from AndersonInstitute Website
The ability to control time in both a forward and backwards direction is possible within the laws of our mathematics and physics. The chart below (click for larger view) compares ten different technologies an methods.
Key characteristics are identified for each and described below.
Under each key characteristic is a column with either a solid or empty circle.
The time control technologies and methods above include the following:
The correct wavelength combined with the proper tunneling barrier makes it possible to pass signals faster than light, backwards in time.
In the diagram above light pulses consisting of waves of various frequencies are shot toward a 10 centimeter chamber containing cesium vapor.
All information about
the incoming pulse is contained in the leading edge of its waves.
This information is all the cesium atoms need to replicate the pulse
and send it out the other side.
This is followed by more detail describing the phenomenon below.
Wave-mechanical tunneling (also called quantum-mechanical tunneling, quantum tunneling, and the tunnel effect) is an evanescent wave coupling effect that occurs in the context of quantum mechanics because the behavior of particles is governed by Schrödinger's wave-equation.
All wave equations exhibit
evanescent wave coupling effects if the conditions are right. Wave
coupling effects mathematically equivalent to those called
"tunneling" in quantum mechanics can occur with Maxwell's
wave-equation (both with light and with microwaves), and with the
common non-dispersive wave-equation often applied (for example) to
waves on strings and to acoustics.
In optics, medium type 1 might be glass, medium type 2 might
be vacuum. In quantum mechanics, in connection with motion of a
particle, medium type 1 is a region of space where the particle
total energy is greater than its potential energy, medium type 2 is
a region of space (known as the "barrier") where the particle total
energy is less than its potential energy.
Depending on the wave equation being used, the leaked
amplitude is interpreted physically as traveling energy or as a
traveling particle, and, numerically, the ratio of the square of the
leaked amplitude to the square of the incident amplitude gives the
proportion of incident energy transmitted out the far side, or (in
the case of the Schrödinger equation) the probability that the
particle "tunnels" through the barrier.
For electrons the thickness of
"medium type 2" (called in this context "the tunneling barrier") is
typically a few nanometers; for alpha-particles tunneling out of a
nucleus the thickness is very much less; for the analogous
phenomenon involving light the thickness is very much greater.
In medium type 1 the kinetic energy would be positive, in medium
type 2 the kinetic energy would be negative. There is no
inconsistency in this, because particles cannot physically be
located at a point: they are always spread out ("delocalized") to
some extent, and the kinetic energy of the delocalized object is
But there never was any convincing experimental evidence that this was true when very small objects and very small distances are involved, and we now know that this viewpoint was mistaken. However, because it is still traditional to teach students early in their careers that particles behave like points, it sometimes comes as a big surprise for people to discover that it is well established that traveling physical particles always physically obey a wave-equation (even when it is convenient to use the mathematics of moving points).
Clearly, a hypothetical classical point particle analyzed according to Newton's Laws could not enter a region where its kinetic energy would be negative. But, a real delocalized object, that obeys a wave-equation and always has positive kinetic energy, can leak through such a region if conditions are right.
An approach to tunneling that avoids mention of the concept of "negative kinetic energy" is set out below in the section on "Schrödinger equation tunneling basics".
tunneling of an electron
An electron approaching a barrier has to be represented as a wave-train.
This wave-train can sometimes be
quite long – electrons in some materials can be 10 to 20 nm long.
This makes animations difficult. If it were legitimate to represent
the electron by a short wave-train, then tunneling could be
represented as in the animation alongside.
Calculations of this kind make the general physical nature of tunneling clear.
One would also like to be able to calculate exact tunneling probabilities for barrier models that are physically more realistic. However, when appropriate mathematical descriptions of barriers are put into the Schrödinger equation, then the result is an awkward non-linear differential equation. Usually, the equation is of a type where it is known to be mathematically impossible in principle to solve the equation exactly in terms of the usual functions of mathematical physics, or in any other simple way.
Mathematicians and mathematical physicists have been working on this problem since at least 1813, and have been able to develop special methods for solving equations of this kind approximately. In physics these are known as "semi-classical" or "quasi-classical" methods. A common semi-classical method is the so-called WKB approximation (also known as the "JWKB approximation").
The first known attempt to use such methods to solve a tunneling problem in physics was made in 1928, in the context of field electron emission.
It is sometimes considered that the first people to get the mathematics of applying this kind of approximation to tunneling fully correct (and to give reasonable mathematical proof that they had done so) were N. Fröman and P.O. Fröman, in 1965. Their complex ideas have not yet made it into theoretical-physics textbooks, which tend to give simpler (but slightly more approximate) versions of the theory.
An outline of one particular semi-classical method is given below.
Three notes may be helpful. In general, students taking physics courses in quantum mechanics are presented with problems (such as the quantum mechanics of the hydrogen atom) for which exact mathematical solutions to the Schrödinger equation exist.
Tunneling through a realistic barrier is a reasonably basic physical phenomenon. So it is sometimes the first problem that students encounter where it is mathematically impossible in principle to solve the Schrödinger equation exactly in any simple way. Thus, it may also be the first occasion on which they encounter the "semi-classical-method" mathematics needed to solve the Schrödinger equation approximately for such problems.
surprisingly, this mathematics is likely to be unfamiliar, and may
feel "odd". Unfortunately, it also comes in several different
variants, which doesn't help.
The precise nature of this wave-like behavior is, however,
a much deeper matter, beyond the scope of this article on tunneling.
For this reason, some writers prefer to call the phenomenon "wave-mechanical tunneling.
