Spherical Cosmologies with Anisotropic Pressure
Description of Problem and Solution
A spherically symmetric distribution of matter can be envisaged which consists
of a large number of dust particles, posessing angular momentum about the
centre of the universe. Spherical symmetry requires the particles to comove
radially on spherical shells, and for there to be a large nmber of particles
moving in every tangential direction at each point on a shell. The tangential
motion produces an effective tangential pressure in the spacetime, while the
assumption that the dust particles only interact gravitationally menas the
radial pressure vanishes. The matter distribution is thus anisotropic and can
also be inhomogeneous. It is possible to solve Einstein's equations of general
relativity for such a system. The static solution was found by Einstein, which
he took to be a model of a cluster. Datta considered the dynamic case, but only
found the solution for the case in which the every dust particle has the same
angular momentum. Bondi then generalised this to the case in which the angular
momentum differs from shell to shell. Evans also investigated the same system a
few years later. None of these authors obtained the metric for the spacetime.
Magli solved for the metric in 1997, using mass-area coordinates, which were
first introduced by Ori for charged dust collapse. Magli and others then
investigated this solution further, but only in the context of gravitational
It is possible to solve this problem in a different way, by using the
coordinates R, which is the comoving coordinate labelling the dust shells, and
tau, the proper time experienced by the dust. These coordinates are more
physical than mass-area, since the latter may both be spacelike in general. The
resulting solution exhibits a wide range of behaviors, with the equation of
motion for the areal radius, r, of each shell of dust being decoupled from the
others. The shells move in an effective potential which is a quintic polynomial
divided by r cubed. This means that an individual shell can undergo one of seven
different types of motion - expansion and recollapse, static circular orbit,
unbounded expansion/collapse, coasting expansion/collapse, oscillations,
'bouncing' and 'hesitating'. The solution has four free functions within it -
the mass within each shell, M(R), the angular momentum of the dust on a shell,
L(R), the 'energy' of a shell, E(R), and the initial distribution of shells -
r(0,R). One of these functions is not truely free, but corresponds to a choice
of radial coordinate. The global evolution of the spacetime depends on the
evolution of each shell within it, which is in turn determined by the choice of
the three functions. This choice is restricted by the
requirement that the shells of dust do not cross.
The general model is relevant for the study of cosmic censorship in
gravitational collapse, and also as a model of a toy cosmology. It is also
possible to obtain self-similar solutions to the equations. It has been shown
that many systems approach a self-similar behavior asymptotically and for this
reason new exact self-similar solutions are of interest. By letting the angular
momentum tend to infinity the system becomes a 'Universe of Light' composed of
photons with angular momentum rather than dust. These solutions also have
interesting behaviors. Finally, by solving the case in which every shell of dust
is at the same distance from the origin, an inhomogeneous generalisation of the
Kantowski-Sachs solution is found, for free streaming matter.
The general solution is described in Gair, J R, 2001, Spherical universes
with anisotropic pressure , Class. Quantum Grav. 18
4897-4919. This is available as gr-qc/0110017.
The self-similar solutions are described in Gair, J R, 2001, Self-similar
spherical metrics with tangential pressure, Class. Quantum
Grav. 19 2079-2106. This is available as a postscript file - click
The solution for null dust is described in Gair, J R, 2002, Some
radiation universes which generalize Vaidya, Class. Quantum Grav.
19 3883-3899. This is available as a postscript file -click
The solution in which every shell of dust is at the same distance
from the origin is described in Gair, J R, 2002, Kantowski-Sachs
universes with counter-rotating dust, Class. Quantum Grav.
19 6345-6358. This is available as a postscript file -click
Other Useful References
Einstein A 1939 Annals of Mathematics 40 921. This paper
describes the static Einstein cluster, which the current work generalises.
Datta B K 1970 Gen. Rel. Grav. 1 19-25. The initial paper on
the dynamic solution. Restricts to L=constant and does not obtain the
Bondi H 1971 Gen. Rel. Grav. 2 321-29. Generalises
Datta's solution to L=L(R), but no solution for the metric.
Evans A B 1977 Gen. Rel. Grav. 8 155. Further work on this
problem. No metric solution.
Magli G 1997 Class. Quantum Grav. 14 1937-53. Obtains the
metric in mass-area coordinates.
Magli G 1998 Class. Quantum Grav. 15 3215-28. Discusses
the solution applied to gravitational collapse.
Ori A 1990 Class. Quantum Grav. 7 985-98. The paper that
introduces mass-area coordinates, for charged dust collapse.
Harada T, Iguchi H and Nakao K 1998 Phys. Rev. D
58 041502. Restrict to a particular form for M, L and E and investigate
naked singularity formation.
Harada T, Iguchi H and Nakao K 1999 Class. Quantum Grav.
16 2785-96. Further investigation of the solution from the previous paper.
Kudoh H, Harada T and Iguchi H 2000 Phys. Rev. D
62 104016. Yet further investigation.
Jhingan S and Magli G 2000 Phys. Rev. D
62 124006. Discussion of some properties of more general solutions,
again primarily concerned with gravitational collapse.
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