In studying Einstein's field equations we are faced with a
problem (see [16,
20,
35
]): 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 § 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 [16,
35
]. 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.
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Hyperbolic methods for Einstein's Equations
Oscar A. Reula http://www.livingreviews.org/lrr-1998-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |