Lyapunov functional approach to raiative metrics

Semi-global existence and convergence of solutions of the Robinson-Trautman (2-dimensional Calabi) equation

“…Following [18] we shall use the Kahler structure of 2 M to show that the righthand side of (A.I) is nonpositive (cf. e.g.…”

“…Changing the orientation of the w-axis if necessary, by a rescaling of u we may always achieve and in the remainder of this paper we shall always assume that this normalization has been chosen. We have the following formulae: 4 The convergence rate of the metric is made precise in Proposition 5.1; the results of that proposition can be sharpened by establishing an asymptotic expansion of λ for large u -these results and their implications for the global structure of the Robinson-Trautman space-times will be discussed elsewhere Equation (2.9) follows from (2.2); (2.10) is the Gauss-Bonnet theorem; (2.11) is the Calabi-LPPS inequality [3,18] ( 2 M i dμ 0 )) can be defined in a standard way. Proposition 3.1 is the key to the global existence proof and is specific to the problem at hand -the rest of the proof, as carried on in the next sections, is rather standard and applies to a quite general class of equations.…”

Some physical consequences of abrupt changes in the multipole moments of a gravitating body

“…These are very special however because if the wave fronts are sufficiently smooth (free of conical singularities) and the field (Riemann tensor) contains no 'wire' or 'directional' singularities then the solutions approach a Schwarzschild limit exponentially in time [2,3]. A limiting case is Penrose's [4] spherical impulsive gravitational wave propagating through flat space-time and similar solutions [5].…”

Asymptotic stability of vacuum twisting type II metrics

“…This seems rather complicated as compared to noting that P −3 also decreases with u. The authors of [5] most probably were unaware that the average mP −3 u=const is the Bondi mass of an asymptotically flat RT space-time and that the RT equation is responsible for the energy loss, but still it is evident that for positive, smoothP and m > 0 that average is positive and that its u-derivative is negative (being the average of the news function). Establishing this is attributed to D. Singleton in [6].…”

“…Asymptotic stability of the Schwarzschild solution within the RobinsonTrautman (RT) class of metrics [4] has been established in [5]. The Lyapunov functional used by Lukács et al was S K 2 , where S = {u = const, r = const} and K is its Gauss curvature 2 , and they had to apply the Laplace operator of S (2P 2 ∂ ξ ∂ξ) to the Robinson-Trautman equation to make it an evolution of K alone.…”