The modification of the field equations outside the constraint sub-manifold, or the 3+1 decomposition point of view

3.2 The modification of the field equations outside the constraint sub-manifold, or the 3+1 decomposition point of view

Another approach to deal with the diffeomorphism freedom of Einstein’s equations is by first removing the diffeomorphism invariance. This is done by prescribing the time foliation along evolution, that is, by prescribing a lapse-shift pair along evolution. This removes the diffeomorphism invariance up to three-dimensional diffeomorphisms at the initial surface. Sometimes four dimensional covariance is also broken by splitting Einstein’s equations, and possibly other supplementary equations, with respect to that foliation, and then recombining the split pieces in a suitable way. The resulting equations are equivalent to the original ones – for it is just a linear combination of them – so they have the same solutions. Notice that here we are doing more than just a 3+1 decomposition, since in general one is recombining space-space components of the Einstein tensor with time-time, and time-space components in a non-covariant way, and taking as equations this combination, or even transforming the equations to first order in derivatives of the variables by defining new variables and equations and modifying them.

After this procedure is done, one obtains a system which is symmetric hyperbolic for most choices of given lapse-shift functions, once they are suitably re-scaled. Subsequent arguments go very much on adding equations for the lapse-shift vector in order to make the whole system well posed, and presumably useful for some application. It is instructive to think of these modifications of the evolution equations from the point of view of the initial value formulation. There one starts by solving the constraint equations, the time-time and time-space components of the Einstein tensor, at the initial surface. With the initial data thus obtained, one finds the solution to the evolution equations which are taken to be the space-space components of the Einstein tensor. Since that evolution preserves the constraints, (The vector field generating the flow in phase-space is tangent to the constraint sub-manifold.), one can forget about the constraint equations and think of the evolution equations as providing an evolution for the whole phase-space. In this sense, the modification one is making affects the evolution vector outside the constraint sub-manifold, leaving the vector intact at it. Uniqueness of solutions, which follows from the well posedness of the system, then implies that the solutions stay on the sub-manifold. Nevertheless, and we shall return to this point, as shown in [31 ], there is no guarantee that the sub-manifold of constraint solutions is stable with respect to the evolution vector field as extended on the whole phase-space. This is an important point for numerical simulations.