Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

Scalar field dark matter 177
rotation of the galaxy does not affect the motion of test particles around
the galaxy, dragging effects in the halo of the galaxy should be too small
to affect the tests particles (stars and dust) traveling around the galaxy.
Hence, in the region of interest we can suppose the space–time to be
static, given that the circular velocity of stars (like the sun) of about
230 Km/s seems not to be affected by the rotation of the galaxy and we
can consider a time reversal symmetry of the space-time.
We start from the general spherically symmetric line element and find
out the conditions on the metric in order that the test particles in the
galaxy possess a flat rotation curve in the region where the scalar field
(the dark matter) dominates.
Assuming that the halo has spherical symmetry and that in such
regions dragging effects on stars and dust are inappreciable, i.e. the
space–time is static, the following line element is the appropriate
where A and B are arbitrary functions of the coordinate
The dynamics of test particles on this space-time can be derived from
the Lagrangian
We have two conserved quantities, the energy the
momentum and the total angular momentum,
with The radial motion equation can thus be
written as:
with the expression for the potential
Notice that, due to the spherical symmetry, we do not need to restrict the
study to equatorial orbits, this last radial motion is valid for any angle
For circular stable orbits, we have the conditions, and
which imply the following expression for the energy and total
momentum of the particles in such orbits: