[ Speed-of-Light Limitation ][ Warp Drive ][ Alcubierre drive
][ Micro-Warp Drive ]
[ Negative Energy / Exotic Matter ]
Why can't we go
As an object approaches the speed of light, its mass (in equivalent energy) gets greater and greater. The mass
increases geometrically: The closer to light speed you get, the faster that your mass-equivalent-energy
approaches infinity. Your mass, however, is not the only thing that increases as you go faster; your inertia
increases right in step with your mass. Inertia is the resistance to any new change in motion, so it gets
harder and harder to increase your speed any further. You need to put in more and more energy to get
out that little bit extra. As it works out, you actually have to put in an infinite amount of energy to
ever reach the speed of light (186,000 miles per second). Because of this, it is thought that no physical
object (in other words, one that has mass) can ever reach or exceed the speed of light.
Twin Paradox: For any travelers aboard a starship which is traveling close to the speed of light, time outside
would appear to pass faster and faster. A stationary observer, on the other hand, would see time pass
slower and slower for the people on the starship. If the stationary observer and the starship traveler were
both twins, the one twin which accelerated off to a distant star, and the other twin which stayed at home
would experience time moving at different rates. When the traveling twin returns home, he would find that
much more time has passed on the Earth than he has experienced. His twin would be much older than he,
and so this is called the "Twin Paradox".
Causal Paradoxes: The
existence of an ultimate speed limit sets up a specific temporal
relation between events.
This defines the concept of cause and effect. Because of the light-speed limit, you can only effect events in
your future that you would have been able to reach using the speed of light or slower. Anything outside of
this distance could not be considered to have been caused by you. Likewise, only events in the past that
could have reached you traveling at the speed of light or slower could have affected you. Thus, the existence
of an ultimate speed limit strictly defines the temporal relation of events--in other words, cause and effect.
To travel faster than light--if one could-- would result in all sorts of causal paradoxes; with events seeming
to precede causes.
A "Warp Drive" is the name given to the engine or mechanism which gets us around the faster-than-light (FTL)
speed limitation. The Warp Drive, which has long been an integral part of science fiction, is based on a
loop-hole in Einstein's General Theory of Relativity.
Space-time: According to
Relativity, space is not simply an empty void. It is a continuum:
space and time
taken together make up a 4-dimensional entity called "Space-time".
In Special Relativity space-time gets treated like a flat, static entity, described by Euclidean geometry.
This rigid geometry might make faster-than-light travel or communications impossible, but that's not
the end of the story: the theory of General Relativity superceeds the Special Theory.
In General Relativity space-time is seen as a manifold -- a dynamic entity that can be twisted and distorted
by concentrations of matter and energy. Space-time in this theory has a unique geometry of curved space
all of its own. From the point of view of General Relativity, Special Relativity is seen as a restricted
sub-theory. It still applies but it does so on a local level -- a "local level" is basically any region of space
that's sufficiently small where its curvature can be neglected. In other words, Special Relativity might
forbid objects to move faster than light speed because it is the local speed limit of flat space-time, but
in the broader context of General Relativity spacetime itself can be warped and distorted so there might
be a way of getting around this local rule. To use an analogy: there might be a speed limit to how fast a
ball bearing can roll across a rubber sheet, but the motion or changes in the rubber sheet itself is a separate
issue and can effect the ball bearing's motion. It might take an enormous amount of matter or energy to
create such distortions in space-time, but it is theoretically possible.
Wormholes: One example of
a special kind of space warp is a wormhole. A wormhole is a
shortcut made by
connecting two widely separated locations through the warping of space (bending the rubber sheet). By
taking this short cut through space an object can travel great distances in a short time. The object could
take its time to move at a low speed through the wormhole, observing the local speed-limit laws all the way.
The effect produced is that of the object traveling at many times the speed of light without it ever having to
actually go faster than light.
Einstein Field Equation
The shape and behavior of space-time is determined by its "metric". One of the fundamental ideas in
general relativity is that matter and energy act to curve space-time, i.e. they tell the metric equation how
to behave. It is Einstein's field equations that describe this mathematically:
Here, g ij represents the metric, R ij is the Ricci tensor (it is
essentially derivatives of the metric), R is
the trace of the Ricci tensor (it is like the radius of curvature of space-time), is the cosmological
constant, G is the gravitational constant, and T ij is the stress-energy tensor. The above equation is a
tensor equation and thus represents several equations. The stress energy tensor describes the
distribution of matter and energy and has diagonal components equal to density and pressure. Thus,
given a distribution of matter and energy the field equations govern the form of space-time (i.e. the
metric). The key then is to discover a metric equation that will be consistent with the field equations.
