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 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 [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 spherically-symmetric Einstein equation, not
necessarily of the Einstein–Vlasov system, for which
, and
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 [11] that a spherically-symmetric static
solution of the Einstein–Maxwell system for which
, and
satisfy
The study in [13] is concerned with the non-charged situation when a positive cosmological constant
is included. The following inequality is derived
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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.
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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