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ORIGINAL RESEARCH published: 25 February 2016 doi: 10.3389/fspas.2016.00005

Reference Ellipsoid and Geoid in Chronometric Geodesy

Sergei M. Kopeikin 1, 2*

1 Department of Physics and Astronomy, University of Missouri, Columbia, MO, USA, 2 Department of Physical Geodesy and Remote Sensing, Siberian State University of Geosystems and Technologies, Novosibirsk, Russia

Chronometric geodesy applies general relativity to study the problem of the shape of celestial bodies including the earth, and their gravitational field. The present paper discusses the relativistic problem of construction of a background geometric manifold that is used for describing a reference ellipsoid, geoid, the normal gravity field of the earth and for calculating geoid’s undulation (height). We choose the perfect fluid with an ellipsoidal mass distribution uniformly rotating around a fixed axis as a source of matter generating the geometry of the background manifold through the Einstein equations. We formulate the post-Newtonian hydrodynamic equations of the rotating fluid to find out the set of algebraic equations defining the equipotential surface of the gravity field. In order to solve these equations we explicitly perform all integrals characterizing the interior gravitational potentials in terms of elementary functions depending on the parameters defining the shape of the body and the mass distribution. We employ the coordinate freedom of the equations to choose these parameters to make the Edited by: Agnes Fienga, shape of the rotating fluid configuration to be an ellipsoid of rotation. We derive Observatoire de la Côte d’Azur, France expressions of the post-Newtonian mass and angular momentum of the rotating fluid Reviewed by: as functions of the rotational velocity and the parameters of the ellipsoid including its Alberto Vecchiato, Osservatorio Astrofisico di Torino - bare density, eccentricity and semi-major axes. We formulate the post-Newtonian Pizzetti INAF, Italy and Clairaut theorems that are used in geodesy to connect the parameters of the Ramesh Jaga Govind, reference ellipsoid to the polar and equatorial values of force of gravity. We expand the University of Cape Town, South Africa post-Newtonian geodetic equations characterizing the reference ellipsoid into the Taylor *Correspondence: Sergei M. Kopeikin series with respect to the eccentricity of the ellipsoid, and discuss the small-eccentricity [email protected] approximation. Finally, we introduce the concept of relativistic geoid and its undulation with respect to the reference ellipsoid, and discuss how to calculate it in chronometric Specialty section: This article was submitted to geodesy by making use of the anomalous gravity potential. Fundamental Astronomy, a section of the journal Keywords: physical geodesy, reference ellipsoid, geoid, relativistic geodesy, reference frames, general relativity Frontiers in Astronomy and Space and gravitation Sciences

Received: 22 November 2015 Accepted: 08 February 2016 1. INTRODUCTION Published: 25 February 2016 Accurate definition, determination and realization of terrestrial reference frame is essential for Citation: fundamental astronomy, celestial mechanics, geophysics as well as for precise satellite and aircraft Kopeikin SM (2016) Reference Ellipsoid and Geoid in Chronometric navigation, positioning, and mapping. The International Terrestrial Reference Frame (ITRF) is Geodesy. materialized by coordinates of a number of geodetic points (stations) located on the Earth’s surface, Front. Astron. Space Sci. 3:5. and spread out across the globe. Definition of the coordinates is derived from the International doi: 10.3389/fspas.2016.00005 Terrestrial Reference System (ITRS) which is a theoretical concept. ITRF is used to measure plate

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 1 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy tectonics, regional subsidence or loading and to describe the University of Jena in Germany (Petroff, 2003; Ansorg et al., Earth’s rotation in space by measuring its rotational parameters 2004; Meinel et al., 2008; Gürlebeck and Petroff, 2010, 2013). (Petit and Luzum, 2010). These papers focused primarily on studying the astrophysical Nowadays, four main geodetic techniques are used to compute aspects of the problem like stability of the rotating stars, the accurate terrestrial coordinates and velocities of stations— the points of bifurcations, exact axially-symmetric spacetimes, GPS, VLBI, SLR, and DORIS, for the realization of ITRF. The emission of gravitational waves, etc. The present paper observations are so accurate that geodesists have to model and focuses on the post-Newtonian effects in physical geodesy. include to the data processing the secular Earth’s crust changes More specifically, we extend the research on the figures to reach self-consistency between the various ITRF realizations of equilibrium into the realm of relativistic geodesy and referred to different epoch. It is recognized that in order to pay attention to the possible geodetic applications for the maintain the up-to-date ITRF realization as accurate as possible adequate processing of the high-precise data obtained by various the development of the most precise theoretical model and geodetic techniques that include but not limited to SLR, LLR, relationships between parameters of the model is of a paramount VLBI, DORIS, and GNSS (Drewes, 2009; Plag and Pearlman, importance. 2009). This model is not of a pure kinematic origin because Important stimulating factor for pursuing more advanced the gravity field plays an essential role in geodetic network theoretical research on relativistic effects in geodesy and the Earth computations (Heiskanen and Moritz, 1967; Hofmann- figure of equilibrium is related to the recent breakthrough in Wellenhof and Moritz, 2006) making a particular theory of manufacturing ring laser gyroscopes (Schreiber, 2013; Beverini gravity a part of definition of the ITRS. Until recently, such et al., 2014; Hurst et al., 2015). Earth rotation and orientation a theory was the Newtonian theory of gravity. However, provide the link between the terrestrial (ITRF) and celestial current SLR/GPS/VLBI techniques allow us to determine reference frames (ICRF). Traditionally, the Earth orientation the transformation parameters between the coordinates and parameters (EOPs) are observed by the International Earth velocities of the collocation points of the ITRF realization Rotation Service (IERS) that operates the global VLBI network. with the precision approaching to 1 mm and 1 mm/year, VLBI technique determines EOPs off-line after accumulating ∼ ∼ respectively (Petit and Luzum, 2010, Table 4.1). At the same the data over sufficiently long period of time. Moreover, time, general relativity predicts that the post-Newtonian effects VLBI depends on suitable models of Earth’s troposphere and in measuring the ITRF coordinates, as compared with the geophysical factors which are difficult to predict. Ring laser Newtonian theory of gravity, is of the order of the Earth’s gyroscopes provide direct access to the Earth rotation axis and gravitational radius that is about 1 cm (Kopejkin, 1991; Müller a high resolution in the short-term. Currently, the modern et al., 2008). This general-relativistic effect distorts the coordinate quantum optics has matured to a point where the ring laser 12 grid of ITRF at the noticeable level which can be detected with gyroscopes can resolve rotation rates of 10− rad/s after 1 h of the currently available geodetic techniques and, hence, should be integration and demonstrate an impressive stability over several taken into account. month (Schreiber, 2013). The combination of VLBI and ring ITRS is specified by the Cartesian equatorial coordinates laser measurements offers an improved sensitivity for the EOPs rotating rigidly with the Earth. For the purposes of geodesy and in the short-term and the direct access to the Earth rotation gravimetry the Cartesian coordinates are often converted to a axis. Another application of the ring laser gyroscopes is to geographical system of elliptical coordinates (h,θ,λ) referred to measure the Lense-Thirring effect predicted by Einstein’s general an international reference ellipsoid which is a non-trivial solution relativity (Ciufolini and Wheeler, 1995), in a terrestrial laboratory of the Newtonian gravity field equations found by Maclaurin environment (Beverini et al., 2014). (Chandrasekhar, 1967). It describes the figure of a fluid body Another, rapidly emerging branch of relativistic geodesy is with a homogeneous mass density rotating around a fixed z-axis called chronometric geodesy (Petit et al., 2014). This new line with a constant angular velocity ω. The post-Newtonian effects of development is pushed forward by the fascinating progress deforms the shape of Maclaurin’s ellipsoid (Chandrasekhar, 1965; in making up quantum clocks (Poli et al., 2013), ultra-precise Bardeen, 1971) and modify the basic equations of classic geodesy time-scale dissemination over the globe (Kómár et al., 2014), (Müller et al., 2008; Kopeikin et al., 2011, 2016). These relativistic and geophysical applications of the clocks (Bondarescu et al., effects must be properly calculated to ensure the adequacy of 2012, 2015). Clocks at rest in a gravitational potential tick slower the geodetic coordinate transformations at the millimeter level than clocks outside of it. On Earth, this translates to a relative 16 of accuracy. A pioneering study of relativistic effects in geodesy frequency change of 10− per meter of height difference (Falke have been carried out by Bjerhammar (1985). et al., 2014). Comparing the frequency of a probe clock with a Post-Newtonian equilibrium configurations of uniformly reference clock provides a direct measure of the gravity potential rotating fluids have been discussed in literature by researchers difference between the two clocks. This novel technique has been from USA (Chandrasekhar, 1965, 1967a,b,c, 1971a,b; Bardeen, dubbed chronometric leveling. It is envisioned as one of the most 1971; Chandrasekhar and Elbert, 1974, 1978; Chandrasekhar promising application of the relativistic geodesy in a near future and Miller, 1974), Lithuania (Bondarenko and Pyragas, 1974; (Mai and Müller, 2013; Mai, 2014; Petit et al., 2014). Optical Pyragas et al., 1974, 1975), USSR (Tsirulev and Tsvetkov, frequency standards have recently reached stability of 2.2 16 × 1982a,b; Tsvetkov and Tsirulev, 1983; Galtsov et al., 1984) 10− at 1 s, and demonstrated an overall fractional frequency and, the most recently, scientists from the Fridrich Schiller uncertainty of 2.1 10 18 (Nicholson et al., 2015) which enables × −

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 2 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy their use for chronometric geodesy at the absolute level of one approximation. Finally, Section 14 introduces the concept of centimeter. the post-Newtonian geoid’s undulation and its relation to the The chronometric leveling directly measures the equipotential current problem of the measurement of the global average sea surface of gravity field (geoid) without conducting a complicated level. Appendix contains technical details of calculations of the gravimetric survey and solving the differential equations for the mathematical integrals which appear in the main part of the anomalous gravity potentials (Stoke’s or Molodensky’s problem paper. Hofmann-Wellenhof and Moritz, 2006; Torge and Müller, The following notations are used throughout the paper: 2012). Combining the data of the chronometric leveling with the Greek indices α, β, ... run from 0 to 3, those of the conventional geodetic techniques will allow us • the Roman indices i, j, ... run from 1 to 3, to determine the normal heights of reference points with an • repeated Greek indices mean Einstein’s summation from unprecedented accuracy (Bondarescu et al., 2012). An adequate • 0to3, physical interpretation of this type of measurements requires repeated Roman indices mean Einstein’s summation from more precise theoretical definition of geoid (Kopeikin et al., • 1to3, 2015; Oltean et al., 2015) and is inconceivable without an the unit matrix (also known as the Kroneker symbol) is accompanying development of the corresponding mathematical • denoted by δ δij, algorithms accounting for the major relativistic effects in the ij = the fully antisymmetric symbol Levi-Civita is denoted as ε description of the equipotential level surface and its, so-called, ijk • ijk = normal) gravity field (Hofmann-Wellenhof and Moritz, 2006; ε with ε123 1, = + 1 2 3 i 1 2 3 Torge and Müller, 2012). the bold letters a (a , a , a ) (a ), b (b , b , b ) • i = ≡ = ≡ The objective of the present paper is to suggest a solution (b ), and so on, denote spatial 3-dimensional vectors, a dot between two spatial vectors, for example a b a1b1 of Einstein’s gravity field equations that gives rise to the post- • = + a2b2 a3b3 δ aibj, means the Euclidean dot product, Newtonian equipotential surface described by the second-order + = ij the cross between two vectors, for example (a b)i εijkajbk, polynomial which exactly coincides with the shape of the • × ≡ Maclaurin ellipsoid in the Newtonian gravity. Our previous paper means the Euclidean cross product, (Kopeikin et al., 2016) has explored the post-Newtonian case of a we use a shorthand notation for partial derivatives ∂α • α = homogeneous density distribution of a uniformly rotating perfect ∂/∂x , covariant derivative with respect to a coordinate xα is denoted fluid. We have shown that the post-Newtonian effects inevitably • as α; distort the Maclaurin ellipsoid of rotation to a surface described ∇ the Minkowski (flat) space-time metric ηαβ by the fourth-order polynomial which cannot be reduced to • = diag( 1, 1, 1, 1), the ellipsoidal surface of the second order by doing coordinate − + + + gαβ is the physical spacetime metric, transformations. In the present paper we assume that the mass • gαβ is the background spacetime metric describing the distribution is not homogeneous but an ellipsoidal one with • a free parameter, and prove that under this assumption the gravitational field of the reference ellipsoid, the Greek indices are raised and lowered with the metric gαβ , equipotential level surface can be made as an exact ellipsoid of • G is the universal gravitational constant, the second order which may be found more convenient for doing • c is the speed of light in vacuum, the reduction of high-precise geodetic measurements with taking • ω is a constant rotational velocity, into account relativistic corrections. • ρ is a bare constant density of the fluid, The paper consists of several sections, bibliography and • a is a semi-major axis of the ellipsoid of revolution, appendix. Section 2 explains briefly the principles of the post- • b is a semi-minor axis of the ellipsoid of revolution, Newtonian approximations and describes the post-Newtonian • ρ is a constant bare density of matter, metric tensor. In Section 3 we give definition of the post- • 0 κ πGρ a2/c2 is a dimensional parameter characterizing the Newtonian geoid. Section 4 discusses the post-Newtonian • ≡ 0 ellipsoid which generalizes the Maclaurin ellipsoid and is, in strength of gravitational field, a bar above any quantity indicates that it belongs to the general, the surface of the fourth order. It also introduces • background spacetime manifold with the metric gαβ . the reader to the concept of the post-Newtonian coordinate freedom and shows how this freedom can be used to simplify Other notations are explained in the text as they appear. the mathematical description of the PN-ellipsoid. Sections 5– 7 calculate respectively the Newtonian and post-Newtonian gravitational potentials inside the rotating PN-ellipsoid. Sections 2. POST-NEWTONIAN SPACETIME AND 8, 9 give the post-Newtonian definitions of the conserved mass METRIC TENSOR and angular momentum. Section 10 derives the post-Newtonian equations of the equipotential level surface. Sections 11, 12 Discussion of the chronometric geodesy starts from the provide the reader with the relativistic generalization of the construction of the spacetime manifold for the case of a Pizzetti and Clairaut theorem of classical geodesy (Pizzetti, 1913). rotating Earth. We shall employ Einstein’s general relativity to Section 13 approximates the relativistic formulas which can be build such a manifold though some other alternative theories used in practical applications of relativistic geodesy and shows of gravity discussed, for example in Will (1993), can be how to build the exact reference-ellipsoid in the post-Newtonian used as well. Einstein’s field equations represent a system

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 3 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy of ten non-linear differential equations in partial derivatives following form (Kopeikin et al., 2011) for the ten components of the metric tensor, gαβ , which represent gravitational potentials. Because the equations are 2V 2 2 6 g00 1 V O c− , (2) difficult to solve exactly due to their non-linearity, we apply = − + c2 + c4 − + the post-Newtonian approximations (PNA) for their solution i 4V 5

g0i O c− , (3) (Chandrasekhar, 1965). = − c3 + The PNA are applied in case of slowly-moving matter having 2V
4 a weak gravitational field. This is exactly the situation in the solar gij δij 1 O c− , (4) = + c2 + system which makes PNA highly appropriate for constructing  
relativistic theory of reference frames (Soffel et al., 2003) and where the gravitational potentials entering the metric, satisfy the relativistic celestial mechanics in the solar system (Soffel, 1989; Poisson equations, Brumberg, 1991; Kopeikin et al., 2011). The PNA are based on the assumption that a Taylor expansion of the metric tensor can V 4πGρ , (5) be done in the inverse powers of the speed of gravity c that = − Vi 4πGρvi , (6) is equal to the speed of light in vacuum in general relativity. = − Exact mathematical formulation of a set of basic axioms required 3p 4πGρ 2v2 2V , (7) for doing the post-Newtonian expansion was given by Rendall = − + + + ρ   (1990). Practically, it requires to have several small parameters characterizing the source of gravity. They are: ε v /c, ε with p and vi being pressure and velocity of matter respectively, i ∼ i e ∼ v /c, and η U /c2, η U /c2, where v is a characteristic and is the internal energy of matter per unit mass. We e i ∼ i e ∼ e i velocity of motion of matter inside a body, v is a characteristic emphasize that ρ is the local mass density of baryons per e velocity of the relative motion of the bodies with respect to each a unit of invariant (3-dimensional) volume element dV = other, U is the internal gravitational potential of each body, and gu0d3x, where u0 is the time component of the 4-velocity of i − U is the external gravitational potential between the bodies. matter. The local mass density, ρ, relates in the post-Newtonian e p If one denotes a characteristic radius of a body as L and a approximation to the invariant mass density ρ gu0ρ, ∗ = − characteristic distance between the bodies as R, the internal and namely (Kopeikin et al., 2011) p external gravitational potentials will be U GM/L and U i ≃ e ≃ GM/R, where M is a characteristic mass of the body. Due to the ρ 1 2 ρ∗ ρ v 3V . (8) virial theorem of the Newtonian gravity (Chandrasekhar, 1965) = + c2 2 +   the small parameters are not independent. Specifically, one has 2 2 ε ηi and ε ηe. Hence, parameters εi and εe are sufficient in The internal energy, , is related to pressure, p, and the local i ∼ e ∼ doing post-Newtonian approximations. Because within the solar density, ρ, through the thermodynamic equation system these parameters do not significantly differ from each other, we shall not distinguished them. Quite often we shall use 1 d pd 0 , (9) notation κ πGρa2/c2 to mark the powers of the fundamental + ρ = ≡ speed c in the post-Newtonian terms.   We assume that physical spacetime has the metric tensor and the equation of state, p p(ρ). = denoted as gαβ . This spacetime is well-approximated by a We shall further assume that the background matter rotates background manifold with the metric tensor denoted gαβ . The rigidly around fixed z axis with a constant angular velocity ω. This perturbation of gravitational field is denoted with ̹αβ and makes the background spacetime stationary with the background is called the anomalous gravity potential. We, first, build the metric being independent on time. In the stationary spacetime, metric tensor of the background manifold by solving Einstein’s the mass density ρ∗ obeys the exact equation of continuity equations in harmonic coordinates xα (x0, xi), where 0 = i x ct, and t is the coordinate time. The class of the ∂i ρ∗v 0. (10) = = harmonic coordinates is defined by imposing the de Donder
gauge condition on the metric tensor (Fock, 1964; Weinberg, Velocity of rigidly rotating fluid is 1972), vi εijkωjxk , (11) αβ = ∂α gg 0. (1) − = where ωi (0, 0,ω) is the constant angular velocity. p  = Einstein equations for the metric tensor are a complicated non- Replacing velocity vi in Equation (10) with Equation (11), and linear system of differential equations in partial derivatives. differentiating, reveals that Because gravitational field of the solar system is weak and i motion of matter is slow, we focus on the first post-Newtonian v ∂iρ 0 , (12) approximation of general relativity. We also assume that the = source of the background metric is a perfect fluid. Under these which means that velocity of the fluid is tangent to the surfaces of assumptions the post-Newtonian background metric has the a constant density ρ.

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 4 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy

α β 3. RELATIVISTIC GEOID Because dτ/dt gαβ x x , where the dot means the total = − ˙ ˙ time derivative with respect to the coordinate time t, Equation The Newtonian concept of Earth’s geoid was extended to the post- q (16) becomes, Newtonian approximation of general relativity in Soffel (1989) and Kopejkin (1991). Exact concept of the relativistic geoid in 1/2 2 2 i 1 i j general relativity that is not limited to the first post-Newtonian W(x) c 1 g00 g0iv gijv v . (17) ≡ − − − c − c2 approximation has been discussed in Kopeikin et al. (2015) and "   # Oltean et al. (2015). This matches the Newtonian definition of the geoid after We consider the Earth as made up of the background decomposition of the metric tensor in the post-Newtonian series matter with small perturbations which reflect the actual mass and discarding all post-Newtonian terms. distribution, stresses, and velocity flow of Earth’s matter with respect to the rigidly rotating frame of reference. The angular Definition 2: The relativistic a-geoid represents a two- velocity, ω, of Earth’s rotation also slowly changes because of dimensional surface at any point of which the direction precession, nutation, polar motion and variations in the length- of a plumb line measured by the terrestrial observer, is of-day. The physical metric, gαβ , depends on time and spatial orthogonal to the tangent plane of geoid’s surface. coordinates. It is decomposed into an algebraic sum of the In order to derive equation of a-geoid, we notice that the background metric gαβ , and its perturbation, ̹αβ , direction of the plumb line is given by a four-vector of 2 the physical acceleration of gravity, gα c aα where β ≡ − g g ̹ . (13) aα u β uα is a four-acceleration of the terrestrial αβ αβ αβ = ∇ = + observer being orthogonal to uα. In case of a stationary In the present paper, we shall neglect dependence of the rotating configuration of matter, we get (Lightman et al., 1975, Problems 10.14 and 16.17) perturbation ̹αβ on time because it produces very tiny relativistic effects that are currently unobservable. 2 W gα c ∂α ln 1 , (18) Terrestrial reference frame is formed by the world lines of = − − c2 observers being fixed with respect to the crust of the Earth. Each   α 1 α observer moves in spacetime with four-velocity u c− dx /dτ where W is calculated in Equation (17). We consider an α 0 1 2 3 = α α β where x x , x , x , x are coordinates of the observer, and arbitrary displacement, dx h β dx , on the spatial = ≡ τ is its proper time defined in terms of the metric tensor Equation hypersurface being orthogonal⊥ to uα everywhere, and make  (13) as follows, a scalar product of dxα with the direction of the plumb line. It gives, ⊥ 2 2 α β c dτ gαβ dx dx . (14) = − α α 2 W dx gα dx gα c d ln 1 . (19) = = − − c2 Physical space of observers at each instant of time is represented ⊥   by a three-dimensional hypersurface of constant proper time that From definition of a-geoid the left-hand side of Equation is orthogonal everywhere to the world lines of the observers. The (19) must vanish due to the condition of orthogonality metric tensor on this hypersurface is denoted as hαβ and is given α of the two vectors, dx and gα. Therefore, it makes by Zelmanov et al. (2006) and Landau and Lifshitz (1975) d ln 1 W/c2 0, which⊥ means the constancy of the − = gravity potential W on the three-dimensional surface of the
hαβ gαβ uαuβ . (15) a-geoid. Thus, the surface of a-geoid coincides with that of ≡ + u-geoid. It is used to measure the spatial distances. Rising and lowering Calculation of the geoid is a difficult mathematical problem of Greek indices of geometric objects residing on the perturbed of geodesy. It is solved by introducing a reference surface of manifold are done with the help of the full metric gαβ . reference ellipsoid. Geoid’s surface is defined in terms of the geoid Similarly to classic geodesy, general relativity offers two undulation (height) with respect to the reference ellipsoid. definitions of relativistic geoid (Soffel, 1989; Kopejkin, 1991)

