The Casimir Effect

So far, I have had the pleasure of talking to you about dark energy and extra dimensions. Next, I would like to talk about a phenomonon that I have been fascinated by for some time; namely, the Casimir Effect. And later, I will discuss how all these subjects could be related.

Typically, everything that happens in the universe is a consequence of one of the four basic forces: gravity, electromagnetism, the weak nuclear force, and the strong nuclear force. However, in 1948, the Dutch physicist Hendrik B. Casimir published a profound paper in which he predicted that two parallel conducting plates, if placed in a perfect vacuum, would attract each other with a force proportional to one over the fourth power of the separation. Casimir also predicted that this force is not explainable in terms of the four known forces, but is due to the modification of the quantum vacuum itself.

First, let’s discuss what we mean by the term `quantum vacuum’. The most successful physical theory that physicists have developed is called quantum field theory (QFT). QFT accurately desribes the subatomic realm with unprecedented precision, and has been verified by particle accelerators for decades. QFT has also predicted the existence of an array of previously unknown particles, which have since been proven to exist; one such particle is the top quark. The large hadron collider (LHC) being built on the Swiss-Geneva border is being built primarily to try and prove the existence of the Higgs boson, which is another prediction of QFT.

The basic `building block’ in QFT is a quantum field. This means that everything in the universe (electrons, quarks, photons, neutrinos etc.) can be described fundamentally as a field with a specific frequency and wavelength. These fields can be described mathematically as an oscillator occupying every point in space and time. One way to visualize this is to picture a tiny pendulum occupying all points in the universe, and the different amplitudes and frequencies of this pendulum’s vibration correspond to the matter and energy we observe.

One interesting prediction that came out of quantum mechanics is the Heisenberg Uncertainty Principle (HUP), which tells us that no field oscillator can ever be completely at rest. What this means is that these `pendulums’ which occupy all points in the universe always have some residual vibration, and are never perfectly still. Because of this, these pendulums are constantly emitting radiation, much like when an electron accelerates. This radiation is called `zero point radiation’. Another prediction of QFT is that this radiation is free to take all frequency values from zero all the way up to infinity, and so what we understand to be the vacuum is not the classical picture of emptiness, but instead a sea of zero point oscillations.

Now that we have acquired some insight into the vacuum, let us now turn to understanding the Casimir effect. The basic idea is this: when two parallel conducting plates are placed in a vacuum, their very presence modifies the quantum vacuum fluctuations. Only resonant modes, or `standing waves,’ are free to exist within the boundaries. These are waves that fit exactly within the confines. This is most easily pictured by looking at Figure 1.

 

We see that on the exterior region of the plates, the vacuum energy is free to take all possible frequencies, but on the interior region of the plates, the vacuum is restricted to standing waves. This assymmetry of the vacuum creates an energy gradient which generates a force of attraction between the plates. This energy gradient is purely quantum in nature. One interesting feature of this Casimir energy is that it is negative. In classical physics negative energies are strictly forbidden.

This force, first predicted in 1948, remained unknown for many years. However, from the 1970’s onward, the Casimir effect received increasing attention, and over the last decade it has become very popular. Many attempts have been made at measuring this force, and the most accurate-to-date was performed by Steven Lamoreaux at the University of Washington in the late 1990’s, who used an atomic force microscope. The experiment validated the theoretical predictions to within 1% accuracy (see Figure 2).

The solid black line shows the predicted force of attraction using the Casimir equations. The data points acquired using the AFM closely follow the predicted values.

The Casimir effect is far more than just a theoretical curiosity, and is studied in the context of a wide variety of fields in physics. These include gravitation and cosmology, condensed matter physics, atomic and molecular physics, quantum field theory and even nanotechnology. My own interest in the Casimir effect is its possible role in explaining dark energy. Another interest of mine regards the stability of compact extra dimensions. In previous articles we have discussed the fact that extra dimensions, if they exist, must be very small. One important question regarding these dimensions is, what keeps them small and stable? Why do they not perpetually shrink, or conversely, why do they not expand like our familiar macroscopic dimensions? It is possible that the Casimir effect plays a role here. I hope to continue to discuss these compelling ideas with you in future articles.

