6.3.2 The Central Pressure of a Relativistic Star

Fig. 6.6 In both pictures the thiner line corresponds to the exact relativistic solution for the pressure behavior in the case of a uniform density spherical star while the thicker line depicts the Newtonian solution for the same values of the radius R and of the mass M. Measuring distances in units of the Schwarzschild radius, the only relevant parameter is R, the mass being unity by definition. In the first picture the radius R is very small and close to the critical value

R0 = 9/4. Here we note a big difference between the Newtonian and the relativistic behavior. In the second picture the radius R is not too close to the critical value R0 and the relativistic behavior, at R = 50, is already almost coinciding with the Newtonian one

Hence for normal stars, whose radius is much bigger than the Schwarzschild radius, the Newtonian theory is an extremely accurate description of their behavior and relativistic effects are completely negligible. On the other hand for small stars relativistic effects become very important and there is a qualitatively different new feature. Indeed it is evident from (6.3.63) that the central pressure becomes infinite (there is a pole) when the ratio between the star radius R and its Schwarzschild radius GMc2 approaches the following critical value