We examine the Einstein equation coupled to the Klein-Gordon equation for a
complex-valued scalar field. These two equations together are known as the
Einstein-Klein-Gordon system. In the low-field, non-relativistic limit, the
Einstein-Klein-Gordon system reduces to the Poisson-Schr\"odinger system. We
describe the simplest solutions of these systems in spherical symmetry, the
spherically symmetric static states, and some scaling properties they obey. We
also describe some approximate analytic solutions for these states.
The EKG system underlies a theory of wave dark matter, also known as scalar
field dark matter (SFDM), boson star dark matter, and Bose-Einstein condensate
(BEC) dark matter. We discuss a possible connection between the theory of wave
dark matter and the baryonic Tully-Fisher relation, which is a scaling relation
observed to hold for disk galaxies in the universe across many decades in mass.
We show how fixing boundary conditions at the edge of the spherically symmetric
static states implies Tully-Fisher-like relations for the states. We also
catalog other "scaling conditions" one can impose on the static states and show
that they do not lead to Tully-Fisher-like relations--barring one exception
which is already known and which has nothing to do with the specifics of wave
dark matter.Comment: This is a posting of my PhD thesis (Duke University Department of
Mathematics, 2015). 73 pages, 17 figures, 3 table