The Quantum Vacuum

One fundamental feature of quantum field theory (QFT) is the notion that empty space is not really empty. Emptiness has been replaced by the concept of the vacuum ground state. It is this ground state which is responsible for a ubiquitous energy density that is ultimately believed to act as a contribution to the cosmological
constant appearing in Einstein’s field equations from 1917 onwards,

In 1916 Nernst, who was originally inspired by the new ideas of quantum theory and Plancks law for the radiation from a black body, put forward the proposition that the vacuum of spacetime is not empty but is, in fact, a medium filled with radiation containing a large amount of energy. One feature of this model was that the energy density of the vacuum was infinite, and even when a modest cutoff was proposed, the total energy content was still large. Nernst’ ideas about the vacuum were never used for any cosmological models as his interests were in chemistry, and in forming a model of the water molecule.

A more solid foundation for speculations of the energy density of the vacuum became available with the developments in QFT in which all the fields in nature are treated as a collection of quantized harmonic oscillators. The various amplitudes and frequencies of oscillation represent the different boson and fermion species we observe in nature.

The vacuum has all the quantum properties a particle may acquire, for example energy, spin and polarization. These quantities on average cancel each other out, with the exception of the vacuum expectation value of the energy . A consequence of the Heisenberg Uncertainty principle is that no field oscillator can ever be completely at rest. There will always be some residual `zero-point energy.’ Naively one can see this in

.

For n=0 we are left with

.

One problem that arises frequently arises with regards to the ground state of the vacuum are the huge energies that are found. Pauli was concerned with the \textit{gravitational} effects of the zero-point energy.  Pauli’s calculation (mid-late 1920s) demonstrated that if the gravitational effect of the zero-point energies was taken into account  the radius of the universe would be smaller than the distance from the Earth to the Moon. Pauli’s calculation invloved applying a cut-off energy at the classical electron radius, which was considered to be a natural cut-off in his day. Pauli’s concern was readdressed by Straumann in 1999, who found that indeed, the radius of the universe would be . Setting the calculation reads

Inserting the appropriate cutoff,

and plugging into

where a is the radius of curvature obtained from solving Einsteins equation for a static dust filled universe. One obtains

One way to reconcile this inconsistency with the known size of the universe, as noted in Pauli’s \textit{Handbuch der Physik} is that it is more consistent to begin from the ansatz that the zero-point energy does not interact with the gravitational field. Indeed, the speculations of Dirac regarding the huge vacuum energy and also the final version of quantum electrodynamics (QED) constructed by Schwinger, Feynman and others never prompted any interest in the \textit{gravitational} consequences of these theories. This is not surprising when one considers that theoretical landscape of QED which was plagued with divergences in higher order calculations which preoccupied the community.

Extra Dimensions

Below is an article on extra dimensions I wrote for a German magazine Freie Honnefer:

A common theme in theoretical physics is the idea of extra dimensions which has been highly popularized by string theory. This article aims to explain how and why extra dimensions play a role in physics.

It was the great mathematician Riemann with his development of differential geometry in the nineteenth century who gave the tools necessary to study higher dimensional descriptions of the world. Riemann held the belief that three-dimensional space was not enough to provide an adequate description of nature. Improvements in physics led to Maxwell’s unified theory of electricity and magnetism and Einstein had formulated both GR and had unified space and time with Special Relativity (SR). Inspired by these unifications physicists of the early twentieth century wanted to unify gravity and electromagnetism.

The first to attempt this was Nordstrom in 1914 and later Weyl and Kaluza followed two separate paths. Weyl’s attempt involved trying to alter the geometry of space-time in four dimensions. His early attempts had physical consequences which did not match experimental data, however his work was extended by Einstein and Schrodinger independently who arrived at the Einstein-Schrödinger non-symmetric field theory which is widely regarded as the most advanced unified field theory based on classical physics.

It was Theodore Kaluza who was the first to introduce an additional dimension into Einstein’s equations. With this simple addition Kaluza was able to build a theory which included both gravity and the electromagnetic field. Kaluza contacted Einstein in 1919 about his idea, but the introduction of an extra dimension was considered so radical at the time that he was unsure and it wasn’t until 1921 that Einstein encouraged him to publish. As well as unifying electromagnetism and gravity one additional field physicists call a ’scalar’ field was predicted. This was an embarrassment to Kaluza at the time as no scalar field had ever been observed in nature. That problem coupled with the fact that we clearly live in three spatial dimensions caused the theory to draw heavy criticism from the physics community.

In 1926 Oskar Klein suggested that the fifth dimension compactifies so as to have the geometry of a circle of extremely small radius. One way to envisage this space-time is to imagine a hosepipe. From a long distance it looks like a one dimensional line but a closer inspection reveals that every point on the line is in fact a circle. Because the space had a circular topology, the higher dimensional periodicity allowed for mathematical solutions which included charge quantization, something that was becoming important in the developing theory of quantum mechanics.

Seen up close a seemingly one dimensional object has a deeper structure.

 

Theories of extra dimensions lay dormant for over forty years however in 1968 the physicist Gabriele Venziano who at the time was a research fellow at CERN suggested that certain properties of the strong nuclear force could be mathematically modeled by a one dimensional string. Prior to this all elementary particles were considered to be point like particles (effectively zero dimensional). Many research papers followed this exciting observation and bosonic string theory was born. Within a few years it was discovered that certain vibrations of the string had the characteristics that matched the graviton, the gravitational forces messenger particle. This was an extremely exciting discovery at the time because physicists believed they may be on the correct track to formulate a quantum theory of gravity, the holy grail of theoretical physics. The theory had two problems however. One was that it contained a particle with imaginary mass called a Tachyon, and the other was that the theory was only mathematically consistent in 26 dimensions, although bosonic string theories answer was similar to Klein’s idea which was to wrap them up into a very small `compact’ space.

As string theory developed it branched into a number of seemingly separate theories named Type I, Type IIA, Type IIB, Heterotic and $E_8\times E_8$. Each theory had a different predictive power, for example type IIA string theory describes massless fermions. These five theories required not 26 but only 10 space-time dimensions. Although string theory showed much promise it was a problem that there were five theories. In the mid-nineties by a stroke of genius the physicist Ed Witten was able to show that these five seemingly separate theories could in fact be unified into a single theory he called M-theory which requires 11 space-time dimensions. M-theory is still an active field of research and although it is not without its critics, it is still the most promising theory we have to date which promises to unify all the forces of nature.

The common thread in this discussion is the theme of unification of the forces of nature. Kaluza-Klein theory attempted to unify gravity and electromagnetism, and now M-theory are efforts to provide a complete description of nature at the subatomic level which includes gravity, electromagnetism and the strong and weak nuclear forces. Although string and M-theory are the most well known of theories which involve extra dimensions there are in fact a number of popular theories in existence which utilize the rich and tantalizing possibilities offered to physics by extra dimensions which will be discussed in future articles.

Dark Energy

Below is an article I wrote for a German magazine Freie Honnefer:

I spent some time thinking about what I might write for my first article, hoping to generate some excitement and interest amongst the readers of FreieHonnefer. As a physicist, I spend a lot of time thinking about some of the tantalizing ideas emerging in cosmology and particle physics and so I finally decided that it might be a good idea to introduce the current state of theoretical physics from a historical perspective which would aid the interested reader in understanding how we arrived at some of the conclusions we now take for granted. Over the next few articles I plan to introduce some of the exciting ideas in cosmology, gravity and particle physics and show you how the two are gradually merging into the unified research field of string theory.

Einstein is commonly referred to as the grandfather or modern physics and his General theory of Relativity (GR) is a good place to start. Einstein had gained a huge amount of credibility in the physics community after his research regarding the photo-electric effect (for which he won the noble prize) and his special theory of relativity. This credibility gave him the freedom to pursue what would become his most significant theory, the General theory of relativity, which was completed in 1915. GR is commonly believed to be a theory of gravity, but I think that it is more useful to think of GR as a theory of space-time, from which the phenomenon we call gravity ultimately emerges.  Although this magazine is written for the interested reader and assumes no mathematical knowledge I would like to include one single equation to demonstrate to you the beautiful simplicity of Einstein’s theory. Einstein’s field equation is

As remarkably simple as this might look, hidden within this powerful equation are in fact ten equations. Each equation can take many pages of calculations to solve. The left hand side of this equation describes the geometry of space-time, and the right hand side describes the matter and energy content of space-time. To solve this formula one inserts into the right side of the equation how much mass and energy we believe to be in the universe (stars, planets electromagnetic radiation etc) and by solving Einstein’s equation we are told what the geometry of the universe does as a consequence of that matter energy content. Space-time can do one of three possible things, and each possibility is dependent on your initial guess as to the total mass-energy content of the universe. It can contract, remain static or it can expand. Einstein’s initial discovered after solving the field equations implied that the universe is expanding. He disliked this result immensely as it was his profound belief that the universe was static, unchanging and eternal. To reconcile his disbelief in the predictions of his equation he added what physicists call a `fudge’ factor. He simply added an extra term to his equation `by hand’ that forced the universe to be static and unchanging. He called this term the `Cosmological Constant’.

