PostNewtonian Reference Frames for Advanced Theory of the Lunar Motion and a New Generation of Lunar Laser Ranging
Abstract
We construct a set of postNewtonian reference frames for a comprehensive study of the orbital dynamics and rotational motion of Moon and Earth by means of lunar laser ranging (LLR) with the precision of one millimeter. We work in the framework of a scalartensor theory of gravity depending on two parameters, and , of the parameterized postNewtonian (PPN) formalism and utilize the concepts of the relativistic resolutions on reference frames adopted by the International Astronomical Union (IAU) in 2000. We assume that the solar system is isolated and spacetime is asymptotically flat at infinity. The primary reference frame covers the entire spacetime, has its origin at the solarsystem barycenter (SSB) and spatial axes stretching up to infinity. The SSB frame is not rotating with respect to a set of distant quasars that are assumed to be at rest on the sky forming the International Celestial Reference Frame (ICRF). The secondary reference frame has its origin at the EarthMoon barycenter (EMB). The EMB frame is locallyinertial with its spatial axes spreading out to the orbits of Venus and Mars, and is not rotating dynamically in the sense that equation of motion of a test particle moving with respect to the EMB frame, does not contain the Coriolis and centripetal forces. Two other local frames – the geocentric (GRF) and the selenocentric (SRF) frames – have their origins at the center of mass of Earth and Moon respectively and do not rotate dynamically. Each local frame is subject to the geodetic precession both with respect to other local frames and with respect to the ICRF because of the relative motion. The advantage of dynamically nonrotating local frames is in a more simple mathematical description. The set of the global and three local frames is introduced in order to fully decouple the relative motion of Moon with respect to Earth from the orbital motion of the EarthMoon barycenter as well as to connect the coordinate description of the lunar motion, an observer on Earth, and a retroreflector on Moon to directly measurable quantities such as the proper time and the roundtrip laserlight distance. We solve the gravity field equations and find out the metric tensor and the scalar field in all frames, which description includes the postNewtonian definition of the multipole moments of the gravitational field of Earth and Moon. We also derive the postNewtonian coordinate transformations between the frames and analyze the residual gauge freedom imposed by the scalartensor theory on the metric tensor. The residual gauge freedom is used for removal spurious, coordinatedependent postNewtonian effects from the equations of motion of Earth and Moon.
pacs:
04.20.Gz, 04.80.y, 95.55.Br, 96.15.VxContents
 1 Introduction
 2 The ScalarTensor Theory of Gravity
 3 Theoretical Principles of the PostNewtonian Celestial Mechanics
 4 PostNewtonian Reference Frames
 5 PostNewtonian Transformations Between the Reference Frames
1 Introduction
1.1 Background
The tremendous progress in technology, which we have witnessed during the last 30 years, has led to enormous improvements of precision in the measuring time and distances within the boundaries of the solar system. Further significant growth of the accuracy of astronomical observations is expected in the course of time. Observational techniques like lunar and satellite laser ranging, radar and Doppler ranging, very long baseline interferometry, highprecision atomic clocks, gyroscopes, etc. have made it possible to start probing not only the static but also kinematic and dynamic effects in motion of celestial bodies to unprecedented level of fundamental interest. Current accuracy requirements make it inevitable to formulate the most critical astronomical dataprocessing procedures in the framework of Einstein’s general theory of relativity. This is because major relativistic effects are several orders of magnitude larger than the technical threshold of practical observations and in order to interpret the results of such observations, one has to build physicallyadequate relativistic models. Many current and planned observational projects and specialized space missions can not achieve their goals unless the relativity is taken into account properly. The future projects will require introduction of higherorder relativistic models supplemented with the corresponding parametrization of the relativistic effects, which will affect the observations.
The dynamical modeling for the solar system (major and minor planets), for deep space navigation, and for the dynamics of Earth’s satellites and Moon must be consistent with general relativity. Lunar laser ranging (LLR) measurements are particularly important for testing general relativistic predictions and for advanced exploration of other laws of fundamental gravitational physics. Current LLR technologies allow us to arrange the measurement of the distance from a laser on Earth to a cornercube reflector (CCR) on Moon with a precision approaching 1 millimeter Battat et al. (2007a); Murphy et al. (2008). There is a proposal to place a new CCR array on Moon Currie et al. (2008), and possibly to install other devices such as microwave transponders Bender et al. (1990) for multiple scientific and technical purposes. Successful human exploration of the Moon strongly demands further significant improvement of the theoretical model of the orbital and rotational dynamics of the EarthMoon system. This model should inevitably be based on the theory of general relativity, fully incorporate the relevant geophysical processes, lunar libration, tides, and should rely upon the most recent standards and recommendations of the IAU for data analysis Soffel et al. (2003).
This paper discusses relativistic reference frames in construction of the highprecise dynamical model of motion of Moon and Earth. The model will take into account all the classical and relativistic effects in the orbital and rotational motion of Moon and Earth at the millimeter level. Although a lot of efforts has been made in this field of dynamic astronomy, there are some controversial issues, which obscure the progress in better understanding of the fundamental principles of the relativistic model of the EarthMoon system (see, for example, recent discussion Kopeikin (2007a); Murphy et al. (2007a, b)). It is one of the goals of our investigation and,particularly, this paper to clarify these incomprehensible issues in order to allow the upcoming millimeter LLR to perform one of the most precise fundamental tests of general relativity in the solar system.
From Newton’s time, many mathematical astronomers have attempted to create lunar theories capable to yielding predictions as accurate as the best observed positions of Moon. The first attempt was undertaken by Newton himself Cohen (1975); subsequent contributions, in the framework of his theory of gravitation Newton (1686), were given by many remarkable astronomers including Euler, Laplace, Delaunay, Newcomb, Brown, Hill, and others Euler (1991); Clairaut (1754); d’Alembert (2002); Poisson (1808); Laplace (2005); Damoiseau (1828); Plana (1832); Lubbock (1834); de Pontécoulant (1846); Delaunay (1858); Hill (1879); Hansen (1862); Airy (1874); Tisserand (1990); Brown (1896); Adams (1900); Newcomb (1903); Poincaré (2005); Plummer (1960); Eckert and Eckert (1967); Deprit et al. (1970). The theories can be grouped into three categories: analytic, numericalanalytic and numerical. The most impressive example of a purely analytic theory is given by Delaunay Delaunay (1858) whose elaboration took over twenty years. Because of its completely analytic nature, it can be applied to any threebody problem. Examples of semianalytic theories are those by Laplace Laplace (2005) and BrownHill Hill (1879); Brown (1896). For example, the BrownHill theory includes 1500 separate terms and was used in the Apollo program of the human exploration of the Moon Gutzwiller (1998). An entirely numerical theory was devised by Airy Airy (1874).
After Einstein published his theory of gravitation Einstein (1916), its effects on the lunar dynamics were worked out by a number of authors: de Sitter de Sitter (1916), Chazy Chazy (2005), Eddington Eddington (1975), Brumberg Brumberg (1991), Baierlein Baierlein (1967a), Lestrade and Bretagnon J.F. and Bretagnon (1982), Mashhoon and Theiss Mashhoon and Theiss (2001), Soffel et al Soffel et al. (1986). Modern lunar ephemerides fully including the postNewtonian effects of general relativity are the ELP (Institut de Mécanique Céleste et de Calcul des Éphémérides) ChaprontTouze and Chapront (1983), LE (Jet Propulsion Laboratory, NASA) Standish (1998), EPM (Institute of Applied Astronomy, Russian Academy of Sciences) Pitjeva (2005) and PMO (Purple Mountain Observatory, Chinese Academy of Sciences) Li et al. (2008).