Classically, the particle is confined to the
nucleus because of the high energy requirement to escape the very
strong potential. Under this system, it takes an enormous amount of
energy to pull apart the nucleus. In quantum mechanics, however,
there is a probability the particle can tunnel through the potential
and escape. Gamow solved a model potential for the nucleus and
derived a relationship between the half-life of the particle and the
energy of the emission.
Today the theory of tunneling is even applied to the early cosmology of the universe.
Quantum tunneling was later applied to other situations, such as the cold emission of electrons, and perhaps most importantly semiconductor and superconductor physics.
Phenomena such as field emission, important to flash memory, are
explained by quantum tunneling. Tunneling is a source of major
current leakage in Very-large-scale integration (VLSI) electronics,
and results in the substantial power drain and heating effects that
plague high-speed and mobile technology.
It has even been shown, in the enzyme glucose oxidase,
that oxygen nuclei can tunnel under physiological conditions.
The closer to the speed of light, the further into the future the travel.
The key characteristics of the application of near-lightspeed travel for time control and time travel are presented in the picture below.
This is followed by more detail describing the effect below.
The ship can ride the wave to accelerate to high speeds and time travel.
The Alcubierre drive, also known as the
Alcubierre metric or Warp Drive, is a mathematical model of a
spacetime exhibiting features reminiscent of the fictional "warp
drive" from Star Trek, which can travel "faster than light"
(although not in a local sense - see below).
This is followed by more detail describing the effect below.
Alcubierre Warp Drive Description
In 1994, the Mexican physicist Miguel Alcubierre proposed a method of stretching space in a wave which would in theory cause the fabric of space ahead of a spacecraft to contract and the space behind it to expand.
The ship would ride this wave inside a region known as a warp bubble of flat space. Since the ship is not moving within this bubble, but carried along as the region itself moves, conventional relativistic effects such as time dilation do not apply in the way they would in the case of a ship moving at high velocity through flat spacetime.
Also, this method of travel does not actually involve moving faster than light in a local sense, since a light beam within the bubble would still always move faster than the ship; it is only "faster than light" in the sense that, thanks to the contraction of the space in front of it, the ship could reach its destination faster than a light beam restricted to travelling outside the warp bubble.
Thus, the Alcubierre drive
does not contradict the conventional claim that relativity forbids a
slower-than-light object to accelerate to faster-than-light speeds.
The Alcubierre Metric defines the so-called warp drive spacetime.
This is a Lorentzian manifold which,
if interpreted in the context of general relativity, exhibits
features reminiscent of the warp drive from Star Trek: a warp bubble
appears in previously flat spacetime and moves off at effectively
superluminal speed. Inhabitants of the bubble feel no inertial
effects. The object(s) within the bubble are not moving (locally)
faster than light, instead, the space around them shifts so that the
object(s) arrives at its destination faster than light would in
where α is the lapse function that gives the interval of proper time between nearby hypersurfaces, βI is the shift vector that relates the spatial coordinate systems on different hypersurfaces and γij is a positive definite metric on each of the hypersurfaces.
The particular form that Alcubierre studied is defined by:
...with R > 0 and σ > 0 arbitrary parameters. Alcubierre's specific form of the metric can thus be written;
With this particular form of the metric, it can be shown that the energy density measured by observers whose 4-velocity is normal to the hypersurfaces is given by
where g is the determinant of the metric tensor. Thus, as the energy density is negative, one needs exotic matter to travel faster than the speed of light.
The existence of exotic matter is not theoretically ruled out, the Casimir effect and the accelerating universe both lending support to the proposed existence of such matter.
However, generating enough exotic matter and sustaining it to perform feats such as faster-than-light travel (and also to keep open the 'throat' of a wormhole) is thought to be impractical.
Low has argued that within the context of general relativity, it is impossible to construct a warp drive in the absence of exotic matter.
It is generally believed that a consistent
theory of quantum gravity will resolve such issues once and for all.
For those familiar with the effects of special relativity, such as Lorentz contraction and time dilation, the Alcubierre metric has some apparently peculiar aspects.
In particular, 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 g-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.
Alcubierre interpreted his "warp bubble" in terms of a
contraction of "space" ahead of the bubble and an expansion behind.
But this interpretation might be misleading, since the contraction
and expansion actually refers to the relative motion of nearby
members of the family of ADM observers.
This practice means that the solution can violate various energy conditions and require exotic matter.
The need for exotic matter leads to questions about whether it is actually possible to find a way to distribute the matter in an initial spacetime which lacks a "warp bubble" in such a way that the bubble will be created at a later time. Yet another problem is that, according to Serguei Krasnikov, it would be impossible to generate the bubble without being able to force the exotic matter to move at locally FTL speeds, which would require the existence of tachyons.
Some methods have
been suggested which would avoid the problem of tachyonic motion,
but would probably generate a naked singularity at the front of the
Significant problems with the metric of this form stem from the fact that all known warp drive spacetimes violate various energy conditions.
It is true that certain experimentally verified quantum phenomena, such as the Casimir effect, when described in the context of the quantum field theories, lead to stress-energy tensors which also violate the energy conditions and so one might hope that Alcubierre type warp drives could perhaps be physically realized by clever engineering taking advantage of such quantum effects.
However, if certain quantum inequalities conjectured by Ford and Roman hold, then the energy requirements for some warp drives may be absurdly gigantic, e.g. the energy -1067gram equivalent might be required to transport a small spaceship across the Milky Way galaxy. This is orders of magnitude greater than the mass of the universe.
Counterarguments to these apparent problems have been offered, but not everyone is convinced they can be overcome.
Chris Van Den Broeck, in 1999, has tried to address the potential issues.