Much to Einstein’s dismay, in 1929 the American astronomer Edwin Hubble made a profound discovery. He found that wherever he looked in the night sky, in general the galaxies seemed to be receding from Earth. Not only that, more distant galaxies appeared to be moving away much faster than the closer galaxies. This observation provided evidence to support the recently discovered `big-bang’ theory. Hubble’s discovery was based on something called the cosmological red-shift, which is similar to the well known Doppler-effect on Earth. As a speeding ambulance drives past you, one notices a change in the pitch of the sound of the siren due to a kind of stretching of the sound waves. A similar effect happens to light and it is this phenomenon that reveals to us the motion of the galaxies.

What is particularly strange about this discovery is that on first glance it implies something very strange. If all the galaxies are moving away from us then that makes us somewhat \textit{special}? It implies that Earth is located in some location that is of great cosmic significance.

Of course, Earth is not cosmically significant and the way to understand what is actually going on is to picture the following. Imagine drawing small dots on the surface of a deflated balloon. Each of the dots represents a galaxy in our universe. Now, as you blow up that balloon it stretches and all of the dots move further and further away from each other. From the perspective of a little creature situated on any of the dots it appears as though all the others dots (galaxies) are rushing away from you. And so, we can imagine space-time as being a fabric upon which all matter and energy is woven upon, and this fabric is currently expanding.

 

The idea that the universe is expanding has been cosmological dogma for nearly a century but in 1999 astronomers discovered something perhaps more profound than and expanding universe. By observing the cosmological red-shift of distant supernova it was discovered that the universe is actually accelerating! That is was not only expanding, but that the rate of expansion is getting larger.

What is particularly profound about this discovery is that some kind of exotic anti-gravitational field is necessary to account for this acceleration because common gravitational energy, which is ubiquitous throughout the universe, would cause the universe to decelerate. A contemporary term for this exotic field is ‘dark energy’. Research into dark energy is very popular today and numerous theoretical explanations exist as to its fundamental nature, but as of yet no commonly accepted theory exist.

My own research involves trying to uncover the nature of dark energy, and I plan to discuss both my ideas, and other popular theories regarding the nature of dark energy in later articles but I hope this short article has given you a glimpse into the exciting field of cosmology.

Create a Matlab Movie Showing the Growth of a Spacetime Warp Bubble

Copy and Paste the Following Code into Matlab to create a nice movie.

I modified the original Alcubierre metric to include an ‘engine parameter’ which simulates the gradual creation of a warp bubble after the engine is activated.

% Formation or Alcubierre Warp Bubble

% Richard Obousy. Oct 2007

% Create Mesh

[x, y] = meshgrid([-15:.5:15],[-15:.5:15]);
% Prepare for Movie
axis tight
set(gca,’nextplot’,'replacechildren’);

% Record the movie

% CYCLE 1 – The generation of the Bubble
for k = 1:75     % k embodies the ‘Engine Paramter’
   
 x=x-.001*k    ;  
for i=1:length(x);
    for j=1:length(x);
z(i,j)=-1/2*(tanh (1*sqrt(abs(x(i,j)^2+y(i,j)^2-58)-10)+2.5)-tanh(1*sqrt(abs(x(i,j)^2+y(i,j)^2-58)-10)-2.5))…
*tanh(16)*(exp(k/75)-1)*x(i,j);
    end
end

% Plot

mesh(x,y,z)
axis([-12 12 -12 12 -15 15 -7 7])       % Adjust Perspective
view([133,22])
colormap(jet)
F(k) = getframe;
 pause(.05)
end

