What follows are descriptions of the problem of hyperbolicity in general relativity and of the main approaches that have been proposed to deal with the problem.
In studying Einstein’s field equations we are faced with a problem (see [20, 34, 29]): While the theory of partial differential equations has developed as a theory for tensor components in a given coordinate system, or at best for tensor fields in a given metric space, Einstein’s equations acquire their full meaning – and the characteristics which distinguish them from all other physical theories – when they are viewed as equations for geometries, that is, for equivalent classes of metric tensors. The object of the theory is not a metric tensor, but the whole equivalence class to which it belongs – all other metrics related to the first one by a smooth diffeomorphism. This fact is contained in the equations, for they are invariant under those diffeomorphisms. To clarify the concept, and see the problem, let us assume we have a solution to Einstein equations in a given region of a manifold. Take a space-like hypersurface across it, , and a small “lens shaped” region which can be foliated by smooth space-like surfaces starting at . If Einstein’s equations were hyperbolic for the metric tensor, then uniqueness of the solution (the metric tensor) in the lens shaped region would follow from the standard theory once proper initial data at is given. But we know that if we apply a diffeomorphism to the original metric tensor solution, which is different from the identity only in a region inside the lens shaped one but which does not intersect the initial slice, the resulting metric would also satisfy Einstein’s equations, thus contradicting uniqueness, and so the possibility that the system be hyperbolic.
Since, as shown in Section 2, hyperbolicity is equivalent to the existence of norms which are bounded under evolution, we see that for Einstein’s equations there cannot be such norms on the space of metric tensors. Norms are not only important for well posedness, but also for other related issues which often appear in general relativity, in particular when one tries to see whether some approximation schema is indeed an approximation. Examples of this appear in very unrelated cases, for instance, in numerical algorithms and post-Newtonian approximations. Thus, a method is needed to find relevant norms on metric tensors, that is, to break the diffeomorphism invariance. The norms thus obtained are not natural, and so by themselves do not imply any physical closeness of metrics in numerical values. They have to be considered in their topologically equivalent class. Physically relevant notions of closeness can still be obtained by building, out of the metric tensor and its derivatives, diffeomorphism invariant quantities and making the comparisons with then.
Can we avoid this detour into tensors and make a theory of diffeomorphism invariant objects? It is not clear whether this can be done. Some attempts in this direction have been made by trying to build norms which have some partial diffeomorphism invariance. Here the norms are made out of scalars built out of curvature tensor components of the metric, in particular see [20, 29]. But I think a fully geometrical theory needs other types of mathematics than the theory of partial differential equations, a theory which might be emerging from parallel questions in quantum gravity.
© Max Planck Society and the author(s)