4.2 The frame representation
Another way of dealing with Einstein equations is through the frame representation. There,
instead of using the metric tensor as the basic building block of the theory, a set of frame fields
is used. At first this representation seems to be even less economical than the metric tensor
representation of geometries, since, on top of the diffeomorphism freedom, one has the freedom
to choose the frame vector fields. In short, one has sixteen variables instead of ten. But the
antisymmetry of the connection coefficients compared with the symmetry of the Christoffel
symbols levels considerably the difference, ending, after adding the second fundamental form and
others, with twenty-eight variables, instead of the thirty of the conventional system. But this
count is not entirely correct. As mentioned above, in order to close the system of equations one
has to add the evolution equations for the electric and magnetic part of the Weyl tensor, thus
ending with a total of thirty four variables. Actually one can close the system with the twenty
four variables at the expense of making the equations into second order wave equations, (See
for instance [55].), so effectively adding more variables when re-expressing it as a first order
system.
The more important application of the frame representation has been the conformal system obtained by
Friedrich [27, 26] (see Section 1), where in a fixed gauge he got a symmetric hyperbolic system which
allowed him to study global solutions. Later, using similar techniques and spinors, he found a symmetric
hyperbolic system with the remarkable property that lapse and shift appear in an undifferentiated form,
allowing for greater freedom in relating them to the geometry without hampering hyperbolicity [28].
In [29
], he introduces new symmetric systems for frame components where one can arbitrarily prescribe the
gauge functions, which in this case does not only include the equivalent to the lapse-shift pair, but also a
three by three matrix fixing the rotation of the frame. In this case, these gauge functions enter up to first
derivatives. This compares very favorably with the ADM representation schema where the
lapse-shift entered up to second order derivatives. Friedrich also finds a symmetric hyperbolic
system with the generalized harmonic time condition. Contrary to the systems in the ADM
formalism, where the issue is rather trivial, these systems do not seem to allow for a writing in flux
conservative form. We do not consider that a serious drawback. The structure of Einstein’s
equations is very different than those of fluids, where the unavoidable presence of shocks makes it
important to write them that way. Indeed the reason fluids have shocks can be attributed to their
genuinely non-linear character [43], a property not shared by Einstein’s theory. (More about this in
Section 4.)
