PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Einstein's field equations
The Geometry of Gravity, Ch FRW cosmology as an exact solution
Problem
A toy cosmological slice is homogeneous, so it has all three translational Killing vectors. Its matter content is a uniform magnetic field pointing along a fixed axis, so of the three rotational Killing vectors possible about a point, only the one that rotates about the field's own axis survives; the other two are broken by the field's preferred direction.
((a)) How many Killing vectors does this slice have in total?
((b)) The maximum possible number of Killing vectors for a three-dimensional space is n(n+1)/2 with n=3. How many more Killing vectors would this slice need to become maximally symmetric (homogeneous and isotropic)?