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# Secrets of Warp Drive

by Michael j. Pfenning.P.D

Department of Physic and Astronomy
University of Guelph
Guelph, Ontario  N1H 8H5

Imagine being the captain on the bridge of your own starship, presently in orbit around Earth. You sense the vibrations of the ship's idling engines.  The computer screens are flashing with data and the crew is going about their routine.  You give the command to jump to faster than light speed.  The engines surge with power and the ship accelerates.  The star field through the front window shifts and becomes bluer .  Eventually you  accelerate right past the speed of light .  You have reached warp speed one.  The ship keeps accelerating reaching warp two , then three, then four.  After a few minutes the computer issues the standard stop command.  The ship begins to slow; the star field returns to normal.  Eventually the ship comes to rest.  The ship is now in orbit around the second planet of the star alpha centauri.  Such is the long sought after dream of warp drive.

In both the scientific community and pop culture, humans have been fascinated with the prospect of being able to travel between the stars within their own lifetimes. Within the framework of Einstein's Theory of Relativity,  a space-going traveler may locally travel with any velocity up to, but not including, the speed of light.  For the  moving astronauts, relativity also says that there should be a slowing down of the passage of time relative to stationary clocks on earth.  This allows the astronauts to make the round trip from earth to any star and back to earth in an arbitrarily short elapsed time.  However, upon returning to earth the astronauts would find that their family and friends would have aged considerably.  This is the well-known twin paradox  [1]  (although it called a paradox, it really isn't).  Let us look at an example:

A group of astronauts board a state of the art spacecraft and fly at 99.9999% of the speed of light to the nearest star to our sun called Proxima Centauri, which is 4.3 light-years away.  According to the clocks onboard the spacecraft, the total duration of the flight takes 1 day 14 hours 4 minutes and 44 seconds.  The astronauts spend 21 days taking data and performing experiments while in orbit of Proxima Centauri.  They then fly home at the same speed.  For the astronauts, the whole trip takes 24 days 4 hours 9 minutes and 28 seconds.   However,  on Earth,  a total time of 8.66 years has passed, 8 years 240 days 3 hours 40 minutes and 35 seconds, to be exact.  The difference in the passage of time is due to the time dialation effect when the astronauts are moving so fast.  While they were in motion, time on board the spaceship passed at a rate of just 0.14% that of the time on Earth.

For the astronauts, space travel with very fast rockets doesn't seem so bad.   However,  if the astronauts had to travel  to stars much futher away, say hundreds of thousands of light-years distant, then when they return to Earth they would find that so much time had passed on Earth that all their friends and relatives would be dead. Several generations of humans could have lived and died.   For this reason many scientist have searched for alternative forms of transportation to travel to distant world.

In 1994,  Dr. Miguel Alcubierre [2], then at the University of Wales, Cardiff, proposed a spacetime, fondly called the "Warp Drive" after it's science fiction conterpart,  in which superluminal (faster than light travel) is possible.  In this spacetime a spaceship can travel to a  star a distance D away and return home, such that the elapsed time for the stationary observers on earth would be less than 2D/c, where c is the velocity of light.  What is most surprising about this spacetime is that the proper time of the space going traveler's trip is identical to the elapsed time on earth.     That is to say,  if the astrounauts take a trip using a warp drive moving at 100 times the speed of light  that lasted for 1 year, then when they return to Earth, the would find that only 1 year had passed on earth as well.  So, this spacetime overcomes the "problem" of the time dilation of special relativity,  while additionally allowing the observer to travel a distance of 50 light-years away and back in just a single year.

You may now be asking, "Doesn't this violate Einstein's statement that no material object can move faster than the speed of light?"  The answer is no.   You see, the spaceship never locally travels faster than the speed of light.  In fact, the spaceship can sit at rest with respect to the interior of the warp bubble.  The ship is carried along by the spacetime, much in the same way that the galaxies are receding away from each other at extreme speeds due to the expansion of the universe, while locally they are at rest.  The warp drive utilizes this type of expansion and contraction in order to achieve the ability of faster than light  travel .

Let us explore how warp drive works.  Imagine a rocket ship out in space, far away from any other objects.  Surround the ship with a thick spherical bubble.   The spacetime is now divided into three different volumes,   the exterior space outside the bubble,  the insides of the thick bubble wall, and the interior of the bubble.   Now warp drive works because it causes the spacetime in the forward wall of the bubble to contract .  In addition it causes the spacetime in the backward half of the bubble wall to expand.  The region inside the bubble is then transported forward, carrying anthing inside the bubble along with it.  Since there is no limit on how fast empty space can contract or expand, we can cause the interior volume of the bubble to moved through space at any velocity, including speeds greater than that of the speed of light.