Definition 1: The relativistic u-geoid represents a two- 4. POST-NEWTONIAN REFERENCE dimensional surface at any point of which the rate of the ELLIPSOID proper time, τ, of an ideal clock carried out by a set of fiducial terrestrial observers is constant. In classical geodesy the reference figure for calculation of geoid’s undulation is the Maclaurin ellipsoid which is a surface of the The u-geoid is determined by equation W const., where the second order formed by a rigidly rotating fluid of constant = physical gravity potential density ρ. Maclaurin’s ellipsoid is a surface of the second order (Chandrasekhar, 1969)

2 2 2 2 dτ x y z W(x) c 1 . (16) + 1 , (20) = − dt 2 2   a + b =

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 5 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy where a and b are semi-major and semi-minor axes of the z r , of the PN-ellipsoid by the condition, σ 0. Equation = p = ellipsoid. We also assume a > b, and define the eccentricity of (22) yields the Maclaurin ellipsoid as 1 rp b 1 κB2 . (25) √a2 b2 = + 2 e − . (21)   ≡ a2 The equatorial and polar radii of the PN-ellipsoid should Physically, the ellipsoidal shape of rotating, self-gravitating fluid be used in the post-Newtonian approximation instead of the is formed because the Newtonian gravity potential is a scalar equatorial and polar radii of the Maclaurin reference-ellipsoid for function represented by a polynomial of the second order with calculation of observable physical effects like the force of force, respect to the Cartesian spatial coordinates, and the differential etc. We characterize the “oblateness” of the PN-ellipsoid by the Euler equation defining the equilibrium of the gravity and post-Newtonian eccentricity pressure is of the first order partial different equation which leads to the quadratic (w.r.t. the coordinates) equation of the level 2 2 re rp surface. ǫ − . (26) q We shall demonstrate in the following sections that in the ≡ re post-Newtonian approximation the gravity potential, W, of the rotating fluid is a polynomial of the fourth order as was first It differs from the eccentricity Equation (21) of the Maclaurin noticed by Chandrasekhar (1965). Hence, the post-Newtonian ellipsoid by relativistic correction equation of the level surface of a rigidly-rotating fluid is expected to be a surface of the fourth order. We shall assume 1 e2 ǫ e κ − (B2 B1) . (27) that the surface remains axisymmetric in the post-Newtonian = − 2e − approximation and dubbed the body with such a surface as PN-ellipsoid. The PN-ellipsoid vs. Maclaurin’s ellipsoid is visualized in We shall denote all quantities taken on the surface of the PN- Figure 1. ellipsoid with a bar to distinguish them from the coordinates Theoretical formalism for calculation of the post-Newtonian outside of the surface. Let the barred coordinates xi x, y, z level surface can be worked out in arbitrary coordinates. For = { } denote a point on the surface of the PN-ellipsoid with the axis of mathematical and historic reasons the most convenient are symmetry directed along the rotational axis and with the origin harmonic coordinates which are also used by the IAU (Soffel located at its post-Newtonian center of mass 1. Let the rotational et al., 2003) and IERS (Petit and Luzum, 2010). The class of axis coincide with the direction of z axis. Then, the most general the harmonic coordinates is selected by the gauge condition equation of the PN-ellipsoid is Equation (1). Different harmonic coordinates are interrelated by coordinate transformations which are not violating the gauge σ 2 z2 condition 1. This freedom is known as a residual gauge (or 1 κF(x) , (22) a2 + b2 = + coordinate) freedom. The field Equations (5)–(7) and their solutions are form-invariant with respect to the residual gauge where κ πGρa2/c2, σ 2 x2 y2, is the post-Newtonian transformations. ≡ ≡ + parameter which is convenient in the calculations that follows, The residual gauge freedom is described by a post-Newtonian coordinate transformation, σ 2 z2 σ 4 z4 σ 2z2 F(x) K K B B B , (23) α α α 1 2 2 2 1 4 2 4 3 2 2 x′ x κξ (x) , (28) ≡ a + b + a + b + a b = + α and K1, K2, B1, B2, B3 are arbitrary numerical coefficients. where functions, ξ , obey the Laplace equation, Each cross-section of the PN-ellipsoid being orthogonal to the rotational axis, represents a circle. The equatorial cross- ξ α 0. (29) section has an equatorial radius, σ r , being determined from = = e Equation (22) by the condition z 0. It yields = Solution of the Laplace equation which is convergent at the origin of the coordinate system, is given in terms of the 1 re a 1 κB1 . (24) harmonic polynomials which are selected by the conditions = + 2   imposed by the statement of the problem. In our case, the problem is to determine the shape of the PN-ellipsoid which The meridional cross-section of the PN-ellipsoid is no longer an has the surface described by the polynomial of the fourth ellipse (as it was in case of the Maclaurin ellipsoid) but a curve order Equation (32) with yet unknown coefficients B1, B2, B3. of the fourth order. Nonetheless, we can define the polar radius, The form of the Equation 32 does not change (in the post- α 1Post-newtonian definitions of mass, center of mass, and other multipole moments Newtonian approximation) if the functions ξ in Equation (28) can be found, for example, in Kopeikin et al. (2011). are polynomials of the third order. It is straightforward to show

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 6 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy

FIGURE 1 | Meridional cross-section of the PN-ellipsoid (a red curve in on-line version) vs. the Maclaurin ellipsoid (a blue curve in on-line version). The left panel represents the most general case with arbitrary values of the shape parameters B1, B2, B3 when the equatorial, re, and polar, rp, radii of the PN-ellipsoid differ from the semi-major, a, and semi-minor, b, axes of the Maclaurin ellipsoid, re a, rp b. The right panel shows the most important physical case of = = B B 0 when the equatorial and polar radii of the PN-ellipsoid and the Maclaurin ellipsoid are equal. The angle ϕ is the geographic latitude ( 90 ϕ 90 ), 1 = 2 = − ◦ ≤ ≤ ◦ and the angle θ is a complementary angle (co-latitude) used for calculation of integrals in appendix of the present paper (0 θ π). In general, when B B 0, ≤ ≤ 1 = 2 = the maximal radial difference (the “height” difference) between the surfaces of the PN-ellipsoid and the Maclaurin ellipsoid depends on the choice of the post-Newtonian coordinates, and may amount to a few cm (see Section 13). Carefully operating with the residual gauge freedom of the post-Newtonian theory allows us to make the difference between the two surfaces much less than that. that the admissible harmonic polynomials of the third order have radius r a, the eccentricity of the PN-ellipsoid ǫ e, and e = = the following form function σ 2z2 1 x 2 2 F(x) B . (32) ξ hx p σ 4z , (30a) 2 2 = + a2 − ≡ a b y ξ 2 hy p σ 2 4z2
, (30b) Let xi x, y, z is any point inside the PN-ellipsoid. We 2 = { } = + a − introduce a quadratic polynomial z ξ 3 σ 2
2 , kz q 2 3 2z (30c) = + b − σ 2 z2
C(x) 1. (33) ≡ a2 + b2 − where h, k, p, and q are arbitrary constant parameters. Polynomials Equations (30a)–(30c) represent solutions of the This polynomial vanishes on the boundary surface of the PN- Laplace Equation (29). We choose ξ 0 0 because we consider = ellipsoid in the Newtonian approximation. However, in the stationary spacetime so all functions are time-independent. post-Newtonian approximation we have on the boundary the i Coordinate transformation Equation (28) with ξ taken from following condition Equations (30a)–(30c) does not violate the harmonic gauge condition Equation (1) but it changes the numerical post- C(x) κF(x) , (34) Newtonian coefficients in the mathematical form of Equations = (22) and Equation (23) as a consequence of Equation (22). In terms of the polynomial C(x) function F(x) in Equation (32) can be formally recast to K K 2h , (31a) 1 → 1 + K K z2 z2 2 2 2k , (31b) F(x) B 1 C(x) . (35) → + = b2 + − b2 B1 B1 2p , (31c)   → + B B 2 2 4q , (31d) The term being proportional to C(x) can be discarded on → − b2 a2 the boundary of the PN-ellipsoid. Now we shall calculate the B3 B3 8p 6q , (31e) → − a2 + b2 gravitational potentials of the rotating PN ellipsoid which enter the metric tensor Equations (2)–(4). Thus, it makes evident that only one out of the five coefficients ¯ K1, K2, B1, B2, B3 is algebraically independent while the four 5. NEWTONIAN POTENTIAL V others can be chosen arbitrary. To simplify our calculations and eliminate the gauge-dependent terms from mathematical Newtonian gravitational potential V satisfies the inhomogeneous equations we choose K K B B 0. The constant Poisson’s equation 1 = 2 = 1 = 2 = B B is left free. It will be fixed later on. With the accepted 3 ≡ choice of the coordinates the polar radius r b, the equatorial V(x) 4πGρ(x) , (36) p = = −

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 7 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy inside the mass. Its particular solution is given by We make replacement of variable x in Equation (37) to r ′ = x x′, and use spherical coordinates to perform integration with 3 − ρ(x′)d x′ respect to the radial coordinate r x x from the point r 0 V(x) G , (37) =| − ′| = = V x x to the point lying on the surface of the PN ellipsoid, r R (θ, λ) ′ = − Z | − | or r R (θ, λ). For the internal point, the angular integration = + where V is the volume occupied by the matter distribution. We in the remaining integral over the surface of the PN ellipsoid is shall assume that the density has an ellipsoidal distribution inside performed over the solid angle 4π, and the integration with the the body which differs from the homogeneous density, ρ0, only in point R (θ, λ) equals to that with the radial direction R (θ, λ). the post-Newtonian approximation. Moreover, we shall consider This observation− makes the angular integration easier beca+use the the most simple, linearized case of the ellipsoidal distribution integral Equation (37) can be written down in the following form (Chandrasekhar, 1969) σ 2 z2 ρ(x) ρ0 1 κA , (38) = + a2 + b2 1 2 2    V Gρ0 R R d κAI1 , (44) = 4 2 + + IS + − where ρ0 is a constant bare density, and A is as yet
undetermined numerical constant. This assumption allows us where R and R are defined in Equation (42), and the integral + − to perform integration in Equation (37) explicitly. Inside the 3 2 2 mass the integral Equation (37) can be calculated by making d x′ σ ′ z′ I1 Gρ0 (45) use of spherical coordinates. The procedure is as follows = V x x a2 + b2 Z ′   (Chandrasekhar, 1969). | − | Let us consider a point xi x, y, z inside the PN-ellipsoid takes into account the ellipsoidal distribution of the density. After = { } Equation (22). It is connected to the point xi on the surface of the making use of Equation (42) and expanding the integrand in Equation (44) with respect to the post-Newtonian parameter κ, ellipsoid by vector Ri xi xi where Ri Rℓi, R δ RiRj, = − = = ij the Newtonian potential takes on the following form and a unit vector, ℓi sin θ cos λ, sin θ sin λ, cos θ . Weq have = { } 2B2 AC κ i i i 1 −2 [F F x x ℓ R , (39) A + 2A + + − A V Gρ0 B d κ I1 , = + 2 = 2 S ( 2 (F F ) ) + I − √B AC + − − Substituting Equations (39)–(22) yields a quadratic equation − i (46) where all post-Newtonian terms of the higher order in κ have AR2 2BR C κF (x ℓR) , (40) + + = + been discarded, and integration is performed over a unit sphere S2 with respect to the angles λ and θ, d sin θdθdλ is the ≡ where element of the solid angle. Now, we expand F in polynomial w.r.t. R , sin2 θ cos2 θ A , ± ± ≡ a2 + b2 2 3 F α0 α1 cos θR α2 (cos θR ) α3 (cos θR ) ± ± ± ± sin θ x cos λ y sin λ z cos θ = + 4 + + B + , α4 (cos θR ) , (47) 2 2 + ± ≡ a
+ b σ 2 z2 C 1. (41) where the coefficients ≡ a2 + b2 − z2 z2 α0 B 1 , (48) We solve Equation (40) by iterations by expanding R R = b2 − b2 2 = ˆ +   c− R, where R is either R or R corresponding to two 2 ˆ ˆ + ˆ − 2Bz z solutions of the quadratic Equation (40) with the right hand side α1 1 2 (49) = b2 − b2 being nil. The bare solution R is used, then, to calculate the right   ˆ 2 side of Equation (40) for performing the second iteration. We B z α2 1 6 (50) obtain = b2 − b2   4Bz 2 α , B B AC κAF 3 4 (51) R − + ± , (42) = − b ± = −A ± A B p α , 4 4 (52) where = −b

2 4 are the polynomials of z only. We also notice that on the surface z cos θR z cos θR B ˆ B ˆ of the PN-ellipsoid, F(x) α , as follows from Equation (35) F + ± + ± , (43) = 0 ± ≡ b ! − b ! where we can use, C(x) 0, in the post-Newtonian terms. = Replacing Equations (47) in (46) transforms it to where the term C(x) AR2 2BR C vanishes because of = + + Equation (40). V V κV κAI , (53) = N + pN + 1

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 8 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy where 6. POST-NEWTONIAN VECTOR V¯ I B2 C POTENTIAL VN Gρ0 d (54) = 2 A2 − 2A i IS   Vector potential V obeys the Poisson equation 2 α0 B 2 2B C VpN Gρ0 α1 cos θ α2 cos θ i i = 2 2A − A2 + A3 − 2A2 V 4πGρ(x)v (x) , (62) IS    = − 3 3 2B BC 2α3 cos θ which has a particular solution − A4 − A3   4 2 2 i 4 8B 6B C C i ρ(x′)v (x′) 3 α4 cos θ d , (55) V G d x′ . (63) + A5 − A4 + 2A3 = V x x   Z | − ′| Equations (53)–(55) describe the Newtonian potential exactly For a rigidly rotating configuration, vi(x) εijkωjxk so that both on the surface of the PN-ellipsoid and inside it. = These equations cannot be extended to the external space k Vi εijkωjD , (64) which requires a separate integration which will be discussed = somewhere else. where The integrals entering V in Equations (54), (55) are discussed ρ x i 3 in Appendix A in Supplementary Material. The integrals I1 Di ( ′)x′ d x′ G . (65) appears besides Equation (53) also in the post-Newtonian = x x′ potentials of the gravitational field in Equation (80), and its Z | − | discussion is postponed to section 7. After evaluating the It can be recast to the following form integrals in the internal point and reducing similar terms, i i i i ρ(x′)(x′ x ) 3 potentials VN and VpN take on the following form: D x VN G − d x′ , (66) = + x x′ 2 2 Z | − | 2 z z .(where V is the Newtonian potential given in Equation (56 (56) , 1ג(C(x 1ג 3 1 0ג VN πGρ0a 1 = − b2 − − b2 − N      Because the potential Vi appears only in the post-Newtonian V πGρ a2 F (z) b2F (z)C(x) b4F (z)C2(x) , (57) pN 0 1 2 3 terms we can consider the density ρ as constant, ρ ρ0. In this = + + = case the second term in the right hand side of Equation (66) can where   be integrated over the radial coordinate, yielding (58) ג2α z ג F (z) α 1 0 0 1 1 ρ i i ρ = − 2+ 2 (x′)(x′ x ) 3 0 3 3 i 2 z 3z − d x′ R R l d . (67) − + + V x x = 6 S2 2ג 1 1ג 2α2b 1 − b2 − − b2 Z | − ′| I     
2 2 2 z 5z After making use of Equation (42) to replace R and R , we − + 3ג 3 2ג 4α3b z 3 1 − − b2 − − b2 + obtain       2 2 2 4 i i 3 4 z z z ρ(x′ x ) 3 4 B BC i , ρ 3ג 5 6 1 2 2ג 6α4b 1 2 2 4 − d x′ 0 3 2 l d (68) " − b − − b + b V x x = S2 −3 A + A     Z | − ′| I   z2 35 z4 where we have omitted the post-Newtonian terms (with κ) since , ג 1 10 2 4 4 + − b + 3 b the vector potential Vi itself appears only in the post-Newtonian    terms. Integrals entering Equation (68) are given in Appendix A (59) (3ג3 2ג4α3z (2 (2ג2 1ג) F2(z) α2 = − −2 − +2 2 z 3z in Supplementary Material. Calculation reveals 3ג 1 3 2ג 6α4b 1 − b2 − − b2     2 2 ρ(x′)(x′ x) 3 2 z 0ג 5z − d x′ πρ0a x 1 V x x = − − b2 , 4ג 1 2 + − b2 Z | − ′|      2 2 3z 5z ג ג ג 2ג 1 1ג F3(z) α4 ( 2 6 3 6 4) , (60) 2 1 = − + − − b2 + − b2      α , α , α , α , α 2 (69) , (2ג 1ג) (and the polynomial coefficients 0 1 2 3 4 are given in xπρ0a C(x Equations (48)–(52). It is worth noticing that VpN obeys the + −2 ρ(x′)(y′ y) 3 2 z 0ג Laplace equation − d x′ πρ0a y 1 V x x = − − b2 Z | − ′|   V 0 , (61) 2 2 pN 3z 5z 2ג 1 1ג 1 2 = − − b2 + − b2 and Equation (57) represents the harmonic polynomial of the      (70) , ( ג ג) (fourth order. yπρ a2C(x + 0 1 − 2

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ρ(x )(z z) z2 where we have introduced two integrals ג πρ 2 3 ′ ′ − d x′ 4 0a z 1 2 1 V x x′ = − − b Z | − |   3 2 2 5z2 d x′ σ ′ z′ (I1 Gρ0 , (80 . ( ג2 ג) (2zπρ a2C(x ג 1 2 2 0 1 2 = V x x a2 + b2 − − 3b − − Z | − ′|      2 (71) Gρ0 z′ 3 I2 d x′ . (81) = b2 V x x Z | − ′| Substituting this result to Equation (66) and making use of Notice that the integral I has already appeared in the calculation Equation (56) yields 1 of the Newtonian potential in Equation (45) while the integral I 2 i x y z is a new one. D D , D , D xD1, yD1, zD2 , (72) ≡ = The integrals can be split in several algebraic pieces,  
where σ 2 z2 z2 I 2D V 2 D D 1 2 2 1 N 2 2 1 3z2 5z2 = a + b − + b −   (73), ג(C(x ג 1 ג D πGρ a2 1 1 0 2 1 2 2 2 Gρ
sin2 θ cos2 θ
≡ − b − − b − 0 4 4 ,      R R 2 2 d (82) 2 2 2 + 8 S2 + + − a + b 2 z z 5z I  
2 2ג 1 4 1ג 7 5 0ג D2 πGρ0a 1 ≡ − b2 − − b2 + − 3b2 z Gρ0 4 4 2       I2 2D2 VN R R cos θd ,(83) − + + b2 − + 8b2 2 = ג ג C(x) (4 2 3 1) , (74) IS + −

 where the integrals 7. POST-NEWTONIAN SCALAR ¯ POTENTIAL 1 sin2 θ cos2 θ 4 4 R R 2 2 d 8 S2 + + − a + b Potential is defined by equation I   2B
4 B2C C2 , 3 2 2 d (84) π ρ x φ x , = S2 A − A + 4A 4 G ( ′) ( ′) (75) I   = − 1 R4 R4 cos2 θd where function 8 2 + + − IS 4
2 2 2B B C C 2 2 2 p 2 cos θd , (85) φ(x′) 2ω σ 3 2V . (76) = 2 A4 − A3 + 4A2 ≡ + ρ + N IS   We use the results of Appendix A in Supplementary Material to Potential enters only the post-Newtonian equations. calculate these integrals, and obtain Therefore, everywhere in calculation of we can assume the density ρ being approximated as constant. The pressure p 4 2 2 2B B C C inside the massive body can be calculated from the hydrostatic 2 d (86) 2 A3 − A2 + 4A equilibrium equation with a constant value of the bare density IS   2 ρ ρ . The solution is Chandrasekhar (1969) 3 z2 z2 z4 ג ג π 2 0 = a 1 2 0 2 1 6 2 5 4 1 = 2 " − b − − b + b p     2 4 2 ( ג ג) π ρ 2 G 0a C(x) 0 2 1 . (77) z 35 z 2 z 0ג πa 1 2ג ρ0 = − − 1 10 + − b2 + 3 b4 + − b2      Making use of Equation (77) and Equation (56) we can write z2 z2 (C(x 2ג 5 1 3 1ג 3 1 4 down function φ(x′) as − − b2 + − b2      3 2 ג ג 2 2 2 2 2 σ z πa 1 2 C (x) 2 − − (1ג4 0גφ(x′) a 2ω πGρ0 (3 = − − a2 + b2      2 2 2 z 2 . (1ג6 0גπGρ0a (5 (1ג3 0ג) 2a ω πGρ0 − − − b2 + − 2B4 B2C C2 cos2 θ 2 d (87)   (78) 2 A4 − A3 + 4A2 b2 IS   2 2 2 4 3 2 z z z 2ג 5 6 1 2 1ג Particular solution of Equation (75) can be written as πa 1 = 2 − b2 − − b2 + b4 "    4 2 ג 2ω2a2 I I πGρ a2 3I 2I 5V 1 2 0 1 2 N 0 z 35 z 3ג 10 1 − + − − = b2 + 3 b4 − + (79) , ג 2I 3I 3V 2 + 1 + 2
− N 1 
  


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2 2 2 z z 8. POST-NEWTONIAN MASS 2ג 3 1 4 1ג πa 1 + − b2 − − b2     2 The post-Newtonian conservation laws have been discussed by a z 2 3 2 ;C (x) number of researchers the most notably in textbooks (Fock, 1964 3ג 2ג C(x) πa 3ג 5 1 3 + − b2 + − 2      Will, 1993; Kopeikin et al., 2011). General relativity predicts that the integrals of energy, linear momentum, angular momentum Substituting these results in Equations (82) and (83) yields and the center of mass of an isolated system are conserved 4 2 2 in the post-Newtonian approximation. In the present paper 1 2 z z 5 z we deal with a single isolated body so that the integrals of (88) 1ג 1 6 0ג I1 πGρ0a 1 = 2 − b4 − b2 − 3 b2     the center of mass and the linear momentum are trivial and z2 35 z4 we can always choose the origin of the coordinate system at 2ג 10 1 − − b2 + 3 b4 the center of mass of the body with the linear momentum    2 2 being nil. The integrals of energy and angular momentum are 2 3z z 1 2C(x) C(x) , less trivial and requires detailed calculations which are givenג 2ג 5 1 1ג πGρ0a − b2 + − b2 + 2     below. 2 2 2 4 2 z z 3 z 25 z The law of conservation of energy yields the post-Newtonian (89) 1ג 12 0ג I2 πGρ0a 1 = − b2 b2 + 2 − b2 + 2 b4 mass of a rotating fluid ball that is defined as follows (Fock, 1964;     z2 85 z4 3 z2 35 z4 Will, 1993; Kopeikin et al., 2011) 3ג 15 2ג 26 3 − − b2 + 3 b4 + 2 − b2 + 2 b4      1 2 2 M MN MpN , (94) 2 z z = + c2 2ג 5 1 4 1ג πGρa 1 6 + − b2 − − b2     where V is the volume of PN-ellipsoid, and z2 3 .(C(x) C(x 3ג 2ג 3ג 5 1 3 + − b2 + − 2 3      MN ρ(x)d x , (95) = V Z Replacing these expressions to Equation (79) results in is the rest mass of baryons comprising the body, and 2 0 1C(x) 2C (x) , (90) = + + 2 5 3 MpN ρ(x) v VN d x , (96) = V + + 2 where Z   is the post-Newtonian correction. ג ג ג ג 2ג 4 2 2 2 1 0 π G ρ0 a 7 0 3 0(8 1 5 2 2 3) (91) In order to calculate the rest mass, MN, we introduce = 2 − − + normalized spherical coordinates r,θ,φ related to the Cartesian 2גπ 2G2ρ2a4 7 ( ג9 ג20  ג15) ג2 + 1 1 − 2 + 3 − 0 0 (harmonic) coordinates x, y, z as follows, ג ג ג ג ג ג ג 0( 60 1 67 2 30 3) 2 1(51 1  88 2 + − 2 + − + − x ar sin θ cos φ , y ar sin θ sin φ , z br cos θ . z 1 2 2 2 4 2 = = = (97) 2ג445 1ג288 )0ג גπ G ρ a 21 (3ג45 + b2 + 6 0 0 + − + In these coordinates the integral Equation (95) is given by   z4 (3ג63 2ג116 1ג57)1ג10 (3ג210 − + − + b4 r(θ) π 2π 2 2 2 ג ג ג  ג ג ג ג 2 4 πGρ0a ω [ 0 3 1 5 2 3 3 2 ( 0 9 1 21 2 MN a bρ0 1 κAr r sin θdrdθdφ , (98) + 2 − + − − 4− + = + z z Z0 Z0 Z0
, (3ג7 2ג9 1ג3)5 0ג (3ג15 − b2 + − − + b4  where r(θ) describes the surface of the PN-ellipsoid defined above (92) ג6)
ג2 ( ג6 ג11 ג7 ) ג π 2G2ρ2a4 1 = 0 0 − 1 + 2 − 3 + 1 1 in Equations (22), (32) ( ג30 ג55 ג21) ג π 2G2ρ2a4 ( ג9 ג14 − 2 +  3 + 0 0 1 − 2 + 3 2 B 2 2 2 r (θ) 1 κ sin cos θ . (99)  z 4 2 = + 1ג) 2πGρ0a ω (3ג45 2ג70 1ג24)1ג2 − − + b2 − Integration in Equation (98) results in  2  z ג ג ג ג ג 3 2 3 3) 3( 1 5 2 5 3) 2 4π 2 κ − + − − + b MN ρ0a b 1 (A B) . (100)  = 3 + 5 + 1 4 h i πGρ0(3ג1ג18 3ג0ג6 2ג1ג8 2ג0ג ) πGρ0a 2 = 2 − + + − The post-Newtonian contribution, MpN, to the rest mass reads 2 (ω , (93(3ג 2ג)6 − − 1 π 2π 4 2 2 2 5 2 2 r cos θ 1 0ג MpN ρ0a b ω r sin θ πGρ0  This finalizes the calculation of the post-Newtonian potentials = + 2 − Z Z Z  inside the mass. 0 0 0 

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 11 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy r2 1 3cos2 θ r2 sin θdrdθdφ where r(θ) is defined in Equation (99). After integration we ג − 1 − obtain, 8π 4 2 (101) . 0גGρ0 
ρ0a b ω 5π = 15 + 8π 4 κ SN a bρ0ω 1 (5A 2B) . (109)
= 15 + 7 + The total mass Equation (94) becomes h i Replacing the density ρ0 by the total mass M with the help 2 4π 2 κ 2ω of Equation (102), makes the Newtonian part of the angular (102) . 0גM ρ0a b 1 A B 10 = 3 + 5 πGρ + + + momentum take on the following form,   0 

The inverse relation is used to convert the density ρ0 to the total 2 2 2 κ 14ω (110) . 0גSN Ma ω 1 2A 3B 70 mass, = 5 − 35 − + + πGρ   0  2 3M κ 2ω i It is straightforward to prove that x and y components of S (103) . 0גρ0 1 A B 10 = 4πa2b − 5 + + + πGρ pN   0  also vanish due to the axial symmetry, and only its z component, Sz S , remains. We notice that The total post-Newtonian mass, M, depends on the sum of the pN ≡ pN parameters, A B. + ρ(x) x V z d3x ρ(x)D (x v)z d3x , (111) × = 1 × 9. POST-NEWTONIAN ANGULAR Z Z
MOMENTUM where D1 is taken from Equation (73). Therefore, Vector of the post-Newtonian angular momentum, Si 2 p z 3 x y z = S ρ(x) v 6V 4D (x v) d x . (112) (S , S , S ), is defined by Fock (1964), Will (1993), and Kopeikin pN = + N + ρ − 1 × et al. (2011) Z   Making transformation to the coordinates Equation (97) and i i 1 i S SN SpN , (104) approximating ρ ρ , yields = + c2 = 0 i i 1 π 2π where SN and SpN are the Newtonian and post-Newtonian ג4 ג) ג6 גcontributions respectively, S ωa6bρ ω2r2 sin2 θ 7 pN = 0 + 0 − 1 − 0 + 1 Z0 Z0 Z0 i i 3  r2 cos2 θ ( ג10 ג15 גr2 2 (3 ( גSN ρ(x) (x v) d x , (105) 4 = × − 2 − 0 − 1 + 2 Z 4 3 i 2 p i 3 r sin θdrdθdφ (113) SpN ρ(x) v 6V (x v) d x = + + ρ × 4 2 2 . (1ג16 0גZ   Ma ω 2ω πGρ0 (19 = 35 + − 4 ρ(x) x V i d3x , (106) − ×   Z The total angular momentum becomes
and vector-potential V Vi has been given in Equations (64) 2 ≡ 2 2 κ 6ω . (1ג4 0גand (72). S Ma ω 1 2A 3B 40 (3 = 5 − 35 − − − − πGρ It can be checked up by inspection that in case of an   0  axisymmetric mass distribution with the ellipsoidal density (114) distribution Equation (38), the only non-vanishing component of the angular momentum is S3 Sz S. Indeed, v (ω x), = ≡ = × and (x v)i (x (ω x))i ωi(x2 y2 z2) xiωz. It 10. POST-NEWTONIAN EQUATION OF THE × = × × = + + − results in LEVEL SURFACE

y Sx ω ρxzd3x , S ω ρyzd3x , The figure of the rotating fluid body is defined by the boundary N = − N = − Z condition of vanishing pressure, p 0. The surface p 0 R = = Sz ω ρ(x2 y2)d3x . (107) is called the level surface. Relativistic Euler equation derived N = + for the rigidly rotating fluid body tells us that the level Z surface coincides with the equipotential surface of the post- By making use of the coordinates Equation (97) the reader can y Newtonian gravitational potential W which is expressed in terms easily confirm that Sx S 0, and Sz S reads N = N = N ≡ N of the centrifugal and gravitational potentials by the following expression (Kopejkin, 1991; Kopeikin et al., 2011, 2015) r(θ) π 2π 4 4 3 SN a bω ρ(x)r sin θdrdθdφ , (108) 1 2 2 1 1 4 4 = W ω σ VN κ VpN AI1 ω σ Z0 Z0 Z0 = 2 + + + + c2 8 

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3 1 (122) , (3ג1ג18 3ג0ג6 2 2 2 2 2 ω σ VN 4ω σ D1 VN , (115) + − + 2 − − 2 + 1 3 (123) ( ג20 ג36 ג16 ג) P ω2a4 κω2a2  1 = − 2 − 0 − 1 + 2 − 3 2 2 4c 2 where κ πGρ0a /c , and the potentials VN, VpN, I1, D1, 2 2 2ג1ג140 2ג0ג55 ג45 1ג0גκπGρ0a 20 ≡ have been explained in Sections 5–7. After substituting these + − 1 − + ג51 ג25 גκπGρ a2 [2 (3 [ ג ג90 ג ג30 potentials to equation Equation (115) it can be presented as a + 0 3 −  1 3 − 0 1 − 2 + 3 . [A ( ג5 גB (3 ( גquadratic polynomial with respect to function C(x), 30 − 4 + 1 − 2 2 W(x) W W C(x) W C (x) , (116) Coefficient W2 in Equation (116) is constant, = 0 + 1 + 2 1 4 4 1 2 2 (124) (3ג6 2ג2 1גwhere the coefficients of the expansion are the polynomials of the W2 ω a κω a (3 z coordinate only. In particular, the coefficient W is a polynomial = 8c2 − 2 − − 0 of the fourth order, 2 1 2Aג B (4ג6 3ג6 2ג) κπGρ0a − − + + 2 z2 z4 " W K K K , (117) 0 0 1 2 2 4 ג ג ג ג ג ג ג ג 2ג b + b 1 + = 1 0 2 8 1 2 6 0 3 18 1 3 . + 2 + − − + # where
On the level surface of the PN-ellipsoid2 we have all three 1 2 2 2 (118) (1ג 0ג) K0 ω a πGρ0a = 2 + − coordinates interconnected by Equation (34) of the PN-ellipsoid, C(x) κα (z), so that Equation (116) becomes 1 4 4 1 2 2 0 = (3ג6 2ג18 1ג17 0גω a κω a (5 + 8c2 + 2 − + − W W0 κW1α0(z) , (125) 1 2 2 ≡ + (3ג6 2ג15 1ג22)0ג גκπGρ0a 6 0 2 + 2 − − + and the term with W2 O(κ ), is discarded as negligibly small. 2 ∼ After reducing similar terms, the potential W on the level surface 2ג4 1ג) κπGρ0a [2 (3ג18 2ג 40 1ג29)1ג + − + + − is simplified to the polynomial of the fourth order, 1  , A (2ג 0ג) B (4ג3 3ג6 + − + 2 −  z2 z4 , 1 2 2 2 W K0 K1 2 K2 4 (126) b + b + = (119) (1ג3 0ג) K1 ω a πGρ0a = −2 − − 1 4 4 2 2 where (3ג30 2ג66 1ג40 0גω a κω a (5 − 4c2 − − + − (K0 K0 , (127 ג ג ג ג 2ג ג ג 2ג 2 κπGρ0a 6 0 56 0 1 99 1 67 0 2 176 1 2 = − − + + − 1 2 2 2 (B , (128 1ג K1 K1 κ ω a πGρ0a ג78 ג18 ג)] κπGρ a2 ( ג ג90 ג ג30 − 0 3 + 1 3 + 0 0 − 1 + 2 = + 2 −   A ג ג B ג ג 120 3 60 4) (3 1 5 2) ] , 1 (B . (129 גK K κ ω2a2 πGρ a2 − − + − 1 4 4 1 2 2 2 2 0 1 − 2 − = (120) (3ג70 2ג130 1ג63 0גK2 ω a κω a (5 = 8c2 + 2 − + −   1 2 2 Because the potential W is to be constant on the level surface 2ג89 1ג54)0ג5 גκπGρ0a 18 + 6 0 − − (Kopeikin et al., 2015), the numerical coefficients K and K must 1 2 vanish. The first condition, K 0, yields a relation between ( ג630 ג1160 ג 543) ג ( ג42 3 1 1 2 3 1 = + + − + the angular velocity of rotation, ω, and the eccentricity, e, of the B ( ג70 ג140 ג90 ג20 ג)] κπGρ a2 − 0 0 − 1 + 2 − 3+ 4 rotating fluid ellipsoid: 1 35 . A 2ג 1ג10 0ג + 2 − + 3 ω2 ω2a2 (B ג ג ג ג ) κ    1 2 2 5 0 40 1 66 2 30 3 2πGρ0 + 2c + − + − − Coefficient W1 in Equation (116) is a polynomial of the second   ג ג176 ג ג67 2ג99 ג ג56 2גκ 6 ג ג3 order, = 1 − 0 − 0 − 0 1 + 1 + 0 2 − 1 2 2 h 2ג78 1ג18 0ג) A (2ג5 1ג3) 3ג1ג90 3ג0גz 30 W1 P P1 , (121) − + + − − − + = + b2 (B . (130 (4ג60 3ג120 where − + i Equation (130) generalizes the famous result that was first 1 2 2 1 2 2 obtained by Colin Maclaurin in 1742, from the Newtonian theory (3ג12 2ג28 1ג18 0גP ω a 1 ω a κ (3 = 2 + 2c2 + − + −   of gravity to the realm of general relativity. B ( ג12 ג18 ג8 ג)] κπGρ a2 גπGρ a2 0 1 0 1 2 3 4 2We remind the reader that the coordinates on the surface of the PN-ellipsoid are − + 2 − 2 + − .denoted as x, y, z ג ג28 ג ג11 ג11 ג גA] κπGρ a 6 ג − 2 − 0 0 1 − 1 − 0 2 + 1 2

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The second condition, K 0, yields an algebraic equation 11. POST-NEWTONIAN PIZZETTI’S 2 = for the two coefficients A, B, THEOREM

The force of gravity on the equipotential surface of a massive 1 35 ω2 (rotating body is defined by equation (Kopeikin et al., 2011 2ג90 1ג21 0ג A 2ג 1ג10 0ג 2 − + 3 + 2πGρ + − +    0 ω4 ω2 γ j ∂ W , (134) i = i j x x ג63 גB (5 ( ג70 ג140 − 3 + 4 = π 2 2ρ2 + π ρ 0 − 1 = 8 G 4 G   where ∂ ∂/∂xi, the post-Newtonian gravity potential W is 1 2 i ≡ (3ג42 2ג89 1ג54)0ג5 ג18 (3ג70 2ג130 + − + 6 0 − − + given in Equation (116), and (131) . (3ג630  2ג1160 1ג543)1ג + − + j ij 1 1 i j i δ 1 VN v v , (135)  = − c2 − 2c2   Equation (131) imposes one constraint on two coefficients A, B defining the shape of the PN-ellipsoid and the law of distribution is the matrix of transformation from the global coordinates to of mass density. One more constraint is required to fix these the local inertial frame of observer, vi (ω x)i is velocity = × coefficients. We can use, for example, the Maclaurin relation of the observer with respect to the global coordinates, and VN Equation (130) to set an additional (geodetic) constraint on is the Newtonian potential Equation (56). We emphasize that the parameters by demanding that the post-Newtonian part of we, first, take a derivative in Equation (134), and then, take the Equation (130) vanishes. spatial coordinates, x, on the equipotential surface described by We explored another possibility to impose self-consistent the equation of the post-Newtonian ellipsoid Equation (22). Velocity vi is orthogonal to the gradient ∂ W everywhere, constraints on the shape parameters of the PN-ellpsoid directly i related to the gravimetric measurements in geodesy. Newtonian that is theory of rotating reference-ellipsoid connects the shape vi∂ W 0. (136) parameters of the Maclaurin ellipsoid with the measured values i = of its gravity force at the pole and equator of the ellipsoid through Indeed, it is easy to prove that the Pizzetti and Clairaut theorems (Pizzetti, 1913; Torge and Müller, 2012). Let us denote the force of gravity measured at the vi∂ W ω x∂ W y∂ W . (137) i y x pole of the ellipsoid by γp γi(x 0, y 0, z b), and the = − ≡ = = =
force of gravity measured on equator by γe γi(x 0, y Partial derivatives of W are calculated from Equation (116), ≡ = = a, z 0). Due to the rotational symmetry of the ellipsoid the = equatorial point can be, in fact, chosen arbitrary. The classic form dW dW 2x dW dW 2y ∂xW ∂xC(x) , ∂yW ∂yC(x) . of the Pizetti theorem is = dC = dC a2 = dC = dC a2 (138) Replacing the partial derivative from Equations (138) to (137) γe γp 3GM 2 2ω2 , (132) yields Equation (136). After accounting for Equations (136), a + b = a2b − (134) is simplified to while the Clairaut theorem states 1 γ (x) 1 V ∂ W . (139) i 2 N i = − c x x    = 3 √ 2 γe γp 3GM 3e e 3 1 e arcsin e 2 − − − ω . (133) We shall denote the force of gravity on the pole by γp γi(x 2 3 ≡ = a − b = 2a b e + 0, y 0, z r ), and the force of gravity on the equator by = = p γ γ (x 0, y r , z 0) where the equatorial r and polar r e ≡ i = = e = e p These two theorems were crucial in geodesy of XIX-th century radii are defined in Equations (24) and (25) respectively. Taking the partial derivative from W in Equation (139), yields for deeper understanding that the geometric shape of Earth’s figure can be determined not only from the measurements of 2 2 2 the geodetic arcs but, independently, from rendering the intrinsic 2πGρ0a ω a (140) (3ג2 2ג3 1ג) κ 16 (1ג2 0ג) γp measurements of the force of gravity on its surface. The gravity- = b − + b − + 2 geometry correspondence led to the pioneering idea that the 2πGρ0a B (4ג32 3ג76 2ג60 1ג17 0ג)] κ gravity force and geometry of space must be always interrelated— + b − + − + the idea which paved a way in XX-th century to the development 2 8 4πGρ0a 2ג56 1ג27) 0ג] A κ 2ג 1ג4 0ג of the general theory of relativity by A. Einstein. We derive the + − + 3 + 3b − post-Newtonian analogs of the Pizzetti and Clairaut theorems    , ג ג ג ג ג in the next two sections and explore what kind of natural 24 3) 2 1 (33 1 74 2 36 3)] + − −4 3 + limitations (if any) they can set on the shape parameters of the 2 ω a (141) 2ג28 1ג18 0ג1Gπρ0 ω κ [ 3גγe a 2 post-Newtonian ellipsoid. = − − 2c2 + − + −

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GMω2 1 ג ג ג 2 ג ג ג 12 3 ( 0 1)] ω a 2κπGρ0a [( 1 8 2 18 3 e( 525 1045e2 730e4 234e6) + + − − − + + 20e7b c2 − + − + (3ג6 2ג11 1ג5) 0ג] 2A] 2κπGρ0aג B (4ג12 − + + − + 35 93e2  92e4 42e6 8e8 arcsin e + − + − 15 . [( ג9 ג14 ג5) ג2 1 1 2 3 + √1 e2 − − + −  9 G2M2 105 347e2 420e4 218e6 40e8 We form a linear combination − + − + arcsin2 e − 16 c2 e10a2b2 2 2 2 3 G M 2 4 γe γp a 2 1 4 2 e 3675 7525e 4850e + − 2ω ω a − 320e8a2b2 c2 (1ג2 0ג) 1ג2πGρ0 2 2 a + b = + b2 − − − c2    √1 e2 2 1000e6 48e8 15 − (7 4e2) 35 40e2 a − − − e − − (142) 1ג) 8 3ג12 2ג28 1ג17 0ג2κ 2 − − + − − b2 4 B  8e arcsin e 2 + G2M2 19 9 3ג36 2ג16 1גω 2κπGρ0 [2 [(3ג2 2ג3 − + − − +
 e 7 e2 2e4 a2 + 16 a2b2e5c2 − 3 −    B (4ג32 3ג76 2ג60 1ג17 0ג) 4ג24 − − b2 − + − + 2 2 A  1 e 7 4e arcsin e . 2 − − − a 8 p
i A 2ג 1ג4 0ג 2ג2κπGρ0 2 − + b2 − + 3 This equation represents the post-Newtonian extension of the    .(classical Pizzetti theorem (132 ג5) ג2 ( ג6 ג11 ג5) ג] 4κπGρ + 0 0 1 − 2 + 3 − 1 1 4a2 2ג56 1ג27)0ג κπGρ0 [(3ג9 2ג14 − + + 3b2 − 12. POST-NEWTONIAN CLAIRAUT’S THEOREM . (3ג 36 2ג74 1ג33)1ג2 (3ג24 + − − +  Making use of integrals given in Appendix A in Supplementary In order to derive the post-Newtonian analog of the Clairaut Material, we can check that Equation (142) is simplified to theorem Equation (133) we follow its classic derivation given, for example, in Pizzetti (1913). To this end we form a linear γe γp 2 1 4 2 difference between the forces of gravity measured at the pole and 2 4πGρ0 2ω ω a (143) a + b = − − c2 on the equator of the PN-ellipsoid by making use of EquationS κ (139) and (140). We get e(1 e2)(105 104e2 42e4) − 3e7 − − + 2 3 1 e2(5 4e2)(7 6e2 2e4) arcsin e ω2 γe γp a (145) (1ג2 0ג) 1ג 2πGρ0 − − − − + a − b = − + b2 − κπpGρ0 i   7 4e2 5e(21 31e2 10e4) 1 − 12e9 − − + ω2 ω4a2 + + 2c2
 2 4 ג12 ג28 ג17 גe2(35 40e 8e ) arcsin e B κ [2 1 3 − − − + + 0 − 1 + 2 − 3 i 2 κπpGρ0 19 2 4 a 2 ω [(3ג 2 2ג3 1ג) e 7 e 2e 16 + e5 − 3 − + b2 − + ג24 ג36 ג16 גκπGρ [ 2    2 2 A 0 1 2 3 4 1 e 7 4e arcsin e − 2 − + − + ג ג ג ג ג 2a − − − i ( 0 17 1 60 2 76 3 32 4) B pκπGρ0 2
2 4 6 − b2 − + − + 10 1 e 315e 621e 250e  + 3e − − + − a2 8 A 2ג 1ג4 0ג 2ג2κπGρ0 2 6 4 2 2
8 24e 2e 1 e (315 516e 169e 18e ) − − b2 − + 3 + + − −2 +4 6− 2    ( ג6 ג11 ג5 ) ג arcsin e p3(105 242e 178e 40e ) arcsin e . 2κπGρ − − + − + 0 0 − 1 + 2 − 3 ( ג9 ג14 ג5) ג2  In order to compare Equation (143) with its classic + 1 1 − 2 + 3 2 counterpart Equation (132), we convert, then, the density 4a  (3ג24 2ג56 1ג27)0ג κπGρ0 ρ0, to the total relativistic mass of the PN-ellipsoid by + 3b2 − + . ( ג36 ג74 ג33) גmaking use of Equation (103). It allows us to recast Equation 2 (143) to − 1 1 − 2 + 3  2 4 We use the results of Appendix A in Supplementary Material to γ γp 3GM a ω 2 e 2ω2 (144) a + b = a2b − − c2 replace the integrals entering the right side of Equation (145), with their explicit expressions given in terms of the eccentricity e 3√1 e2 G2M2 − e 1 e2 315 621e2 of the Maclaurin ellipsoid Equation (20). It yields + 16e9a2b2 c2 − − + 250e4 24e6 h 2p315 831 e2 685e4 − + + − + γe γp 6πGρ0 6 8 e 1 e2 arcsin e (146) 187e 42e
arcsin e 3 − + a − b = e − −
  p 

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1 2πGρ ω2 ω4a2 13. REFERENCE ELLIPSOID IN − + + 2c2 SMALL-ECCENTRICITY APPROXIMATION κ 3 5 7 7 75e 5e 122e 42e + 6e − − + − Let us apply the formalism of previous sections to derive 3 1 e2(25 10e2 34e4 8e6) arcsin e ω2 practically useful post-Newtonian relationships used for + − + − + processing geodetic and gravimetric measurements on the pκ 2 4 6 πGρ0 e 525 1075e 662e 112e − 24e9 − + − surface of Earth and in space. To this end we notice that the flattening, f of the terrestrial reference ellipsoid is about 2  2 4 6
B 3 1 e (175 300e 144e 16e ) arcsin e f e2/2 1/298 0.0034 (Petit and Luzum, 2010, Section − − − + − ≃ ≃ = κp 2 4 i 1) that can be used as a small parameter for expanding all πGρ0 e 15 29e 6e − 6e5 − + post-Newtonian terms in the convergent Taylor series with respect to f . We shall keep the Newtonian expressions as they are 3 1 e2 5 8e2 arcsin e A
− − − without expansion, and take into account only terms of the order κp 2
i 2 4 6 of e2 in the post-Newtonian parts of equations by systematically πGρ0(1 e ) e(225 351e 58e 24e ) − 6e9 − − − + discarding terms of the order of e4, and higher. Because, ∼ 2 1 e2(225 276 e2 85e4 18e6) arcsin e according to Maclaurin’s relation Equation (130), the square of − − − − + the angular velocity, ω2 e2, (see Pizzetti, 1913; Chandrasekhar, pκ 2 2 4 6 i ≃ πGρ0(1 e ) 75 142e 38e 40e 1969; Torge and Müller, 2012 for more detail), we shall also − 2e10 − − + + discard terms of the order of ω4 and ω2e2. This procedure shall 2
arcsin e. dubbed as a small-eccentricity approximation. This form of the Clairaut theorem apparently depends on the Now, we have to make a decision about what kind of A B shape parameters A, B in different combinations and can be constraints on the model parameters of the PN ellipsoid , used to impose a constraints on one of them in addition to the would be more preferable for practical applications. We have considered the case of A 0, B 0 in our previous work constraint given by the level surface Equation (131). For example, = = (Kopeikin et al., 2016). We have proved that B O e4 , and we could demand that all post-Newtonian terms in Equation ≃ can be ignored in all geodetic equations of the small-eccentricity (146) vanish.
approximation under assumption of the homogeneous density The last step is to replace the density ρ0 in Equation (146) with the total mass of the PN-ellipsoid by making use of the distribution. In the present paper we explore the complementary case A 0, B 0. This constraint makes the level surface gauge-invariant expression Equation (103). We get = = of the post-Newtonian ellipsoid coinciding exactly with the 3e e3 3√1 e2 classic ellipsoid Equation (20) which means that the ellipsoidal − − − γ γ 3GM arcsin e a2ω4 coordinate system, having been ubiquitously used in geodesy, is e p ω2 a − b = 2a2b e3 + + 2c2 not deformed by the relativistic corrections to the gravity field at all. This requires to adopt that the mass density inside the e(375 25e2 682e4 234e6) ellipsoid has an inhomogeneous ellipsoidal distribution given + − + 3√1 e2(125 50e2 194e4 40e6) by Equation (38) depending on the parameter A which is − − + − + arcsin e GM determined from the equation of the level surface Equation (131). ω2 − 40be7 c2 We solve Equation (131) under this constraint with respect to A, and find out that it can be approximated with e(2625 5375e2 3310e4 − + − 704e6 48e8) √1 e2(875 + − − 8 4A 472 4 11152 6 O 8 3 1500e2 720e4 128e6) arcsin e G2M2 e e e e , (148) − + − B 315 = 4725 + 363825 + − 640 a3be9 c2
that yields e 15 29e2 6e4 3√1 e2 − +2 − − 2 2 3 5 8e arcsin e G M 2 −
A 59 1394e 2 2 5 2 A , (149) − 32 a b
e c = 15 + 1155 e√1 e2(225 351e2 58e4 − − − Post-Newtonian correction to the mass of the Earth, M , 24e6) 2(225 501e2 191e4 ⊕ + − − + + can be evaluated from Equation (102). Because the mass couples 3 175e6 42e8) arcsin e G2M2 − with the universal gravitational constant G, it contributes to the − 160 a4e9 c2 numerical value of the geocentric gravitational constant GM ⊕ = 3.986004418 1014 m3s 2. Numerical values for the geopotential 9 G2M2 75 142e2 38e4 88e6 × − 2 and the semi-major axis, a , of reference ellipsoid are given in 2 − +4 10 + arcsin e (147) − 32 c a e Petit and Luzum (2010, Table⊕ 1.1.) They yield, GM /c2a ⊕ ⊕ = This is the post-Newtonian extension of the classical Clairaut 6.7 10 10. After expansion of the right side of Equation (102) × − theorem Equation (133). with respect to the eccentricity e, the relativistic variation in the

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GM 59 GM 11 value of GM is e2 ω2 . (157) ⊕ + ac2 25 a3 + 14   δ(GM )pN GM 9 (150) , 10− 2.08 ⊕ 3 0ג2κ ⊕ GM ≃ ≃ c2a ≃ × The post-Newtonian corrections to the gravitational field ⊕ ⊕ entering the the Pizzetti and Clairaut theorems Equations (156), where the numerical value of κ was calculated on the basis of (157) are not so negligibly small, amount to the magnitude of 2 2 2 relationship κ πGρ0a (3/4)GM /c a . The current approximately 3 Gal (1 Gal = 1 cm/s ), and are to be taken into = ≃ ⊕ ⊕ uncertainty in the numerical⊕ value of GM is δGM 8 account in calculation of the parameters of the reference-ellipsoid 5 3 2 ⊕ ⊕ = × 10 m s− (Petit and Luzum, 2010, Table 1.1) which gives the from astronomical and gravimetric data in a foreseeable future. fractional uncertainty Equations (148)–(151) extrapolates the Newtonian relations adopted by IERS and IUGG for definition of the Earth reference δ(GM ) 9 ellipsoid and for connecting its parameters to the angular velocity ⊕ 2.0 10− . (151) GM ≃ × ω and gravity field, to the realm of general relativity. ⊕ This is comparable with the relativistic contribution Equation 14. POST-NEWTONIAN GEOID’S (150) that must be taken into account in the reduction of precise geodetic data processing which are currently based on the UNDULATION Newtonian theory. We define the anomalous (disturbing) gravity potential T T(x) The Maclaurin relation Equation (130) takes on the following ≡ as the difference between the physical gravity potential, W approximate form ≡ W(x), and the background gravity potential, W(x), of the PN ω2 √1 e2(3 2e2) arcsin e 3e(1 e2) 99 GM reference ellipsoid 2 , − − 3 − − 2 e 2πGρ0 = e + 175 ac T W W . (158) (152) = − which, after replacing ρ0 with the help of Equation (103) and The gravity potential W is defined by the Einstein equations expansion, can be recast to with the sources that take into account the real distribution of density, pressure, etc. of the real Earth. We don’t need to solve 3GM (3 2e2) arcsin e 3e√1 e2 99 GM the full Einstein equations again in order to find out T. Because ω2 − − − e2 . 3 3 2 the difference between W and W is very small, we can consider = 2a " e + 175 ac # (153) merely a linear perturbation of the background metric tensor [see This can be further decomposed and presented as follows Equation (13)]

̹αβ gαβ (x) gαβ (x). (159) 297 GM = − ω ω 1 , (154) = ⊕ + 280 ac2 The gravity potential W is constant on the perturbed   equipotential surface, and is defined by where dτ 1/2 2 , 3GM (3 2e2) arcsin e 3e√1 e2 W c 1 (160) ω , = − dt 3 − 3 − − (155)   ⊕ = 2a e ! 1 α β where dτ c− gαβ dx dx is the proper time of observer in is the currently used value for the Earth angular velocity of = − 10 1 the real physicalq spacetime. Making use of Equations (160) and rotation, ω 729211.5 10− rad s− (Petit and Luzum, ⊕ = × (16) allows us to recast Equation (158) to 2010). Contribution from the post-Newtonian term in the right 10 side of Equation (154) amounts to 7.1 10− which makes dτ dτ × T c2 , (161) it evident that the currently adopted value for ω should be = dt − dt corrected by the IUGG. ⊕   The approximate version of the post-Newtonian Pizzetti which can be further simplified by noticing that theorem Equation (144) reads 2 2 dτ α β dτ γ γp 3GM 4GM 47 GM 1 1 u u ̹αβ , (162) 2 e 2ω2 3 e2 ω2 . dt = − dt a + b = − + a2b + ac2 + 25 a3 + 4       
(156) where uα is the unperturbed four-velocity of observer which The post-Newtonian corrections in the Clairaut theorem coincides with the four-velocity of matter of the rotating Equation (147), after they are expanded with respect to the reference ellipsoid. Accounting for definition Equation (161), we eccentricity, yield get the anomalous gravity potential in the form,

2 √ 2 2 γe γp 2 3GM e(3 e ) 3 1 e arcsin e c α β ω − − − T u u ̹αβ , (163) a − b = + 2a2b e3 = 2

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 17 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy which is fully sufficient for practical applications. We emphasize In terms of the scalar q the anomalous gravity potential Equation that ̹αβ is the difference between the actual physical metric, gαβ , (163) reads and the metric gαβ of the background manifold, and as such, c2 should not be confused with the post-Newtonian expansion of T q , (171) the metric gαβ around a flat spacetime with the Minkowski metric = − 2 ηαβ diag( 1, 1, 1, 1) like it is shown in Equations (2)–(4). Our where we have used the property ̹ l. Taking from both sides = − = next task is to derive the differential equation for the anomalous of Equation (171) the covariant Laplace operator yields gravity potential T. Let us assume that inside the Earth the deviation of the real c2 2T 2q , (172) matter distribution from its unperturbed value is described by the ≡ − 2 symmetric energy-momentum tensor where 2q q| is to be calculated from Equation (167). αβ α β αβ ≡ | c2T e u u s , (164) We notice that according to Kopeikin and Petrov (2013, = + m Equations 148–150) Fαβ is directly proportional to the where uα is four-velocity, e is the energy density, and sαβ is thermodynamic quantities of the background matter and, the symmetric stress tensor of the perturbing matter. The stress thus, vanishes in the exterior (with respect to the background m tensor includes the isotropic pressure (diagonal components) and matter) space. Hence, we can drop off Fαβ in (167) in the shear (off-diagonal components), and is orthogonal to uα, that is exterior-to-matter domain. After contracting Equation (167) α αβ sαβ u 0. The energy density of the matter perturbation with g , and accounting for Equation (170) we obtain = 2 e ρ c P , (165) α α 8πG = + 2q p| α A α T , (173) − | − | = − c2 where ρ is the mass density of the perturbation,
and P is the internal (compression) energy of the perturbation. For further where all terms depending on the Ricci tensor Raβ cancel out, αβ α calculations, a more convenient metric variable is T g Tαβ , and we still have terms with the gauge field A . ≡ Now, we use the gauge freedom of general relativity to simplify 1 (173). More specifically, we impose the gauge condition lαβ ̹αβ gαβ ̹ , (166) ≡ − + 2 Aα pα , (174) αβ where ̹ g ̹αβ . The dynamic field theory of manifold = − ≡ α αβ perturbations leads to the following equation for lαβ (Kopeikin where p g ∂β p. This gauge allows us to eliminate function ≡ and Petrov, 2013, 2014), p from Equation (173) and, after making use of Equation (172), reduce it to to a simple form of a covariant Poisson equation lαβ | gαβ A 2Aα β R αlβ R β lα | + | − | − − 2T ν m 16πG 4πGT . (175) 2Rανβ l 2F Tαβ , (167) = − + αβ = c4 In the Newtonian approximation the trace of the energy- α αβ where A l β is the gauge vector function, depending on momentum tensor is reduced to the negative value of the matter ≡ | the choice of the coordinates, Rανβ is the Riemann (curvature) density of the perturbation, T ρ, while 2T T. Hence, ≃ − ≃ N tensor of the background manifold depending on the metric Equation (175) matches its Newtonian counterpart. Outside the ν tensor gαβ , its first and second derivatives, Rαβ g Rανβ — mass distribution the master equation for the anomalous gravity m = the Ricci tensor, and Fαβ is the tensorial perturbation of the potential is reduced to the covariant Laplace equation background matter induced by the presence of the perturbation Tαβ (see Kopeikin and Petrov, 2013, Equations 148–150 for 2T 0. (176) = particular details). In what follows, we focus on derivation of the master equation Equations (175), (176) extend equations of classic geodesy to the for the anomalous gravity potential T in the exterior space that realm of the post-Newtonian chronometric geodesy. The main is outside of the background matter of the reference ellipsoid. difference is that the covariant Laplace operator in Equations This assumption is normally used in geodesy (Torge and Müller, (175), (176) is taken in curved space with the metric gαβ . The 2012). To achieve our goal, we introduce two auxiliary scalars, explicit form of the covariant Laplace operator applied to a scalar T in coordinates xi x1, x2, x3 , reads (Lightman et al., 1975, = { } α β l Problem 7.7.) q u u lαβ , (168) ≡ + 2 αβ α β 1 ij p g u u lαβ , (169) 2T ∂i gg ∂jT , (177) ≡ + ≡ g − −   where
p where the repeated indicesp mean the Einstein summation, aβ i l g lαβ 2(p q). (170) ∂ ∂/∂x is the partial derivative with respect to the spatial ≡ = − i ≡

Frontiers in Astronomy and Space Sciences | www.frontiersin.org 18 February 2016 | Volume 3 | Article 5 Kopeikin Reference Ellipsoid and Geoid in Chronometric Geodesy coordinates, and we omitted all time derivatives because the We introduce the relativistic geoid height, N, by making background spacetime is stationary. For the geodetic purposes use of relativistic generalization of Bruns’ formula (Heiskanen it is convenient to choose a spherical coordinate system, and Moritz, 1967, Equations 2–144). We consider a set of x1, x2, x3 r,θ,λ co-rotating rigidly along with the equipotential surfaces defined by constant values of the gravity { } = { } reference ellipsoid and with z-axis directed along the axis of the potential W. Let a point Q lie on an equipotential reference S α P rotation. With a sufficient post-Newtonian accuracy the metric surface 1 and has coordinates xQ, and a point lie on another tensor of the background spacetime in this coordinate system S α equipotential surface 2, and has coordinates xP. The height reads difference, N, between the two surfaces is defined as the absolute value of the integral taken along the direction of the plumb line 1 2 2 2 3 g00 1 ω r sin θ 2VN O c− , (178) passing through the points Q and P, = − + c2 + + 1 2 2 3

P g0λ ωr sin θ O c− , (179) dxα = c + N nα dℓ , (188) = Q dℓ 2VN 3
Z grr 1 O c− , (180) = + c2 + where nα gα/ g is the unit (co)vector along the plumb 2VN 2
3 ≡ | | gθθ 1 r O c− , (181) line, gα is the relativistic acceleration of gravity Equation (18), = + c2 + aβ 1/2   g (h gαgβ ) , and ℓ is the proper length defined in space by
| | ≡ 2VN 2 2 3 Zelmanov et al. (2006) and Landau and Lifshitz (1975) gλλ 1 r sin θ O c− , (182) = + c2 + 2 α β   dℓ hαβ dx dx , (189)
= where V is the Newtonian gravitational potential defined in where hαβ gαβ uαuβ is the metric tensor in three- Equation (56). Determinant of the metric ≡ + dimensional space. In case, when the height difference is 2 small enough, we can use the second mean value theorem for 4 2 2V g det[gαβ ] r sin θ 1 , (183) integration (Hobson, 1909) and approximate the integral in ≡ = − + c2   Equation (188) as follows αβ After calculating the inverse metric g , substituting it to P α 2 P 3 gα(x)dx c W α Equation (177), and discarding all terms od the order of O(c− ), N ∂α ln 1 dx = Q g(x) = −gQ Q − c2 the equation for the anomalous gravity potential takes on the Z | | Z   following form W(Q) 1 c2 2 2 2T − c ω ∂ ln P , (190) T , (184) = gQ W( ) = c2 ∂λ2 1 − c2 where where gQ g(Q) denotes the magnitude of the relativistic = | | 2 1 1 1 acceleration of gravity taken on the equipotential surface S1. ∂rr ∂r ∂θθ ∂θ ∂λλ , (185) ≡ + r + r2 + r2 tan θ + r2 sin2 θ Equation (190) is exact. Separation of the height N in the Newtonian part and the post-Newtonian corrections depends on is the Laplace operator in the spherical coordinates. The post- how we define the reference equipotential surface S1. Newtonian equation Equation (184) can be solved by iterations Let us choose the reference surface by equation W(Q) W = by expanding the distrubing potential in the post-Newtonian where W is the normal (relativistic) gravity field of the reference series ellipsoid. Then, expanding the logarithm in Equation (190) with respect to the ratio W/c2 and making use of definition Equation 1 4 T TN TpN O(c− ) , (186) T = + c2 + (158) of the anomalous gravity potential , we obtain from (190) where TN is the Newtonian disturbing potential expressed, T(P) N | | , (191) for example, by the Stokes integral formula (Heiskanen and = γQ Moritz, 1967, Equations 2–163a), and TpN is the post-Newtonian correction which we are searching for. Substituting Equation where the disturbing potential, T, is measured at the point P on (186) in Equation (184) yields the inhomogeneous differential the geoid surface W, and the acceleration of gravity γQ gQ is ≡ equation for TpN, measured at point Q on the reference surface W. Relativistic Bruns’ formula Equation (191) yields geoid’s ∂2T T ω2 N , (187) undulation with respect to the unperturbed reference level pN = ∂λ2 surface, W const., in general relativity. Because we have = which can be solved by means of standard mathematical defined this surface as an equipotential surface W which is the techniques for known TN. post-Newtonian solution of the Einstein equations, the height

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2 N does not represent the undulation of the relativistic geoid W (VN/c ) R 1 cm which looks pretty small but crucially × ⊕ ≃ with respect to the Newtonian equipotential surface defined by important in the current study of the changes in the global the surface of the constant Newtonian gravity potential, V average sea level which is now rising twice as fast as it did over N = const. Expansion of the height N in Equation (191) in the the past century, providing clear evidence of global warming on post-Newtonian series around the value of the surface W, yields a short time scale. The measurements are done with the help of the the satellite laser altimetery which is so precise that allows 1 4 us to measure the change in the global average sea level with the N N NpN O c− , (192) = + c2 + uncertainty 1 mm/year (Fu and Haines, 2013; Ablain et al., ≤
2015) which is a factor of 10 larger than the post-Newtonian where N is the classic definition of the geoid height given in terms effects in the determination of the equipotential level surface! of the Newtonian disturbing potential, V V as explained, for N − N example, in textbook (Vanícekˇ and Krakiwsky, 1986, chapter 21). The post-Newtonian correction NpN to the height N is caused by ACKNOWLEDGMENTS the difference between the post-Newtonian terms in W and W, and has a magnitude of the order N (V /c2) N, where We thank the referees for valuable comments and suggestions for pN ≃ N × VN is the Newtonian gravitational potential of the Earth. Because improving the presentation of the manuscript. The work of SK the largest undulation of the Newtonian geoid of the Earth has been supported by the grant No 14-27-00068 of the Russian does not exceed 100 m (Torge and Müller, 2012), the explicit Science Foundation (RSF). post-Newtonian correction NpN to the undulation is exceedingly 6 small, NpN 7 10− cm. SUPPLEMENTARY MATERIAL ≃ × Thus, the main relativistic effects in terrestrial geodesy enter through the difference between the equipotential surface of the The Supplementary Material for this article can be found relativistic potential of the gravity force, W, and one of the online at: http://journal.frontiersin.org/article/10.3389/fspas. Newtonian gravity potential VN. This difference amounts to 2016.00005

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