Post-Newtonian Gravitational Radiation

Let us declare that the most important devoir of any physical theory is to draw firm predictions for the outcome of laboratory experiments and astronomical observations. Unfortunately, the devoir is quite difficult to fulfill in the case of general relativity, essentially because of the complexity of the Einstein field equations, to which only few exact solutions are known. For instance, it is impossible to settle the exact prediction of this theory when there are no symmetry in the problem (as is the case in the problem of the gravitational dynamics of separated bodies). Therefore, one is often obliged, in general relativity, to resort to approximation methods. It is beyond question that approximation methods do work in general relativity. Some of the great successes of this theory were in fact obtained using approximation methods. We have particularly in mind the test by Taylor and collaborators [1–3] regarding the orbital decay of the binary pulsar PSR 1913+16, which is in agreement to within 0.35% with the general-relativistic post-Newtonian prediction. However, a generic problem with approximation methods (especially in general relativity) is that it is non trivial to define a clear framework within which the approximation method is mathematically well-defined, and such that the results of successive approximations could be considered as theorems following some precise (physical and/or technical) assumptions. Even more difficult is the problem of the relation between the approximation method and the exact theory. In this context one can ask: What is the mathematical nature of the approximation series (convergent, asymptotic, . . .)? What its “reliability” is (i.e., does the approximation series come from the Taylor expansion of a family of exact solutions)? Does the approximate solution satisfy some “exact” boundary conditions (for instance the no-incoming radiation condition)? Since the problem of theoretical prediction in general relativity is complex, let us distinguish several approaches (and ways of thinking) to it, and illustrate them with the example of the prediction for the binary pulsar. First we may consider what could be called the “physical” approach, in which one analyses the relative importance of each physical phenomena at work by using crude numerical estimates, and where one uses only the lowest-order approximation, relating if necessary the local physical quantities to observables by means of balance