1.2 Lunar Laser Ranging
Lunar laser ranging (LLR) is a technique based on a set of laser stations on Earth and corner retroreflectors (CCR) located on a visible (near) side of Moon Bender et al. (1973); Alley (1983) making the natural reference frame to a mutual study of geophysical and selenophysical processes. Indeed, LLR technique is currently the most effective way to study the interior of Moon and dynamics of the MoonEarth system. The most important contributions from LLR include: detection of a molten lunar core Williams (2007) and measurement of its influence on Moon’s orientation Williams et al. (2001a) and tidal dissipation Williams et al. (2001b); Williams et al. (2008); detection of lunar free libration along with the forced terms from Venus Williams et al. (1996a) and the internal excitation mechanisms Rambaux et al. (2008); an accurate test of the strong principle of equivalence for massive bodies Williams et al. (1976); Müller and Nordtvedt (1998) also known as the Nordtvedt effect (Will, 1993, Section 8.1); and setting of a stringent limit on time variability of the universal gravitational constant and (non)existence of longrange fields besides the metric tensor Nordtvedt (2001). LLR analysis has also given access to more subtle tests of relativity Mueller et al. (1991); Müller et al. (1996); Williams et al. (2004a); Bailey and Kostelecký (2006); Battat et al. (2007b); Soffel et al. (2008), measurements of Moon s tidal acceleration Calame and Mulholland (1978); Xu and Jin (1994); Chapront et al. (2002) and geodetic precession of the lunar orbit Bertotti et al. (1987); Dickey et al. (1989), and has provided ordersofmagnitude improvements in the accuracy of the lunar ephemeris Newhall et al. (1996); Kudryavtsev (2007); Li et al. (2008); Kopeikin et al. (2008); Standish (2008) and its threedimensional rotation Chapront et al. (1999); Williams et al. (2003). On the geodesy front, LLR contributes to the determination of Earth orientation parameters, such as nutation, precession (including relativistic geodetic precession), polar motion, UT1, and to the longterm variation of these effects Müller et al. (2008a, b). LLR also contributes to the realization of both the terrestrial and selenodesic reference frames Huang et al. (1996, 1999). The Satellite Laser Ranging (SLR) realization of a dynamicallydefined inertial reference frame Standish and Williams (1990) in contrast to the kinematicallyrealized frame of VLBI (Walter and Sovers, 2000, Section 6), offers new possibilities for mutual crosschecking and confirmation Müller et al. (2008b) especially after the International Laser Ranging Service (ILRS) was established in September 1998 to support programs in geodetic, geophysical, and lunar research activities and to provide the International Earth Rotation Service (IERS) with products important to the maintenance of an accurate International Terrestrial Reference Frame (ITRF) Pearlman and Bosworth (2002).
Over the years, LLR has benefited from a number of improvements both in observing technology and data modeling, which led to the current accuracy of postfit residuals of 2 cm (see, for example, Meyer et al. (2002) and (Pearlman and Carey, http://ilrs.gsfc.nasa.gov/reports/ilrsreports/ilrsar2003.html, Section 11)) Recently, subcentimeter precision in determining range distances between a laser on Earth and a retroreflector on Moon has been achieved Battat et al. (2007a); Murphy et al. (2008). As precision of LLR measurements was gradually improving over years from a few meters to few centimeters, enormous progress in understanding evolutionary history of the Earth Moon orbit and the internal structure of both planets has been achieved. With the precision approaching 1 millimeter and better, accumulation of more accurate LLR data will lead to new, fascinating discoveries in fundamental gravitational theory, geophysics, and physics of lunar interior Murphy et al. (2007c) whose unique interpretation will intimately rely upon our ability to develop a systematic theoretical approach to analyze the subcentimeter LLR data Kopeikin et al. (2008).
1.3 EIH Equations of Motion in Nbody Problem
Nowadays, the theory of the lunar motion should incorporate not only the numerous Newtonian perturbations but has to deal with much more subtle relativistic phenomena being currently incorporated to the ephemeris codes ChaprontTouze and Chapront (1983); Standish (1998); Pitjeva (2005); Li et al. (2008). Theoretical approach, used for construction of the ephemerides, accepts that the postNewtonian description of the planetary motions can be achieved with the EinsteinInfeldHoffmann (EIH) equations of motion of pointlike masses Einstein et al. (1938), which have been independently derived by Petrova Petrova (1949) and Fock (Fock, 1959, Section 6) for massive fluid balls as well as by Lorentz and Droste Lorentz and Droste (1917a, b, 1937) under assumptions that the bodies are spherical, homogeneous and consist of incompressible fluid. These relativistic equations are valid in the barycentric frame of the solar system with time coordinate and spatial coordinates .
Due to the covariant nature of general theory of relativity the barycentric coordinates are not unique and are defined up to the spacetime transformation Brumberg (1972, 1991); Soffel (1989)
(1.1)  
(1.2) 
where summation goes over all the massive bodies of the solar system (); is the universal gravitational constant; is the fundamental speed in the Minkowskian spacetime; a dot between any spatial vectors, , denotes an Euclidean dot product of two vectors and ; is mass of a body B; and are coordinates and velocity of the center of mass of the body B; is a relative distance from a field point to the body B; and are constant, but otherwise free parameters being responsible for a particular choice of the barycentric coordinates. We emphasize that these parameters can be chosen arbitrary for each body B of the solar system. Standard textbooks Brumberg (1972, 1991); Soffel (1989) (see also (Will, 1993, section 4.2)) assume that the coordinate parameters are equal for all bodies, that is and . These simplifies the choice of coordinates and their transformations, and allows one to identify the coordinates used by different authors. For instance, corresponds to harmonic or isotropic coordinates Fock (1959), and realizes the standard coordinates used in the book of Landau and Lifshitz Landau and Lifshitz (1975) and in PPN formalism Will (1993). The case of corresponds to the GullstrandPainlevé coordinates Painlevé (1921); Gullstrand (1922). We prefer to have more freedom in transforming EIH equations of motion and do not equate the coordinate parameters for different massive bodies. Physically, it means that the spacetime around each body is covered locally by its own coordinate grid, which matches smoothly with the other coordinate charts of the massive bodies in the buffer domain, where the different coordinate charts overlap.
If the bodies in Nbody problem are numbered by indices B, C, D, etc., and the coordinate freedom is described by equations (1.1), (1.2), EIH equations have the following form (compare with (Brumberg, 1972, equation 88))
(1.3) 
where the Newtonian force
(1.4) 
the postNewtonian perturbation
and is velocity of the body B, is its acceleration, , are relative distances between the coordinates of the bodies.
EIH equations (1.3)–(1.3) differ from the equations of the PPN formalism (Estabrook, 1969, equation 3) employed in particular at JPL for actual calculation of the ephemerides of the major planets by the fact that the right side of equation (1.3) has been resolved into radiusvectors and velocities of the massive bodies and does not contain second derivatives (accelerations). This elimination of the highorder time derivatives from a perturbed force is a standard practice in celestial mechanics for calculation of the perturbed motion.
Barycentric coordinates and velocities of the center of mass of body are adequate theoretical quantities for description of the worldline of the body with respect to the center of mass of the solar system. However, the barycentric coordinates are global coordinates covering the entire solar system. Therefore, they have little help for efficient physical decoupling of the postNewtonian effects existing in the orbital and rotational motions of a planet and for the description of motion of planetary satellites around the planet. The problem stems from the covariant nature of EIH equations, which originates from the fundamental structure of spacetime manifold and the gauge freedom of the general relativity theory.
This freedom is already seen in the postNewtonian EIH equations (1.3) as it explicitly depends on the choice of spatial coordinates through parameters . At the same time the EIH force does not depend on parameters , which means that transformation (1.1) of the barycentric coordinate time does not affect the postNewtonian equations of motion of the solar system bodies. Each term, depending explicitly on and in equation (1.3), has no direct physical meaning as it can be eliminated after making a specific choice of these parameters. In many works on experimental gravity and applied relativity researches fix parameters , which corresponds to working in harmonic coordinates. Harmonic coordinates simplify EIH equations to large extent but one has to keep in mind that they have no physical privilege anyway, and that a separate term or a limited number of terms from EIH equations of motion can not be measured Brumberg (1991). This is because the coordinate description of motion of the bodies does not exist independently of observable quantities and must be connected to them via equations of light propagation.
EIH equations of motion can be recast to another form proposed by Brumberg Brumberg (1991). It is based on a simple property of decomposition of a vector of relative distance between any two bodies in an algebraic sum of two vectors connecting the two bodies with any other body of the Nbody system. Let us take as an example a 4body problem. Radiusvectors connecting each pair of the bodies are: , , , , , . However, only three of the six vectors are algebraically independent. Indeed, if one takes the first three vectors as independent the others can be expressed in terms of them: , , . Analogous reasoning is valid for any number of the bodies in the Nbody problem. This property allows us to reshuffle terms in the original EIH equation and to recast it to the following form Brumberg (1991)
(1.6) 
where is the relative velocity between the bodies, the dot over function denotes a time derivative, and the coefficients of the postNewtonian acceleration are
Equations (1.6)–(1.3) have been derived by Brumberg (Brumberg, 1991, pages 176177).
1.4 Gravitoelectric and Gravitomagnetic Forces
Brumberg’s form of EIH equations of motion can be further modified to separate the, socalled, gravitoelectric and gravitomagnetic forces in Nbody problem Nordtvedt (1988). Straightforward rearrangement of the terms depending on velocities reveal that equations (1.6)–(1.3) can be represented in the form being similar to the Lorentz force in electrodynamics
(1.8) 
where is called the gravitoelectric force, and the terms associated with the cross products and are referred to as the gravitomagnetic force Nordtvedt (1988).
The gravitoelectric force is given by
(1.9) 
where the first term is the Newtonian force of gravity and the postNewtonian correction
The gravitomagnetic force is given by equation
(1.11) 
where the dot means a time derivative. As one can see, the gravitomagentic force is proportional to the Newtonian force multiplied by the factor of , where is the relative velocity between two gravitating bodies. Equation (1.11) can be also obtained by making use of a linearized Lorentz transformation from the static to a moving frame of the body Nordtvedt (1988); Kopeikin and Fomalont (2007). Similar arguments work in electrodynamics for physical explanation of the origin of magnetic field of a uniformly moving charge (Landau and Lifshitz, 1975, Section 24).
Recently, there was a lot of discussions about whether LLR can measure the gravitomagnetic field Murphy et al. (2007a, b); Williams et al. (2004a); Kopeikin (2007a); Soffel et al. (2008). The answer to this question is subtle and requires more profound theoretical consideration involving the process of propagation of the laser pulses in a curved spacetime of the EarthMoon system. We are hoping to discuss this topic in an other publication. Nevertheless, what is evident already now is that equation (1.8) demonstrates a strong correlation of the gravitomagnetic force of each body with the choice of coordinates. For this reason, by changing the coordinate parameter one can eliminate either the term or from EIH equations of motion (1.8). It shows that the strength of the factual gravitomagnetic force is coordinatedependent, and, hence, a great care should be taken in order to properly interpret the LLR ”measurement” of such gravitomagnetic terms in consistency with the covariant nature of the general theory of relativity and the theory of astronomical measurements in curved spacetime outlined in papers Brumberg (1981); Synge (1962), in the textbooks by Synge Synge (1964), by Infeld and Plebansky Infeld and Plebański (1960), and by Brumberg Brumberg (1972).
1.5 The Principle of Equivalence in the EarthMoon System
Let us discuss in this section the case of the EarthMoon system moving in the gravitational field of Sun neglecting other planets of the solar system. This is a threebody problem, where two bodies  Earth (index E) and Moon (index M)  form a bounded binary system perturbed by the tidal gravitational field of a third body  Sun (index S). Brumberg Brumberg (1972, 1958) extended the HillBrown theory of motion of Moon to the postNewtonian approximation by making use of an Euclidean translation of the barycentric coordinates of Moon to the geocenter (see also Baierlein Baierlein (1967b))
(1.12) 
and introducing a vector of the Newtonian center of mass of the EarthMoon system, , in such a way that the distance EarthSun – , and that MoonSun – , are given by the Newtonianlike equations
(1.13) 
In these new variables EIH equations (1.8) for the geocentric motion of Moon and the centerof mass of the EarthMoon system, assumes the form
(1.14)  
(1.15) 
where functions , , , , , , , depend on relative coordinates , of the bodies and their velocities , . Exact analytic form of these functions is notoriously sophisticated and can be found, for example, in the book of Brumberg (Brumberg, 1991, Section 5). Let us neglect postNewtonian corrections to the gravitational field of the planets, Earth and Moon, and leave only the Schwarzschild gravitational field of Sun. Then, the main terms in these functions read
(1.17)  
(1.18)  
(1.19)  
(1.20)  
(1.21)  
(1.22)  
(1.23) 
where parameter describes the gauge freedom in choosing coordinates of the Schwarzschild’s problem for Sun Brumberg (1972, 1991).
The reader should notice that the equations (1.14)(1.23) are still EIH equations of motion in the solar barycentric coordinates expressed in terms of the relative distances between the bodies. Newtonian part of equation (1.14) of the orbital motion of Moon around Earth couples with vector of the EarthMoon center of mass only through the tidal terms. This can be seen by expanding the second term in right hand side of equation (1.14) in powers of :
(1.24) 
where dots denote small terms of the higher order of magnitude. Comparing with the Newtonian term one can confirm that the Newtonian tidal perturbation (1.24) is smaller than the Newtonian term by a factor of .
More important is to note that the Newtonian tidal perturbation (1.14) of the lunar orbit is a coupling of the second (quadrupole) derivative of the Newtonian gravitational potential of Sun with vector of the lunar orbit
(1.25) 
where here and everywhere else the repeated (dummy) indices mean the Einstein summation rule, for example, , , and so on. Equation (1.25) elucidates that the first derivatives of the solar potential does not perturb the lunar orbit in the Newtonian approximation. The first derivatives of the potential are associated with the affine connection (the Christoffel symbols) of the spacetime manifold in a metric theory of gravity Landau and Lifshitz (1975); Misner et al. (1973); Landau and Lifshitz (1975). Hence, their disappearance from the Newtonian equations of the relative motion of Moon around Earth is a consequence of the principle of equivalence. This principle states that the Christoffel symbols of the background gravitational field can be eliminated on the world line of a particle falling freely in this field. The EarthMoon system can be considered in a first approximation as such a particle, composed of Earth and Moon and located at the EarthMoon barycenter, which falls in the field of Sun in accordance with equation (1.15).