By contracting the 3+1 dimensional
surface area of the 'bubble' being transported by the drive, while
at the same time expanding the 3 dimensional volume contained
inside, Van Den Broeck was able to reduce the total energy needed to
transport small atoms to less than 3 solar masses. Later, by
slightly modifying the Van Den Broeck metric, Krasnikov reduced the
necessary total amount of negative energy to a few milligrams.
It is only able to travel routes which, like a railroad, have first been equipped with the necessary infrastructure.
However, it is
necessary to place devices along the route in advance, and since the
pilot cannot do this while "in transit", the bubble cannot be used
for the first trip to a distant star. In other words, to travel to
Vega (which is 26 light-years from the Earth) one first has to
arrange everything so that the bubble moving toward Vega with a
superluminal velocity would appear and these arrangements will
always take more than 26 years.
Coule argues that an analogous objection will
apply to any proposed method of constructing an Alcubierre Drive.
As the same time, special relativity states that this would require infinite energy.
Faster-than-light (also superluminal or FTL) communications and travel refer to the propagation of information or matter faster than the speed of light.
special theory of relativity, a particle (that has mass) with
subluminal velocity needs infinite energy to accelerate to the speed
of light, although special relativity does not forbid the existence
of particles that travel faster than light at all times.
This is followed by more detail describing the effect below.
Outside of mainstream physics, others have speculated on mechanisms that might allow FTL travel to be achieved, often relying on new conjectures of physics of their own invention, but their ideas have not gained significant acceptance in the physics research community.
Fictional depictions of superluminal travel and the mechanisms of achieving it are also a staple of the science fiction genre.
In the context of this article, FTL is transmitting information or matter faster than c, a constant equal to the speed of light in a vacuum, 299,792,458 meters per second, or about 186,282 miles per second.
This is not quite the same as traveling faster than light, since:
Neither of these phenomena violates special relativity or creates
problems with causality, and thus neither qualifies as FTL as
Faster-than-light communication is, by Einstein's theory of relativity, equivalent to time travel.
According to Einstein's theory of special relativity, what we measure as the speed of light in a vacuum is actually the fundamental physical constant c. This means that all observers, regardless of their relative velocity, will always measure zero-mass particles such as photons traveling at c in a vacuum. This result means that measurements of time and velocity in different frames are no longer related simply by constant shifts, but are instead related by Poincaré transformations.
These transformations have important implications:
Radically Curve Spacetime Using Slip String Drive
There is one way that doesn't violate Relativity. Andrew L. Bender's Slip String Drive.
Bender proposes traveling by completely isolating a region of spacetime from the rest of our universe using Einstein's gravity waves. These compression waves of spacetime are generated by a ship, which emits them from its hull in all directions until it is completely isolated from the rest of our universe. Then, by emitting more gravity waves behind the ship, it stretches out its isolated bubble into an egg-shape, causing external spacetime to squeeze in on the bubble unevenly, propelling the craft forward at speeds no longer limited by relativity.
Time passes normally within the isolated region,
eliminating the possibility of paradox or time travel.
This option is popular particularly in science fiction. However, empirical and theoretical evidence strongly supports Einstein's theory of special relativity as the correct description of high-speed motion, which generalizes the more familiar Galilean relativity, which is actually an approximation at conventional (much less than c) speeds.
Similarly, general relativity is an overwhelmingly supported and experimentally verified theory of gravitation, except in the regime of very high energy densities over very short distances, where an as-yet-undeveloped theory of quantum gravity is necessary. Special relativity, however, is incorporated easily into quantum field theories.
Therefore, even in the broader contexts of general
relativity and quantum mechanics, conventional acceleration from
subluminal to superluminal speeds is not possible.
That is, it will be the same from any frame of reference moving at a constant speed. The equations do not specify any particular value for the speed of the light, which is an experimentally determined quantity for a fixed unit of length.
Since 1983, the unit of length (the meter) has been defined using the speed of light.
However, the vacuum we know is not the only possible vacuum which can exist. The vacuum has energy associated with it, called the vacuum energy. This vacuum energy can perhaps be changed in certain cases. When vacuum energy is lowered, light itself has been predicted to go faster than the standard value 'c'. This is known as the Scharnhorst effect.
Such a vacuum can be produced by bringing two perfectly smooth metal plates together at near atomic diameter spacing. It is called a Casimir vacuum. Calculations imply that light will go faster in such a vacuum by a minuscule amount: a photon traveling between two plates that are 1 micrometer apart would increase the photon's speed by only about one part in 1036.
Accordingly there has as yet been no experimental verification of the prediction. A recent analysis argued that the Scharnhorst effect cannot be used to send information backwards in time with a single set of plates since the plates' rest frame would define a "preferred frame" for FTL signaling.
However, with multiple pairs of plates in motion relative to one another the authors noted that they had no arguments that could "guarantee the total absence of causality violations", and invoked Hawking's speculative chronology protection conjecture which suggests that feedback loops of virtual particles would create "uncontrollable singularities in the renormalized quantum stress-energy" on the boundary of any potential time machine, and thus would require a theory of quantum gravity to fully analyze.
Other authors argue that Scharnhorst's original analysis which
seemed to show the possibility of faster-than-c signals involved
approximations which may be incorrect, so that it is not clear
whether this effect could actually increase signal speed at all.
Nimtz told New Scientist magazine:
However, other physicists say that this phenomenon does not allow information to be transmitted faster than light.
Aephraim Steinberg, a quantum optics expert at the University
of Toronto, Canada, uses the analogy of a train traveling from
Chicago to New York, but dropping off train cars at each station
along the way, so that the center of the train moves forward at each
stop; in this way, the speed of the center of the train exceeds the
speed of any of the individual cars.