Miguelle Alcubierre worked out how a warp drive might be constructed within the restrictions of General
Relativity, our present "standard model" of gravity. When theoretical physicists use general relativity, their
normal procedure is to start with some distribution of massive objects and to calculate the "metric"-- a
mathematical description of the space-time curvature that the masses would produce. Alcubierre reversed
this procedure. Without worrying about how it might be formed, he worked backwards to construct a metric
which described a volume of flat space containing a spaceship, that would move at superluminal (FTL)
speeds. Alcubierre discovered a metric that was a solution of Einstein’s equations of General Relativity
and would be consistent with Einstein's theories.
Alcubierre envisioned placing a volume of flat space inside a
"bubble’ of highly curved space. In his warp-
drive, this moving section of flat spacetime is created by generating a distortion in the space-time
continuum -- "a space warp", both in front of and behind the volume, that propels it faster than light.
Specifically, space-time is contracted in front of the ship and expanded behind it. In front of the spacecraft
you have to deform space-time to attract it (positive gravitation) using ordinary mass (like the Earth or our
sun). But behind your spacecraft you have to deform space-time to repulse it (negative gravitation).
In the picture: the spacecraft will be in the middle of the small plateau and not moving at all, just warping
the space around it. The spaceship itself
would not be accelerating, and experiences
no time dilation. Only the plateau will be
Using this warp drive is a kind of like
"space-time-surfing"; Within the "warp field" the ship travels like a
surfer does on a wave, and with no speed limit. Although Special Relativity might forbid objects to move
faster than light within spacetime, it is unknown how fast spacetime itself can move.
The idea of expanding spacetime is not new. As the universe expands, new space is created between any
two separated objects. The objects may each be at rest in their local space-time, but nevertheless the
distance between them may grow at a rate that is much greater than the speed of light. According to the
current standard model of cosmology, most of the universe is receding from us at FTL speeds and therefore
is completely isolated from us. And the "Inflationary Universe" theory, depends on the idea that the
universe has expanded faster than light speed during the early moments of the Big Bang.
Alcubierre’s metric uses an analogous expansion of space to drive the warp bubble forward, so by the same
thinking, why shouldn't it be possible to move the flat space-time plateau of the warp drive faster than
light-speed as well?
There are added benefits to this theory. Since the ship within
the warp bubble is at rest in its local space, the
astronauts would have zero acceleration relative to the space around them. They will feel no acceleration
forces when the forward speed of the bubble changes. The ship could slow down or speed up as fast as its
pilots pleased, without the fear that they would be reduced to paste by the inertial forces. Nor will they
experience the "usual" relativistic effects of mass increase and time dilation. No matter how many times the
speed of light or how far they travel, the time outside the ship will be the same as inside. To both the
occupants and outside observers, if the trip takes one year, one year has passed, no more and no less.
Problems with the theory:
1) Manipulating Space-Time - We don't know how to manipulate space-time without using
gravitational mass. But here we have some newer findings indicating the possibility of quantum-gravity
and some yet unresolved phenomena indicating gravity-shielding or possibly anti-gravity effects (e.g.
the Podklednov experiment).
2) Violations of Energy Conditions - It violates
the strong, dominant, and weak energy conditions (WEC)
of General Relativity. To create this effect, you’ll need a ring of negative energy wrapped
around the ship, and lots of it too. The net energy of the warp bubble, as it turns out, is extremely large
and negative. For example, a warp bubble 100 meters in radius that might contain a space ship of
reasonable size would have to have a net negative energy that is roughly ten times larger in magnitude
than the entire (positive) energy of the visible universe.
Although Classical Physics says negative energy cannot exist, Quantum Physics allows it as a
possibility. Exotic matter is a hypothetical form of matter that has "negative energy".
But there is no known physical process to generate it and/or control it.
3) Building Materials - The walls of the bubble
would have to be so thin that they could not be
constructed with matter, even "collapsed matter" of nuclear density.
4) Horizons: (Causally Disconnected portions of Space-time)
- You’ll need a way to control thhis effect
and to turn it on and off at will, but most of the warp bubble is disconnected from a sizable part of
the external negative energy region. Therefore, the surface part of the bubble could not be carried
along and would not be able to be controlled from the ship. It would have to be continuously
generated externally. The drive therefore can not be self-contained or self-operated.
5) Doppler blueshifts - the Alcubierre solution
is unstable against Doppler blueshifts, rendering this
solution impossible. The infinite Doppler blueshift in the so-called light cone horizon, was also
found to be a problem.