Structure of the postNewtonian force in equation (1.14) seems to violate the principle of equivalence because it contains terms, which are explicitly proportional to the Christoffel symbols, which are the first derivatives of the solar gravitational potential , coupled with velocities of Sun and Moon. However, the principle of equivalence is exact, and must be valid not only in the Newtonian theory but in any approximation beyond it. The contradiction can be resolved if one investigates the residual gauge freedom of the postNewtonian terms in equations of motion (1.14)(1.23) more carefully.
1.6 The Residual Gauge Freedom
The primary gauge freedom of EIH equations of motion is associated with the transformations (1.1)(1.2) of the barycentric coordinates of the solar system, which are parameterized by parameters and . We have noticed that the postNewtonian perturbations in the lunar equations of motion are made up of the Christoffel symbols, which admit a certain freedom of coordinate transformations. This freedom remains even after fixing the coordinate parameters and in equations 1.14), (1.15). It is associated with the fact that the EarthMoon system moves in tidal gravitational field of Sun and other planets, which makes the local background spacetime for this system not asymptoticallyflat. The residual freedom remains in making transformations of the local coordinates attached to the EarthMoon system. It induces the gauge transformation of the metric tensor and the Christoffel symbols and changes the structure of the postNewtonian terms in EIH equations of motion of the EarthMoon system. The residual gauge freedom is explicitly revealed in the linear dependence of the postNewtonian force in equation (1.14) on the orbital velocity of the EarthMoon system with respect to Sun. This dependence seems to point out to violation of the principle of relativity according to which an observer can not determine one’s velocity of motion with respect to an external coordinate frame by making use of local measurements that are not sensitive to the curvature of spacetime (the second derivatives of the solar gravitational potential). LLR is a local measurement technique, which does not observe Sun directly, and, hence, should not be able to determine velocity of the EarthMoon system with respect to it as it appears in equations (1.14) because those velocitydependent terms are not gaugeinvariant and have no absolute physical meaning.
Thus, we face the problem of investigation of the residual gauge freedom of the lunar equations of motion, which goes beyond the choice of the barycentric coordinates by fixing a specific value of the gauge parameter in equations (1.5)(1.21). This freedom is actually associated with the choice of the local coordinates of the EarthMoon barycentric frame as well as the geocentric and selenocentric reference frames. Proper choice of the local coordinates removes all nonphysical degrees of freedom from the metric tensor and eliminates spurious (nonmeasurable) terms from the postNewtonian forces in the relative equations of motion of Moon. If one ignores the residual gauge freedom and operates, for example, with the Newtonian definitions (1.12)(1.13) of the relative coordinates between the bodies, the gaugedependent terms will infiltrate the equations of motion causing possible misinterpretation of LLR observations. This problem is wellknown in cosmology where the theory of cosmological perturbations is designed essentially in terms of the gaugeindependent variables so that observations of various cosmological effects are not corrupted by the spurious, coordinatedependent signals Mukhanov et al. (1992). Similarly to cosmology, the residual gauge degrees of freedom existing in the relativistic threebody problem, can lead to misinterpretation of various aspects of gravitational physics of the EarthMoon system Kopeikin (2008, 2007a), thus, degrading the value of extremely accurate LLR measurements for testing fundamental physics of spacetime and deeper exploration of the lunar interior Kopeikin et al. (2008).
The residual gauge freedom of the three body problem (SunEarthtest particle) was studied by Brumberg and Kopeikin Brumberg and Kopejkin (1989a), Klioner and Voinov Klioner and Voinov (1993), and Damour, Soffel and Xu Damour et al. (1994). They found that the postNewtonian equations of motion of a test body (artificial satellite) can be significantly simplified by making use of a fourdimensional spacetime transformation from the solar barycentric coordinates , to the geocentric coordinates
(1.26)  
(1.27) 
where the gauge functions , , are polynomials of the geocentric distance of the field point from Earth’s geocenter, which barycentric coordinates are . Coefficients of these polynomials are functions of the barycentric time that are determined by solving a system of ordinary differential equations, which follow from the gravity field equations and the tensor law of transformation of the metric tensor from one coordinate chart to another Kopejkin (1988a).
Contrary to the test particle, the Moon is a massive body, which makes the exploration of the residual gauge freedom of the lunar motion more involved. This requires introduction of one global (SSB) frame and three local reference frames associated with the EarthMoon barycenter, the geocenter, and the center of mass of Moon (selenocenter). It should be clearly understood that any coordinate system can be used for processing and interpretation of LLR data since any viable theory of gravity obeys the Einstein principle of relativity, according to which there is no preferred frame of reference Landau and Lifshitz (1975); Fock (1959); Misner et al. (1973). For this reason, we do not admit a privileged coordinate frame in rendering analysis of the LLR data irrespectively of its accuracy. It means that our approach is insensitive to PPN parameters , , , etc., which describe the preferred frame and preferred location effects in gravitational physics. Accepting the Einstein principle of relativity leads to discarding any theory of gravity based on a privileged frame (aether) Eling et al. (2006) or admitting a violation of the Lorentz invariance Kostelecky (2008). The class of scalartensor theories of gravity, which have two PPN parameters  and Will (1993); Damour and EspositoFarese (1992), is in agreement with the principle of relativity and it will be used in this paper.
The principle of relativity also assumes that a randomly chosen, separate term in the postNewtonian equations of motion of massive bodies and/or light can not be physically interpreted as straightforward as in the Newtonian physics. The reason is that the postNewtonian transformations (1.1)(1.2) and (1.26)(1.27) of the barycentric and local coordinates, change the form of the equations of motion so that they are not forminvariant. Therefore, only those postNewtonian effects, which do not depend on the frame transformations can have direct physical interpretation. For example, the gauge parameters and entering transformations (1.1)(1.2) and EIH equations (1.14)(1.23) can not be determined from LLR data irrespectively of their accuracy because these parameters define the barycentric coordinates and can be fixed arbitrary by observer without any relation to observations. This point of view has been argued by some researchers who believe that separate terms in the barycentric EIH equations of motion of Moon do have direct physical meaning, at least those of them, which are associated with gravitomagnetism Nordtvedt (1988); Soffel et al. (2008); Nordtvedt (2001). These gravitomagnetic terms can be easily identified in quations (1.14)–(1.15) as being proportional to the velocity of motion of Moon, and that of the EarthMoon barycenter . Such orbital velocitydependent terms in equations of motion of gravitating bodies are associated with the extrinsic gravitomagnetic field as opposed to the intrinsic gravitomagnetism caused by rotational currents of matter Kopeikin (2006). It is remarkable that all the orbital velocitydependent terms can be eliminated from the orbital equations of motion (1.14)–(1.15) by choosing the GullstrandPainlevé (GP) coordinates Painlevé (1921); Gullstrand (1922) with , which makes the equation coefficients . It means that the extrinsic gravitomagnetic force, which is directly caused by the orbital motions of Earth and Moon, can not be measured by LLR technique Kopeikin (2007a), – only the tidal extrinsic gravitomagnetic field of Sun can be measured Ciufolini (2008). Papers Ciufolini (2008); Iorio (2008) discuss whether the intrinsic gravitomagnetism can be measured with LLR or not.
