Closed Form Solutions To Bosonic Perturbations In General Relativity
We present some results regarding metric perturbations in general relativity and other metric theories of gravity. In particular, using the Newman Penrose variables, we write down and discuss the equations which govern tensor field perturbations of ranks $0, \pm 1$ and $\pm 2$ (scalar,vector,tensor bosonic perturbations) over certain space-times that admit specific background metrics expressible in isotropic coordinates. Armed with these equations, we are able to develop the Hadamard series which can be associated with the fundamental solution of the equations, wherein we introduce an inhomogeneous singularity at the physical space-time point of the perturbing particle. The Hadamard series is then used to generate closed form solutions by making choices for an appropriate ansatz solution. In particular, we solve for the spin-weighted electrostatic potential for the Reissner-Nordstrom black hole and for the fully dynamical potential for the Friedmann-Robertson-Walker cosmological solution.