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Asymptotic expansions for relativistic celestial mechanics

The method of asymptotic expansions is used to build an approximation scheme relevant to celestial mechanics in relativistic theories of gravitation. A scalar theory is considered, both as a simple example and for its own sake. This theory is summarized, then the relevant boundary problem is seen to be the full initial-value problem. It is shown that, with any given system of gravitating bodies, one may associate a one-parameter family of similar systems, the parameter measuring the gravitational field-strength. After a specific change of units, the derivation of asymptotic expansions becomes straightforward. Two hypotheses could be made as to which time variable has to be used in the expansion. The first one leads to an"asymptotic"post-Newtonian approximation (PNA) with instantaneous propagation, differing from the standard PNA in that, in the asymptotic PNA, all fields are expanded. The second hypothese could lead to an"asymptotic"post-Minkowskian approximation (PMA) allowing to describe propagation effects, but it is not compatible with the Newtonian limit. It is shown that the standard PNA is not compatible with the application of the usual method of asymptotic expansions as envisaged here.


Introduction
In order to evaluate the predictions of relativistic theories of gravitation, e.g. for the motion of celestial bodies in the solar system, it is necessary to use some approximation method. In Einstein's general relativity (GR), the most widely used method for that purpose is the so-called "post-Newtonian approximation" (PNA), elaborated notably in Refs. [1][2][3][4][5]. The PNA is the method used to consolidate one of the key tests of GR, namely that it predicts so accurately Mercury's residual advance in perihelion. The textbook derivation of that prediction is based on the ideal situation in which Mercury would be a test particle orbiting in the static, spherically symmetric gravitational field of the isolated Sun. Using the PNA, one finds that the same residual advance is predicted if one assumes a more realistic situation where, in particular, the perturbations due to the other planets are accounted for [1,3,5]. More generally, the aim of the PNA was to build a consistent relativistic celestial mechanics and to compare its predictions with astronomical observations. Basically, the standard PNA consists in defining a small parameter ε that characterizes the weakness of the gravitational field [1,4], in expanding the gravitational field in powers of ε, and in deriving approximate equations from this expansion and the exact field equations. A first question arises: in which sense does the solution of the approximate equations actually approach the relevant solution of the exact equations? This question is complicated, in GR, by the special structure of that theory, in particular by the necessity of imposing a gauge condition [6][7]. To the author's knowledge, this question has not received a clear answer today. A rigorous derivation of a PNA scheme, resulting from the assumed existence of a one-parameter family of metrics and mass tensors obeying the Einstein equations, has yet been proposed by Rendall [8]. But, as noted by Rendall, his scheme differs from the standard PNA in that, in his scheme, all variables, thus gravitational fields as well as matter fields, are expanded. 1 In contrast, in the standard PNA, only the gravitational field is expanded (the leading expansion being that of the space-time metricγ γ which gives the expansions of the Christoffel symbols and the Ricci tensor). 2 As a consequence, Rendall's work [8] does not seem to provide a direct justification for the standard PNA. It rather gives motivation for developing the application of the method of ordinary asymptotic expansion, even if it leads to difficulties in the comparison with observations [12]. Moreover, Rendall [8] does not attempt todescribe how the one-parameter family of metrics, that he considers, could be related to a family of physically relevant boundary value problems.
