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3.1 Stationary solutions

Many of the results on global solutions of the Einstein equations involve considering classes of spacetimes with Killing vectors. A particularly simple case is that of a timelike Killing vector, i.e. the case of stationary spacetimes. In the vacuum case there are very few solutions satisfying physically reasonable boundary conditions. This is related to no hair theorems for black holes and lies outside the scope of this review. More information on the topic can be found in the book of Heusler [181] and in his Living Review [182] (see also [55] where the stability of the Kerr metric is discussed). Anderson [34] has proved uniqueness theorems for stationary vacuum spacetimes under very weak assumptions. The case of phenomenological matter models has been reviewed in [298]. The account given there will be updated in the following.

The area of stationary solutions of the Einstein equations coupled to field theoretic matter models has been active in recent years as a consequence of the discovery by Bartnik and McKinnon [37Jump To The Next Citation Point] of a discrete family of regular, static, spherically symmetric solutions of the Einstein-Yang-Mills equations with gauge group SU (2). The equations to be solved are ordinary differential equations, and in [37] they were solved numerically by a shooting method. The first existence proof for a solution of this kind is due to Smoller, Wasserman, Yau, and McLeod [324] and involves an arduous qualitative analysis of the differential equations. The work on the Bartnik-McKinnon solutions, including the existence theorems, has been extended in many directions. Recently, static solutions of the Einstein-Yang-Mills equations that are not spherically symmetric were discovered numerically [217]. It is a challenge to prove the existence of solutions of this kind. Now the ordinary differential equations of the previously known case are replaced by elliptic equations. Moreover, the solutions appear to still be discrete, so that a simple perturbation argument starting from the spherical case does not seem feasible. In another development, it was shown that a linearized analysis indicates the existence of stationary non-static solutions [71]. It would be desirable to study the question of linearization stability in this case, which, if the answer were favourable, would give an existence proof for solutions of this kind. It has, however, been argued that solutions of this kind should not exist [343].

Now we return to phenomenological matter models, starting with the case of spherically symmetric static solutions. Basic existence theorems for this case have been proved for perfect fluids [307], collisionless matter [279271Jump To The Next Citation Point], and elastic bodies [265]. All these theorems demonstrate the existence of solutions that are everywhere smooth and exist globally as functions of area radius for a general class of constitutive relations. The physically significant question of the finiteness of the mass of these configurations was only answered in these papers under restricted circumstances. For instance, in the case of perfect fluids and collisionless matter, solutions were constructed by perturbing about the Newtonian case. Solutions for an elastic body were obtained by perturbing about the case of isotropic pressure, which is equivalent to a fluid. Further progress on the question of the finiteness of the mass of the solutions was made in the case of a fluid by Makino [240Jump To The Next Citation Point], who gave a rather general criterion on the equation of state ensuring the finiteness of the radius. Further information on this issue was obtained in [174Jump To The Next Citation Point] using dynamical systems methods. Makino’s criterion was generalized to kinetic theory in [281]. This resulted in existence proofs for various models that have been considered in galactic dynamics and which had previously been constructed numerically (cf. [58320] for an account of these models in the non-relativistic and relativistic cases, respectively). In the non-relativistic case dynamical systems methods were applied to the case of collisionless matter in [173]. Most of the work quoted up to now refers to solutions where the support of the density is a ball. For matter with anisotropic pressure the support may also be a shell, i.e. the region bounded by two concentric spheres. The existence of static shells in the case of the Einstein-Vlasov equations was proved in [274Jump To The Next Citation Point].

In the case of self-gravitating Newtonian spherically symmetric configurations of collisionless matter, it can be proved that the phase space density of particles depends only on the energy of the particle and the modulus of its angular momentum [38]. This is known as Jeans’ theorem. It was already shown in [271] that the naive generalization of this to the general relativistic case does not hold if a black hole is present. Recently, counterexamples to the generalization of Jeans’ theorem to the relativistic case, which are not dependent on a black hole, were constructed by Schaeffer [318]. It remains to be seen whether there might be a natural modification of the formulation that would lead to a true statement.

For a perfect fluid there are results stating that a static solution is necessarily spherically symmetric [234]. They still require a restriction on the equation of state, which it would be desirable to remove. A similar result is not to be expected in the case of other matter models, although as yet no examples of non-spherical static solutions are available. In the Newtonian case examples have been constructed by Rein [275Jump To The Next Citation Point]. (In that case static solutions are defined to be those in which the particle current vanishes.) For a fluid there is an existence theorem for solutions that are stationary but not static (models for rotating stars) [172]. At present there are no corresponding theorems for collisionless matter or elastic bodies. In [275], stationary, non-static configurations of collisionless matter were constructed in the Newtonian case.

Two obvious characteristics of a spherically symmetric static solution of the Einstein-Euler equations that has a non-zero density only in a bounded spatial region are its radius R and its total mass M. For a given equation of state there is a one-parameter family of solutions. These trace out a curve in the (M, R) plane. In the physics literature, pictures of this curve indicate that it spirals in on a certain point in the limit of large density. The occurrence of such a spiral and its precise asymptotic form have been proved rigorously by Makino [241] for a particular choice of equation of state. An approach to these spirals which leads to a better conceptual understanding can be found in [174].

The existence of cylindrically symmetric static solutions of the Einstein-Euler system has been proved in [56]. For some remarks on the question of stability of spherically symmetric solutions see Section 4.1.

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