Causal configurations of homogeneous energy density in general relativity

If the causality condition [the speed of sound always remains less than that of light in vacuum, i. e., $v \leq c = 1$] is imposed on the spheres of homogeneous energy density, the `ratio of the specific heats', $\gamma \leq 2.59457$, constraints the compaction parameter, $u [\equiv (M/a)$, mass to size ratio in geometrized units] of the dynamically stable configurations $\leq 0.34056$ [corresponding to a surface redshift $(z_a) \leq 0.771$]. Apparently, The maximum value of $u$ obtained in this manner belongs to an absolute upper bound, and gives: (i) The maximum value for static neutron star masses as $5.4 M_\odot$, if we substitute the density at the surface of the configuration equal to the average nuclear density, $E = 2 \times 10^{14}$ g cm$^{-3}$ [e.g. {\it Nature}, {\bf 259}, 377 (1976)]. (ii) However, if the density of the static configuration is constrained to the value $1.072 \times 10^{14}$ g cm$^{-3}$, by imposing the empirical result that the minimum rotation period of the fastest rotating pulsar known to date, PSR 1937 + 21, is 1.558 ms, the maximum mass value for static neutron stars exceed upto $7.4 M_\odot$. These masses have important implications for the massive compact objects like Cyg X-1, Cyg XR-1, and LMC-X3 etc., which may not, necessarily, represent black holes. (iii) The minimum rotation periods for a static $1.442 M_\odot$ neutron star to be 0.3041 ms. (iv) A suitable stable model of ultra-compact objects [$u > (1/3)$] which has important astrophysical significance.