Covariant Equations of Motion of Extended Bodies with Arbitrary Mass and Spin Multipoles Sergei M. Kopeikin Department of Physics & Astronomy, University of Missouri, 322 Physics Bldg., Columbia, Missouri 65211, USA∗ Gravitational wave detectors allow to test general relativity and to study the internal structure and orbital dynamics of neutron stars and black holes in inspiraling binary systems with a potentially unlimited rigor. Currently, analytic calculations of gravitational wave signal emitted by inspiralling compact binaries are based on the numerical integration of the asymptotic post-Newtonian expan- sions of the equations of motion in a pole-dipole approximation that includes masses and spins of the bodies composing the binary. Further progress in the accurate construction of gravitational-wave templates of the compact binaries strictly depends on our ability to significantly improve theoretical description of gravitational dynamics of extended bodies by taking into account the higher-order (quadrupole, octupole, etc.) multipoles in equations of motion of the bodies both in the radiative and conservative approximations of general relativity and other viable alternative theories of gravity. This paper employs the post-Newtonian approximations of a scalar-tensor theory of gravity along with the mathematical apparatus of the Cartesian symmetric trace-free tensors and the Blanchet- Damour multipole formalism to derive translational and rotational equations of motion of N extended bodies having arbitrary distribution of mass and velocity of matter. We assume that spacetime manifold can be covered globally by a single coordinate chart which asymptotically goes over to the Minkowskian coordinate chart at spatial infinity. We also introduce N local coordinate charts adapted to each body and covering a finite domain of space around the body. Gravitational field in the neighborhood of each body is parametrized by an infinite set of mass and spin multipoles of the body as well as by the set of tidal gravitoelectric and gravitomagnetic multipoles of external N − 1 bodies. The origin of the local coordinates is set moving along the accelerated worldline of the center of mass of the corresponding body by an appropriate choice of the internal and external dipole moments of the gravitational field. Translational equations of motion of the body’s center of mass and rotational equations of motion for its spin are derived by integrating microscopic equations of motion of body’s matter and applying the method of asymptotic matching technique to splice together the post-Newtonian solutions of the field equations of the scalar-tensor theory of gravity for the metric tensor and scalar field obtained in the global and local coordinate charts. The asymptotic matching is also used for separating the post-Newtonian self-field effects from the external gravitational environment and constructing the effective background spacetime man- ifold. It allows us to present the equations of translational and rotational motion of each body in covariant form by making use of the Einstein principle of equivalence. This relaxes the slow- motion approximation and makes the covariant post-Newtonian equations of motion of extended bodies with weak self-gravity applicable for the case of relativistic speeds. Though the covariant equations of the first post-Newtonian order are still missing terms from the second post-Newtonian approximation they may be instrumental to get a glimpse of the last several orbital revolutions of stars in ultra-compact binary system just before merging. Our approach significantly general- izes the Mathisson-Papapetrou-Dixon covariant equations of motion with regard to the number of body’s multipoles and the post-Newtonian terms having been taken into account. The equations of translational and rotational motion derived in the present paper include the entire infinite set of covariantly-defined mass and spin multipoles of the bodies. Thus, they can be used for much more accurate prediction of orbital dynamics of tidally deformed stars in inspiraling binary systems and arXiv:1810.11713v3 [gr-qc] 6 Apr 2019 construction of templates of gravitational waves at the merger stage of coalescing binary when the strong tidal distortions and gravitational coupling of higher-order mass and spin multipoles of the stars play a dominant role in the last few seconds of the binary life. PACS numbers: 04.20.Cv,04.25.