A second difficulty is that the approximate equations for the gravitational field, obtained by using the PNA, turn out to be Poisson equations for the Newtonian potential and some additional, "post-Newtonian" (PN) potentials. Thus, whereas an essential new feature of relativistic theories of gravitation is that gravitation propagates with speed c, it is found that gravitation propagates instantaneously at the PNA; this is not only true at the first (1/c 2 ) iteration 3 , but also at the second (1/c 4 ) one [13]: only at the "2½" (1/c 5 ) iteration of the PNA does one find out propagation effects, including reaction effects to the emission of gravitational waves [14]. But, as noted by Ehlers et al. [6], the determination of the lowest-order time-odd terms in the expansion of the space-time metric, which is crucial in the derivation of the 2½ iteration of the PNA, was not achieved in a self-consistent way in Ref. 14: this determination used results obtained with a different approximation method, namely the standard linearized theory of gravitational radiation; in brief, the statement of the 2½ PNA in Ref. 14 lacks the straightforward character that one might a priori expect from an iterative approximation scheme. The "post-Minkowskian" approximation (PMA) schemes differ from the PNA in that the approximate equations for the gravitational field in the PMA involve a d'Alembert (wave) operator in the place of Laplace's operator, thus automatically maintaining the effects of propagation [15][16]. Damour & Schmidt [17] gave existence proofs of one-parameter families of metrics solutions of the Einstein equations, and showed that the successive coefficients in the Taylor expansion (with respect to the parameter) of a such family satisfy the hierarchy of equations encountered in the successive iterations of the PMA. As for Ref. 8 (devoted to the PNA), they do not link the appearance of their family of metrics with a family of physically relevant boundary value problems. Moreover, they study Einstein's vacuum field equations, whereas celestial mechanics should consider a system of extended bodies [9]. In vacuo, there is only the gravitational field to be expanded, hence the work [17] does not apply directly when the standard PMA, consisting in expanding the gravitational field only [9,[15][16], is used [9] for extended bodies.
The aim of the present paper is to build a general approximation scheme based on the usual method of asymptotic expansions and to obtain the explicit expansions of the fields, equations and boundary conditions up to the order 1/c 2 included. Preliminary results have been One must expand all unknown fields if one searches for asymptotic expansions, in the usual sense, of solutions to systems of partial differential equations. Typical examples can be found in transonic gas dynamics [10] and in the homogenization theory as applied, e.g., to porous media or to composite materials [11] 2 A kind of expansion of some matter variables such as the components of the energy-momentum tensor T is used in presented in Ref. 18. In this whole work, we shall consider a scalar, preferred-frame theory of gravitation, proposed in Refs. 19-20. (Cf. Ref. 21 for a review including recent results.) This choice has the following advantages: (i) in the scalar theory, no gauge condition is needed, so that the comparison between the proposed "asymptotic" scheme and the standard PNA will be unbiased; (ii) since the scalar theory assumes a "prior-geometrical" flat metric and since its exact field equation may be formulated in terms of this flat metric, the "flat space-time operators" needed to obtain a tractable approximation scheme occur naturally and do not raise any difficulty of interpretation, in contrast to the situation in GR [6][7]; (iii) the scalar theory is mathematically much simpler than GR, in particular the derivation of asymptotic expansions is very easy. Thus, the choice of this theory provides a simple illustration of the application of the usual method of asymptotic expansion and of its difference with the standard PNA and PMA. Moreover, this theory predicts just the same effects of gravitation on light rays as does GR, thus in the gravitational field of a static spherical body, and this is also true if, as one expects, the body is in a translation with respect to the preferred frame of the theory [22]. Due to the importance of this first experimental check, it is worth to test the scalar theory further and, for that goal, to develop a celestial mechanics according to that theory; the present paper may also be considered as a step in this direction.
The plan is as follows: in Section 2, we summarize the scalar theory which is considered. In Section 3, we derive the exact equations for a perfect fluid. In Section 4, we show that the relevant boundary-value problem is the full initial-value problem. In Section 5, we describe how to associate, with the given gravitating system S, a one-parameter family (S ε ) of similar systems. In our opinion, this is needed in order to define a meaningful asymptotic approach in a physically relevant situation: if we do not have such family at our disposal, it is rather in a formal sense that we may use an asymptotic method for solving the given physical problem. Of course, one is often led to use formal asymptotics in physics, but the presentation of a framework really suitable for an asymptotic method seems to be a progress. In Section 6, we derive the asymptotic expansion of the field equation and the local equations of motion for a perfect fluid, using a PNA and expanding all field variables, and we compare the equations thus obtained, with the equations obtained when one expands the gravitational field only. In Section 7, we discuss the boundary conditions for the expanded fields. We finish the paper by the Conclusion (Section 8) and a bibliographical note (Section 9).