Another approach is to accept special relativity, but to posit that mechanisms allowed by general relativity (e.g., wormholes) will allow traveling between two points without going through the intervening space.
While this gets around the infinite acceleration problem, it still would lead to closed timelike curves (i.e., time travel) and causality violations. Causality is not required by special or general relativity, but is nonetheless generally considered a basic property of the universe that cannot be sensibly dispensed with. Because of this, most physicists expect (or perhaps hope) that quantum gravity effects will preclude this option.
An alternative is to conjecture that,
while time travel is possible, it never leads to paradoxes; this is
the Novikov self-consistency principle.
This is understood to be due to the
expansion of the space between the objects, and general relativity
still reduces to special relativity in a "local" sense, meaning that
two objects passing each other in a small local region of spacetime
cannot have a relative velocity greater than c, and will move more
slowly than a light beam passing through the region.
The best-known attempt is doubly-special relativity, which posits that the Planck length is also the same in all reference frames, and is associated with the work of Giovanni Amelino-Camelia and João Magueijo.
One consequence of this theory is a variable speed of
light, where photon speed would vary with energy, and some zero-mass
particles might possibly travel faster than c. However, even if this
theory is accurate, it is still very unclear whether it would allow
information to be communicated, and appears not in any case to allow
massive particles to exceed c.
If confirmed, this would
imply special relativity is an approximation to a more general
theory, but since the relevant comparison would (by definition) be
outside the observable universe, it is difficult to imagine (much
less construct) experiments to test this hypothesis.
A very popular option in space opera is to assume the existence of some other realm (typically called hyperspace, subspace, or slipspace) which is accessible from this universe, in which the laws of relativity are usually distorted, bent, or nonexistent, facilitating rapid transport between distant points in this universe, sometimes with acceleration differences - that is, not requiring as much energy or thrust to go faster.
accomplish rapid transport between points in hyperspace/subspace,
special relativity is often assumed not to apply in this other
realm, or that the speed of light is higher. Another solution is to
posit that distant points in the mundane universe correspond to
points that are close together in hyperspace.
Although the theory of special relativity forbids objects to have a relative velocity greater than light speed, and general relativity reduces to special relativity in a local sense (in small regions of spacetime where curvature is negligible), general relativity does allow the space between distant objects to expand in such a way that they have a "recession velocity" which exceeds the speed of light, and it is thought that galaxies which are at a distance of more than about 14 billion light years from us today have a recession velocity which is faster than light.
Miguel Alcubierre theorized that it would be possible to create an Alcubierre drive, in which a ship would be enclosed in a "warp bubble" where the space at the front of the bubble is rapidly contracting and the space at the back is rapidly expanding, with the result that the bubble can reach a distant destination much faster than a light beam moving outside the bubble, but without objects inside the bubble locally traveling faster than light.
However, several objections raised against the Alcubierre drive appear to rule out the possibility of actually using it in any practical fashion. Another possibility predicted by general relativity is the traversable wormhole, which could create a shortcut between arbitrarily distant points in space.
As with the Alcubierre drive,
travelers moving through the wormhole would not locally move faster
than light which travels through the wormhole alongside them, but
they would be able to reach their destination (and return to their
starting location) faster than light traveling outside the wormhole.
Cleaver said positive dark energy is currently responsible for
speeding up the expansion rate of our universe as time moves on.
In 1977, a controversial paper on Heim theory theorized that it may be possible to travel faster than light by using magnetic fields to enter a higher-dimensional space, and the paper received some media attention in January 2006.
due to the many unproven assumptions in the paper, there have been
few serious attempts to conduct further experiments.
This limit can be used to determine a
minimum time quantization of 5.391×10−44 seconds, which corresponds
to a beam of light with a wavelength approaching the Planck length.
This means that there is a physical limit to how much blue shift a
beam of light can endure. According to general relativity there is
no limit to this shift, and an infinitesimally small space can
exist, but according to well accepted quantum theory these limits do
(According to general relativity, the space-time
distortions caused by gravity are fundamentally identical to
space-time distortions caused simply by accelerating your reference
In special relativity, while it is impossible to accelerate an object to the speed of light, or for a massive object to move at the speed of light, it is not impossible for an object to exist which always moves faster than light.
The hypothetical elementary particles that have this property are called tachyons. Their existence has neither been proven nor disproven, but even so, attempts to quantize them show that they may not be used for faster-than-light communication.
Physicists sometimes regard the
existence of mathematical structures similar to Tachyons arising
from theoretical models and theories as signs of an inconsistency or
that the theory needs further refining.
General relativity was developed after special relativity to include concepts like gravity.
It maintains the principle that no object can accelerate to the speed of light in the reference frame of any coincident observer. However, it permits distortions in spacetime that allow an object to move faster than light from the point of view of a distant observer. One such distortion is the Alcubierre drive, which can be thought of as producing a ripple in spacetime that carries an object along with it.
Another possible system is the wormhole, which connects two
distant locations as though by a shortcut. Both distortions would
need to create a very strong curvature in a highly localized region
of space-time and their gravity fields would be immense. To
counteract the unstable nature, and prevent the distortions from
collapsing under their own 'weight', one would need to introduce
hypothetical exotic matter or negative energy.
One theory states that stable wormholes are possible, but that any attempt to use a network of wormholes to violate causality would result in their decay. In string theory Eric Gimon and Petr Hořava have argued that in a supersymmetric five-dimensional Gödel universe quantum corrections to general relativity effectively cut off regions of spacetimes with causality-violating closed timelike curves.
