Ingeneral relativity,Buchdahl's theorem, named afterHans 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 thegravitational constant and${\displaystyle c}$ is thespeed of light. This inequality is often referred to asBuchdahl's bound. The bound has historically also been called Schwarzschild's limit as it was first noted byKarl Schwarzschild to exist in the special case of a constant density fluid.^[2] However, this terminology should not be confused with theSchwarzschild radius which is notably smaller than the radius at the Buchdahl bound.

Given a static, spherically symmetric solution to theEinstein equations (withoutcosmological constant) with matter confined to a real radius${\displaystyle R}$ that behaves as aperfect fluid with adensity that does not increase outwards. (An areal radius${\displaystyle R}$ corresponds to a sphere of surface area${\displaystyle 4\pi R^2}$. In curved spacetime the proper radius of such a sphere is not necessarily${\displaystyle R}$.) Assumes in addition that the density and pressure cannot be negative. The mass of this solution must satisfy

For his proof of the theorem, Buchdahl uses theTolman-Oppenheimer-Volkoff (TOV) equation.

The Buchdahl theorem is useful when looking for alternatives toblack holes. Such attempts are often inspired by theinformation paradox; a way to explain (part of) thedark matter; or to criticize that observations of black holes are based on excluding known astrophysical alternatives (such asneutron stars) rather than direct evidence. However, to provide a viable alternative it is sometimes needed that the object should be extremely compact and in particular violate the Buchdahl inequality. This implies that one of the assumptions of Buchdahl's theorem must be invalid. A classification scheme can be made based on which assumptions are violated.^[3]

The special case of the incompressible fluid or constant density,${\displaystyle \rho (r)={\rho }_{\ast }}$ for, is a historically important example as, in 1916, Schwarzschild noted for the first time that the mass could not exceed the value${\displaystyle \frac{4R{c}^2}{9G}}$ for a given radius${\displaystyle R}$ or the central pressure would become infinite. It is also a particularly tractable example. Within the star one finds,^[4]

and using the TOV-equation

such that the central pressure,${\displaystyle p(0)}$, diverges as${\displaystyle R\to 9GM/4c^2}$.

Extensions to Buchdahl's theorem generally either relax assumptions on the matter or on the symmetry of the problem. For instance, by introducing anisotropic matter^[5][6] or rotation.^[7] In addition one can also consider analogues of Buchdahl's theorem in other theories of gravity^[8][9]

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