1.7 Towards a New Lunar Ephemeris
Existing computerbased theories of the lunar ephemeris ChaprontTouze and Chapront (1983); Standish (1998); Pitjeva (2005); Li et al. (2008) consist of three major blocks:

the Newtonian rotational equations of motion of Moon and Earth;

the barycentric postNewtonian equations of motion for light rays propagating from laser to CCR on Moon and back in standard coordinates with the gauge parameters , .
This approach is straightforward but it does not control gaugedependent terms in EIH equations of motion associated with the choice of the gaugefixing parameters and . Particular disadvantage of the barycentric approach in application to the lunar ephemerides is that it mixes up the postNewtonian effects associated with the orbital motion of the EarthMoon barycenter around Sun with those, which are attributed exclusively to the relative motion of Moon around Earth. This difficulty is also accredited to the gauge freedom of the equations of motion in threebody problem and was pointed out in papers Brumberg and Kopejkin (1989a); Damour et al. (1994); Tao et al. (2000). Unambiguous decoupling of the orbital motion of the EarthMoon barycenter from the relative motion of Moon around Earth with apparent identification of the gaugedependent degrees of freedom in the metric tensor and equations of motion is highly desirable in order to make the theory more sensible and to clean up the LLR data processing software from the fictitious coordinatedependent perturbations, which do not carry out any physicallyrelevant information and may accumulate errors in numerical ephemerides of Moon.
This goal can be rationally achieved if the postNewtonian theory of the lunar motion is consistently extended to account for mathematical properties offered by the scalartensor theory of gravity and the differential structure of the spacetime manifold. Altogether it leads us to the idea that besides the global barycentric coordinates of the solar system one has to introduce three other local reference frames. The origin of these frames should be fixed at the EarthMoon system barycenter, Earth’s center of mass (geocenter), and Moon’s center of mass (selenocenter). We distinguish the EarthMoon barycenter from the geocenter because Moon is not a test particle, thus, making the EarthMoon barycenter displaced from the geocenter along the line connecting Earth and Moon and located approximately 1710 km below the surface of Earth. Mathematical construction of each frame is reduced to finding a metric tensor by means of solution of the gravity field equations with an appropriate boundary condition Fock (1957, 1959). The gauge freedom of the threebody problem is explored by means of matching the set of the metric tensors defined in each reference frame in the overlapping domains of their applicability associated with the specific choice of boundary conditions imposed in each frame on the metric tensor. This matching procedure is an integral part of the equations defining the local differential structure of the manifold Eisenhart (1947); Dubrovin et al. (1984), which proceeds from a requirement that the overlapping spacetime domains covered by the local reference frames, are diffeomorphic.
The primary objective of the multiframe postNewtonian theory of the lunar ephemeris is the development of a new set of analytic equations to revamp the LLR data processing software in order to suppress the spurious gaugedependent solutions, which may overwhelm the existing barycentric code at the millimeter accuracy of LLR measurements, thus, plunging errors in the interpretation of selenophysics, geophysics and fundamental gravitational physics. Careful mathematical construction of the local frames with the postNewtonian accuracy will allow us to pin down and correctly interpret all physical effects having classical (lunar interior, Earth geophysics, tides, asteroids, etc.) and relativistic nature. The gauge freedom in the threebody problem (EarthMoonSun) should be carefully examined by making use of a scalartensor theory of gravity and the principles of the analytic theory of relativistic reference frames in the solar system Kopejkin (1988a); Brumberg and Kopejkin (1989b); Damour et al. (1991) that was adopted by the XXIVth General Assembly of the International Astronomical Union Soffel et al. (2003); Kopeikin (2007b) as a standard for data processing of highprecision astronomical observations.
The advanced postNewtonian dynamics of the SunEarthMoon system must include the following structural elements:

construction of a set of astronomical reference frames decoupling orbital dynamics of the EarthMoon system from the rotational motion of Earth and Moon with the full account of the postNewtonian corrections and elimination of the gauge modes;

relativistic definition of the integral parameters like mass, the center of mass, the multipole moments of the gravitating bodies;

derivation of the relativistic equations of motion of the centerofmass of the EarthMoon system with respect to the barycentric reference frame of the solar system;

derivation of the relativistic equations of motion of Earth and Moon with respect to the reference frame of the EarthMoon system;

derivation of the relativistic equations of motion of CCR on the Moon (or a lunar orbiter that is deployed with CCR) with respect to the selenocentric reference frame;

derivation of the relativistic equations of motion of a laser with respect to the geocentric reference frame.
These equations must be incorporated to LLR data processing software operating with observable quantities, which are proper times of the round trip of the laser pulses between the laser on Earth and CCR on Moon. The computational advantage of the new approach to the lunar ephemeris is that it separates clearly physical effects from the choice of coordinates. This allows us to get robust measurement of true physical parameters of the LLR model and give them direct physical interpretation. The new approach is particularly useful for comparing different models of the lunar interior and for making the fundamental test of general theory of relativity.
There is a practical consideration when we do LLR computer model improvements  a change in the LLR code must have some advantages either for computation, or accuracy, or a more complete and detailed model including adding solution parameters. Spacecraft missions use the output of the orbit integrator and that imposes another practical matter. The output ephemeris must be consistent with the conventions used in the spacecraft orbit determination program Moyer (2003). That means that the new LLR code must be compatible with the solar system barycentric frame, scale and time.
One should also make a distinction between analytical models, which play an important role in understanding of fundamental gravitational physics, and models for numerical computation/prediction of astronomical events and phenomena. For the numerical computations it basically does not matter if there are gaugedependent terms that cancel so long as the computations are internally consistent. For understanding what is going on analytically and how gravitational physics actually works, it does matter what terms cancel and what does not. The analytic LLR model, which we are going to work out, pursues mostly the goals of the fundamental physics. It will refine our understanding of the test of general relativity in the EarthMoon system and the physics of the lunar interior that are the primary concerns of the scientific exploration.
1.8 Main Objectives of The Present Paper
This paper deals with the precise analytic construction of the relativistic reference frames in the EarthMoon system moving in the field of Sun and other planets of the solar system. We shall also identify the postNewtonian gauge modes and eliminate them from the solutions of the gravity field equations. Although our final goal is to develop a practical LLR code having accuracy of one millimeter, the overall development will be as close to the covariant spirit of modern physical theories as possible.
First of all, we discuss the scalartensor theory of gravity in Section 2. We formulate the field equations for the metric tensor and the scalar field and describe the model of matter used in our analytic calculations. Powerful mathematical approach developed for calculation of motion of compact astrophysical objects, like neutron stars and/or black holes, employs the model of matter in the form of the ”multipole moments”, which are the integrals over the volume of the bodies from the unspecified ”effective” tensor of energymomentum Blanchet (2002). The matter in this approach is ”skeletonized” to push calculations as much forward as possible to the nonlinear regime of the gravity field equations. Similar matter ”skeleton” is used in a covariant derivation of equations of motion proposed by Dixon Dixon (1979). These approaches are useless for development of the LLR model because one has to know the internal motion of matter inside Earth and Moon in order to describe the motion of the laser station and CCR with respect to Earth and to Moon respectively. Hence, we use the tensor of energymomentum specified by a continuous distribution of matter’s density, current, and stress.