In particular, in the
quantum theory a smeared supertube is present that cuts the
spacetime in such a way that, although in the full spacetime a
closed timelike curve passed through every point, no complete curves
exist on the interior region bounded by the tube.
This rotational frame-dragging effect is also known as the Lense-Thirring effect. The rotation of an object alters space and
time, dragging a nearby object out of position compared to the
predictions of Newtonian physics. The predicted effect is
small - about one part in a few trillion.
This is followed by more detail describing the science below.
Frame Dragging Effect Basics
Under this effect, the frame of reference in which a clock
ticks the fastest is one which is rotating around the object as
viewed by a distant observer. This also means that light traveling
in the direction of rotation of the object will move around the
object faster than light moving against the rotation as seen by a
distant observer. It is now the best-known effect, partly thanks to
the Gravity Probe B experiment.
The effect is an increase in inertia of a body when other masses are placed nearby. While not strictly a frame dragging effect, it is also derived from the same equation of general relativity. It is a tiny effect that is difficult to confirm experimentally.
where rs is the Schwarzschild radius
and where the following shorthand variables have been introduced for brevity
In the non-relativistic limit where M (or, equivalently, rs) goes to zero, the Kerr metric becomes the orthogonal metric for the oblate spheroidal coordinates
We may re-write the Kerr metric in the following form
This metric is equivalent to a co-rotating reference frame that is rotating with angular speed Ω that depends on both the radius r and the colatitude θ
In the plane of the equator this simplifies to:
Thus, an inertial reference frame is
entrained by the rotating central mass to participate in the
latter's rotation; this is frame-dragging. Frame-dragging occurs
about every rotating mass and at every radius r and colatitude θ.
These include the mass of the rotating body, its rotation speed, relative orientation of the two point to the axis of rotation, and the medium and distance between the localized and distant points in space. More simply, it is a function of the degree of inertial frame-dragging and the characteristics of the medium through which the Time Reactor must operate between the two regions to open a "discharge path."
Also, the amount of energy that is accessed, or time-warped fields generated, can be controlled in several ways through phasing and other characteristics of the emitter and power collector array.
time control we can attempt to generate this large energy level or,
as an alternative, access and channel the energy already existing
and inherent in natural processes and the basic makeup or fabric of spacetime surrounding our planet.
This technology itself does not create the energy
levels required for time control and time travel. Instead, it relies
on and operates using the energy stored within twisted spacetime
around a rotating body that is created by the inertial
frame-dragging effect. With only a small amount of system input
power, time-warped field theory shows how enormous power levels can
These fields of closed-timelike
curves are concentrated and controllable and can permit both forward
and backwards time control.
A number of interesting post-Newtonian phenomena are known to occur for rotating distributions of matter in Einstein’s general theory of relativity.
Inertial frame dragging,
for example, is a consequence of the weak gravitational field of a
slowly rotating massive sphere. In addition, exact solutions of the
Einstein field equations indicate the presence of closed timelike
lines for rotating Kerr black holes, van Stockum rotating dust
cylinders, and the rotating universe of Gödel.
This is followed by more detail describing the approach below.
Recently, Ronald L. Mallett solved the linearized Einstein field equations to obtain the gravitational field produced by the electromagnetic radiation of a unidirectional ring laser.
It was shown that a massive spinning neutral particle at the center of the ring laser exhibited inertial frame dragging.
Their first experiment will be to trap light in a crystal and observe the reaction of a neutron inside the circle.
Mallett will insert polarized neutrons (neutrons that all spin in one direction) into the center of the circulating light.
If he sees a change in their spin he will know
that space is indeed being twisted inside of the crystal. Should
this experiment prove successful, the team will apply for funding to
conduct studies to see if time bending is evident inside the circle
If, at the end of the experiment, one sample had decayed further than the other, Mallett's theories of time travel would be supported.
Where the experiments will go from there is unclear.
There is a vast difference between slowing the decay rate of a radioactive particle and sending a human back in time. Science aside, sending people through time creates philosophical issues as well as physical ones. Consider the "Grandparent Paradox" in which a time traveler goes back in time and kills her grandparents, thus negating her entire existence. If she were never born, then she couldn't go back in time in the first place.
explains paradoxes such as these with a parallel-universe theory. He
believes that with every decision we make, another version of us
makes the opposite decision and splits off into a parallel universe.
Thus the time traveler was born in the universe where she did not
kill her grandparents.
But he explains that the difference between physics and philosophy is experiment.
True, the parallel-universe theory has
not been directly supported by experiment, but Mallett uses the
Heisenberg Uncertainty Principle to explain
why the parallel universe theory is probable.
Without this principle,
A hydrogen atom, one of the building blocks of our universe, consists of a proton and an electron. Since the proton and electron have opposite charges they should be attracted to each other, collide, and destroy the atom.
But if that happened, we would know both the position of the electron (the point of impact with the proton) and its spin (none); therefore it is impossible for them to collide.
Therefore the parallel-universe theory works well.
What will happen next can't be predicted because in fact, everything
This may also apply to
a long but finite circulating cylinder of light.
Wormholes are hypothetical areas
of warped spacetime with great energy that can create tunnels
through spacetime. if traversable would allow a traveler to quickly
move through great distances in space and also travel through time.
The difficulty lies in keeping the wormhole open while the traveler
makes his journey: If the opening snaps shut, he will never survive
to emerge at the other end.
But recent research, especially by the U.S. physicist Kip Thorne, suggests that it could be done using exotic materials capable of withstanding the immense forces involved. Even then, the time machine would be of limited use – for example, you could not return to a time before the wormhole was created.
Using wormhole technology would also require a society so technologically advanced that it could master and exploit the energy within black holes.
