3 Global symmetric solutions2 Local existence2.4 Matter fields

2.5 Free boundary problems

  In applying general relativity one would like to have solutions of the Einstein-matter equations modelling material bodies. As will be discussed in section  3.1 there are solutions available for describing equilibrium situations. However dynamical situations require solving a free boundary problem if the body is to be made of fluid or an elastic solid. We will now discuss the few results which are known on this subject. For a spherically symmetric self-gravitating fluid body in general relativity a local in time existence theorem was proved in [109]. This concerned the case where the density of the fluid at the boundary is non-zero. In [140] a local existence theorem was proved for certain equations of state with vanishing boundary density. These solutions need not have any symmetry but they are very special in other ways. In particular they do not include small perturbations of the stationary solutions discussed in section  3.1 . There is no general result on this problem up to now.

Remarkably, the free boundary problem for a fluid body is also poorly understood in classical physics. There is a result for a viscous fluid  [154] but in the case of a perfect fluid the problem was wide open until very recently. Now a major step forward has been taken by Wu [168], who obtained a result for a fluid which is incompressible and irrotational. There is a good physical reason why local existence for a fluid with a free boundary might fail. This is the Rayleigh-Taylor instability which involves perturbations of fluid interfaces which grow with unbounded exponential rates. (Cf. the discussion in [18].) It turns out that in the case considered by Wu this instability does not cause problems and there is no reason to expect that a self-gravitating compressible fluid with rotation in general relativity with a free boundary cannot also be described by a well-posed free boundary value problem.

One of the problems in tackling the initial value problem for a dynamical fluid body is that the boundary is moving. It would be very convenient to use Lagrangian coordinates, since in those coordinates the boundary is fixed. Unfortunately, it is not at all obvious that the Euler equations in Lagrangian coordinates have a well-posed initial value problem, even in the absence of a boundary. It was, however, recently shown by Friedrich [73] that it is possible to treat the Cauchy problem for fluids in general relativity in Lagrangian coordinates.

3 Global symmetric solutions2 Local existence2.4 Matter fields

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