Theoretical principles of the postNewtonian celestial mechanics of Nbody system are formulated in Section 3. We explain the need of separation of the problem of motion in the internal and external counterparts and the postNewtonian approximation scheme. Current mathematical knowledge of the postNewtonian approximations is rather outstanding Blanchet et al. (2005) and we rely upon it to secure the consistency of our derivation.
PostNewtonian reference frames are constructed in Section 4. They include the solar system barycentric (SSB) frame, the EarthMoon barycentric (EMB) frame, the geocentric (GRF) frame, and the selenocentric (SRF) frame. Each of these frames is associated with the world line of the center of mass of the corresponding gravitating system or a gravitating body. The hierarchical structure of the reference frames corresponds to the hierarchy of masses in the problem under consideration. Each frame has its own region of mathematical applicability, which is reflected in a specific mathematical structure of solutions of the field equations describing behavior of the metric tensor and the scalar field in the corresponding coordinate charts.
The postNewtonian coordinate transformations between the frames are derived in Section 5. The derivation is based on a simple fact that the coordinate charts of the frames overlap in a ceratin region of the spacetime manifold. Hence, the metric tensor and the scalar field expressed in different coordinates, must admit a smooth tensor transformation to each other. This transformation of the physical fields establishes a system of ordinary differential and algebraic equations for the functions entering the coordinate transformation. The overall procedure is called the method of matched asymptotic expansions, which was used in general relativity for the first time by D’Eath D’Eath (1975a, b) and applied in the theory of astronomical reference frames in our work Kopejkin (1988a).
Gaugeindependent derivation of the postNewtonian equations of motion of Moon and Earth in various reference frames as well as a systematic postNewtonian algorithm of LLR data processing with the precision of 1 millimeter will be given elsewhere.
2 The ScalarTensor Theory of Gravity
PostNewtonian celestial mechanics describes orbital and rotational motions of extended bodies on a curved spacetime manifold described by the metric tensor obtained as a solution of the field equations of a metricbased theory of gravitation in the slowmotion and weakgravitational field approximation. Class of viable metric theories of gravity ranges from the canonical general theory of relativity Misner et al. (1973); Landau and Lifshitz (1975) to a scalarvectortensor theory of gravity recently proposed by Bekenstein Bekenstein (2007) for description the motion of galaxies at cosmological scale. It is inconceivable to review all these theories in the present paper and we refer the reader to Will (2006) for further details. We shall build the theory of lunar motion and LLR in the framework of a scalartensor theory of gravity introduced by Jordan Jordan (1949, 1959) and Fiertz Fierz (1956), and rediscovered independently by Brans and Dicke Brans and Dicke (1961); Dicke (1962a, b). This theory extends the Lagrangian of general theory of relativity by introducing a long range scalar field minimally coupled with gravity field causing a deviation of metric gravity from pure geometry. The presence of the scalar field highlights the geometric role of the metric tensor and makes physical content of the gravitational theory more rich. Equations of the scalartensor theory of gravity have been used in NASA Jet Propulsion Laboratory (JPL) and other international space centers for construction of the barycentric ephemerides of the solar system bodies ChaprontTouze and Chapront (1983); Standish (1998); Pitjeva (2005); Li et al. (2008). We adopt the scalartensor theory of gravity for developing the advanced postNewtonian dynamics of the EarthMoon system.
2.1 The Field Equations
Gravitational field in the scalartensor theory of gravity is described by the metric tensor and a longrange scalar field loosely coupled with gravity by means of a function . The field equations in the scalartensor theory are derived from the action Will (1993)
(2.1) 
where the first, second and third terms in the right side of equation (2.1) are the Lagrangian densities of gravitational field, scalar field and matter respectively, is the determinant of the metric tensor , is the Ricci scalar, indicates dependence of the matter Lagrangian on the matter fields, and is the coupling function, which is kept unspecified for the purpose of further parametrization of the deviation from general relativity. This makes the theory, we are working with, to be sufficiently universal.
For the sake of simplicity we postulate that the selfcoupling potential of the scalar field is identically zero so that the scalar field does not interact with itself. This is because this paper deals with a weak gravitational field and one does not expect that this potential can lead to measurable relativistic effects within the boundaries of the solar system Will (2006). However, the selfcoupling property of the scalar field leads to its nonlinearity, which can be important in strong gravitational fields of neutron stars and black holes, and its inclusion to the theory may lead to interesting physical consequences Damour and EspositoFarese (1992, 1993).
Field equations for the metric tensor are obtained by variation of action (2.1) with respect to . It yields Will (1993)
(2.2) 
where
(2.3) 
is the LaplaceBeltrami operator Misner et al. (1973); Eisenhart (1947), and is the tensor of energymomentum (TEM) of matter comprising the Nbody (solar) system. The variational principle defines it by equation Landau and Lifshitz (1975)
(2.4) 
Equation for the scalar field is obtained by variation of action (2.1) with respect to . After making use of the contracted form of equation (2.2) it yields Will (1993)
(2.5) 
In what follows, we shall also utilize another version of the Einstein equations (2.2) which is obtained after conformal transformation of the metric tensor Damour and EspositoFarese (1992)
(2.6) 
Here denotes the background value of the scalar field that may be a graduallychanging function of time due to the cosmological expansion Will (1993). It is worth noting that the determinant of the conformal metric tensor relates to the determinant of the metric as . The conformal transformation of the metric tensor leads to the conformal transformation of the Christoffel symbols and the Ricci tensor. Denoting the conformal Ricci tensor by , one can reduce the field equations (2.2) to more simple form Damour and EspositoFarese (1992)
(2.7) 
The metric tensor is called the physical (JordanFierz) metric Damour and EspositoFarese (1992) because it is used for making real measurements of time intervals, angles, and space distances. The conformal metric is called the Einstein metric Damour and EspositoFarese (1992). Technically, it is more convenient for doing mathematical calculations than the JordanFierz metric. Indeed, if the last (quadratic with respect to the scalar field) term in equation (2.7) is omitted, it becomes similar to the Einstein equations of general relativity. In this paper, we prefer to construct the parameterized postNewtonian theory of the lunar motion directly in terms of the physical JordanFierz metric. The conformal metric will be used in discussing propagation of light and the lunar laser ranging somewhere else.
2.2 The EnergyMomentum Tensor
Gravitational field and matter, which is a source of this field, are tightly connected via the Bianchi identity of the field equations for the metric tensor Landau and Lifshitz (1975); Misner et al. (1973). The Bianchi identity makes four of ten components of the metric tensor fully independent so that they can be chosen arbitrary. This freedom is usually fixed by picking up a specific gauge condition, which imposes four restrictions on four components of the metric tensor but no restriction on the scalar field. The gauge condition is associated with a specific class of coordinates, which are used for solving the field equations. The Bianchi identity also imposes four limitations on the tensor of energymomentum of matter, which are microscopic equations of motion of the matter Landau and Lifshitz (1975); Misner et al. (1973). Thus, in order to find the gravitational and scalar fields, and determine motion of the gravitating bodies in Nbody (solar) system one has to make several steps:

to specify a model of matter composing of the Nbody system,

to specify the gauge condition imposed on the metric tensor ,

to simplify (reduce) the field equations by making use of the gauge freedom,

to solve the reduced field equations,

to derive equations of motion of the bodies from the conditions of compatibility of the reduced field equations with the gauge conditions.