Superluminal Subway (Krasnikov's proposal):
One problem with Alcubierre's original model is that the interior of the warp bubble is causally
disconnected from its forward edge. A starship captain on the inside cannot steer the bubble or
turn it on or off; some external agency must set it up ahead of time. To get around this problem,
Krasnikov proposed the idea of a "superluminal subway," a tube of modified space-time (not
the same as a wormhole). It could connect Earth and a distant star. Within the tube, superluminal
travel in one direction is possible. During the outbound journey at sublight speed, a spaceship
crew would create such a tube. On the return journey, they could travel through it at warp speed.
Like warp bubbles, the subway involves negative energy. In fact, it has since been shown that
any scheme for faster-than-light travel will require the use of negative energy.
Van Den Broeck
Modification (Micro-warp drive, "Ship in a Bottle", TARDIS-drive):
Dr. Van Den Broeck has proposed an improvement on Alcubierre’s scheme that appears to solve many
of its problems. Van Den Broeck observed that most of the undesirable effects of Alcubierre’s drive
are related to the volume or surface area of the warp bubble. His solution is simply to make the radius
of the warp bubble so small that the problems go away.
But the problem arises that although one can make the external surface of the bubble arbitrarily small, the
interior of the warp bubble has to remain large enough to hold a ship. To do this, Van Den Broeck again
made use of curved space from General Relativity. The interior volume of a region of space bounded by
a closed surface, because of the space curvature allowed in General Relativity, can be made much larger
than the flat-space volume bounded by its surface. This is sort of like the TARDIS of the sci fi series
"Dr.Who", where the inside of the ship is much larger than the outside telephone-booth volume that it
The new metric of the Van Den Broeck
warp bubble is like a bulls-eye target of
4 concentric rings.
Region 1: the center (Region 1) surrounded
by three concentric rings (Regions 2 to 4).
The central sphere in Region 1 is flat space
large enough to hold a spaceship.
Region 2 is a spherical shell containing distorted
space that connects the large interior volume
of Region 1 to an exterior region that is
smaller in radius by a factor of 1/alpha.
Region 3 is a transition region of flat space, a
spherical shell with a volume much less
than that of Region 1.
Region 4 is a spherical shell that is Alcubierre’s
warp bubble, but now with a very small radius.
Van Den Broeck makes the radius of Region 1 about 100 meters (big
enough for our starship), and he
sets alpha to 10^34. This determines that Region 4 will be about 3x10^-32 meters in radius. With
such a small radius, if the warp bubble travels at 10 times the velocity of light the amount of negative
mass-energy required is only about –0.06 grams. And even if it travels at 100 times the velocity of light,
it still only requires about –56 kilograms of negative mass-energy. A quantity of negative mass-energy
is also required to create Region 2, where the volume of space is compressed from inside to outside,
but Van Den Broeck calculates that only about –4 grams is needed here. These small quantities of
negative energy eliminate many of the problems of Alcubierre’s original concept.
Problems with the theory:
1) Energy Density Requirement - While the magnitude of energy required to form a warp bubble
becomes more reasonable in Van Den Broeck’s warp drive, the energy density requirement (both
positive and negative) still remains unphysically large.
2) Staying in Our Universe - Van Den Broeck’s warp
drive is a large volume of flat space that is
connected to normal space by a tiny "neck". It therefore resembles the more familiar General Relativity
topologies of wormholes or "baby universes" and perhaps has a similar behavior. This raises the issue
of how the neck is prevented from pinching off altogether, isolating our space travelers in a new
universe of their own rather than transporting them to a new part of the old one.
3) Seeing and Steering - Since the diameter of
the warp bubble is many orders of magnitude smaller than
a wavelength of visible light (about 4x10^-7 meters) there would be no possibility of seeing out from
inside the bubble. Any trip would be a blind one, with no possibility of seeing or steering.
4) The Planck Length (1.62 x 10^
-35 meters) - the minimum length-scale of the universe. As any
measurement of any part of the warp drive approaches this limit, quantum effects become unavoidable.
Region 4, whose radius measures 3x10^-32 meters is much smaller than a proton and is getting close to
5) Quantum Gravity - Van den Broeck's calculations were
performed on the basis of General Relativity
which does not take into account the quantum effects at "small" dimensions. In particular, extra space
dimensions affecting gravity may be rolled up into loops about a millimeter in diameter. If this is the
case, it would modify General Relativity at the millimeter scale and would almost certainly render
Van Den Broeck’s metric unachievable.
6) Entering and Exiting the Bubble - Van Den Broeck’s
calculations indicate that slowing the bubble to
a near stop might permit it to be expanded to any desired size. However, such an expansion would
decrease the wall thickness, and it is not clear what would happen if the wall thickness became smaller
than the Planck length.