A short summary of the scalar theory
Space-time is assumed to be equipped with both a flat metricγ γ and a curved metricγ γ, the latter being more directly related to physical measurements (the "effective metric"). The only equations of this Section that will be used in the remainder are those defining the effective metric in terms of the scalar field f [Eqs.(2.3) and (2.5)], the field equation (2.6) for f, and the dynamical equations (2.9)-(2.10). Obviously, they are independent and may be taken axiomatically, independently of the physical status that one wishes to attribute to that theory. The preferred frame Eis assumed to be an inertial frame for the flat metric. By this, we mean that there are Galilean coordinates (x a )forγ γ 0 [i.e., in that coordinates,(γ 0 µν ) =(η µν ) ≡ diag(1, −1, −1, −1)], that are adapted to the frame E. ("Adapted coordinates" are such that any particle bound to the given frame has constant space coordinates [23].) The inertial time T ≡ x 0 /c in the preferred frame is called the "absolute time". The scalar field of the theory is the contraction factorβ affecting the time interval dt x measured by a clock fixed at point x bound to the frameE(due to the dilation of the clock period in the gravitational field), or the square f ≡ β The latter defines the γ 00 component of the physical space-time metricγ γin any coordinates(\ ),though with y 0 = cT, that are adapted to the frameE: The field f is also a potential for the assumed gravity acceleration g [20]: 4 where(g 0 ij ) is the inverse matrix of the component matrix (g LM)of g 0 , withg the spatial part of the flat metric γ γ 0 in the frame E (thus g 0 is an invariable Euclidean metric defined on the "preferred reference body" M, i.e. the set of the particles bound toE). The fieldβstill defines the dilation of an infinitesimal distance dl measured in the direction of vector g, as compared with the distance dl 0 evaluated using the Euclidean metric: dl = dl 0 /β,while the infinitesimal distances in directions perpendicular to g are unaffected. (This makes the physical space metric g in the frameE discontinuous at the isolated points where g = 0. Butg is merely the effective metric and, moreover, it remains bounded in the neighborhood of any such point: no theoretical difficulty occurs [22].) This gives the following expression for metricg in the frameE(in any coordinates (y µ ) adapted toE) : The space-time metric writes, in any coordinates (y µ ) adapted toEand such that y 0 = cT : Thus, the γ 0i components are always zero in the ether frame E. This is the expression of the assumption that this frame admits a global simultaneity defined with the absolute time T. Except for cosmological problems, the field equation [19] may be written as [20]: with ∆ the usual Laplace operator, defined with the Euclidean metric g 0 , and where G is Newton's gravitation constant, andσ≡ (T 00 ) E is the mass-energy density in the ether frame. This equation, which is merely space-covariant, implies that gravitation propagates with the velocity of light c (as measured with physical standards affected by gravitation).
With the gravity acceleration g assumed (in the preferred frame E) in the theory [Eq.(2.
2)], it implies Einstein's motion along geodesics of metric γ γ, but only for a static gravitational field; however, Newton's second law (2.7) may be defined in any reference frame, and also for metric theories: for those, one adds to the g of Eq. (2.2) a certain velocity-dependent term, which gives geodesic motion in the general case [24]. In Eq.(2.7), F 0 is the non-gravitational force,vis the modulus of the velocity v of the test particle (relative to the considered arbitrary frameF). The velocity v is measured with the local time t x synchronized along the given trajectory [25] (t x is given by Eq.(2.1) if the preferred frame of the present theory is considered, thus ifF = E) and its modulusvis defined with the space metrichin the frameF (thush=g given by Eq. and D/Dt x is the derivative of a spatial vector appropriate to the case where the Riemannian spatial metric h varies with time. In particular, with this derivative, Leibniz' rule for the time derivative of a scalar product v.w =h(v,w) is satisfied [20,24].