% CYCLE 2 – The Motion of the Bubble

[x, y] = meshgrid([-15:.5:15],[-15:.5:15]);

for kk = 1:75
   
 x=x-.0013*kk    ;   % Term encodes ‘acceleration’
for i=1:length(x);
    for j=1:length(x);
z(i,j)=-1/2*(tanh (1*sqrt(abs(x(i,j)^2+y(i,j)^2-58)-10)+2.5)-tanh(1*sqrt(abs(x(i,j)^2+y(i,j)^2-58)-10)-2.5))…
*tanh(16)*(exp(75/75)-1)*x(i,j);
    end
end
 

axis tight
set(gca,’nextplot’,'replacechildren’);
mesh(x,y,z)
axis([-12 12 -12 12 -15 15 -7 7])
view([133,22])
colormap(jet)
G(kk) = getframe;
 pause(.02)
end

movie2avi(F,’warpmotion3′,’compression’,'cinepak’)
movie2avi(G,’warpmotion4′,’compression’,'cinepak’)

%———————————————————————-

You can see the interior region of the bubble remains flat throughout the generation of the bubble. Any spacecraft located within this region would experience no acceleration effects. Hypothetically a craft and its crew could reach unprecedented speeds whilst escaping the ultra-high g-forces such an acceleration would normally produce.

This assymetric warp at the front and aft of the ship reflect the ’shrinking’ and ’stretching’ of space respectively. This interstellar propulsion proposal works by effectively expanding and contracting spacetime locally to reach the destination. To read more about this take a look at the following paper here which is written for the educated laymen (no equations). For the more mathematically inclined you can take a look at the paper which includes the calculations here.

Alcubierre Warp Drive Metric in Matlab

Here is some code I wrote to generate the asymmetric spacetime found in the Alcubierre metric. Copy and paste into matlab.

% Warp Drive Metric.
% Richard Obousy August 2007.

k=0

[x, y] = meshgrid([-10:.3:10],[-10:.3:10]);

for i=1:length(x);
    for j=1:length(x);
z(i,j)=-1*(tanh (2*sqrt(abs(x(i,j)^2+y(i,j)^2-16)-5)+3)-tanh(2*sqrt(abs(x(i,j)^2+y(i,j)^2-16)-5)-3))…
*tanh(6)*x(i,j);
    end
   
end

 

mesh(x,y,z)
axis([-10 10 -10 10 -10 10 -10 10])
view([158,26])
colormap(hsv)

 

 

 

 

 

 

 

 

Tow Launch Qualification Obtained Today

Today I took a trip to Lake Conroe in Texas to work on my hang gliding tow launch qualification.

There are a number of ways to get a hang glider flying. One way is to launch yourself off the side of a hill or mountain. In Texas there aren’t too many mountains so I have been tow training.

Just after launch

A tow launch involves being pulled behind some motor vehicle, a boat or a truck for example. For my training I am pulled by a speed boat. I am attached to a winch on the boat which releases the tow line until I get to about 2000 feet from which point the boat stops. When the boat stops I can see there is no wake from behind the boat and so this is my cue to release the tow line and glide.

Gaining altitude!

This is my second day of lessons, although my first day was a couple of months ago. I have am qualified to launch solo off of a mountain (foot launch) but not tow launch and as there is a number of additional safety requirements I had to fly tandem with an experienced instructor several times before my first solo.

I took 4 tandem flights my first day, and 2 tandem flights today making a total if 6 tandem flights. My instructor felt I was ready to fly solo today so I got my first tow launch solo in, and am now qualified.

Whats great about this is that it means I can do a lot more gliding now. Gregg (my instructor) can take me up for $20 a time now so I can get plenty of airtime in for not too much expense. This will help me build confidence and experience and expose me to a lot more airtime so that I can begin to work towards my next qualification, the hang 3 license.

Way up there!

Popular Articles Discussing Recent Warp Drive Model

I was flattered to see that a number of big name sites chose to write articles on the new warp drive concept I recently introduced. It was great for me to see that there is so much public interest in the possibilities this kind of research generates.

The Discovery Channel wrote an interesting article which you can find here.

Fox news also wrote an article which you can find here.

MSNBC wrote an article here.

Yahoo’s article can be found here.

Spacedaily’s article can be found here.

Space-travel wrote an article here.

Warp Drive Basics

Warp drive is a name borrowed from science-fiction and applied to theoretical propulsion mechanisms which have the remarkable ability to propel a spacecraft faster than the speed of light. I have long been interested in this concept and have had the privilidge to spend some time working on a novel form of warp drive. I published a paper on the internet in 2005 – the Supersymmetry Breaking Casimir Warp Drive. I did not submit the paper for peer review as I felt there were still a lot of details to be worked out. However, the paper caught the attention of the organizers of the STAIF conference and I was invited to speak. You can watch my STAIF presentation here. Late in 2007 I was invited to talk at the British Interplanetary society at the Warp Drive Symposium. The conference encouraged me to divert some time from my PhD research to developing the warp drive concept I had had two years earlier. As a result of this I wrote a more refined paper that has been accepted for publication in a peer reviewed journal.

The rest of this page is laymans review of warp drive theory.

What is a Warp Drive?

The term ‘warp drive’ originates from science fiction, and for many people it conjures images of Captain Kirk and the starship Enterprise. However, a 1994 paper by the theoretical physicist Miguel Alcubierre placed the concept of a warp drive on a more scientific foundation. Alcubierre’s paper demonstrated that a certain class of solution to the equations of general relativity could “stretch” space in a way such that space itself would expand behind a hypothetical spacecraft, while contract in front of the craft, creating the effect of motion.

One motivation for creating a warp drive is that the universe huge, and the todays propulsion technology restricts us to the exploration of our own solar system. Visiting even the nearest star systems would take us many tens of thousands of years at best. A persuasive reason for why we might actually want to visit other stars is the recent evidence of “extrasolar planets,” which are planets orbiting stars other than our sun. To date, we know of at least 250 extrasolar planets. Even more exciting is the possibility that some of these planets may be “Earth-like.” If we wanted to visit extrasolar worlds in time-frames on the order of a human lifespan a warp drive would need to be developed.

One particularly appealing aspect of this approach to propulsion is that a spacecraft could theoretically travel faster than the speed of light. Although Special relativity forbids objects from moving through space at or above the speed of light, the fabric of space is not restricted in any way. Indeed, it is believed that during the inflationary period of the universe immediately after the big bang, that spacetime inflated at many thousands of times the speed of light.

Negative Energy and the Quantum Vacuum

The warp drive concept is based on Einstein’s general theory of relativity (GR), an accepted and well-tested physical theory. One of the necessary components of a warp drive is a form of energy called negative energy, which would have to be produced in large amounts for propulsion of a spacecraft to occur. Negative energy has been experimentally verified in a famous experiment called the “Casimir Effect”.

The Casimir Effect is one of the most exciting physical manifestations of quantum vacuum fluctuations. In its simplest form, it is the interaction between a pair of neutral, parallel conducting planes which modifies the ground state of the quantum vacuum in the interior portion of the plates, creating a force which attracts the plates to each other.

The interpretation of this phenomenon is that a negative energy state exists in the interior region of the plates. In theory, the Casimir Effect could be used to create the negative energy required for a warp drive. From this perspective, there is nothing that prevents the creation of warp drive.

Even if we could generate the required negative energies, how could we interact directly with spacetime to cause the required expansion and contraction? In this case nature herself can provide a level of insight. Spacetime is already expanding, albeit very slowly. In 1929, Hubble’s observation of galactic redshifting cemented the paradigm of an expanding spacetime in physical cosmology.

The energy responsible for the current expansion of spacetime is usually termed the “cosmological constant,” or equivalently, the “quantum vacuum energy” (both terms will be used interchangeably in this article). On a local scale, space is expanding incredibly slowly: around a billion billionth of a meter per second per meter. Clearly, to build a warp drive would require spacetime to be stimulated in some way to expand (and contract) at a far higher rate, but the fact that it is already expanding gives us a primitive research direction.

Understanding the cosmological constant could be the key to warp drive. However, there are currently several models which attempt to explain the physical origin of this phenomenon. One model suggests that the vacuum energy of graviton fluctuations in the extra dimensions may ultimately be responsible.

Extra Dimensions

The concept of extra dimensions may sound fantastical, but the idea is not new. It was originally introduced by the physicist Theodore Kaluza in 1919, in an effort to unify the laws of gravitation and electromagnetism. Although the theory did contain flaws, it was the first hint that extra-spatial dimensions may play an important role in physics.

One way to envisage an extra dimension of spacetime is to imagine a hosepipe. From a long distance it looks like a one dimensional line but a closer inspection reveals that every point on the line is in fact a circle.

Extra dimensions have now become an accepted component of contemporary theoretical physics. M-theory is a theory that attempts to unify all known physics under a single mathematical and conceptual framework, and predicts the existence of extra spatial dimensions. Another other popular extra-dimensional model is the Arkani-Hamed-Dimopoulos-Dvali (ADD) theory of large extra dimensions. The theory attempts to explain the observation that gravity is far weaker than the other known forces. One way to think about the ADD model is to picture gravity as being free to propagate in all dimensions (including the extra ones), while other forces are restricted to our familiar three spatial dimensions. Thus, gravity is, in a sense, diluted, which results in its being much weaker than the other forces.

How are higher dimensions and the vacuum energy related to the cosmological constant? Well, the vacuum energy is precisely the energy that causes the plates in the Casimir effect to attract one another, as discussed earlier. This vacuum energy should also exist in higher dimensions In the ADD model it is the size (or radius) of the extra dimension that directly effects the higher dimensional vacuum energy which ultimately regulates the magnitude of the cosmological constant, and therefore the expansion of spacetime.

Higher Dimensional Warp Drive

I recently investigated the plausibility of locally adjusting the size of the extra dimension to locally adjust the cosmological constant (by local, we mean in the vicinity of a spacecraft); this would theoretically modify spacetime around a craft and could be tuned to acquire the characteristics of the Alcubierre warp bubble. The basic idea is that by altering the radius of an extra dimension, it would be possible, in principle, to adjust the energy density of spacetime which relates directly to the cosmological constant which ultimately controls the inflation/contraction of space itself. The laymens version of the paper can be found here, of for the more mathematically minded please click here (note, when you follow the links you will then need to download the pdf file in the top right of the page).

We studied the idea from two angles. The first approach involved utilizing the physics of quantum field theory, and the another involved the physics of Einstein’s theory of general relativity. The equations of both theories gave complimentary results indicated that the physics of the extra dimensional space effects the expansion rate of “normal” space by a “dimensional shearing” effect. The equations of GR demonstrated that shrinking the extra dimension would inflate our space, and that expanding the extra dimension would contract our space. In this way, a bubble of expanding/contracting spacetime could be created at the rear/front of a spacecraft.

Preliminary calculations using quantum field theory indicated that superluminal propulsion could be achieved at an estimated energy cost of 10E45 Joules, or roughly the total mass-energy contained within the planet Jupiter after using the famous relation E=mc2. Although this number may appear enormous, it is certainly an improvement on earlier calculations, which indicated that the warp drive would require more mass-energy than is contained in the entire observable universe.

Further calculations revealed an upper bound on the velocity a warp drive might obtain. If the cosmological constant is indeed a consequence of the size of the extra dimension, then there would be a minimum size to this dimension; this would be the Planck length, which is the believed to be the smallest length possible. If the extra dimension were shrunk to the Planck length, then our calculations reveal the limit on warp drive velocity to be 10E32c (where c is the speed of light). This number is a theoretical bound, as our calculations regarding the energy required to reach this velocity indicate that significantly more mass-energy than is available in the observable universe would be required.

One important aspect of future research would have to involve studying how to locally manipulate an extra dimension. String theory suggests that dimensions are globally held compact by strings wrapping around them; if this is the case, then it may be possible to locally modify the string tension, or perhaps even counter the effects of some of the string winding modes. This would achieve the desired effect of changing the size of the extra dimensions, which would theoretically lead to propulsion at greater than lightspeed.

This novel approach to warp drive, although only theoretical at this stage, gives us a glimpse as to how we might address the problems associated with the vast distances involved in interstellar travel, and how we may one day reach the stars.

About Me

Richard Obousy is a growth-oriented individual with interests spanning numerous fields. Professionally he is working on a Ph.D in theoretical physics with an interest in cosmology, quantum gravity and the Casimir Effect. Click here for more about his research. He has published papers on faster than light propulsion , SETI, string theory and astrobiology and has written and published a one week review course for the MCAT physics exam. He also enjoys writing science fiction. During the summer of 2007 Richard was accepted on to a ten week summer internship working as a quantitve analyst for a Houston based hedge fund where worked on mathematical models for the commodities market.

Richard also enjoys  teaching and has helped many undergraduates though two semesters of physics.

Richard likes to take up new challenges. He has several years of kick-boxing experience, has jumped out of an airplane multiple times (solo), holds a PADI scuba diving certification and holds a USHPA hang 2 hang gliding license. He flew his first mountain solo in Georgia this January (08). He enjoys reading science fiction and playing chess in his spare time.
 
He also loves to travel and has seen much of Western Europe, Thailand, North Africa, The Unites States, Canada, Guatemala and Belize.