A wormhole has at least two mouths that are connected to a single throat or tube. If the wormhole is traversable, then matter can 'travel' from one mouth to the other by passing through the throat.
While there is no observational evidence for wormholes, spacetime containing wormholes are known to be valid solutions in general relativity.
However, the idea of wormholes had
already been theorized in 1921 by the German mathematician Hermann Weyl in connection with his analysis of mass in terms of
electromagnetic field energy.
This is followed by more detail describing the science below.
This solution was
discovered by Albert Einstein and his colleague Nathan Rosen, who
first published the result in 1935. However, in 1962 John A. Wheeler
and Robert W. Fuller published a paper showing that this type of
wormhole is unstable, and that it will pinch off instantly as soon
as it forms, preventing even light from making it through.
The possibility of traversable wormholes in general relativity was first demonstrated by Kip Thorne and his graduate student Mike Morris in a 1988 paper; for this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, is referred to as a Morris-Thorne wormhole. Later, other types of traversable wormholes were discovered as allowable solutions to the equations of general relativity, including a variety analyzed in a 1989 paper by Matt Visser, in which a path through the wormhole can be made in which the traversing path does not pass through a region of exotic matter.
However in the pure Gauss-Bonnet theory exotic matter is not needed
in order for wormholes to exist- they can exist even with no matter.
A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al., in which it was proposed
that such wormholes could have been naturally created in the early
However, it has been said a time traversing wormhole cannot take you back to before it was made but this is disputed.
Special relativity only applies locally.
Wormholes allow superluminal (faster-than-light) travel by ensuring that the speed of light is not exceeded locally at any time. While traveling through a wormhole, subluminal (slower-than-light) speeds are used. If two points are connected by a wormhole, the time taken to traverse it would be less than the time it would take a light beam to make the journey if it took a path through the space outside the wormhole.
However, a light beam traveling through the wormhole would always beat the traveler. As an analogy, running around to the opposite side of a mountain at maximum speed may take longer than walking through a tunnel crossing it.
You can walk slowly while reaching your destination more quickly because the distance is smaller.
A wormhole could allow time travel.
This could be accomplished by accelerating one end of the wormhole to a high velocity relative to the other, and then sometime later bringing it back; relativistic time dilation would result in the accelerated wormhole mouth aging less than the stationary one as seen by an external observer, similar to what is seen in the twin paradox.
However, time connects differently through the wormhole
than outside it, so that synchronized clocks at each mouth will
remain synchronized to someone traveling through the wormhole
itself, no matter how the mouths move around. This means that
anything which entered the accelerated wormhole mouth would exit the
stationary one at a point in time prior to its entry.
After being taken on a trip at relativistic velocities, the accelerated mouth is brought back to the same region as the stationary mouth with the accelerated mouth's clock reading 2005 while the stationary mouth's clock read 2010. A traveler who entered the accelerated mouth at this moment would exit the stationary mouth when its clock also read 2005, in the same region but now five years in the past.
Such a configuration of wormholes would allow for a particle's world line to form a closed loop in spacetime, known as a closed timelike curve.
It is thought that it may not be possible to convert a wormhole into a time machine in this manner; some analyses using the semi-classical approach to incorporating quantum effects into general relativity indicate that a feedback loop of virtual particles would circulate through the wormhole with ever-increasing intensity, destroying it before any information could be passed through it, in keeping with the chronology protection conjecture.
This has been called into question by the suggestion that radiation would disperse after traveling through the wormhole, therefore preventing infinite accumulation. The debate on this matter is described by Kip S. Thorne in the book Black Holes and Time Warps. There is also the Roman ring, which is a configuration of more than one wormhole.
This ring seems to allow a closed time loop with stable wormholes when analyzed using semi-classical gravity, although without a full theory of quantum gravity it is uncertain whether the semi-classical approach is reliable in this case.
One type of non-traversable wormhole metric is the Schwarzschild solution:
Wing Commander ships
It is common for the creators of a fictional universe to decide that faster-than-light travel is either impossible or that the technology does not yet exist, but to use wormholes as a means of allowing humans to travel long distances in short periods. Military science fiction (such as the Wing Commander games) often uses a "jump drive" to propel a spacecraft between two fixed "jump points" connecting stellar systems.
Connecting systems in a network like this results in a fixed "terrain" with choke points that can be useful for constructing plots related to military campaigns. The Alderson points used by Larry Niven and Jerry Pournelle in The Mote in God's Eye and related novels are an example, although the mechanism does not seem to describe actual wormhole physics.
David Weber has also used the device in the Honorverse and other books such as those based upon the Starfire universe. Naturally occurring wormholes form the basis for interstellar travel in Lois McMaster Bujold's Vorkosigan Saga. They are also used to create an Interstellar Commonwealth in Peter F. Hamilton's Commonwealth Saga.
In Jack L. Chalker's The Rings of the Master series, interstellar class
spaceships are capable of calculating complex equations and punching
Wormholes in the fabric of the Universe in order to enable rapid
Or the Privateer Remake, a remake of Wing Commander: Privateer.
Several examples appear in the Star Trek franchise, including the Bajoran wormhole in the Deep Space Nine series. In 1979's Star Trek: The Motion Picture the USS Enterprise was trapped in an artificial wormhole caused by an imbalance in the calibration of the ship's warp engines when it first achieved faster-than-light speed.
In the Star Trek: Voyager
series, the cybernetic species the Borg use what, in the Star Trek
universe, are referred to as transwarp conduits, allowing ships to
move nearly instantaneously to any part of the galaxy in which an
exit aperture exists. Although these conduits are never described as
"wormholes", they appear to share several traits in common with
A trip through the black hole turns theological, abandoning scientific rationale.
