A modified Tolman mass-energy formula

A modified Tolman mass (energy) formula is derived for spherically symmetric, time-independent systems. In the absence of surfaces of discontinuity, the modified formula has, in contradistinction to the original formula, the following desirable properties: (i) it always gives the correct mass of the system as a whole; (ii) it always gives the correct mass of any portion of the system which is surrounded by vacuum; and (iii) it remains invariant under a rescaling of the time coordinate of the formt Ct, C=constant. In the presence of surfaces of discontinuity the Tolman mass formula is further modified by the addition of the Israel mass associated with each surface. The resulting formula also has the above three properties. A new exact solution of Einstein's equations which is well behaved everywhere and is, in a sense, a generalization of Florides' new interior Schwarzschild solution is also presented.This result explains why, takingT₁¹=[C] in Section 2, we were able to obtain a solution free of surfaces of discontinuity.