-g,04.25.Nx,95.10.Ce ∗ E-mail: [email protected] 2 CONTENTS I. Introduction 4 II. Primary Concepts and Mathematical Notations 8 III. Scalar-Tensor Theory and Post-Newtonian Approximations 12 A. Lagrangian and Field Equations 12 B. Post-Newtonian Approximations 14 C. Post-Newtonian Expansions 15 D. Conformal Harmonic Gauge 17 E. Post-Newtonian Field Equations 18 IV. Parametrized Post-Newtonian Coordinates 19 A. Global Coordinate Chart 21 1. Boundary Conditions 21 2. Scalar Field 22 3. Metric Tensor 22 B. Local Coordinate Chart 24 1. Boundary Conditions 24 2. Scalar Field: internal and external solutions 26 3. Metric Tensor: internal solution 26 4. Metric Tensor: external solution 27 5. Metric Tensor: the coupling component 28 6. Body-Frame Internal Multipoles 29 V. Matched Asymptotic Expansions and Coordinate Transformations 30 A. Basic Principles 30 B. Transition Functions 32 C. Matching Solution 33 1. Post-Newtonian Coordinate Transformation 33 2. Body’s Self-Action Force and Bootstrap Effect 34 3. World Line of the Origin of the Local Coordinates 34 4. Body-Frame External Multipoles 35 VI. Post-Newtonian Equations of Motion of an Extended Body in the Local Coordinates 36 A. Microscopic Equations of Motion of Matter 37 B. Post-Newtonian Mass of a Single Body 38 C. Post-Newtonian Center of Mass and Linear Momentum of a Single Body 39 D. Post-Newtonian Spin of a Single Body 41 E. Translational Equation of Motion of the Center of Mass of a Single Body 42 F. Rotational Equations of Motion of Spin of a Single Body 44 VII. Multipolar Expansion of External Potentials in the Global Coordinates 45 A. Multipolar Expansion of Potential U¯ 45 B. Multipolar Expansion of Potential U¯ i 51 C. Multipolar Expansion of Potential Ψ¯ 51 D. Multipolar Expansion of Potential χ¯ 52 VIII. Multipolar Expansion of External Multipoles in the Global Coordinates 53 A. Scalar-Field Multipoles PL 53 B. Gravitoelectric Multipoles QL 54 C. Gravitomagnetic Multipoles CL 55 IX. Translational Equations of Motion of Bodies in the Global Coordinates 56 A. Computation of Gravitational Force 56 1. Reduction of Similar Terms 56 2. STF Derivatives from the Scalar Potentials W¯ and U¯ 59 3. STF Derivatives from the Scalar Potential Ω¯ 59 3 4. STF Derivatives from the Vector Potential Ω¯ i 61 pN 5. Post-Newtonian Local Acceleration Qi 63 i 6. Complementary Vector Function Ic. Adjustment of the Center of Mass. 66 B. Explicit Formula for Gravitational Force 66 1. Newtonian Force 67 2. Post-Newtonian Force 68 C. Reduced Post-Newtonian Force 69 1. Mass Multipole Coupling Force 70 2. Spin Multipole Coupling Force 72 3. Precession Multipole Coupling Force 73 D. Comments 73 X. Rotational Equations of Motion of Spin in the Global Coordinates 74 A. Computation of Torque 75 B. Explicit Formula for Torque 79 1. Newtonian Torque 79 2. Post-Newtonian Torque 80 3. Fermi-Walker Torque 81 C. Reduced post-Newtonian Torque 81 1. Mass Multipole Coupling Torque 82 2. Spin Multipole Coupling Torque 84 3. Precession-Multipole Coupling Torque 85 XI. Covariant Equations of Motion of Extended Bodies with All Multipoles 85 A. The Mathisson Variational Dynamics 85 B. The Effective Background Manifold 89 C. Geodesic Worldline and 4-Force on the Background Manifold 92 D. Four-dimensional Form of Multipoles 94 1. Internal Multipoles 94 2. External Multipoles 95 E. Covariant Translational Equations of Motion 98 F. Covariant Rotational Equations of Motion 100 Acknowledgments 101 A. Auxiliary Mathematical Properties of STF Tensors 102 B. Comparison with the Racine-Vines-Flanagan Equations of Motion 102 C. The Dixon multipole moments 105 D. Comparison with Mathisson-Papapetrou-Dixon Equations of Motion 107 1. Comparison of Dixon’s and Blanchet-Damour multipole moments 107 2. Comparison of translational equations of motion 110 3. Comparison of rotational equations of motion 111 References 111 4 I. INTRODUCTION Mathematical problem of derivation of relativistic equations of motion of extended bodies has been attracting theorists all the time since the discovery of general relativity. An enormous progress in solving this problem has been reached for the case of an isolated gravitating system consisting of spinning massive bodies in the, so-called, pole-dipole particle approximation [1–3] that was originally discussed by Mathisson [4, 5], Papapetrou [6] and Dixon [7–11] (see also papers of the other researchers [12–15] and references therein). These type of equations of motion is used for a comprehensive study of the nature of gravity through the monitoring orbital and rotational motion of bodies in the solar system [16, 17], binary pulsars [18–21], and inspiralling compact binary systems made of neutron stars and/or black holes [22]. A new branch of relativistic astrophysics – gravitational wave astronomy can test general relativity in a strong field, fast-motion regime of coalescing binaries to unprecedented accuracy and probe the internal structure of neutron stars by measuring their Love numbers [23–27] through the gravitational response of their internal multipoles subject to the immense strength of the tidal gravitational field of inspiralling binary just before the merger [28]. Therefore, more advanced study of the dynamics of relativistic N-body system is required to take into account gravitational perturbations generated by higher-order multipoles of extended bodies (quadrupole, octupole, etc.) that can significantly affect the orbital motion of the pole-dipole massive particles [29–35]. The study of these perturbations is also important for improving the solar system experimental tests of various gravity theories [36, 37] and for building more precise relativistic models of astronomical data processing [38–41].