The round trip, which to Ellie lasts 18 hours, passes by in a
fraction of a second on Earth, making it appear she went nowhere. In
her defense, Foster mentions an Einstein-Rosen bridge and tells how
she was able to travel faster than light and time. Analysis of the
situation by Kip Thorne, on the request of Sagan, is quoted by
Thorne as being his original impetus for analyzing the physics of
In the latter series, the devices were discovered in Egypt by an archeologist, and were built by aliens known as the Ancients or the Alterans. In the science fiction series Sliders, a wormhole (or vortex, as it is usually called in the show) is used to travel between parallel worlds, and one is seen at least once or twice in every episode.
In the pilot episode it was referred to as an "Einstein-Rosen-Podolsky bridge".
Although no one has actually found a cosmic string, astronomers have suggested that they may explain strange effects seen in distant galaxies.
A cosmic string is a 1-dimensional (spatially) topological defect in various fields.
Cosmic strings are hypothesized to form when the field undergoes a phase change in different regions of spacetime, resulting in condensations of energy density at the boundaries between regions. This is somewhat analogous to the imperfections that form between crystal grains in solidifying liquids, or the cracks that form when water freezes into ice.
The phase changes that produce cosmic strings may have occurred
in the earliest moments of the universe's evolution.
Cosmic strings, if they exist, would be extremely thin with diameters on the same order as a proton.
They would have immense density, however, and so would represent significant gravitational sources. A cosmic string 1.6 kilometers in length may be heavier than the Earth. However general relativity predicts that the gravitational potential of a straight string vanishes: there is no gravitational force on static surrounding matter.
The only gravitational effect of a straight cosmic string is
a relative deflection of matter (or light) passing the string on
opposite sides (a purely topological effect). A closed loop of
cosmic string gravitates in a more conventional way. During the
expansion of the universe, cosmic strings would form a network of
loops, and their gravity could have been responsible for the
original clumping of matter into galactic superclusters.
It was once thought that the gravitational influence of cosmic strings might contribute to the large-scale clumping of matter in the universe, but all that is known today through galaxy surveys and precision measurements of the cosmic microwave background fits an evolution out of random, Gaussian fluctuations.
These precise observations therefore tend to
rule out a significant role for cosmic strings.
In 2003 a group led by Mikhail Sazhin reported the accidental discovery of two seemingly identical galaxies very close together in the sky, leading to speculation that a cosmic string had been found. However, observations by the Hubble Space Telescope in January 2005 showed them to be a pair of similar galaxies, not two images of the same galaxy.
A cosmic string would produce a similar duplicate image of fluctuations in the cosmic microwave background, which might be detectable by the upcoming Planck Surveyor mission.
A second piece of evidence supporting cosmic string theory is a phenomenon observed in observations of the "double quasar" called Q0957+561A,B.
Originally discovered by Dennis
Walsh, Bob Carswell, and Ray Weymann in 1979, the double image of
this quasar is caused by a galaxy positioned between it and the
Earth. The gravitational lens effect of this intermediate galaxy
bends the quasar's light so that it follows two paths of different
lengths to Earth. The result is that we see two images of the same
quasar, one arriving a short time after the other (about 417.1 days
Schild and his team believe that the only explanation for this
observation is that a cosmic string passed between the Earth and the
quasar during that time period traveling at very high speed and
oscillating with a period of about 100 days.
There is no direct connection between string theory and the theory of cosmic strings (the names were chosen independently by analogy with ordinary string).
However, work in string theory revived interest in cosmic strings in the early 2000s. In 2002 Henry Tye and collaborators observed the production of cosmic strings during the last stages of brane inflation. It was also pointed out by string theorist Joseph Polchinski that the expanding Universe could have stretched a "fundamental" string (the sort which superstring theory considers) until it was of intergalactic size.
Such a stretched string would exhibit many of the properties of the old "cosmic" string variety, making the older calculations useful again. Furthermore, modern superstring theories offer other objects which could feasibly resemble cosmic strings, such as highly elongated one-dimensional D-branes (known as "D-strings").
As theorist Tom Kibble remarks,
Older proposals for detecting cosmic strings could now be used to investigate superstring theory.
It would emerge thousands, even billions, of years from its starting point and possibly several galaxies away. There are problems, though. For the mathematics to work properly, Tipler’s cylinder has to be infinitely long. Also, odd things happen near the ends and you need to steer well clear of them in your timeship.
However, if you make the device as long as you can, and stick to paths close to the middle of the cylinder, you should survive the trip!
The Tipler cylinder, also called a
Tipler time machine, is a hypothetical object theorized to be a
potential mode of time travel - an approach that is conceivably
functional within humanity's current understanding of physics,
specifically the theory of general relativity, although later
results have shown that a Tipler cylinder could only allow time
travel if its length would appear infinite.
Tipler showed in his 1974 paper, "Rotating Cylinders and the Possibility of Global Causality Violation" that in a spacetime containing a massive, infinitely long cylinder which was spinning along its longitudinal axis, the cylinder should create a frame-dragging effect.
This frame-dragging effect warps spacetime in such a way that the light cones of objects in the cylinder's proximity become tilted, so that part of the light cone then points backwards along the time axis on a space time diagram.
Therefore a spacecraft accelerating sufficiently in the appropriate direction can travel backwards through time along a closed timelike curve or CTC.
an unnerving habit of appearing in some of the most important exact
solutions in general relativity, including the Kerr vacuum (which
models a rotating black hole) and the van Stockum dust (which models
a cylindrically symmetrical configuration of rotating pressureless
fluid or dust).