Equations (2.2) and (2.7) define Newton's second law in the preferred frame, for any mass particle. It is thereby defined also for a dust, since dust is a continuum made of coherently moving, non-interacting particles, each of which conserves its rest mass. It then mathematically implies, independently of the assumed form for the space-time metricγ γprovided it satisfiesγ 0i = 0 in the preferred frame, the following dynamical equation for the dust, in terms of its mass tensor T µν = ρ*U µ U ν (withρ* the proper rest-mass density and U µ = dy µ /ds the 4-velocity) [26]: [Indices are raised and lowered with metric γ γ, unless mentioned otherwise. Semicolon means covariant differentiation using the Levi-Civita connection associated with metric γ γ. The derivation of Eqs.(2.9)-(2.10) [26] is outlined in Ref. 21; they are valid in any coordinates (y µ ) that are adapted to the frameE and such that y 0 = φ(T) with T the absolute time.] Equation (2.9), with the definition (2.10), is assumed to hold for any material continuum: accounting for the mass-energy equivalence, this is the expression of the universality of gravitation.

Exact dynamical equations for a perfect fluid
With the help of the identity ( ) [in which theΓ νλ µ 's are the Christoffel symbols of metricγ γ], selecting coordinates such thatγ≡ detγ µν = -1, and using Eq.(2.5), one may generally rewrite the spatial part of Eq.(2.9) as ( ) in which the Γ jk i 's are the Christoffel symbols of metricg and (g ij ) ≡ (g ij ) -1 . Note thatγ= -1 in Cartesian coordinates for the Euclidean metric g 0 (i.e., coordinates in which g 0 ij = δ ij ), since Eq.(2.5) implies thatγ = fgwith g≡ detg ij and since the assumed relation betweeng 0 and the physical space metricg implies that Let us apply Eq.(3.2) to a perfect fluid. The mass tensor is then where µ* is the volume density of the rest-mass plus internal energy in the proper frame, expressed in mass units, We note that and that, due to Eq. Using this, it is straightforward to check that, for a perfect fluid, Eq.(3.2) takes the form As to the "time" part of Eq.(2.9), one gets by lowering index µ in (2.9) and by writing the µ = 0 component in Cartesian coordinates:

The question of the relevant boundary value problem for a perfect fluid
To be definite, we assume henceforth that any matter is in the form of barotropic perfect fluids (one fluid per astronomical body), which means thatµ * in Eq.(3.5), hence alsoρ * and Π, depend only on the pressure p, withΠbeing given by [1] The integral must be finite.) Note that the source of the gravitational field [Eq.(2.6)] is given by and is hence determined by p, f and u. The field equation (2.6), together with the dynamical equations (3.8) and (3.10), make a system of five independent partial differential equations for the five independent unknowns p, f and u (orψ, f and u; orσ, f and u). All of these equations are nonlinear. The dynamical equations (3.8) and (3.10) make a nonlinear first-order system of the same nature as, though more complicated than, It is thus a hyperbolic equation, the characteristic equation to be satisfied on a characteristic hypersurfaceφ(x µ ) = 0 being [27]A µν φ µ φ ν =0, i.e. here ( ) The hypersurface T = 0 (or T = Const), i.e.φ(x µ ) ≡ x 0 -Const = 0, cannot be characteristic, for this would mean that 1/f(x,0) 2 = 0. Therefore, the Cauchy problem, in which one assumes an arbitrary initial data f(x,T=0) and ∂ T f(x,T=0) for all x, is very plausibly a well-posed problem, and this should be the case for a generic sourceσ =σ(x,T). On the other hand, in NG, one also may consider the source ρx, T) as given: but Poisson's equation is an elliptic linear second-order equation for U N , so the problem is well-posed when one assumes, at any time T, a spatial boundary condition for U N ; typically the condition at infinity, U N = O(1/r) and ∇U N = O(1/r 2 ) as r → ∞, is assumed. Now, in NG, the natural boundary value problem for the full system of equations is a mixed initial/ spatial-boundary problem, in which one assumes a spatial boundary condition at any time for U, as just mentioned, plus an initial data for the velocity u and the densityρ(or the pressure p). Since (i) the dynamical equations (3.8) and (3.10) have the same structure as the system made of the continuity equation plus Euler's equation in NG, and (ii) the natural boundary condition for f is an initial condition, whereas the natural boundary condition for U N is a spatial boundary condition (due to the hyperbolic nature of Eq. (2.6), as opposed to the elliptic nature of Poisson's equation), we are led to consider the full initial-value problem as the relevant boundary value problem in the scalar theory. That is, one should assume the data σx,0) [or u(x,0), f(x,0) and ∂ T f(x,0), for all x.

