Buchdahl's theorem

In general relativity, Buchdahl's theorem, named after Hans Adolf Buchdahl,^[1] makes more precise the notion that there is a maximal sustainable density for ordinary gravitating matter. It gives an inequality between the mass and radius that must be satisfied for static, spherically symmetric matter configurations under certain conditions. In particular, for areal radius ${\displaystyle R}$, the mass ${\displaystyle M}$ must satisfy

where ${\displaystyle G}$ is the gravitational constant and ${\displaystyle c}$ is the speed of light. This inequality is often referred to as Buchdahl's bound. The bound has historically also been called Schwarzschild's limit as it was first noted by Karl Schwarzschild to exist in the special case of a constant density fluid.^[2] However, this terminology should not be confused with the Schwarzschild radius which is notably smaller than the radius at the Buchdahl bound.