No uniform density star in general relativity

As per general relativity (GR), there cannot be any superluminal propagation of energy. And thus, the sound speed in a continuous medium, $c_s=\sqrt{dp/d\rho}$, must be subluminal. However, if one would conceive of a {\em homogeneous} fluid, one would have $c_s=\infty$ unless pressure too would be homogeneous. Thus it is universally accepted that the maiden GR interior solution obtained by Schwarzschild, involving a homogeneous fluid having a boundary, is unphysical. However no one has ever shown how this exact solution is in reality devoid of physical reality. Also, this solution is universally used for approximate modelling of general relativistic stars and compact objects. But here first we show that in order that the Kretschmann scalar is continuous, one should have $\rho=0$ for strictly homogeneous static stars. Further, by invoking the fact that in GR, given one time label $t$ one can choose another time label $t_*=f(t)$ {\em without any loss of generality}, we obtain the same result that for a static homogeneous sphere $\rho=0$. Consequently, it is eventually found that the static homogeneous sphere having a boundary is just part of the vacuum where $c_s=0$ rather than $\infty$. Therefore all general relativistic stars must be inhomogeneous.