The situation suitable to derive asymptotic expansions for weak gravitational fields
We consider a gravitating system S made of N separated bodies labeled a, b or c (with a, b, c = 1,  2, ..., N), each of which being a spatially compact body made of some material medium, so that the velocity field v [Eq.(2.8)] and the field of proper rest-mass density ρ * are unambiguously defined. It is for such a given system, in particular for the solar system, that the celestial mechanics based on some theory of gravitation has to be built and physically tested. Obviously, an agreement between observations and predictions based on arbitrary approximations would prove neither the correctness of the theory nor that of the approximation method. One therefore has to introduce some definite method of approximation, which should have been mathematically tested, if not in the theory of gravitation, at least in other domains. We choose the method of asymptotic expansions, which has been quite extensively tested in many domains (see e.g. Kevorkian & Cole [28], Awrejcewicz et al. [29], and references therein). Strictly speaking, a consistent asymptotic expansion may be defined and tested only if we have a family (S ε ) of gravitating systems at our disposal, depending on a parameter ε which should take values having 0 as an accumulation point. Hence, we should associate a such (virtual) family to the given real system S, with that system corresponding to a given, small valueε 0 of the parameter. Of course,ε should characterize the weakness of the gravitational field. As is usually done, we shall associate the weak-field limit with a situation in which NG is a good first approximation (indeed an extremely good one, in the solar system). We shall try to introduce this link without logical circularity. As in Section 4, we consider that any matter is in the form of a barotropic perfect fluid. However, we do not need that the state equation, sayρ * = F(p), is the same for each of the N bodies: we merely assume (for simplicity) that F is the same function inside any given body; since the bodies are separated, it is unnecessary to mark this by an index a, etc.

The weak-gravitational-field limit in Newtonian gravity
In NG, there are five independent fields: p, the Newtonian potential U N , and the "flat space-time" velocity, u ≡ dx/dT. The five independent equations for those fields are: Poisson's equation (of course, there is no distinction in NG between the proper rest-mass density ρ * and the density of rest-mass in the given reference frame, and we shall note both simplyρ): and Euler's equation, Let these equations be satisfied for a "Newtonian gravitating system" and, for anyε !VHWξ= εε 0 (where ε 0 will be defined hereafter) and define new fields thus: Note thatU Nε 0 = U N ,p ε 0 = p andu ε 0 = u. Moreover, in defining (U Nmax is assumed finite; this will be the case, e.g., for an "insular mass distribution" quasiperodic in time; if necessary, one may restrict the time domain to a compact interval.) From (5.4) 1 , it follows that [(U Nε ) max /c 2 ] 1/2 =ε. Thus, with any Newtonian system, we may associate a one-parameter family of systems. All systemsS ε have just the same geometry, only the time scale increases asε -1 (accordingly, the velocity decreases asε ), the density and the potential decrease asε 2 , and the pressure asε 4 , asεtends towards zero. This is the weak-gravitational-field limit in NG. (Yet the family extends to strong Newtonian fields, forεis arbitrary.) This result explains order-of-magnitude estimates partially justified by virial theorems (see e.g. Will [5], p. 89) and allows to use them in a definite asymptotic framework.

