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The Alcubierre Warp Drive
by John G. Cramer
Alternate View Column AV-81
Keywords: Alcubierre Warp Drive FTL spacewarp solution Einstein's
equations general relativity
Published in the November-1996 issue of Analog Science Fiction &
This column was written and submitted 4/15/96 and is copyrighted ©1996
by John G. Cramer.
All rights reserved. No part may be reproduced in any form without
the explicit permission of the author.
The theoretical physicist Miguel Alcubierre was born
in Mexico City, where he lived until 1990 when he traveled to Cardiff
in the UK to enter graduate school at the University of Wales. He
received his PhD from that institution in 1993 for research in
numerical general relativity, solving Einstein's gravitational
equations with fast computers. He continues to work in this field,
devising numerical techniques for describing the physics of orbiting
black holes that spin down to collision.
Two years ago Alcubierre published a remarkable paper which grew
from his work in general relativity, the current "standard model" for
space-time and gravitation. His paper describes a very unusual
solution to Einstein's equations of general relativity, described in
the title as a "warp drive", and in the abstract as "a
modification of space time in a way that allows a space ship to travel
at an arbitrarily large speed". In this Alternate View column, I
want to explore Alcubierre's work and its implications.
Let's start by considering the well-known velocity-of-light speed
limit, as viewed by special relativity and by general relativity. In
the context of special relativity, the speed of light is the absolute
speed limit of the universe for any object having a real mass (i.e.,
everything but the semi-mythical tachyon), for two reasons. First,
giving a fast object even more kinetic energy has the main effect of
causing an increase in mass-energy rather than speed, with mass-energy
going infinite as speed snuggles up to the velocity of light. By this
mechanism, relativistic mass increase limits massive objects to sub-light
There is also a second faster than light (FTL) prohibition supplied
by special relativity. Suppose a device like the "ansible" of LeGuin
and Card were discovered that permitted faster-than-light or
instantaneous communication. Special relativity is based in the
treatment of all reference frames (i.e., coordinate system moving at
some constant velocity) with perfect even-handedness and democracy.
Therefore, FTL communication is implicitly ruled out by special
relativity because it could be used to perform "sumultaneity tests" of
the readings of separated clocks which would reveal the preferred or "true"
reference frame of the universe. The existence of such a preferred
frame is in conflict with special relativity.
General relativity treats special relativity as a restricted sub-theory
that applies locally to any region of space sufficiently small that
its curvature can be neglected. General relativity does not forbid
faster-than-light travel or communication, but it does require that
the local restrictions of special relativity must apply . In other
words, light speed is the local speed limit, but the broader
considerations of general relativity may provide an end-run way of
circumventing this local statute. One example of this is a wormhole [see
my AV columns in Analog, June-1989 and May-1990] connecting
two widely separated locations in space, say five light-years apart.
An object might take a few minutes to move with at low speed through
the neck of a wormhole, observing the local speed-limit laws all the
way. However, by transiting the wormhole the object has traveled five
light years in a few minutes, producing an effective speed of a
million times the velocity of light.
Another example of FTL in general relativity is the expansion of
the universe itself. As the universe expands, new space is being
created between any two separated objects. The objects may be at rest
with respect to their local environment and with respect to the cosmic
microwave background, but the distance between them may grow at a rate
greater than the velocity of light. According to the standard model of
cosmology, parts of the universe are receding from us at FTL speeds,
and therefore are completely isolated from us. As the rate of
expansion of the universe diminishes due to the pull of gravity,
remote parts of the universe that have been out of light-speed contact
with us since the Big Bang are coming over the lightspeed horizon and
becoming newly visible to our region of the universe.
Alcubierre has proposed a way of beating the FTL speed limit that
is somewhat like the expansion of the universe, but on a more local
scale. He has developed a "metric" for general relativity, a
mathematical representation of the curvature of space, that describes
a region of flat space surrounded by a "warp" that propels it forward
at any arbitrary velocity, including FTL speeds. Alcubierre's warp is
constructed of hyperbolic tangent functions which create a very
peculiar distortion of space at the edges of the flat-space volume. In
effect, new space is rapidly being created (like an expanding universe)
at the back side of the moving volume, and existing space is being
annihilated (like a universe collapsing to a Big Crunch) at the front
side of the moving volume. Thus, a space ship within the volume of the
Alcubierre warp (and the volume itself) would be pushed forward by the
expansion of space at its rear and the contraction of space in front.
Here's a figure from Alcubierre's paper showing the curvature of space
in the region of the travelling warp.
