We have established a two-parameter class of solutions of the Einstein
gravitational field equation, corresponding to the presence of static
fluid matter with the equation of state , for all values of
in the relevant interval
. There are in fact three
parameters in play, the equation of state parameter
, the
Schwarzschild radius
and the radius
of the sphere
containing essentially all the matter, where we could put our arbitrary
radial reference point
, but the solutions depend only on
and on a ratio such as
. The fluid matter is strongly
localized, and the Schwarzschild solution is indeed the
asymptotic limit of the solutions we found here.
Two main physical facts can be abstracted from the results. The first is
that the central space in the interior of a dense conglomeration of matter
is necessarily shrunk, with actual radial lengths that are smaller, and in
some circumstances much smaller, than the corresponding variations of the
coordinate . It is even possible that these lengths go all the way to
zero in the limit in which a black hole with an event horizon would form.
In any case, this reduces the available volume of space under the matter,
and therefore compresses the matter even further. This seems to indicate
that the apparent volume of a very dense star or black hole is therefore
much larger than the actual physical one.
The second is that it is possible to have a static solution with a
localized conglomeration of pure energy, that is, of matter with the pure
radiation equation of state , matter that consists, therefore, of
pure relativistic radiation. Of course this is, in part, a consequence of
the fact that we are ignoring the outward radiation of energy from the
surface of the fluid matter to asymptotic infinity. However, close to the
limit in which a black hole would form the loss of energy by this
mechanism is strongly contained by the red shift effect upon outgoing
radiation originating from close to the position of the Schwarzschild
radius, thus justifying the static hypothesis as an approximation.
Therefore, we can only claim that this type of static or quasi-static
solution with pure radiation exists in the case of distributions of fluid
matter which are very close to a black-hole configuration.
In the limit in which we make
, where
is
the position of the inflection point of
, we seem to have
that
, where
is the radial position of the
sphere that contains essentially all the fluid matter, and
is the
Schwarzschild radius of the corresponding mass
. At the same time we
seem to have that
for almost all
, so
that all radial lengths within the matter distribution are going toward
zero. In this limit one does seem to obtain a black hole, in the usual
sense given to that term, or at least something that is indistinguishable
from it when viewed from the outside. This is not, however, a ``naked''
black hole as is usually thought to be the case, but a ``dressed'' one,
with plenty of matter and a definite, if unexpected, geometry within it.
It is very interesting to observe that this situation establishes an unexpected connection with G. 't Hooft's ideas about quantum mechanics and black holes [#!GTHooftNoInterior!#]. In describing his studies involving quantum entanglement around black holes, that author has used the figure of speech that the interior of the black hole ``is not really there'', motivated by quantum correlations between antipodal points of the surface of the black hole. Well, one way to have this situation realized is to have the internal volume of the black hole be zero, and the distances between these diametrically opposed points also be zero, just like all other radial lengths within the region containing the fluid matter. It is interesting that while that author's conclusions come from a quantum theory involving black holes, our conclusions here emerge from a theory which is, in so far as one can currently see, entirely classical.
We end with a few words about the possible routes for the continuation of
this line of work. It is to be noted that this is a very simple model, in
so far as the hypotheses about the matter are concerned, and one should
not expect it to have immediate applicability in all realistic physical
cases of interest. For realistic stars, which are known to have an
internal structure consisting of layers, it would probably be necessary to
extend the model to include equations of state that depend on , that
is, to exchange the constant
for a function
. Also
with the intent of making the model more realistic, a possibly simple
extension would be to the time-dependent case, still with spherical
symmetry. This would concern only the solution strictly within the matter,
since by the Jebsen-Birkhoff [#!JebsenTheorem!#,#!BirkhoffTheorem!#] theorem
the Schwarzschild solution outside would not change in a significant way.
It might be sufficient, though, to simply accommodate slowly changing
solutions due to outward radiation from the surface of the fluid matter
distribution.
It would certainly be interesting to extend the results obtained here to
the case of the Kerr metric, so that the extended results would be
applicable to rotating stars and black holes. However, this is likely to
turn out to be quite a difficult enterprise. Another possible extension,
as an alternative to having a fully non-homogeneous equation of state with
a function , would be to fluid matter with a discretely
non-homogeneous equation of state, possibly fluid matter with layered
distributions, with different equations of state in each layer, each with
a different constant
, tied up together by the imposition of
appropriate boundary conditions at the interfaces between layers. Although
it is not at all clear that this is actually possible, it seems like a
possibility worthwhile examining. Instabilities in these configurations
could possibly lead to fast transitions such as those which are
characteristic of some astrophysical phenomena.
Since what we presented here is a previously unknown class of solutions of
the Einstein gravitational field equation, it is likely that these new
solutions will find applications in the study of stars, as well as of
other denser astrophysical objects. Further study of the non-linear
differential equation show in Equations (77) for the
function , and in Equations (79)
and (86) for the function
, may turn out to
be useful to elucidate the actual physical structure of extremely dense
objects such as neutron stars and black holes. In particular, further
numerical studies aimed at clarifying the behavior of various aspects of
the geometry in the limit in which
, where
is the position of the inflection point of
, a
limit which we might loosely describe as the black-hole limit, would
probably be quite interesting. This is currently being worked on, and will
be included in the aforementioned forthcoming paper.