The well-posedness of the worldtube-nullcone problem for Einstein’s equations has not yet been established. Rendall [249] established the well-posedness of the double null version of the problem where data is given on a pair of intersecting characteristic hypersurfaces. He did not treat the characteristic problem head-on but reduced it to a standard Cauchy problem with data on a spacelike hypersurface passing through the intersection of the characteristic hypersurfaces. Unfortunately, this approach cannot be applied to the null-timelike problem and it does not provide guidance for the development of a stable finite-difference approximation based upon characteristic coordinates.
Another limiting case of the nullcone-worldtube problem is the Cauchy problem on a characteristic cone, corresponding to the limit in which the timelike worldtube shrinks to a nonsingular worldline. Choquet-Bruhat, Chruลciel, and Martín-García established the existence of solutions to this problem using harmonic coordinates adapted to the null cones, thus avoiding the singular nature of characteristic coordinates at the vertex [81]. Again, this does not shed light on numerical implementation in characteristic coordinates.
A necessary condition for the well-posedness of the gravitational problem is that the corresponding
problem for the quasilinear wave equation be well-posed. This brings our attention to the Minkowski space
wave equation, which provides the simplest version of the worldtube-nullcone problem. The treatment of
this simplified problem traces back to Duff [103], who showed existence and uniqueness for the case of
analytic data. Later, Friedlander extended existence and uniqueness to the
case for the linear wave
equation on an asymptotically-flat curved-space background [112, 111].
The well-posedness of a variable coefficient or quasilinear problem requires energy estimates for the
derivatives of the linearized solutions. Partial estimates for characteristic boundary-value problems were
first obtained by Müller zum Hagen and Seifert [213]. Later, Balean carried out a comprehensive study of
the differentiability of solutions of the worldtube-nullcone problem for the flat space wave equation [22, 23].
He was able to establish the required estimates for the derivatives tangential to the outgoing null
cones but weaker estimates for the time derivatives transverse to the cones had to be obtained
from a direct integration of the wave equation. Balean concentrated on the differentiability
of the solution rather than well-posedness. Frittelli [121] made the first explicit investigation
of well-posedness, using the approach of Duff, in which the characteristic formulation of the
wave equation is reduced to a canonical first-order differential form, in close analogue to the
symmetric hyperbolic formulation of the Cauchy problem. The energy associated with this first-order
reduction gave estimates for the derivatives of the field tangential to the null hypersurfaces but, as
in Balean’s work, weaker estimates for the time derivatives had to be obtained indirectly. As
a result, well-posedness could not be established for variable coefficient of quasilinear wave
equations.
The basic difficulty underlying this problem can be illustrated in terms of the one(spatial)-dimensional wave equation
where As a result, the standard technique for establishing well-posedness of the Cauchy problem fails. For
Equation (3), the solutions to the Cauchy problem with compact initial data on
are
square integrable and well-posedness can be established using the
norm (4
). However, in
characteristic coordinates the one-dimensional wave equation (5
) admits signals traveling in the
-direction with infinite coordinate velocity. In particular, initial data of compact support
on the characteristic
admits the solution
, provided that
. Here
represents the profile of a wave, which travels from past null infinity
(
) to future null infinity (
). Thus, without a boundary condition at past null
infinity, there is no unique solution and the Cauchy problem is ill posed. Even with the boundary
condition
, a source of compact support
added to Equation (5
), i.e.,
On the other hand, consider the modified problem obtained by setting ,
The modification in going from (7) to (8
) leads to an effective modification of the standard energy for
the problem. Rewritten in terms of the original variable
, Equation (11
) corresponds to the
energy
This technique was introduced in [193] to treat the worldtube-nullcone problem for the
three-dimensional quasilinear wave equation for a scalar field in an asymptotically-flat curved space
background with source
,
http://www.livingreviews.org/lrr-2012-2 |
Living Rev. Relativity 15, (2012), 2
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