For those familiar with usual rules of special relativity, with its
Lorentz contraction, mass increase, and time dilation, the Alcubierre
warp metric has some rather peculiar aspects. Since a ship at the
center of the moving volume of the metric is at rest with respect to
locally flat space, there are no relativistic mass increase or
time dilation effects. The on-board spaceship clock runs at the same
speed as the clock of an external observer, and that observer will
detect no increase in the mass of the moving ship, even when it
travels at FTL speeds. Moreover, Alcubierre has shown that even when
the ship is accelerating, it travels on a free-fall geodesic. In other
words, a ship using the warp to accelerate and decelerate is always in
free fall, and the crew would experience no accelerational gee-forces.
Enormous tidal forces would be present near the edges of the flat-space
volume because of the large space curvature there, but by suitable
specification of the metric, these would be made very small within the
volume occupied by the ship.
All of this, for those of us who would like to go to the stars
without the annoying limitations imposed by special relativity,
appears to be too good to be true. "What's the catch?" we ask. As it
turns out, there are two "catches" in the Alcubierre warp drive scheme.
The first is that, while his warp metric is a valid solution of
Einstein's equations of general relativity, we have no idea how to
produce such a distortion of space-time. Its implementation would
require the imposition of radical curvature on extended regions of
space. Within our present state of knowledge, the only way of
producing curved space is by using mass, and the masses we have
available for works of engineering lead to negligible space curvature.
Moreover, even if we could do engineering with mini black holes (which
have lots of curved space near their surfaces) it is not clear how an
Alcubierre warp could be produced.
Alcubierre has also pointed out a more fundamental problem with his
warp drive. General relativity provides a procedure for determining
how much energy density (energy per unit volume) is implicit in a
given metric (or curvature of space-time). He shows that the energy
density is negative, rather large, and proportional to the
square of the velocity with which the warp moves forward. This means
that the weak, strong, and dominant energy conditions of general
relativity are violated, which can be taken as arguments against the
possibility of creating a working Alcubierre drive. Alcubierre,
following the lead of wormhole theorists, argues that quantum field
theory permits the existence of regions of negative energy density
under special circumstances, and cites the Casimir effect as an
example. Thus, the situation for the Alcubierre drive is similar to
that of stable wormholes: they are solutions to the equations of
general relativity, but one would need "exotic matter" with negative
mass-energy to actually produce them, and we have none at the moment.
The possibilities for FTL travel or communication implicit in the
Alcubierre drive raise the possibility of causality violations and "timelike
loops", i.e., back-in-time communication and time travel. Alcubierre
points out that his metric contains no such closed causal loops, and
so is free of their paradoxes. However, he speculates that it would
probably be possible to construct a metric similar to the one he
presented which would contain such loops.
A scheme for converting FTL signaling to back-in-time signaling
requires some gymnastics with moving reference frames to invert the
time sequence of the "send" event and the "receive" event in a signal
transmission. I described such a scheme in a recent column on quantum
tunneling and FTL signaling [ Analog, December- 1995]. In the
case of the` Alcubierre drive, this would probably require either
externally moving the warp generating mechanism at near lightspeed
velocities or embedding one warp within the flat-space region of
The implications of the Alcubierre warp drive for science fiction
are fairly clear. If the theoretical and engineering problems outlined
above could be overcome, we would have FTL travel, fully consistent
with general relativity, that is reminiscent of the warp drives of the
good old-time space operas. Remember, however, that using such a drive
would undoubtedly require the manipulation of planet-scale quantities
of energy (positive or negative). The user would also have to be very
careful to avoid the tidal forces of the distorted-space region at the
edges of the flat-space region containing the ship.
And there is also the question of writing the environmental impact
statement. What would happen to external objects (space dust, rocks,
other ships, asteroids, planets, ...) that happened to lie in the path
of an Alcubierre ship and entered the region of distorted space-time
at the leading edge of the warp, where space is rapidly being
collapsed? The nuclei of any matter transiting that region would first
experience enormous compressional forces, probably form a quark-gluon
plasma reminiscent of the first microsecond of the Big Bang, and then
explode in a flood of pi mesons and other fundamental particles when
the compression forces were released, stealing energy from the warp
field in the process.
A ship traveling in an Alcubierre space warp should be equipped
with plenty of radiation shielding. Perhaps that is not a problem,
since the equations for the metric and the energy density of the warp
do not seem to depend on how much mass is placed in the flat-space
region which is given an FTL velocity.
The Alcubierre Warp Drive:
Miguel Alcubierre, Classical and Quantum Gravity, v. 11, pp.
C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation,
W.H. Freeman (1973).
Note: see also a recent paper by Pfenning and Ford
applying quantum limits to the Alcubierre warp drive.
This page was created by John G. Cramer on 7/12/96.
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