In view of the Schwarzschild metric (51), Schwarzschild asked already in 1916 the question: How large can possibly be? He gave the answer [170] in the special case of the Schwarzschild interior solution, which has constant energy density and isotropic pressure. In 1959 Buchdahl [41] extended his result to isotropic solutions for which the energy density is nonincreasing outwards and he showed that also in this case
This is sometimes called the Buchdahl inequality. Let us remark that the Buchdahl inequality can obviously be written as , but since it is the quantity , which appears in the Schwarzschild metric (51), it is common to keep the form of Equation (62). A bound on has an immediate observational consequence since it limits the possible gravitational red shift of a sphericallysymmetric static object.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 nonincreasing energy density, but nevertheless is always found to be less than in the numerical study [22]. Also for other matter models the assumptions are not satisfying. As pointed out by Guven and Ó Murchadha [91], 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 [110] proposed a model of an anisotropic star already in 1933, and Binney and Tremaine [38] explicitly allow for an anisotropy coefficient. Hence, it is an important question to investigate bounds on under less restrictive assumptions.
In [10] it is shown that for any static solution of the sphericallysymmetric Einstein equation, not necessarily of the Einstein–Vlasov system, for which , and
the following inequality holds Moreover, the inequality is sharp and sharpness is obtained uniquely by an infinitely thin shell solution. Note in particular that for Vlasov matter and that the righthand side then equals as in the Buchdahl inequality. An alternative proof was given in [103] 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 [103] is weaker than the result in [10] 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 Einsteinmatter 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 [8], where in particular it is shown that arbitrarily thin shells exist, which are regular solutions of the sphericallysymmetric Einstein–Vlasov system. Using the strategy in [9] it follows thatin 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 sphericallysymmetric 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 [11] that a sphericallysymmetric static solution of the Einstein–Maxwell system for which , and satisfy
Note, in particular, that the inequality holds for solutions of the Einstein–Vlasov–Maxwell system, since the conditions above are always satisfied in this case. This inequality (65) implies that the stability radius is outside the outer horizon of a Reissner–Nordström black hole. In [78] the relevance of an inequality of this kind on aspects in blackhole 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 [16] that there is also another type of solution, which saturates the inequality for which the inner and outer horizon coincide.The study in [13] is concerned with the noncharged 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 nonunique. 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.
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Living Rev. Relativity 14, (2011), 4
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