Tipler's original solution involved a cylinder of infinite length, which is easier to analyze mathematically, and although Tipler suggested that a finite cylinder might produce closed timelike curves if the rotation rate were fast enough, he did not prove this.
A spirallohedron of 6
Hawking's proof appears in his 1992 paper on the chronology protection conjecture, where he examines,
In quantum field theory, the Casimir effect and the Casimir-Polder force are physical forces arising from a quantized field.
The typical example is of two uncharged metallic plates in a vacuum, placed a few micrometers apart, without any external electromagnetic field. In a classical description, the lack of an external field also means that there is no field between the plates, and no force would be measured between them.
When this field
is instead studied using quantum electrodynamics, it is seen that
the plates do affect the virtual photons which constitute the field,
and generate a net force - either an attraction or a repulsion
depending on the specific arrangement of the two plates.
This is followed by more detail describing the effect below.
Although the Casimir effect can be expressed in terms of virtual particles interacting with the objects, it is best described and more easily calculated in terms of the zero-point energy of a quantized field in the intervening space between the objects.
This force has been measured, and is a striking example of an effect purely due to second quantization. However, the treatment of boundary conditions in these calculations has led to some controversy. In fact "Casimir's original goal was to compute the van der Waals force between polarizable molecules" of the metallic plates.
Thus it can be interpreted without any reference to the zero-point energy (vacuum energy) or virtual particles of quantum fields.
The classic form of the experiment, described above, successfully demonstrated the force to within 15% of the value predicted by the theory.
Because the strength of the force falls off rapidly with distance, it is only measurable when the distance between the objects is extremely small.
On a submicrometre scale,
this force becomes so strong that it becomes the dominant force
between uncharged conductors. In fact, at separations of 10 nm - about
100 times the typical size of an atom - the Casimir effect produces
the equivalent of 1 atmosphere of pressure (101.3 kPa), the precise
value depending on surface geometry and other factors.
In a simplified view, a "field" in physics may be envisioned as if space were filled with interconnected vibrating balls and springs, and the strength of the field can be visualized as the displacement of a ball from its rest position.
Vibrations in this field propagate and are governed by the appropriate wave equation for the particular field in question.
The second quantization of quantum field theory
requires that each such ball-spring combination be quantized, that
is, that the strength of the field be quantized at each point in
space. Canonically, the field at each point in space is a simple
harmonic oscillator, and its quantization places a quantum harmonic
oscillator at each point. Excitations of the field correspond to the
elementary particles of particle physics. However, even the vacuum
has a vastly complex structure, all calculations of quantum field
theory must be made in relation to this model of the vacuum.
The quantization of a simple harmonic oscillator states that the lowest possible energy or zero-point energy that such an oscillator may have is
Summing over all possible oscillators at all points in space gives an infinite quantity.
To remove this infinity, one may argue that only differences in energy are physically measurable; this argument is the underpinning of the theory of renormalization. In all practical calculations, this is how the infinity is always handled. In a deeper sense, however, renormalization is unsatisfying, and the removal of this infinity presents a challenge in the search for a Theory of Everything.
Currently there is no compelling explanation for how this infinity should be treated as essentially zero; a non-zero value is essentially the cosmological constant and any large value causes trouble in cosmology.
In this case, the correct way to find the zero point energy of the field is to sum the energies of the standing waves of the cavity. To each and every possible standing wave corresponds an energy; say the energy of the nth standing wave is En. The vacuum expectation value of the energy of the electromagnetic field in the cavity is then
with the sum running over all possible values of n enumerating the standing waves. The factor of 1/2 corresponds to the fact that the zero-point energies are being summed - it is the same 1/2 as appears in the equation...
this way, this sum is clearly divergent; however, it can be used to
create finite expressions.
for the vacuum expectation value. At this point comes an important observation: the force at point p on the wall of the cavity is equal to the change in the vacuum energy if the shape s of the wall is perturbed a little bit, say by δs, at point p.
That is, one has
This value is finite in many practical calculations.
In this case, the standing waves are particularly easy to calculate, since the transverse component of the electric field and the normal component of the magnetic field must vanish on the surface of a conductor. Assuming the parallel plates lie in the x-y plane, the standing waves are
where ψ stands for the electric component of the electromagnetic
field, and, for brevity, the polarization and the magnetic
components are ignored here. Here,
is the wave-vector perpendicular to the plates. Here, n is an integer, resulting from the requirement that ψ vanish on the metal plates. The energy of this wave is
where c is the speed of light. The vacuum energy is then the sum over all possible excitation modes
where A is the area of the metal plates, and a factor of 2 is introduced for the two possible polarizations of the wave. This expression is clearly infinite, and to proceed with the calculation, it is convenient to introduce a regulator (discussed in greater detail below). The regulator will serve to make the expression finite, and in the end will be removed. The zeta-regulated version of the energy per unit-area of the plate is
In the end, the limit
is to be taken. Here s is just a complex number, not to be confused with the shape discussed previously. This integral/sum is finite for s real and larger than 3. The sum has a pole at s=3, but may be analytically continued to s=0, where the expression is finite. Expanding this, one gets
where polar coordinates
were introduced to turn the double integral into a single integral. The q in front is the Jacobian, and the 2π comes from the angular integration. The integral is easily performed, resulting in
The sum may be understood to be the Riemann zeta function, and so one has
But ζ( − 3) = 1 / 120 and so one obtains
The Casimir force per unit area,
The force is negative, indicating that the force is attractive: by moving the two plates closer together, the energy is lowered. The presence of shows that the Casimir force per unit area Fc / A is very small, and that furthermore, the force is inherently of quantum-mechanical origin.