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 collapse.

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.

# References

## My Research

• 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 here.
• 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 here.
• 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 here.
• ## 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 metric.

• 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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