Construction de solutions radiatives approchées des equations d'Einstein

Abstract

In this paper we construct rigorously, without any averaging scheme, an hyperbolic metric

$$g\alpha \beta (x,\omega \varphi ) = \mathop g\limits^0 \alpha \beta (x) + \frac{1}{\omega }\mathop g\limits^{\text{1}} \alpha \beta (x,\omega \varphi ) + \frac{1}{{\omega ^2 }}\mathop g\limits^{\text{2}} \alpha \beta (x,\omega \varphi )$$

((1))

wherex is a point of the space-timeV ₄, ϕ a scalar function onV ₄ (the “phase”) and ω a great parameter (the “frequency”). This metric is an approximate solution of Einstein's equations: it verifies

$$R\alpha \beta = 0(\omega ^{ - 1} )$$

((2))

,$\mathop g\limits^0 \alpha \beta$ and its derivatives,$\mathop g\limits^1 \alpha \beta$ and$\mathop g\limits^2 \alpha \beta$ being bounded, the first and second derivatives of$\mathop g\limits^1 \alpha \beta ,\mathop g\limits^2 \alpha \beta$ of order respectively 0(ω) and 0(ω²).

We first show that the Ricci tensor of (1) stays bounded (when ω increases), with a perturbation$\mathop g\limits^1 \alpha \beta$ physically significant, if and only if ϕ verifies the characteristic equation of the back-ground metric and$\mathop g\limits^1 \alpha \beta$ four algebraic, linear relations, (5.7) and (5.8) in radiative coordinatesx ⁰ = ϕ.

We show afterwards that (1) satisfies (2) if$\mathop g\limits^1 \alpha \beta$ satisfies ordinary differential first order equations (which take a very simple form in radiative coordinates) along the rays of the background, the preceding algebraic relations can be considered as “initial conditions” if the Ricci tensor of the background is of the radiative form

$$R\alpha \beta = (\mathop g\limits^0 \lambda \mu ) = \tau \partial \alpha \varphi \partial \beta \varphi $$

((3))

It is possible to find$\mathop g\limits^1 \alpha \beta$ and$\mathop g\limits^2 \alpha \beta$ with the required conditions of boundedness only if

$$\tau > 0$$

We apply the results to the Vaydya metric (13.1) and show that, by an oscillatory perturbation of this metric one can satisfy Einstein's equations (to the order ω⁻¹) if the coefficientm usually interpreted as the mass is a decreasing function of ϕ =u, which gives, in this context, a proof of the loss of mass by gravitational radiation.