3, 4, 5, 10, 11 dimensions
If you look up at the stars tonight, remember the nearest is over 4 years away (unless you count the sun as a star, in which case 8.3 minutes away). If you quote a distance you are fooling yourself, because everything you see in in the past.
Something you see 1 metre away is 3.3 nanoseconds in the past. Spacetime is real because not only visible light itself, but also the physical forces such as electromagnetism and gravity, travel at the speed of light. So when the 'recession speeds of distant stars increase with distance' you know Hubble is not thinking about the fourth dimension, time.
The increase in speeds is not just with apparent distance, but with time past. Any variation of speed with time is an acceleration, which implies an effective outward force. Anyway, general relativity treats time (multiplied by the speed of light to get distance units) as a dimension, albeit as a Pythagorean spacetime resultant so that its square has an opposite sign to the squares of the three other dimensions. It works mathematically like a fourth dimension.
The fifth dimension is the Kaluza-Klein dimension, which reconciles Maxwell's basic electromagnetism (the light wave, based on the two curl equations) with general relativity, and says the extra dimension is curled up to give a string particle. The first four dimensions can be viewed as a brane on the five dimensional universe, or as a hologram of the five dimensional universe.
It seems that these two differing perspectives are all mathematically equivalent and equally valid physically. The issue then arises that there is a completely different formulation of string theory which is necessary, and involves 10 or 11 dimensional spacetime. This is due to the need to explain supersymmetry in the Standard Model of fundamental particle interactions. Supersymmetry doubles the number of fundamental particles by introducing a 'super partner' for each one.
The original particle and its super partner are related by supersymmetry transformation. This transformation transforms a fermion into its partnered super boson, changing the spin from a half integer to an integer value (in terms of h over twice pi, obviously). Because of supersymmetry, even the virtual particles which produce the 'Higgs field' (or spacetime fabric) have supersymmetric partners.
Unless supersymmetry is broken, the virtual particles have a mass which is the exact opposite of their super partners, so the Higgs field itself has no mass. However there is a slight break of supersymmetry, and this gives rise to mass. The superpartners have very large masses and existing particle accelerators do not have sufficient energy to detect them. Supersymmetry makes the forces described by the Standard Model - the electro-weak and the strong nuclear force - unify into a superforce at an energy of 10^16 GeV.
Without supersymmetry, the Standard Model is less elegant. Therefore 'superstring' theory was developed in 1985 to incorporate supersymmetry. It does so by utilising 10 dimensional spacetime, in which 6 dimensions are compressed into a Calabi-Yai manifold, suggested by Eugenio Calabi and proved as a possibility by Shing-Tung Yau. In 1995 Edward Witten had another success, with M-theory, in which 10 dimensional superstring theory is equivalent to 11 dimensional supergravity.
According to Wikipedia (http://en.wikipedia.org/wiki/Supergravity):
'A supergravity theory is a field theory combining supersymmetry and general relativity. Like any field theory of gravity a supergravity theory contains a spin-2 field whose quantum is the graviton. Supersymmetry requires the graviton field to have a superpartner. This field has spin 3/2 and its quantum is the gravitino. The number of gravitino fields is equal to the number of supersymmetries. Supergravity theories are believed to be the only consistent theories of interacting massless spin 3/2 fields.
'A supergravity theory is generally the zero-length limit of a superstring theory (i.e., the limit in which the string is approximated as having zero length, and treated as a dimensionless point-particle), with the exception of "maximal" 11-dimensional supergravity, which is a limit of M-theory (most likely the limit in which the membranes are treated as having zero volume). The underlying spacetime is a supermanifold and its symmetries are superdiffeomorphisms.'
So we can see why Witten's breakthrough in 1995 was greeted with such enthusiasm that every other approach to gravity was dismissed. Witten had proved that an extra-dimensional quantum theory of gravity, 11-D supergravity, was mathematically equivalent to 10-D superstring theory.
The issue I take is that at each step in this story, the role of pure mathematical induction has increased slightly, and the connection to direct experiment has declined slightly. After sufficient steps, however small, you find yourself in a world of 10/11 dimensional spacetime. It is just a mathematical convenience, or is it a reality which is here to stay?
Peter Woit is concerned that if string theory is allowed to go unchallenged, experiments may become a thing of the past. From the other end of physics, when you look deeply at the electromagnetic field, which is crucual to Maxwell's electromagnetism, you can see that it may give way to a deeper understanding in a different way.
The physical mechanisms of forces, the light speed energy of gauge bosons which actually flow along electric and magnetic field lines to cause forces, can be examined by a new approach, the experimental facts of how a light speed logic pulse is propagated by a transmission line, a pair of wires. What is amazing is the culture clash between the mathematical physicists and some electrical engineering physicists who when presenting a problem or a new approach are treated as crackpots. Who is more in tune with reality, the mathematician or the engineer?
Of course both can turn crackpot when they are outside their area of expertise. Consider the following article, which is very valuable as a source for a new, wave based, approach to physics:
Sadly that article, while brilliantly showing that a capacitor charges up with light speed enegy flowing into it (sucking in electrons to the negative plate at the same time, of course), was deemed politically inexpedient and was suppressed for 27 years.
No wonder Ivor Catt has little patience with modern physics.
I think it weird that in order to point out that a light photon is half negative electric field energy and half positive, and that in pair production a light ray (gamma photon) of at least two electron rest masses transforms into an electron and positron pair, I have to be obscure. Clearly the electron is Poynting vector electromagnetic wave energy, in which the electric field, magnetic field and light speed propagation vectors are all perpendicular, trapped in a circle.
This gives rise to radial electric field at long distances, and magnetic dipole field (inspired by Penrose's twistor picture) like a torus. It therefore has all the known electron properties: wave properties, spin, magnetic moment, radial electric field. It is confined by gravity due to its own equivalent mass. But if I point this out too directly, it looks crackpot, so it must be made a harmless muse on the subject of a mathematician's bible (a very useful book in fact, because it covers all of the standard ideas to a high level) written by Roger Penrose:
http://www.math.columbia.edu/~woit/wordpress/?p=259#comments : Nigel Says: September 19th, 2005 at 8:19 am Penrose’s book reminds me of Maxwell’s Treatise on Electricity and Magnetism, being a mixture of ingenious physical ideas .... Whenever I see the picture of Penrose’s twistor, I automatically wonder what the lines are, electric field, magnetic field, or Poynting vector of energy transfer? The same happens with field lines in electromagnetism. If you have a Poynting vector going around in a simple circle, you get a dipole magnetic field which looks similar to Penrose’s twistor, while the electric field lines spread out radially.