Newtonian limit along a regular asymptotic Newtonian sequence

2.1 Newtonian limit along a regular asymptotic Newtonian sequence

This formulation is based on the observation that any asymptotic approximation of any theory needs a sequence of solutions of the basic equations of the theory [146

, 154 ]. Namely, if we write the equations in abstract form as

for an unknown function $g$ , one would like to have a one-parameter (or possibly multi-parameter) family of solutions,

where $λ$ is some parameter. Asymptotic approximation then says that a function $f (λ )$ approximates $g(λ )$ to order $λp$ if $|f(λ) − g(λ)|∕λp → 0$ as $λ → 0$ . We choose the sequence of solutions with appropriate properties in such a way that the properties reflect the character of the system under consideration.

We shall formulate the post-Newtonian approximation according to the general idea just described. As stated in the introduction, we would like to have an approximation which applies to the systems whose motions are described almost by Newtonian theory. Thus we need a sequence of solutions of the Einstein equations parameterized by $ε$ (the typical velocity of the system divided by the speed of light) which has Newtonian character as $ε → 0$ .

The Newtonian character is most conveniently described by the following scaling law. The Newtonian equations involve six variables, namely the density $ρ$ , the pressure $P$ , the gravitational potential $Φ$ , and the velocity $vi$ , $i = 1,2,3$ ):

&#x2207;2 &#x03A6; &#x2212; 4&#x03C0;&#x03C1; = 0, &#x0028;5 &#x0029; &#x2202;t&#x03C1; + &#x2207;i &#x0028;&#x03C1;vi&#x0029; = 0, &#x0028;6 &#x0029; &#x03C1;&#x2202; vi + &#x03C1;vj &#x2207; vi + &#x2207;iP + &#x03C1;&#x2207;i &#x03A6; = 0, &#x0028;7 &#x0029; t j

supplemented by an equation of state. For simplicity we have considered a perfect fluid.

It can be seen that the variables ${ρ(xi,t),P(xi,t),Φ (xi,t),vi(xj,t)}$ obeying the above equations satisfy the following scaling law:

i 2 i &#x03C1;&#x0028;x ,t&#x0029; &#x2192; &#x03B5; &#x03C1;&#x0028;x ,&#x03B5;t&#x0029;, P&#x0028;xi,t&#x0029; &#x2192; &#x03B5;4P &#x0028;xi,&#x03B5;t&#x0029;, vi&#x0028;xk,t&#x0029; &#x2192; &#x03B5;vi&#x0028;xk,&#x03B5;t&#x0029;, &#x0028;8 &#x0029; &#x03A6;&#x0028;xi,t&#x0029; &#x2192; &#x03B5;2&#x03A6; &#x0028;xi,&#x03B5;t&#x0029;.

One can easily understand the meaning of this scaling by noticing that $ε$ is the magnitude of the typical velocity (divided by the speed of light). Then the magnitude of the gravitational potential will be of order $ε2$ because of the balance between gravity and the centrifugal force. The scaling of the time variable expresses the fact that the weaker gravity is ( $ε → 0$ ) the longer the time scale is.

Thus we wish to have a sequence of solutions of the Einstein equations which has the above scaling as $ε → 0$ . We shall also take the point of view that the sequence of solutions is determined by the appropriate sequence of initial data. This has a practical advantage because there will be no solutions of the Einstein equations which satisfy the above scaling (8 ) even as $ε → 0$ . This is because the Einstein equations are nonlinear in the field variables, so it will not be possible to enforce the scaling everywhere in spacetime. We shall therefore impose it only on the initial data for the solution of the sequence.

Here we first give a general discussion on the formulation of the post-Newtonian approximation independent of any initial value formalism and then present the concrete treatment in harmonic coordinates. The condition is used because of its popularity and some advantages in the generalization to the systems with strong internal gravity.

As the initial data for the matter we take the same data set in the Newtonian case, namely the density $ρ$ , the pressure $P$ , and the coordinate velocity $i v$ . In most of the application, we usually assume a simple equation of state which relates the pressure to the density. The initial data for the gravitational field are $gμν,∂gμν∕∂t$ . Since general relativity is an overdetermined system, there will be constraint equations among the initial data for the field. We shall write the free data for the field as $(Q ,P ) ij ij$ whose explicit forms depend on the coordinate condition one assumes. In any coordinates we shall assume these data for the field vanish since we are interested in the evolution of an isolated system by its own gravitational interaction. It is expected that this choice corresponds to the absence of radiation far away from the source. Thus we choose the following initial data which is indicated by the Newtonian scaling:

i 2 i &#x03C1;&#x0028;t = 0, x,i&#x03B5;&#x0029; = &#x03B5;4a&#x0028;xi&#x0029;, P &#x0028;t = 0, x,&#x03B5;&#x0029; = &#x03B5; b&#x0028;x &#x0029;, vi&#x0028;t = 0,xk,&#x03B5;&#x0029; = &#x03B5;ci&#x0028;xk&#x0029;, &#x0028;9 &#x0029; Qij&#x0028;t = 0, xi,&#x03B5;&#x0029; = 0, P &#x0028;t = 0, xi,&#x03B5;&#x0029; = 0, ij

where the functions $a$ , $b$ , and $ci$ are $C ∞$ functions that have compact support contained entirely within a sphere of a finite radius.

Corresponding to the above data, we have a one-parameter set of spacetime parameterized by $ε$ . It may be helpful to visualize the set as a fiber bundle, with base space $R$ being the real line (coordinate $ε$ ) and fiber $R4$ being the spacetime (coordinates $t,xi$ ). The fiber $ε = 0$ is Minkowski spacetime since it is defined by zero data. In the following we shall assume that the solutions are sufficiently smooth functions of $ε$ for small $ε ⁄= 0$ . We wish to take the limit $ε → 0$ along the sequence. The limit is, however, not unique and is defined by giving a smooth nowhere vanishing vector field on the fiber bundle which is nowhere tangent to each fiber [84 , 146 ]. The integral curves of the vector field give a correspondence between points in different fibers, namely events in different spacetimes with different values of $ε$ . Remembering the Newtonian scaling of the time variable in the limit, we introduce the Newtonian dynamical time,

and define the integral curve as the curve on which $τ$ and $xi$ stay constant² . In fact if we take the limit $ε → 0$ along this curve, the orbital period of the binary system with $ε = 0.01$ is 10 times that of the system with $ε = 0.1$ as expected from the Newtonian scaling. This is what we define as the Newtonian limit. Notice that this map never reaches the fiber $ε = 0$ (Minkowski spacetime). There is no pure vacuum Newtonian limit as expected.

In the following we assume the existence of such a sequence of solutions constructed by the initial data satisfying the above scaling with respect to $ε$ . We shall call such a sequence a regular asymptotically Newtonian sequence. We have to make further mathematical assumptions about the sequence to make explicit calculations. We will not go into details partly because in order to prove the assumptions we need a deep understanding of the existence and uniqueness properties of the Cauchy problem of the Einstein equations with perfect fluids of compact support which are not available at present.