In a sense, warp drive is much like surfing.  If an ocean wave moves at 10 miles an hour, then a surfer who catches the wave will be carried at the same speed of 10 mph.  There would be no disputing that people on the shore would see the surfer moving toward the shore at 10 mph.  However, as far as the surfer is concerned, he has no velocity relative to the wave.  Both are moving at 10 mph in the same direction, so he has a speed relative to the wave of 0 mph.  So globally (relative to the people on the shore) the surfer is moving, but locally (relative to the wave) the surfer is at rest.  The same is true for warp drive.  The warp bubble is a disturbance that moves through the fabric of spacetime just as the ocean wave is a disturbance travelling through the ocean.  A spaceship "catches a ride" in the bubble and gets carried along at whatever speed the bubble is moving.  Globally the spaceship is moving, possibly even superluminally.  Locally, relative to the interior spacetime of the bubble,  the spaceship is at rest.  Since Einstein's statement that no material object can move faster than the speed of light is a local condition, we see that we have not violated any aspect of general relativity.

 You may be wondering what it would look like to see a warp bubble go by.  The movie to the right is a numerical simmulation of such an event.  A computer program was written by researchers at the University of Guelph to calculate the paths of light rays as they travel through the warp drive spacetime.  Each image is a snapshot of the warp bubble at a given moment, which were then glued together  to form the movie. For this simmulation, the bubble is travelling at the speed of light.  The four second duration of the movie is in extreme slow motion.  In reality, all of the images would flash by in under 1.2 millionths of a second.         There is one special note about         If you would like to see images of how the stars would shift if you were standing on the bridge of the starship as it travelled at different warp speeds, see reference [3].

## `Exotic Matter' is not couch potato energy!

Although warp drive sounds appealing, it does have one serious drawback.  In order to achieve warp drive we must use exotic matter, which is a form of matter whose energy per unit volume (energy density) is negative.  Classical  matter has positive energy per unit volume, the value of which can be found from Eintein's famous E = mc2.   A vacuum (in flat space) is defined to have zero energy density.  Negative energy comes about from special states of quantum field theory.  Some examples are highly squeezed states of light,  the Casimir vacuum energy between two uncharged conducting plates and states which are superpositions of particle number.  In all of these examples some region of space has negative energy density which is a  violation of the classical energy conditions.   However, since these are all quantum effects the classical energy conditions really can't be applied.

Now you might be wondering why I may seem concerned about allowing negative energy to exist.   First off, let me say what you could do if you could manipulate large amounts of negative energy.  You could open up wormholes to travel to distant parts of the universe.   You can build faster than light transportation systems.  You could travel backward through time to visit your grandparents.  You could make gravity repulsive.  Even gravity based weapons would be possible.  By themselves, these things don't seem so bad (although I do give a great deal of pause to gravity based weapons), but they do concern physicists because none of these effects have ever been observed.  However, gross effects like this could taken into account by modifying our present theories.  The greatest concern about negative energy comes about because it could be used to violate the second law of thermodynamics.  One could build a negative energy laser which you could shine on an object to suck all the energy out of it, causing the entropy of the system to decrease.  This is a serious attack on one of the fundamental  law of physics and we are forced to ask if there is any resolution.

The solution to our conundrum also comes from quantum field theory in the form of restrictions known as the Quantum Inequalities \cite{F&Ro92, F&Ro95, Ford78, Ford91, F&Ro97}.  These relations do allow negative energy densities to exist . However, they  place serious limitations on their magnitude and duration. Even before the short sampling time expansion of the quantum inequality was developed, researchers had applied the flat space quantum inequality to the curved spacetime geometries of wormholes \cite{F&Ro96} with the restriction that the negative energy be sampled on timescales smaller than the minimum local radius of curvature.  It was argued that over such small sampling times, the spacetime would be locally flat and the inequalities would be valid.  This led to the conclusion that static wormholes must either be on the order of several Planck lengths in size, or there would be large discrepancies in the length scales that characterize the wormhole.

## References

[1]    For an example see "Relativity: hoax or reality" on p. 48 in A Journey into Gravity and Spacetime, John Archibald Wheeler, Scientific American Library,  New York (1990).

[2]    "The warp drive:  hyper-fast travel within general relativity,"  Miguel Alcubierre,  Classical and Quantum Gravity,  vol. 11,  p. L73  (1994).

[3]    "Negative Energy, Wormholes and Warp Drive,"  L. H. Ford and T. A. Roman,  Scientific American,  vol.  282,  p. 46  (January 2000).

[4]    "The unphysical nature of `warp drive',"  M. J. Pfenning and L. H. Ford,  Classical and Quantum Gravity,  vol.  14,  p. 1743  (1997).  gr-qc/9702026

The web page was created by Michael Pfenning, Ph. D.