A convenient choice of units
In expanding the field equations (e.g., of the scalar theory), it will be essential to know the order of the various fields. It turns out that, in adopting a specific variation of the units withε, the situation becomes very simple. The change of units affects merely the time unit, which is multiplied byε -1 for the systemS ε ,and the mass unit, which is multiplied byε 2 . Note that G is not affected by this change. But the time T is changed to T' ≡εT, hence the velocity to u' ≡ε -1 u and, in particular, the velocity of light becomes c' ≡ ε -1 c so that, in the new units, the small parameter may be taken to beε'= 1/c' (withε'=ε if c = 1 in the starting units). The mass density is changed toρ' ≡ε -2 ρ,and the Newtonian potential becomes U N ' ≡ε -2 U N . Finally, the pressure is changed to p' ≡ε -4 p. Therefore, we have in the new units: (5.8) and in the same way u' ε (x, T' ) = u 1 (x, T' ), etc. Thus, in the new units, all fields are independent ofεand they have precisely the values they have (in the starting units) in the particular system S 1 , characterized byε= 1. The spatial, as well as the temporal derivatives, will hence also stay independent ofεin the new units, hence in particular [As noted by Hinch [30], the usual notation ϕ = O(ε k ) implies only that ϕ is of order lower than or equal to k, thus we adopt his notation ϕ = ord(ε k ) when ϕ is of order k exactly.] Note, however, that this situation in which the fields p ε ,u ε , U N ε and ρ ε are just the same (up to a change of units) independently of ε, is special to NG.

Definition of the weak-gravitational-field limit in the scalar theory
As was said at the beginning of this Section, the proposed definition for a weakgravitational-field limit will consist in defining a family (S ε ) of systems, from the data of the given gravitational system of interest. Since, as was seen in Section 4, the natural boundary-value problem in the scalar theory is the initial-value problem, we have to define a family of initial conditions. Let us first state the two conditions that should be fulfilled by the family (S ε ): (i) The curved metricγ γ shouldtend towards the flat metricγ γ 0 as ε→ 0.
(ii) The fieldsρ * , Π,p and u should be of the same order in ε as in the weak-field limit of NG, hence should be ord(ε 0 ) in the varying units of §5.2.
In view of Eqs.(2.3)-(2.4), condition (i) amounts to say that the dimensionless scalar field f = f ε tends towards 1 as ε → 0. Therefore, we are led to assume an expansion We must determine the integer k. To this end, we shall enter this expansion into the field equation (2.6). We first get for the spatial part: because the differentiation with respect to the spatial variables does commute with the expansion with respect to ε, under certain uniformity assumptions for the expansions [ensuring that the spatial derivatives of the remainder term in (5.10) are still O(ε k+1 )]. The two foregoing equations are valid in the fixed units as well as in the varying units, because the length unit is unchanged. To achieve the determination of k, we adopt the varying units, such that the small parameter ε is proportional to ε' =1/c'. But therefore, we have to precise whether the time variable is T ' = εT, or c' T ' = εT/ε' ∝ T. This is exactly the departure point between the PNA (time variable T ' ) and the PMA (time variable c' T ' ), as we shall see. Accordingly, in a PNA (resp. in a PMA), we must assume that the derivative, with respect to T ' (resp. with respect to c'T ' ), of remainder terms like O(ε k+1 ) in (5.10), is still O(ε k+1 ). Under this assumption, we find that the term involving time derivatives in Eq.
Hence the left-hand side of (2.6) is ord(ε k ). On the right-hand side, σ is given by Eq. (4.2). Hence, we get σ ε = ord(ε 0 ) by condition (ii), therefore the right-hand side of (2.6) is ord(ε 2 ). Thus one must have k = 2 in Eq. (5.10). We may set (in any units): (the coefficient -2 is there for convenience; the ratio by c 2 makes V homogeneous to the Newtonian potential). Eq. (5.11) 1 defines V ε , the important point is the estimate (5.11) 2 .
Next, we recall a recent argument [18] showing that the time variable in the varying units must be T ', i.e., a PNA must be chosen. The first point is that the asymptotic expansions will have first of all to be valid at fixed values of the chosen time and space variables. Since we choose to formulate the expansions in the varying units, it is then not equivalent to assume expansions with x' 0 = T ' as the time variable (PN expansions) or with x 0 = c' T ' (PM expansions), because the ratio x' 0 /x 0 = 1/c' = ε /c is proportional to the small parameter: it is easy to convince oneself that a PN expansion can not generally be rewritten as a PM one, and viceversa [18].Moreover, in the varying units, the time variable is T ' = εT where T is the "true" time (i.e. that measured in fixed units), hence c' T ' = cT is (proportional to) the true time. But since the true orbital velocities in the system S ε vary like ε, the true orbital periods are like ε −1 . Hence it is T ', not c' T ' ∝ T, which remains nearly the same, as ε is varied, for one orbital period of a given body (a).In the PMA, the time variable would be the true time T = T '/ε and one thus should have at fixed x and T, if ρ ε is ord(ε 0 ) in the varying units [as imposed by condition (ii)]: (where index 0 means zero-order coefficient in the expansion). But this means that the position of the bodies is approximately independent of ε at a fixed true time, in contradiction with the assumption that the orbital velocities vary like ε. Finally, the reason why the choice of T ' (resp. of c' T ' ) as the time variable in the expansions may be called a PNA (resp. a PMA) is that the first choice leads to Poisson equations for the gravitational potentials, whereas the second one leads to wave equations [see Eqs. (6.6)-(6.8) below]. However, we have just seen that the second choice is not allowed in the "near zone" occupied by the bodies. Now we define the family of systems, i.e. the fieldsf ε , p ε , u ε , andρ * ε ,from the data of one gravitating system of perfect-fluid bodies in the scalar theory: we assume that the fields f, p, u, and ρ* = F(p), are known (for all x∈ ∈M and in some interval I = [0,T 1 ] for T). We provisionally come back to the invariable units for clarity, and we set [cf. Eq. ξ= ε/ε 0 . We definef ε ,p ε , u ε [andρ * ε = F ε (p ε ) withF ε given by (5.5)], as the solution (assumed unique for all x∈ ∈M and in some interval [0,T 2 (ε)] for T) of the initial value problem for the field equations (2.6), (3.8) and (3.10). Equation (5.14) 1 means that, in Eq.(5.11) 1 , This is the simplest way to get the estimate (5.11) 2 . Also, Eqs. (5.15) are the simplest way to ensure that the fields p ε and u ε are ord(ε 4 ) and ord(ε) respectively, as in the weak-field limit of NG. Moreover,u ε = ord(ε) means anε -1 dependence of the characteristic time; if one combines this with Eq.(5.11), one expects the asymptotic validity of Eq.(5.4) 1 for V : V ε x, T ) ~ξ 2 V(x,ξT asε→ 0. (5.17) In uniform conditions, one may differentiate (5.17) with respect to T. This leads to define ∂ T f ε (x, 0) by Eq.(5.14) 2 . Thus, the initial data (5.14)-(5.15) is justified by (5.11), which follows from conditions (i) and (ii), and by the wish that the fieldsp ε andu ε behave as in the weak-field limit of NG, i.e. by condition (ii). In other words, the minimal conditions (i) and (ii) imposed to the weak-field limit lead essentially uniquely to the initial data (5.14)-(5.15). Conversely, one expects that the initial data (5.14)-(5.15) ensures that equation (5.17) holds and that p ε x, T ) ~ξ 4 p(x,ξT ) andu ε (x, T ) ~ξ ux, ξT ) asε→ 0, (5.18) whence also, the state equation beingρ * ε =F ε (p ε ) with F ε given by (5.5), Thus, we have associated with the physically given system S a one-parameter family of similar systems, also governed by the equations of the scalar theory. This provides a framework in which one might test the convergence of the asymptotic expansions derived hereafter. Once again, the essential point is that, in deriving asymptotic expansions, the existence of a one-parameter family (S ε )is implicitly assumed: it seems better to have at least a (very) good candidate for such family. Moreover, the initial data (5.14)-(5.15) will allow us to state initial and boundary conditions for the expanded fields (Section 7). Note that Eqs.(5.10) to (5.19) hold true in the varying units, if one also expresses in these units the velocity of light and the fields V, p, u andρ*,corresponding to the physically given gravitating system (which makes those fields depend on ε ...).