We assume that the solar system is isolated, which means that we neglect any influence of our galaxy on the solar system and ignore cosmological effects. This makes the spacetime asymptoticallyflat so that the barycenter of the solar system can be set at rest. We assume that matter of the solar system is described by tensor of energymomentum of matter with equation of state which is kept arbitrary. There were numerous discussions in early times of development of relativistic celestial mechanics about the role of the energymomentum tensor of matter in derivation of equations of motion of gravitating bodies. There are two basic models of matter  the field singularity and a continuous distribution of matter. The model of bodies as field singularities was advocated by Einstein and his collaborators Einstein et al. (1938); Infeld and Plebański (1960). The model of bodies consisting of a continuous distribution of matter was preferred by Lorentz and Droste Lorentz and Droste (1917a, b, 1937), Fock Fock (1959), Chandrasekhar with collaborators Chandrasekhar and Nutku (1969); Chandrasekhar and Esposito (1970), and others. Damour Damour (1983) and Schäfer Schäfer (1985) succeeded in derivation relativistic equations of motion of selfgravitating bodies, which were modeled by distributions (deltafunctions), up to 2.5 postNewtonian approximation. However, the same equations were derived by Kopeikin Kopeikin (1985) and Grishchuk and Kopeikin Grishchuk and Kopeikin (1983, 1986) for selfgravitating bodies consisting of perfect fluid (see comparison of two approaches in Damour (1989)). It is pretty clear now that any model of matter is appropriate for analysis of the problem of motion of selfgravitating and extended bodies, if mathematical analysis is performed in consistency with physical limitations on the bodies imposed by the field equations. Our goal is to construct a postNewtonian theory of motion of Earth and Moon with respect to each other and with respect to the other bodies of the solar system. This relativistic analysis should match with the classic models of matter adopted in dynamical astronomy and geophysics. For this reason, we shall model the solar system bodies as consisting of the continuous distribution of matter.
Following Fock Fock (1959, 1957) and Papapetrou Papapetrou (1951a, b) we define the energymomentum tensor as
(2.8) 
where and are the density and the specific internal energy of matter in the matter’s comoving frame, is the dimensionless 4velocity of the matter with being the proper time along the world line of matter’s volume element, and is a symmetric stress tensor being orthogonal to the 4velocity of matter
(2.9) 
Equation (2.9) means that the stress tensor has only spatial components in the frame comoving with matter. If one neglects contribution of the offdiagonal components of the stress tensor, it is reduced to a stress tensor of a perfect fluid
(2.10) 
where is an isotropic pressure. Perfectfluid approximation is used, for example, in PPN formalism Will (1993) but it is not sufficient in the Newtonian theory of motion of the solar system bodies because the tidal and dissipative forces affect their orbital and rotational motions (see, for example, Zharkov and Trubitsyn (1978); Markov (1996); Bois and Journet (1993); Christodoulidis et al. (1988); Darwin (1963)). It is not difficult to incorporate the general model of the stress tensor to the postNewtonian approximations (see, for example, Damour et al. (1991); Kopeikin and Vlasov (2004)). Therefore, we discard the model of the perfectfluid and incorporate the anisotropic stresses to the postNewtonian theory of motion of the solar system bodies.
We have noted that due to the Bianchi identity the energymomentum tensor is conserved, that is obeys to the microscopic equation of motion
(2.11) 
where the semicolon denotes the covariant derivative and repeated indices mean the Einstein summation rule. The conservation of the energymomentum tensor leads to the equation of continuity Misner et al. (1973)
(2.12) 
and to the second law of thermodynamics that is expressed as a differential relationship between the specific internal energy and the stress tensor Misner et al. (1973)
(2.13) 
These equations set certain limitations on the structure of the tensor of energymomentum. They will be employed later for solving the field equations and for derivation of the equations of motion of the bodies.
3 Theoretical Principles of the PostNewtonian Celestial Mechanics
3.1 External and Internal Problems of Motion
The postNewtonian theory of motion of extended celestial bodies described in this paper is based on the scalartensor theory of gravity and is a natural extension of PPN formalism for massive pointlike particles as described by Nordtvedt and Will Will and Nordtvedt (1972); Nordtvedt and Will (1972); Will (1993). PPN formalism contains 10 parameters characterizing different type of deviations from general relativity. It also assumes the existence of a privileged PPN coordinate frame violating the principle of relativity for the metric tensor. PPN privileged frame is associated with the isotropy of the cosmic microwave background radiation. Solar system is moving with respect to this frame.
The present paper does not deal with the privilegedframe effects as the scalartensor theory of gravity is Lorentzinvariant. For this reason, we can assume the solar system frame being at rest with the origin located at the solar system barycenter. Heliocentric frame does not coincide with the SSB frame as Sun moves around the SSB at the distances not exceeding two solar radii Hardorp (1985). The SSB frame is global with the gravitational field described by the metric tensor, which approaches the Minkowskian metric at infinity. It means that the global coordinates represent the inertial coordinates of the Minkowskian spacetime at infinity. Harmonic coordinates are particularly useful as they simplify the Einstein equations and reduce them to the hyperbolic system of equations Anderson and Decanio (1975). For this reason, harmonic coordinates were advocated by Fock Fock (1959) who believed in their physical privilege. This point of view was confronted by Infeld Infeld and Plebański (1960) and is currently considered as outdated Misner et al. (1973); Landau and Lifshitz (1975). Adequate physical description of the global SSB frame is the primary goal of the external problem of relativistic celestial mechanics Fock (1959); Damour (1989). However, the global SSB frame is not sufficient for solving the problem of motion of extended bodies at the postNewtonian approximation for two reasons.
First, the motion of matter is naturally split in two components – the orbital motion of the center of mass of each body and the internal motion of matter with respect to the body’s center of mass. The SSB frame is fully adequate for describing the orbital dynamics. However, description of the internal motion of matter demands the introduction of a local frame attached to each gravitating body. If a group of the bodies form a gravitationally bounded subsystem, like Earth and Moon, or the subsystem of satellites of major planets, it is natural to introduce the local frame associated with the center of mass of the subsystem. This will allow us to separate the dynamics of the relative motion of the bodies inside the subsystem from the orbital motion of the center of mass of the subsystem itself with respect to the SSB frame. Adequate physical description of the internal motions at the postNewtonian level of accuracy constitutes the main goal of the internal problem of relativistic celestial mechanics Fock (1959); Damour (1989).
Second, the postNewtonian celestial mechanics is tightly connected to the geometric properties of the spacetime manifold being characterized by the metric tensor, the affine connection (the Christoffel symbol), the curvature tensor and topology. Thus, relativistic description of motion of the celestial bodies is to reflect the diffeomorphic properties of the manifold’s geometric structure associated with the set of overlapping coordinate charts and corresponding transformations between them Dubrovin et al. (1984); Arnold (1995). The metric tensor in a local frame of each body must match with the tidal gravitational field of external bodies, hence, it diverges as distance from the body goes to infinity. Therefore, the local coordinates cover only a limited domain (world tube) in spacetime around the body under consideration, and the process of their construction must be reconciled with the principle of equivalence Kopejkin (1988b); Thorne and Hartle (1985).
