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

Rotating equilibrium configurations in Einstein’s theory of gravitation 71
2. ROTATING DUST CONFIGURATIONS –
THE MACLAURIN DISK
In Newton’s theory of gravitation, it can easily be shown that a rotating,
axially symmetric and stationary dust (Here “dust” means a perfect
fluid with vanishing pressure.) configuration of finite mass must be an
infinitesimally thin disk.
Therefore, the basic equations to be solved are the Poisson equation
and the equation expressing the equilibrium between centrifugal and
gravitational forces in the disk
Here U is the gravitational potential, and are cylindrical coordinates.
It should be noted that for a prescribed rotation law the
surface mass–density cannot be chosen arbitrarily but has to be
calculated from the solution of the resulting boundary–value problem
for the Laplace equation according to
We assume
leading to the boundary condition
In addition to (11) we require at infinity and a bounded surface
mass–density at the rim of the disk The latter requirement leads
to a relation between the parameters and see Eq. (15). This
boundary–value problem has the unique solution
with oblate elliptic coordinates related to by