5.3 Buchdahl-type inequalities

Another aspect of the structure of steady states investigated numerically in [22Jump To The Next Citation Point] concerns the Buchdahl inequality. If a steady state has support in [R0, R1], then the ADM mass M of the configuration is M = m (R ) 1, where the quasi local mass m (r) is given by Equation (50View Equation) in Schwarzschild coordinates.

In view of the Schwarzschild metric (51View Equation), Schwarzschild asked already in 1916 the question: How large can 2M ∕R possibly be? He gave the answer [170] 2M ∕R ≤ 8∕9 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 non-increasing outwards and he showed that also in this case

2M-- 8- R ≤ 9. (62 )
This is sometimes called the Buchdahl inequality. Let us remark that the Buchdahl inequality can obviously be written as M ∕R ≤ 4∕9, but since it is the quantity 2M ∕R, which appears in the Schwarzschild metric (51View Equation), it is common to keep the form of Equation (62View Equation). A bound on 2M ∕R has an immediate observational consequence since it limits the possible gravitational red shift of a spherically-symmetric 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 non-increasing energy density, but nevertheless 2M ∕R is always found to be less than 8∕9 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 2M ∕R under less restrictive assumptions.

In [10Jump To The Next Citation Point] it is shown that for any static solution of the spherically-symmetric Einstein equation, not necessarily of the Einstein–Vlasov system, for which p ≥ 0, and

p + 2pT ≤ Ω ρ, (63 )
the following inequality holds
2m--(r) (1 +-2Ω)2-−-1- r ≤ (1 + 2Ω)2 . (64 )
Moreover, the inequality is sharp and sharpness is obtained uniquely by an infinitely thin shell solution. Note in particular that for Vlasov matter Ω = 1 and that the right-hand side then equals 8∕9 as in the Buchdahl inequality. An alternative proof was given in [103Jump To The Next Citation Point] and their method applies to a larger class of conditions on ρ,p and p T than the one given in Equation (63View Equation). On the other hand, the result in [103Jump To The Next Citation Point] is weaker than the result in [10Jump To The Next Citation Point] 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 2m ∕r arbitrarily close to 8∕9. 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 spherically-symmetric Einstein–Vlasov system. Using the strategy in [9] it follows that
2m (r) 8 sup --r---→ 9, r

in the limit when the shells become infinitely thin.

The question of finding an upper bound on 2M ∕R 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

ds2 = − (1 − 2M-- + Q-)dt2 + (1 − 2M--+ Q-)−1dr2 + r2(d𝜃2 + sin2𝜃dϕ2 ), r r2 r r2

where Q is the total charge of the object. The quantity 1 − 2Mr- + Qr2 is zero when ∘ --------- r± = M ± M 2 − Q2, and r± 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 spherically-symmetric static solution of the Einstein–Maxwell system for which p ≥ 0,p + 2pT ≤ ρ, and Q < M satisfy

√ -- ∘ --------- √ --- --R- R- Q2- M ≤ 3 + 9 + 3R . (65 )
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 (65View Equation) 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 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 (65View Equation), 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 non-charged situation when a positive cosmological constant Λ is included. The following inequality is derived

M 2 ΛR2 2√ ---------- ---≤ --− -----+ -- 1 + 3ΛR2, R 9 3 9

for solutions for which p ≥ 0,p + 2pT ≤ ρ, and 2 0 ≤ ΛR ≤ 1. 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 ΛR2 = 0 and ΛR2 = 1. 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.

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