Newtonian mechanics of Nbody system describes translational motion of the bodies in a single global coordinate frame, , with the origin placed at the center of mass of all bodies. Local coordinates, , are used for description of rotational motion of the bodies, and they are constructed by a simple spatial translation of the global coordinates to the center of mass of each body under consideration. Time in the Newtonian theory is absolute, and, hence, does not change when one transforms the global to local coordinates. Newtonian space is also absolute, which makes the difference between the global and local coordinates physically insignificant.
The theory changes dramatically as one switches from the Newtonian concepts to a consistent relativistic theory of gravity. One still needs a global coordinate frame to describe translational motion of the bodies with respect to one another and the local frames for description of the internal processes inside the bodies. However, there is no longer the absolute time and the absolute space, which are replaced with a Riemannian spacetime manifold and a rather complicated set of relativistic differential equations for geometric (gravitational) variables and other fields. Construction of the postNewtonian global and local frames is now a matter of boundary conditions imposed on the field equations Fock (1959). The principle of relativity should be satisfied when the law of transformation from the global to local coordinates associated with each body (or a subsystem of the bodies) is derived. Not only should it be consistent with the Lorentz transformation but must account for the gauge freedom of the relativistic theory of gravity as well. Time and spatial coordinates are transformed simultaneously making up a class of fourdimensional coordinate transformations Soffel et al. (2003).
3.2 PostNewtonian Approximations
3.2.1 Small Parameters
Field equations (2.2) and (2.5) of the scalartensor theory of gravity represent a system of eleventh nonlinear differential equations in partial derivatives. The challenge is to find their solution for the case of Nbody system represented by Sun and planets which are not considered as test bodies. Exact solution of this problem is not known and may not exist. Hence, one has to resort to approximation methods. Two basic methods are known: the postMinkowskian and the postNewtonian approximations Damour (1989). PostNewtonian approximations assume that matter moves slowly and its gravitational field is weak everywhere – the conditions, which are satisfied within the solar system. For this reason, we use the postNewtonian approximations in this paper.
postNewtonian approximations are based on assumption that expansion of the metric tensor in the near zone of a source of gravity can be done in inverse powers of the fundamental speed that is equal to the speed of light in vacuum. This expansion may be not analytic in higher postNewtonian approximations in a certain class of coordinates Kates and Kegeles (1982); Blanchet and Damour (1986). Exact formulation of a set of basic axioms required for doing the postNewtonian expansion was given by Rendall Rendall (1992). Practically, it requires to have several small parameters characterizing the source of gravity. They are: , , and , , where is a characteristic velocity of motion of matter inside a body, is a characteristic velocity of the relative motion of the bodies with respect to each other, is the internal gravitational potential of each body, and is the external gravitational potential between the bodies. If one denotes a characteristic radius of a body as and a characteristic distance between the bodies as , the internal and external gravitational potentials will be and , where is a characteristic mass of the body. Due to the virial theorem of the Newtonian gravity Landau and Lifshitz (1975) the small parameters are not independent. Specifically, one has and . Hence, parameters and are sufficient in doing postNewtonian approximations. Because within the solar system these parameters do not significantly differ from each other, we shall not distinguish between them when doing the postNewtonian iterations. In what follows, we shall use notation to mark the presence of the fundamental speed in the postNewtonian terms.
Besides the small relativistic parameters and , postNewtonian approximations utilize one more small parameter. This parameter is , and it characterizes dependence of the gravitational field outside the bodies on their internal structure and shape. Parameter has no direct relationship to relativity unless the bodies are not compact astrophysical objects like neutron stars or black holes. This is the case of strong gravitational field when the size of the body approaches its gravitational radius, . In this situation making postNewtonian approximations more laborious.
It is wellknown that in the Newtonian mechanics gravitational field of a sphericallysymmetric body is the same as the field of a single pointlike particle having the same mass as the body Chandrasekhar (1987). This is what Damour calls the effacing principle Damour (1983, 1989). It suggests that for sphericallysymmetric bodies parameter does not play any role in the Newtonian approximation. Our study Kopeikin and Vlasov (2004, 2006) reveals that the effacing principle is violated in the first postNewtonian approximation of the scalartensor theory of gravity so that terms of the order of appear in the translational equations of motion of sphericallysymmetric bodies.
Notice that in general relativity, where the PPN parameter , the effacing principle in equations of motion is violated only by terms of the order of Kopeikin and Vlasov (2004) as all terms of the order can be eliminated after making an appropriate choice of the center of mass of the bodies Kopeikin and Vlasov (2004). For compact relativistic stars , which makes the first postNewtonian approximation for this objects much smaller than 2.5 postNewtonian approximation , where the radiationreaction force due to emission of gravitational waves appears for the first time Damour (1983); Kopeikin (1985); Schäfer (1985). This remark corrects Damour’s consideration on the compatibility of different orders of postNewtonian approximations (see (Damour, 1989, pages 163 and 169)) and fully justifies our result of the postNewtonian calculation of the gravitational radiationreaction force by the FockChandrasekhar method Kopeikin (1985); Grishchuk and Kopeikin (1983, 1986).
If the bodies are not sphericallysymmetric, parameter appears in the Newtonian and postNewtonian approximations as a result of expansion of gravitational field in multipoles. The size of the multipole of multipolarity depends on the parameter of nonsphericity of the body, , related to the elastic properties of matter, which are characterized for a selfgravitating body by Love’s numbers . Generally, they are different for each multipole Zharkov and Trubitsyn (1978); Cheng (1991); Getino (1993). The present paper will account for all gravitational multipoles of the solar system bodies without making finite truncation of the multipolar series.
3.2.2 The PostNewtonian Series
One assumes that the scalar field can be expanded in a power series around its background value , that is
(3.1) 
where is a dimensionless perturbation of the scalar field around its background value. In principle, the background value of the scalar field can depend on time due to cosmological expansion of the universe that may be interpreted as a secular change in the universal gravitational constant (see below). According to theoretical expectations Damour and Nordtvedt (1993) and experimental data Will (1993, 2006) the postNewtonian perturbation of the scalar field must have a very small magnitude, so that we can expand all quantities depending on the scalar field in the Taylor series using the absolute value of as a small parameter. In particular, the postNewtonian decomposition of the coupling function can be written as
(3.2) 
where , , and we impose the boundary condition such that approaches zero as the distance from the solar system grows to infinity.
We look for solution of the field equations (2.5), (2.2) in the form of a Taylor expansion of the metric tensor and the scalar field with respect to parameter such that
(3.3) 
The generic postNewtonian expansion of the metric tensor is not analytic Kates and Kegeles (1982); Blanchet and Damour (1986); Damour (1989). However, the nonanalytic terms emerge only in higher postNewtonian approximations and do not affect results of the present paper since we restrict ourselves by the first postNewtonian approximation. Notice also that the linear, with respect to , terms in the metric tensor expansion (3.3) can be eliminated by coordinate adjustments Thorne and Hartle (1985). These terms correspond to a nonorthogonality of the local coordinate frame and/or a residual rotation of spatial axes Thorne and Hartle (1985). Reference frames with such properties are not used in astronomy and geophysics. Therefore, we assume that all coordinates used in this paper are nonrotating and orthogonal, so that the linear term in expansion (3.3) is absent.
Various components of the metric tensor and the scalar field have in the first postNewtonian approximation the following form
(3.4)  
(3.5)  
(3.6)  
(3.7) 
where and denote terms of the order . In what follows, we shall use notations: