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.