6.2 Fuchsian equations6 Prescribed Singularities6 Prescribed Singularities

6.1 Isotropic singularities 

The existence and uniqueness results discussed in this section are motivated by Penrose's Weyl curvature hypothesis. Penrose suggests that the initial singularity in a cosmological model should be such that the Weyl tensor tends to zero or at least remains bounded. There is some difficulty in capturing this by a geometric condition, and it was suggested in [117] that a clearly formulated geometric condition (which, on an intuitive level, is closely related to the original condition) is that the conformal structure should remain regular at the singularity. Singularities of this type are known as conformal or isotropic singularities.

Consider now the Einstein equations coupled to a perfect fluid with the radiation equation of state tex2html_wrap_inline1969 . Then, it has been shown [181, 182, 88] that solutions with an isotropic singularity are determined uniquely by certain free data given at the singularity. The data that can be given are, roughly speaking, half as much as in the case of a regular Cauchy hypersurface. The method of proof is to derive an existence and uniqueness theorem for a suitable class of singular hyperbolic equations. In [13] this was extended to the equation of state tex2html_wrap_inline1951 for any tex2html_wrap_inline1973 satisfying tex2html_wrap_inline1975 .

What happens to this theory when the fluid is replaced by a different matter model? The study of the case of a collisionless gas of massless particles was initiated in [14]. The equations were put into a form similar to that which was so useful in the fluid case and therefore likely to be conducive to proving existence theorems. Then theorems of this kind were proved in the homogeneous special case. These were extended to the general (i.e. inhomogeneous) case in [12]. The picture obtained for collisionless matter is very different from that for a perfect fluid. Much more data can be given freely at the singularity in the collisionless case.

These results mean that the problem of isotropic singularities has largely been solved. There do, however, remain a couple of open questions. What happens if the massless particles are replaced by massive ones? What happens if the matter is described by the Boltzmann equation with non-trivial collision term? Does the result in that case look more like the Vlasov case or more like the Euler case?

6.2 Fuchsian equations6 Prescribed Singularities6 Prescribed Singularities

image Theorems on Existence and Global Dynamics for the Einstein Equations
Alan D. Rendall
© Max-Planck-Gesellschaft. ISSN 1433-8351
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