifies a particular restricted region of time.  This function is then used to average the energy per unit volume of some quantum field within the time-sampling envelope.

This calculation, which has been performed for a number of fields and sampling functions (Gaussians, Lorentzians, triangles, etc.) leads to the conclusion that the energy per unit volume of an field described by quantum field theory can be no more negative than [−Kh/ (2pc³T⁴)], where h is Planck's constant, c is the speed of light, and K is some constant much less than 1 that depends on which sampling function is used.

Physically, the QI says that the larger the quantity of negative energy existing in some time interval T, the smaller T must be.  An observer cannot see large quantities of negative energy that last for a long time.  A burst of negative energy must be followed in a very short time by an even larger burst of positive energy.  One can think of the negative energy as a "loan" charged to Heisenberg's credit card that must be repaid within a time that becomes shorter as the amount of energy "borrowed" becomes larger.  And the times involved are very short indeed.

The QI is bad news for would-be practitioners of metric engineering.  Taken at face value, the QI says that stable wormholes may be impossible and that a warp drive might, at best, exist for too short a time to go anywhere.  While a wormhole might wink into existence during the short time that the negative energy is present, it would wink out of existence again before any matter could pass through it.  It appears that within the QI conditions, when negative energy is created, it is either too small in magnitude or too brief in duration to do anything interesting.

Is there any escape from these pessimistic conclusions?  Perhaps.  Quantum field theory cannot be trusted in its application to the field-energy situations envisioned by the QI calculations because it attributes far too much positive energy to space-time itself.  The density of "dark energy" deduced from the observations of astronomers investigating Type Ia supernovas is about 6.7 × 10^-10 joules per cubic meter.  The same quantity, as calculated by quantum field theory is about 10⁴⁰ joules per cubic meter.  Thus, quantum field theory missed the mark by abut 50 orders of magnitude.  Therefore, until quantum field theory can accurately predict the energy content of the vacuum, the restrictions that it places on metric engineering cannot be taken too seriously.

Another possible loophole around the QI restrictions comes from alternatives to standard general relativity.  In particular, there have been some "elaborations" of general relativity called R^m gravity theories that attempt to make small changes in the standard theory that are at the same time consistent with all existing observations and also account for the observed accelerated expansion of the universe without the need to invoke dark energy.  There has been a recent study by two Canadian theorists of such R^m theories as they apply to the stability of wormholes.  Their conclusion is that for such theories, stable wormholes can exist that do not require negative energy and that satisfy the WEC.

Thus, as we said during the excruciating election year just passed, all the votes have not yet been counted.  It may be that a proper theory of quantum gravity, when we get one, might rule out wormholes and warp drives, but we do not have such a theory at the moment.  The theories we do have seem to point in several different directions.  As the theorists like to say when they are writing funding proposals, more work is needed in this important area of theoretical physics.

John G. Cramer's 2016 nonfiction book (Amazon gives it 5 stars) describing his transactional interpretation of quantum mechanics, The Quantum Handshake - Entanglement, Nonlocality, and Transactions, (Springer, January-2016) is available online as a hardcover or eBook at: http://www.springer.com/gp/book/9783319246406 or https://www.amazon.com/dp/3319246402.

SF Novels by John Cramer: Printed editions of John's hard SF novels Twistor and Einstein's Bridge are available from Amazon at https://www.amazon.com/Twistor-John-Cramer/dp/048680450X and https://www.amazon.com/EINSTEINS-BRIDGE-H-John-Cramer/dp/0380975106. His new novel, Fermi's Question may be coming soon.

Alternate View Columns Online: Electronic reprints of 212 or more "The Alternate View" columns by John G. Cramer published in Analog between 1984 and the present are currently available online at: http://www.npl.washington.edu/av .

References:

Wormholes and Negative Mass

Lorentzian Wormholes, Matt Visser, AIP Press, Woodbury , NY (1996).

"Some Thoughts on Energy Conditions and Wormholes", Thomas A. Roman, electronic preprint gr-qc/0409090, September 23, 2004 .

Wormholes and R^m Gravity

"Wormhole Throats in R^m Gravity", N. Furey and A. DeBenedictis, electronic preprint gr-qc/0410090088, November 22, 2004 .

Astronomical Search for Negative Mass

J. G. Cramer, R. W. Forward, M. S. Morris, M. Visser, G. Benford, and G. A. Landis, Physical Review D51 3117-3120 (1995).

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