My PhD is in the general area of mathematical physics and specifically is on general relativity in five dimensions. Allow me to explain.
Sir Isaac Newton’s understanding of the universe saw time as absolute and independent of the three spatial dimensions we experience every day. One result of his theory is addition of velocities – if I throw an apple core out of a window of a car travelling at thirty mph, the speed of the apple will be thirty mph plus the speed at which I threw the apple.
So far, so good. But when we try to do the same with light, problems arise. The famous experiment by Michelson and Morley, trying to measure the speed of the ‘ether’ (a substance through which all was meant to move in the universe), found that light moved at the same speed in all directions. This result was already postulated by James Clark Maxwell in his theory of electromagnetism and was adopted by Einstein into his special theory of relativity, published in 1905.
Special relativity does not allow for gravity (inertial frames are assumed to move with constant velocity – hence no accleration and no gravity). It took ten years and lots of help from much better mathematicians than Einstein to produce a theory that does allow for gravity, the general theory of relativity. Gravity is understood as resulting from the geometry of spacetime (note, time is now no longer an absolute), so just as if one places a football on a stretched-out sheet, making the sheet bend, light tries to follow a straight path but ends up following the curvature of spacetime, which in this case is the curvature of the sheet.
Einstein attempted to unite his relativity to electromagnetism by building on the work of Theodor Kaluza and Oskar Klein in the 1920s, who postulated a universe with a fifth (spatial) dimension which is compact (so small we cannot see it). However, their theory could not account for electrons and was soon trumped by quantum theory, which Einstein objected to on theological grounds.
These Kaluza-Klein spacetimes fell out of fashion until the 1970s, when scientists began seeking a way to unite general relativity with quantum mechanics into a ‘quantum theory of gravity’ (as they stand, the two are mathematically incompatible). Stephen Hawking and Gary Gibbons found solutions of Einstein’s equations that contained a finite number of gravitational instantons (think particles). Their equivalents in five dimensional Kaluza-Klein spacetimes are called monopoles; they do not occur in four-dimensional relativity and are as yet unobserved (though the LHC at CERN may change that).
My work looks at instanton and monopole solutions with infinitely many of these objects. If one imagines a periodic distribution, then the resulting potential does not converge (it blows up to infinity, which is not allowed). My thesis looks at various ways to attempt to ensure converge in these contexts (and of black holes, where similar potentials arise) and the problems with these various methods.
That’s a long-winded way of saying that I wrote 207 pages of ‘ha ha, you’re wrong’ to people far cleverer than I will ever be!