The binding energy of a Schwarzschild sphere inD space-time dimensions

Abstract

We calculate the binding energy of a Schwarzschild gas sphere in D space time dimensions. To the post-Newtonian approximation we demonstrate that the interior Schwarzschild solution does not exhibit an instability before the Schwarzschild radius is reached for D = 4. For D = 5, 6 we find binding energy is negativ to the post-Newtonian approximation. Extending the approximation to the post-post-Newtonian approximation we find that the post-post-Newtonian contribution to the binding energy of a Schwarzschild gas sphere is positive for the cases D = 4, 5, 6. The effect of the post-post-Newtonian contribution to the binding energy on the stability of higher dimensional gas spheres is discussed.