In view of the Schwarzschild metric (51), Schwarzschild asked already in 1916 the question: How large can possibly be? He gave the answer  in the special case of the Schwarzschild interior solution, which has constant energy density and isotropic pressure. In 1959 Buchdahl  extended his result to isotropic solutions for which the energy density is non-increasing outwards and he showed that also in this case
The assumptions made by Buchdahl are very restrictive. In particular, the overwhelming number of the steady states of the Einstein–Vlasov system have neither an isotropic pressure nor a non-increasing energy density, but nevertheless is always found to be less than in the numerical study . Also for other matter models the assumptions are not satisfying. As pointed out by Guven and Ó Murchadha , neither of the Buchdahl assumptions hold in a simple soap bubble and they do not approximate any known topologically stable field configuration. In addition, there are also several astrophysical models of stars, which are anisotropic. Lemâitre  proposed a model of an anisotropic star already in 1933, and Binney and Tremaine  explicitly allow for an anisotropy coefficient. Hence, it is an important question to investigate bounds on under less restrictive assumptions.
In  it is shown that for any static solution of the spherically-symmetric Einstein equation, not necessarily of the Einstein–Vlasov system, for which , and and their method applies to a larger class of conditions on and than the one given in Equation (63). On the other hand, the result in  is weaker than the result in  in the sense that the latter method implies that the steady state that saturates the inequality is unique; it is an infinitely thin shell. The studies [10, 103] are of general character and in particular it is not shown that solutions exist to the coupled Einstein-matter system, which can saturate the inequality. For instance, it is natural to ask if there are solutions of the Einstein–Vlasov system, which have arbitrarily close to . This question is given an affirmative answer in , where in particular it is shown that arbitrarily thin shells exist, which are regular solutions of the spherically-symmetric Einstein–Vlasov system. Using the strategy in  it follows that
in the limit when the shells become infinitely thin.
The question of finding an upper bound on can be extended to charged objects and to the case with a positive cosmological constant. The spacetime outside a spherically-symmetric charged object is given by the Reissner–Nordström metric
where is the total charge of the object. The quantity is zero when , and is called the inner and outer horizon respectively of a Reissner–Nordström black hole. A Buchdahl type inequality gives a lower bound of the area radius of a static object and this radius is thus often called the critical stability radius. It is shown in  that a spherically-symmetric static solution of the Einstein–Maxwell system for which , and satisfy the relevance of an inequality of this kind on aspects in black-hole physics is discussed. In contrast to the case without charge, the saturating solution is not unique. An infinitely thin shell solution does saturate the inequality (65), but numerical evidence is given in  that there is also another type of solution, which saturates the inequality for which the inner and outer horizon coincide.
The study in  is concerned with the non-charged situation when a positive cosmological constant is included. The following inequality is derived
for solutions for which , and . In this situation, the question of sharpness is essentially open. An infinitely thin shell solution does not generally saturate the inequality but does so in the two degenerate situations and . In the latter case there is a constant density solution, and the exterior spacetime is the Nariai solution, which saturates the inequality and the saturating solution is thus non-unique. In this case, the cosmological horizon and the black hole horizon coincide, which is in analogy with the charged situation described above where the inner and outer horizons coincide when uniqueness is likely lost.
Living Rev. Relativity 14, (2011), 4
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