Negative Energy /
[ What Neg Energy is NOT ][ Is it even possible? ][ Ways to make it ][ Nature's Constraints on Neg Energy ]
What Negative Energy is
NOT (Negative energy is often confused with other concepts):
Antimatter - Antimatter has positive energy. When an electron and its antiparticle, a positron, collide, they
annihilate. The end products are gamma rays, which carry positive energy. If antiparticles were composed
of negative energy, the positive and negative energy would cancel and the interaction would result in a
final energy of zero, or nothing -- very different from the explosion you actually get when you combine
matter and antimatter.
The energy associated with the cosmological
constant - This energy is postulated for inflatiionary models
of the universe. The cosmological constant represents negative pressure but it is positive energy. (Some
authors also call this exotic matter; but to avoid confusion here, the term "exotic matter" will be reserved
for negative energy densities only.)
How can you have a
The theory of Relativity holds that mass and energy will have different values depending on your frame of
reference. The idea that the mass (or energy) density in any one frame would always be at least equal to or
greater than zero is called the "weak energy condition." An object with negative mass would be less
massive than empty space.
Vacuum Fluctuations - In the old days of classical mechanics the idea of a vacuum was simple. The vacuum
was what remained if you emptied a container of all its particles and lowered the temperature down to
absolute-zero. With the arrival of quantum mechanics, however, our notion of a vacuum has completely
changed. All fields, whether electric, magnetic, or of any other type, have fluctuations. They arise from
Heisenberg's uncertainty principle, which requires that their energy density fluctuates randomly.
In other words at any given moment the actual energy density value will vary around a constant, mean
value. Even if the energy density is zero on average, as it is in a vacuum, it fluctuates. Meaning that even a
perfect vacuum at absolute zero will have fluctuating fields known as "vacuum fluctuations", the mean
energy of which corresponds - in particular for electromagnetic fields - to half the energy of a photon. Thus,
the "quantum vacuum" can never remain empty in the classical sense of the term; it is a roiling sea of
"virtual" particles spontaneously popping in and out of existence. So in Quantum theory, the usual notion
of zero energy has been redefined to mean the vacuum including all its fluctuations. Using this definition, it
turns out that if one can somehow contrive to dampen the undulations, the vacuum will have less energy
than it normally does – in other words, less than zero energy.
The vacuum fluctuations are not some abstraction of a physicist's mind. They have observable
consequences that can be directly visualized in experiments on a microscopic scale. For example, an atom in
an excited state will not remain there infinitely long, but will return to its ground state by spontaneously
emitting a photon. This phenomenon is a consequence of vacuum fluctuations. Its like trying to hold a
pencil upright on the end of your finger. It will stay there if your hand is perfectly stable and nothing
perturbs the equilibrium. But the slightest perturbation will make the pencil fall into a more stable
equilibrium position. Similarly, vacuum fluctuations cause an excited atom to fall into its ground state.
Methods of producing
[ Casimir Effect ][ Scharnhorst Effect ][ Evaporating BlackHoles ][ Squeezed Vacuum ]
The Casimir Effect
- A method for producing negative energgy which arises from introducing
boundaries into space.
All space is filled with vacuum fluctuations. There is,
however, a way of suppressing part of the vacuum-
fluctuations, in effect making "empty" space more empty.
In 1948 Dutch physicist Hendrik B. G. Casimir showed
that two uncharged parallel metal plates can alter the
vacuum-fluctuations. All electromagnetic fields,
including those of the vacuum, have a characteristic
"spectrum" containing many different frequencies. In
a free vacuum all of the frequencies are of equal importance.
Now consider the gap or "cavity" between two plane mirrors
(the uncharged metal plates of Casimir's experiment) which
are placed parallel to one another. Inside the cavity, where
the field is reflected back and forth between the mirrors, the situation is different than in free space.
The field is amplified if integer multiples of half a wavelength can fit exactly inside the cavity. This
wavelength corresponds to a "cavity resonance". At other wavelengths, in contrast, the field is suppressed.
Vacuum fluctuations are suppressed or enhanced depending on whether their frequency corresponds to a
cavity resonance or not. The presence of this pair of conducting walls suppresses all virtual photons with
wavelengths larger than twice the plate separation distance, because the wave structure of such a photon
would intercept the walls in less than half a wavelength and the electric field of the wave would have a
forbidden non-zero value within the conductor. Casimir realized that between the two plates, only those
unseen electromagnetic waves whose wavelengths fit a whole number of times into the gap should be
counted when calculating the vacuum energy. As the gap between the plates is narrowed, fewer waves can
contribute to the vacuum energy and so the energy density between the plates falls below the energy density
of the surrounding space. Thus, the more closely the plates are spaced, the broader becomes the spectrum
of virtual photons that are suppressed, and the vacuum between the plates becomes "emptier" of vacuum-
fluctuations and lower in energy density. The region inside a Casimir cavity actually has a negative
energy density. Since the energy density of a normal "empty space" vacuum is defined to be zero, and
since the energy density between the conductors of a Casimir cavity is less than normal, it must be negative.
In effect, the plates reduce the fluctuations in the gap between them; thus creating negative energy and
pressure, which pulls the plates together. The narrower the gap, the more negative the energy and pressure,
and the stronger is the attractive force pulling them together.
The Casimir (Vacuum-pressure) Force -
The narrower the gap, the more negative the energy and pressure, and the stronger is the attractive force
pulling them together. This physical force can do work (like lifting a weight) on an external system.
Though very weak, it has been measured in the laboratory.
An important physical quantity when discussing the Casimir force is the "field radiation pressure".
Every field - even the vacuum field - carries energy. As all electromagnetic fields can propagate in
space they also exert pressure on surfaces, just as a flowing river pushes on a floodgate. This
radiation pressure increases with the energy - and hence the frequency - of the electromagnetic field.
At a cavity-resonance frequency the radiation pressure inside the cavity is stronger than outside and
the mirrors are therefore pushed apart. In contrast, when out of resonance, the radiation pressure
inside the cavity is smaller than outside and the mirrors are drawn towards each other.
The attractive Casimir force between two plates of area A separated by a distance d can be calculated to be,
pi h c
F = -------- A
It is seen that the force, F, is directly proportional to the
cross-sectional area of the plates and inversely
proportional to the 4th power of the distance between the plates. Apart from these geometrical
quantities the force depends only on fundamental values - Planck's constant, h, and the speed of light, c.
As can be seen, there will be a 16-fold increase in force every time
the distance, d, between the mirrors
is halved. And while the Casimir force is too small to be observed for mirrors that are several metres
apart, it can be measured when the the mirrors are brought within microns of each other. For example,
two mirrors with an area of 1 square cm separated by a distance of 1 µm have an attractive Casimir-
force of about 10-7 N - roughly the weight of a water droplet that is half a millimetre in diameter.
Although this force might appear small, at distances below a micrometre the Casimir force becomes
the strongest force between two neutral objects. Indeed at separations of 10 nm - about a hundred times
the typical size of an atom - the Casimir effect produces the equivalent of 1 atmosphere of pressure.
Particles other than the photon also contribute a small effect but
only the photon force is measurable,
All bosons (a photon is one of the boson-type particles) produce an attractive Casimir force while
fermions make a repulsive contribution, If electromagnetism was supersymmetric there would be
fermionic "photinos" whose contribution would exactly cancel that of the photons and there would
be no Casimir effect, The fact that the Casimir effect exists shows that if supersymmetry exists in
nature it must be a broken symmetry.
According to the theory the total zero point energy in the vacuum is infinite when summed over all the
possible photon modes, and the Casimir effect arises from a difference of energies in which the infinities
cancel. The energy of the vacuum is a puzzle in theories of quantum gravity since it should act
gravitationally and produce a large cosmological constant which would cause space-time to curl up.
The solution to the inconsistency is expected to be found in a theory of quantum gravity.
Problems in calculating the Casimir force
for real objects:
1) Frequency Dependent Reflection - Real plates do not reflect all frequencies perfectly. They reflect
some frequencies well - or even nearly perfectly - while others are reflected badly.&nbssp; In addition, all
mirrors become transparent at very high frequencies. When calculating the Casimir force the frequency-
dependent reflection coefficients of the mirrors have to be taken into account.
2) Thermal Fluctuations - Experiments are never
carried out at absolute zero - as originally envisaged
in Casimir's calculations - but at room temperature. This causes thermal - as well as vacuum -
fluctuations to come into play. These thermal fluctuations can produce their own radiation pressure
and create a bigger Casimir force than expected. For example, the Casimir force between two plane
mirrors 7 µm apart is twice as large at room temperature than at absolute zero. Fortunately, thermal
fluctuations at room temperature are only important at distances above 1 µm, below which the
wavelength of the fluctuations is too big to fit inside the cavity.
3) Variations in Distances - Real mirrors are
not perfectly smooth. the Casimir force is very sensitive to
small changes in distance.
Similar to the Casimir effect, it is predicted that a moving
boundary, such as a moving mirror, could
produce a flux of negative energy.
Scharnhorst Effect: There is another interesting
possibility for breaking the light-barrier by
an extension of the Casimir effect. Light in normal empty space is “slowed” by interactions with the
unseen waves or particles which exist in the quantum vacuum. But within the energy-depleted region
of a Casimir cavity, light should travel slightly faster because there are fewer of these obstacles.
K. Scharnhorst of the Alexander von Humboldt University in Berlin published calculations showing
that, under the right conditions, light can be induced to break the usual light-speed barrier.
Under normal laboratory conditions this increase in speed is incredibly small, but future technology may
find ways of producing a much greater Casimir effect in which light can travel much faster. The effect
that Scharnhorst's paper discusses is independent of frequency and wavelength, so it is not the result of
dispersive (wavelength-dependent) effects. Phase and group velocity are equal and increase together.
Scharnhorst's effect is a consequence of quantum electrodynamics (QED), the quantum theory of light and
electromagnetism. The theory of quantum electrodynamics tells us that empty space, when examined on
a very small distance scale, is not empty at all; it seethes with the fireworks of vacuum fluctuations. Pairs
of "virtual" (energy non-conserving) particles of many kinds continually wink into existence, live briefly
on the energy credit extended by Heisenberg's uncertainty principle, and then annihilate and vanish when
the bill for their energy debts falls due a few picoseconds or femtoseconds later. The seemingly smooth
passage of a photon or electron through space at the QED distance scale, is revealed to be a punctuated chain
of interactions and transformations involving virtual particles. For example, a traveling photon may briefly
be transformed into a virtual electron-positron pair, which moves forward less than one photon wavelength
before annihilating to create a new photon indistinguishable from the old one. Light in normal space is
"slowed" by quantum fluctuations which causes it to spend part of the time as an electron-positron pair.
During the photon's brief
existence as a pair, one of the virtual particles may initiate a "game of catch" using a virtual photon as the
ball, tossing it one or more times to itself or its partner. These QED complications of the smooth passage
of photons through space have the effect of making the photon travel more slowly. In part, this is because
the photon spends a fraction of its existence as an electron-positron pair which can only travel at sub-light
velocity. Scharnhorst has given a new twist to the Casimir effect by considering the velocity v of a photon
travelling across the gap between the plates. If the plates are separated by a gap d, the Casimir effect
suppresses all virtual photons with a wavelength of 2d or greater. Because these virtual photons are absent,
they cannot participate in games of catch between virtual particles. Therefore a real photon travelling
between the plates spends less time as an electron-positron pair because the QED vacuum fluctuations are
suppressed. For this reason, the photon travels faster across the gap. Its speed of travel through normal
vacuum is c, so its speed v in the negative energy vacuum between the plates is greater than c! Einstein's
lightspeed barrier has been broached by a photon! It travels slightly faster in the space of lower (and
negative) energy density between the Casimir plates, where part of the quantum fluctuations are suppressed.
1) Plate gap -
With a plate gap of d, v/c = 1 - (1.6 × 10-60 × d-4). If we make d as small as experimentally possible,
say 1 nanometer (= 1 × 10-9 m) or about ten atomic diameters, we find that (v-c) = 1.6 × 10-24 c.
This is an unmeasurably small change in the velocity of the photon, and only for a very small travel
distance at that.
But even such a small boost in speed comes as a surprise to those who had considered c as the ultimate
speed limit. Moreover, special relativity says that if in one inertial reference frame an object travels only
one part in 1024 times faster than c, one can find another reference frame in which the departure and
arrival times of the object are simultaneous and therefore the velocity is infinite.
Suppose that between two such plates we make a gap on the order of nuclear dimensions, about a
femtometer (10-15 m). If one takes Scharnhorst's equation for index of refraction at face value, c/v goes
to zero and a photon travels at infinite speed when the gap between the plates is decreased to about
1.13 × 10-15 m, or about the diameter of a proton. Of course, the approximations used in the calculation
may not be valid because of higher-order effects at such small distances.
2) Lumpiness of materials - Normal metals are made of atoms
which become very lumpy and non-planar
at the nanometer scale. So something else is needed which is smoother - something non-atomic.
A solution might be to make a pair of Casimir plates from something like superconducting neutronium.
Or perhaps out of the two-dimensional equivalent of cosmic string -- a "cosmic wall". Cosmic walls,
if they exist at all, are supposed to be smooth down to Planck-scale dimensions (10-35 m) and also are
Black Holes -
The concept of negative energy arises in several areas of modern physics. It has an intimate link with
black holes, those mysterious objects whose gravitational field is so strong that nothing can escape from
within their boundary, the event horizon. In 1974 Stephen W. Hawking of the University of Cambridge
made his famous prediction that black holes evaporate by emitting radiation. A
black hole radiates energy at a rate inversely proportional to the square of its mass. Although the
evaporation rate is large only for subatomic size black holes, it provides a crucial link between the laws of
black holes and the laws of thermodynamics. The Hawking radiation allows black holes to come into
thermal equilibrium with their environment.
At first glance, evaporation leads to a contradiction. The horizon
is a one-way street; energy can only flow
inward. So how can a black hole radiate energy outward? Because energy must be conserved, the
production of positive energy - which distant observers see as the Hawking radiation – is accompanied by a
flow of negative energy into the hole. Here the negative energy is produced by the extreme space-time
curvature near the hole, which disturbs the vacuum fluctuations. In this way, negative energy is required
for the consistency of the unification of black hole physics with thermodynamics.
Vacuum States - Researchers in quantum optics have
created special states of fields
in which destructive quantum interference suppresses the vacuum fluctuations. These so-called squeezed
vacuum states involve negative energy. More precisely, they are associated with regions of alternating
positive and negative energy. The total energy averaged over all space remains positive; squeezing the
vacuum creates negative energy in one place at the price of extra positive energy elsewhere. A typical
experiment involves laser beams passing through nonlinear optical materials. The intense laser light induces
the material to create pairs of light quanta, photons. These photons alternately enhance and suppress the
vacuum fluctuations, leading to regions of positive and negative energy, respectively.
Waves of light ordinarily have a positive or zero
energy density at different points in space.
But in a so-called squeezed state, the energy
density at a particular instant in time can become negative
at some locations. To compensate, the peak positive density must increase.
Quantum Inequalities -- Restrictions on the Nature and Durattion of Negative Energy:
Fortunately (or not, depending on your point of view), although quantum theory allows the existence of
negative energy, it also appears to place strong restrictions - known as quantum inequalities - on its
magnitude and duration.
The inequalities bear some resemblance to the uncertainty principle.
They say that a beam of negative
energy cannot be arbitrarily intense for an arbitrarily long time. The permissible magnitude of the negative
energy is inversely related to its temporal or spatial extent. An intense pulse of negative energy can last for
a short time; a weak pulse can last longer.
The Second Law of Thermodynamics:
Negative energy is so strange that one might think it must violate some law of physics. Before and after
the creation of equal amounts of negative and positive energy in previously empty space, the total energy
is zero, so the law of conservation of energy is obeyed. But there are many phenomena that conserve
energy yet never occur in the real world. A broken glass does not reassemble itself, and heat does not
spontaneously flow from a colder to a hotter body. Such effects are forbidden by the second law of
thermodynamics. This general principle states that the degree of disorder of a system–its entropy–cannot
decrease on its own without an input of energy. Thus, a refrigerator, which pumps heat from its cold
interior to the warmer outside room, requires an external power source. Similarly, the second law also
forbids the complete conversion of heat into work.
Negative energy potentially conflicts with the second law. Imagine
an exotic laser, which creates a steady
outgoing beam of negative energy. Conservation of energy requires that a byproduct be a steady stream of
positive energy. One could direct the negative energy beam off to some distant corner of the universe,
while employing the positive energy to perform useful work. This seemingly inexhaustible energy supply
could be used to make a perpetual-motion machine and thereby violate the second law. If the beam were
directed at a glass of water, it could cool the water while using the extracted positive energy to power a
small motor–providing a refrigerator with no need for external power. These problems arise not from the
existence of negative energy per se but from the unrestricted separation of negative and positive energy.
The quantum inequalities prevent violations of the second law. If one tries to use a pulse of negative
energy to cool a hot object, it will be quickly followed by a larger pulse of positive energy, which reheats
the object. Furthermore, the positive pulse that necessarily follows an initial negative pulse must do
more than compensate for the negative pulse; it must overcompensate. The amount of overcompensation
increases with the time interval between the pulses. Therefore, the negative and positive pulses can
never be made to exactly cancel each other. The positive energy must always dominate–an effect
known as quantum interest. If negative energy is thought of as an energy loan, the loan must be
repaid with interest. The longer the loan period or the larger the loan amount, the greater is the interest.
Also, the larger the loan, the smaller is the maximum allowed loan period. Nature is a shrewd banker
and always calls in its debts.
A weak pulse of negative energy could remain separated from its
positive counterpart for a
longer time, but its effects would be indistinguishable from normal thermal fluctuations. Attempts to
capture or split off negative energy from positive energy also appear to fail. One might intercept an energy
beam, say, by using a box with a shutter. By closing the shutter, one might hope to trap a pulse of
negative energy before the offsetting positive energy arrives. But the very act of closing the shutter creates
an energy flux that cancels out the negative energy it was designed to trap.
The larger the magnitude of the negative energy, the nearer must be
its positive energy counterpart.
These restrictions are independent of the details of how the negative energy is produced.
Pulses of negative energy are permitted
by quantum theory but only under three
First, the longer the pulse lasts, the
weaker it must be (a, b).
Second, a pulse of positive energy must
follow. The magnitude of the positive
pulse must exceed that of the initial
Third, the longer the time interval between
the two pulses, the larger the positive one
must be - an effect known as quantum
Inequalities & the Casimir Effect:
In the Casimir effect, the negative energy density between the plates can persist indefinitely, but large
negative energy densities require a very small plate separation. The magnitude of the negative energy
density is inversely proportional to the fourth power of the plate separation. Just as a pulse with a very
negative energy density is limited in time, very negative Casimir energy density must be confined between
closely spaced plates. According to the quantum inequalities, the energy density in the gap can be made
more negative than the Casimir value, but only temporarily. In effect, the more one tries to depress the
energy density below the Casimir value, the shorter the time over which this situation can be maintained.
Inequalities & Space Warps:
When applied to wormholes and warp drives, the quantum inequalities typically imply that such structures
must either be limited to submicroscopic sizes, or if they are macroscopic the negative energy must be
confined to incredibly thin bands. A submicroscopic wormhole would have a
throat radius of no more than about 10-32 meter. This is only slightly larger than the Planck length, 10-35
meter, the smallest distance that has definite meaning. It is possible to have models of
wormholes of macroscopic size but only at the price of confining the negative energy to an extremely thin
band around the throat. For example, in one model a throat radius of 1 meter requires the negative energy
to be a band no thicker than 10^-21 meter, a millionth the size of a proton. It has been estimated that the
negative energy required for this size of wormhole has a magnitude equivalent to the total energy
generated by 10 billion stars in one year. The situation does not improve much for larger wormholes. For
the same model, the maximum allowed thickness of the negative energy band is proportional to the cube
root of the throat radius. Even if the throat radius is increased to a size of one light-year, the negative
energy must still be confined to a region smaller than a proton radius, and the total amount required
increases linearly with the throat size.
It seems that wormhole engineers face daunting problems. They must
find a mechanism for confining
large amounts of negative energy to extremely thin volumes. So-called cosmic strings, hypothesized in
some cosmological theories, involve very large energy densities in long, narrow lines. But all known
physically reasonable cosmic-string models have positive energy densities.
Unfettered negative energy would also have profound consequences for black holes. When a black hole
forms by the collapse of a dying star, general relativity predicts the formation of a singularity, a region
where the gravitational field becomes infinitely strong. At this point, general relativity–and indeed all
known laws of physics–are unable to say what happens next. This inability is a profound failure of the
current mathematical description of nature. So long as the singularity is hidden within an event horizon,
however, the damage is limited. The description of nature everywhere outside of the horizon is
unaffected. For this reason, Roger Penrose of Oxford proposed the cosmic censorship hypothesis: there
can be no naked singularities, which are unshielded by event horizons.
For special types of charged or rotating black holes– known as
extreme black holes–even a small increase
in charge or spin, or a decrease in mass, could in principle destroy the horizon and convert the hole into a
naked singularity. Attempts to charge up or spin up these black holes using ordinary matter seem to fail
for a variety of reasons. One might instead envision producing a decrease in mass by shining a beam of
negative energy down the hole, without altering its charge or spin, thus subverting cosmic censorship. One
might create such a beam, for example, using a moving mirror. In principle, it would require only a tiny
amount of negative energy to produce a dramatic change in the state of an extreme black hole. Therefore,
this might be the scenario in which negative energy is the most likely to produce macroscopic effects.
There are restrictions similar to the Quantum Inequalities on possible violations of cosmic censorship.
A pulse of negative energy injected into a charged black hole might momentarily destroy the horizon,
exposing the singularity within. But the pulse must be followed by a pulse of positive energy, which
would convert the naked singularity back into a black hole - a scenario dubbed "cosmic flashing". The
best chance to observe cosmic flashing would be to maximize the time separation between the negative
and positive energy, allowing the naked singularity to last as long as possible. But then the magnitude
of the negative energy pulse would have to be very small, according to the quantum inequalities. The
change in the mass of the black hole caused by the negative energy pulse will get washed out by the
normal quantum fluctuations in the hole's mass, which are a natural consequence of the uncertainty-
principle. The view of the naked singularity would thus be blurred, so a distant observer could not
unambiguously verify that cosmic censorship had been violated.