The Tolman–Oppenheimer–Volkoff (TOV) equation admits singular solutions in addition to regular ones. Here, we prove the following theorem. For any equation of state that (i) is obtained from an entropy function, (ii) has positive pressure and (iii) satisfies the dominant energy condition, the TOV equation can be integrated from a boundary inwards to the center. Hence, the thermodynamic consistency of the EoS precludes pathological solutions in which the integration terminates at finite radius (because of horizons, or divergences / zeroes of energy density). At the center, the mass function either vanishes (regular solutions) or it is negative (singular solutions). For singular solutions, the metric at the center is locally isomorphic to negative-mass Schwarzschild spacetime. This means that matter is stabilized because the singularity is strongly repulsive. We show that singular solutions are causally well behaved: they are bounded-acceleration complete, and they are conformal to a globally hyperbolic spacetime with boundary. Finally, we show how to modify unphysical equations of state in order to obtain non-pathological solutions, and we undertake a preliminary investigation of dynamical stability for singular solutions.

Tolman–Oppenheimer–Volkoff（TOV）方程除正则解外，还接受奇异解。在这里，我们证明以下定理。对于（i）从熵函数获得，（ii）具有正压且（iii）满足主要能量条件的任何状态方程，可以将TOV方程从边界向内向中心积分。因此，EoS的热力学一致性排除了其中积分在有限半径处终止（由于水平线或能量密度的发散/零）的病理学解决方案。在中心，质量函数消失（正则解）或为负（奇异解）。对于奇异解，中心的度量与负质量Schwarzschild时空局部同构。这意味着物质是稳定的，因为奇异性是强烈排斥的。我们证明了奇异解在因果上表现良好：它们是有界加速完备的，并且它们与带边界的全局双曲时空是共形的。最后，我们展示了如何修改非物理状态方程以获得非病理学解，并对奇异解的动力学稳定性进行了初步研究。