Maximum mass of a spherically symmetric isotropic star

A well known result of a theorem due to Buchdahl, for a regular fluid sphere with a mass density which does not increase outwards, is that the ratio of its gravitational mass M to the coordinate radius R satisfies the inequality $GM/R<~\frac{4}{9}.$ This restriction arises from the condition that the isotropic pressure does not become infinity at the center of the sphere to prevent collapse. Buchdahl has also derived an inequality for the value of the central pressure of the sphere which we use to show that the minimum value for this pressure corresponds to a fluid of constant density. Then, using these results and the energy condition $(|p(r)|<~\ensuremath{\beta}\ensuremath{\rho}(r)/3),$ we find new bounds for the mass to radius ratio given by $2GM/R<~S(\ensuremath{\xi}),$ where $S(\ensuremath{\xi})$ is a nondecreasing function of its argument $\ensuremath{\xi}=\ensuremath{\beta}{\ensuremath{\rho}}_{c}/3\overline{\ensuremath{\rho}},$ where ${\ensuremath{\rho}}_{c}$ is the central density of the star and $\overline{\ensuremath{\rho}}$ its mean density. For a constant density star, and $\ensuremath{\beta}=3$ (which corresponds to the dominant energy condition), we have $S(1)=3/4,$ which implies an upper limit for the gravitational redshift factor for light coming from the surface of the star given by $z<~1.$ We reobtain, for a general model and the values $\ensuremath{\beta}=3,$ ${\ensuremath{\rho}}_{c}\ensuremath{\rightarrow}\ensuremath{\infty},$ Buchdahl's limit; however, a comparison of our results with a previous inequality found by Buchdahl shows that for any values of $\ensuremath{\beta}$ and the ratio ${\ensuremath{\rho}}_{c}/\overline{\ensuremath{\rho}}$ our bound of the mass to radius ratio is more strict.

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