Since Eqs.(5.17)-(5.19) may be assumed to be valid, all fields, e.g. p ε , u ε , V ε and ρ* ε , are ord(ε 0 ) in the specific varying units, in whichε∝1/c. This means that our condition (ii), as well as [entering (5.17) into (5.12)] our condition (i) are indeed verified. It also makes the derivation of asymptotic expansions a routine affair. There is no theoretical difficulty in considering expansions with an arbitrary number n of terms but, of course, the explicit expressions become very involved as n increases. Since, in the solar system, the second iteration, i.e., that immediately following the Newtonian approximation, leads merely to minute corrections, we shall stop at this second iteration. As a last preliminary point before expanding the fields in powers of ε, one has to ask a question about the parity of the expansions. We proved that the first two terms in the expansion of the scalar field are even powers of ε ∝ 1/c [Eq.(5.11)]. But a look at the basic equations for a perfect fluid in the scalar theory, Eqs.(2.6), (3.8) and (3.10), shows that it is only the square 1/c 2 that plays a role (thus, the actual small parameter is 1/c 2 ), hence more generally one may assume even powers in 1/c. In Ref. 18, some test of this assumption has been performed: it has been checked that the first-order expansion in 1/c gives the same results as the zero-order expansion in 1/c 2 -namely, both expansions are equivalent to Newtonian gravity (NG). However, an important comment is in order: polynomials in the effective small parameter (1/c 2 ), i.e. Taylor expansions, are merely the most obvious possibility for asymptotic expansions, corresponding to the most regular behaviour. It may well be that these simple expansions work only until some finite order. Thus, in GR, there is a breakdown in the approximations based on even-powers expansions, either at the order 1/c 4 or at the order 1/c 6 , depending on the gauge condition [8]. May be one should assume then new expansion functions, e.g. odd powers in 1/cas is done for describing the first radiation effects in the standard PNA, where an 1/c 5 power is introduced [14].

Expansion of all fields vs. expansion of the gravitational field only
In this Section, we shall assume that, in the varying units, the fields admit expansions with T (formerly T ' : we omit the prime henceforth) as the time variable; this may be called the "PN hypothesis". We begin with an ordinary asymptotic expansion, then we show that the standard PNA is a different kind of expansion.
Let x 0 be the time variable used in the expansions. (Thus x 0 = T actually, and x 0 = cT for a PM expansion, even though we know from §5.3 that the latter is incompatible with the assumption that the true orbital velocities vary like ε.) We get from (6.1):    The latter shows that the rest-mass is conserved at the PNA of the scalar theory: together with Eq.(6.15), Eq.(6.22) implies that But the rest-mass is not exactly conserved for a perfect fluid. Indeed a creation/destruction rate is predicted by the scalar theory, in a variable gravitational field under a fluid pressure. For a weak field, this rate is ord(ε 4 ) and compatible with the experimental evidence on mass conservation [26]. We also note that the equations for the first PN corrections, i.e. the equations for the order 1/c 2 , Eqs.(6.10), (6.18) and (6.22), are all linear with respect to the "PN fields" A, p 1 , ψ 1 , u 1 . This is typical of a perturbation method.

Standard PNA: gravitational field alone expanded
If we apply to the scalar theory the standard PNA, as exposed by Fock [1], Chandrasekhar [2], Weinberg [3], Misner et al. [4], or Will [5], then we expand the scalar field f, Eq.(6.1), but we leave p and u "unexpanded". Then Eqs.(6.6) and (6.7) still allow to write the l.h.s. of the field equation (2.6) as the l.h.s. of Eq.(6.8). If we wish, we even may express the matter source σ, Eq.(4.2), as a kind of expansion: so that the whole of Eq.(6.8) holds true with this new definition of σ 0 and σ 1 . But since ρ, u and Π are unexpanded and hence (of course) depend on ε, the coefficients of this "expansion", σ 0 and σ 1 , now do depend on ε (this is called a "composite expansion"). As a consequence, we may only rewrite Eq. Clearly, this unique equation for the approximate gravitational fields of the two first orders, U and A, is not enough -the more so, as the same occurs for the dynamical equations, of course. In order to get a solvable approximation scheme at the 1/c 2 level, one must postulate at least one separate equation including 1/c 2 contributions, e.g.