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 [2726] (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 [29Jump To The Next Citation Point], 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.)

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