The Gravitational Field of a Disk

The general solution of the static field equations of general relativity is given for a disk of "counterrotating" dust particles. The only nonvanishing components of the energy-momentum tensor are $T_0^{}{}_{}{}^0$ and $T_{\chi }^{}{}_{}{}^{\chi }$, which are assumed to have $\delta$-function singularities on the disk. Two representative families of solutions are considered, and it is shown that, for these solutions, physical considerations severely limit the strength of the gravitational potentials. The first family has surface density proportional to some power of $1-{\rho }^2$. Th requirement that the velocity of the dust particles should not exceed $c$ places a bound on the gravitational red-shift of $z=1.5803\mathrm{}$ for these models. The second family is that of the uniformly rotating disks defined by $v^2={\rho }^2{\omega }^2{e}^{-4\mathrm{}\phi }$. Bardeen has pointed out that these disks can have arbitrarily large red-shifts without violating the velocity condition. However, it is shown that their red-shift cannot exceed 1.9015 before their binding energy becomes negative. This work suggests that the largest gravitational red-shift to which counter-rotating dust disks can give rise is of order of magnitude 1.