Mathematical Surprises for Physics
I like to think about surprises that mathematics throws up in physics.
Below is a modest list of some that I have thought of or have been suggested to me.
Please write to me to suggest others, or e.g. to point out a historical or mathematical inaccuracy.
Due credit will be given. Perhaps this will asymptotically become a (fun) book.
- The angle of the wake of a duck is constant, and equal to 2arcsin(1/3). In particular, it is does not
depend on the size, mass, or speed of the duck, or the strength of gravity.
- The speed of light is constant for all observers.
- The existence of black holes could be considered one of the greatest such surprises. The
first static, eternal black hole solution to the Einstein field equations was derived by Karl
Schwarzschild in 1915, but was not fully understood to represent a black hole until much later. In 1965
Roger Penrose proved the famous Penrose singularity theorem, showing that black holes can form in
dynamical general relativity provided gravity (matter) is sufficiently concentrated in a small enough region.
- Not only can black holes form in dynamical general relativity, but in fact they can form in vacuum,
from colliding gravitational waves.
- The BMS group is larger than the Poincaré group.
- In bosonic string theory, the vanishing of the Virasoro anomaly (the requirement that there be no
negative-norm states) predicts that the spacetime dimension is 26.
- In superstring theory, after adding fermions, a similar calculation predicts that the spacetime dimension is
10.
- Anti-de Sitter space is expected to be nonlinearly unstable.
- The Banach–Tarski paradox states that a solid ball can be decomposed into a finite number of pieces
in such a way that the pieces may be put together to form two solid balls identical to the original. The
theorem works in dimensions 3 and higher.
