2.1 The worldtube-nullcone problem

A version of the characteristic initial value problem for Einstein’s equations, which avoids caustics, is the worldtube-nullcone problem, where boundary data is given on a timelike worldtube and on an initial outgoing null hypersurface [288Jump To The Next Citation Point]. The underlying physical picture is that the worldtube data represent the outgoing gravitational radiation emanating from interior matter sources, while ingoing radiation incident on the system is represented by the initial null data.

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, Chruล›ciel, 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 [103Jump To The Next Citation Point], who showed existence and uniqueness for the case of analytic data. Later, Friedlander extended existence and uniqueness to the ∞ C 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 [213Jump To The Next Citation Point]. 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

(∂2&tidle;t − ∂2&tidle;x)Φ = 0, (3 )
where &tidle; (t,x&tidle;) are standard space-time coordinates. The conserved energy
1 ∫ ( ) &tidle;E (&tidle;t) = -- d&tidle;x (∂ &tidle;tΦ )2 + (∂x&tidle;Φ )2 (4 ) 2
leads to the well-posedness of the Cauchy problem. In characteristic coordinates (t = &tidle;t − x&tidle;, x = &tidle;t + &tidle;x), the wave equation transforms into
∂t∂xΦ = 0. (5 )
The conserved energy on the characteristics t = const.,
∫ &tidle; 2 E (t) = dx (∂x Φ) , (6 )
no longer controls the time derivative ∂tΦ.

As a result, the standard technique for establishing well-posedness of the Cauchy problem fails. For Equation (3View Equation), the solutions to the Cauchy problem with compact initial data on &tidle;t = 0 are square integrable and well-posedness can be established using the L2 norm (4View Equation). However, in characteristic coordinates the one-dimensional wave equation (5View Equation) admits signals traveling in the +x-direction with infinite coordinate velocity. In particular, initial data of compact support Φ (0,x) = f(x ) on the characteristic t = 0 admits the solution Φ = g(t) + f(x), provided that g(0) = 0. Here g (t) represents the profile of a wave, which travels from past null infinity (x → − ∞) to future null infinity (x → + ∞). 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 Φ (t,− ∞ ) = 0, a source of compact support S(t,x) added to Equation (5View Equation), i.e.,

∂t∂x Φ = S, (7 )
produces waves propagating to x = + ∞ so that, although the solution is unique, it is still not square integrable.

On the other hand, consider the modified problem obtained by setting ax Φ = e Ψ,

−ax ∂t(∂x + a)Ψ = F , Ψ (0,x) = e f (x), a > 0 (8 )
where F = e −axS. With the boundary condition Ψ (t,− ∞ ) = 0, the solutions to (8View Equation) vanish at x = +∞ and are square integrable. As a result, the Cauchy problem (8View Equation) is well posed with respect to an L2 norm. For the simple example where F = 0, multiplication of (8View Equation) by 1 (2aΨ + ∂xΨ + 2∂tΨ) and integration by parts gives
∫ ∫ ∫ 1- ( 2 2 2) a- ( 2) a- 2 2∂t dx (∂xΨ ) + 2a Ψ = 2 dx 2(∂tΨ )∂xΨ − (∂tΨ ) ≤ 2 dx(∂xΨ ) . (9 )
The resulting inequality
∂E ≤ const. E (10 ) t
for the energy
∫ 1 ( 2 2 2) E = 2- dx (∂xΨ ) + 2a Ψ (11 )
provides the estimates for ∂xΨ and Ψ, which are necessary for well-posedness. Estimates for ∂tΨ, and other higher derivatives, follow from applying this approach to the derivatives of (8View Equation). The approach can be extended to include the source term F and other generic lower differential order terms. This allows well-posedness to be extended to the case of variable coefficients and, locally in time, to the quasilinear case.

The modification in going from (7View Equation) to (8View Equation) leads to an effective modification of the standard energy for the problem. Rewritten in terms of the original variable Φ = eaxΨ, Equation (11View Equation) corresponds to the energy

∫ 1 −2ax( 2 2 2) E = -- dx e (∂xΦ ) + a Φ . (12 ) 2
Thus, while the Cauchy problem for (8View Equation) is ill posed with respect to the standard L2 norm it is well posed with respect to the exponentially weighted norm (12View Equation).

This technique was introduced in [193] to treat the worldtube-nullcone problem for the three-dimensional quasilinear wave equation for a scalar field Φ in an asymptotically-flat curved space background with source S,

ab c g ∇a∇b Φ = S(Φ, ∂cΦ,x ), (13 )
where the metric gab and its associated covariant derivative ∇ a are explicitly prescribed functions of c (Φ, x ). In terms of retarded spherical null coordinates a x = (u = t − r,r,๐œƒ,ฯ•), the initial-boundary value problem consists of determining Φ in the region (r > R, u > 0 ) given data Φ (u,R, ๐œƒ,ฯ•) on the timelike worldtube r = R and Φ(0,r,๐œƒ,ฯ• ) on the initial null hypersurface u = 0. It was shown that this quasilinear wave problem is well posed on a domain extending to future null infinity subject to smoothness and asymptotic falloff conditions on the data. The treatment was based upon energy estimates obtained by integration by parts with respect to the characteristic coordinates. As a result, the analogous finite-difference estimates obtained by summation by parts [191Jump To The Next Citation Point] do provide guidance for the development of a stable numerical evolution algorithm. The corresponding worldtube-nullcone problem for Einstein’s equations plays a major underlying role in the CCM strategy. Its well-posedness appears to be confirmed by numerical simulations but the analytic proof remains an important unresolved problem.

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