First Meeting of IAG JWG2.1 Sergei Kopeikin University of Missouri-Columbia, USA Theoretical problems of relativistic geodesy • Reference ellipsoid • Normal gravity field • Relativistic equations of motion • Relativistic GNSS • Relativistic IERS • Relativistic TAI • Relativistic time transfer and clock synchronization • Beyond 1 Post-Newtonian approximation • Very advanced topics - gravitational wave astronomy with clocks connected by optical fiber links, local measurement of the Hubble constant with clocks, etc. First Meeting of IAG JWG2.1, 5/15/2017 2 Hannover, Germany Post-Newtonian Metric First Meeting of IAG JWG2.1, 5/15/2017 3 Hannover, Germany First Meeting of IAG JWG2.1, 5/15/2017 4 Hannover, Germany Post-Newtonian reference ellipsoid (interior solution) Level surface of uniformly rotating perfect fluid of homogeneous density � takes on the form of the Maclaurin biaxial ellipsoid "# &# + = 1 where �+ ≡ �+ + �+ $# '# PN theory tells us that the shape of such a uniformly rotating fluid body is to be a spheroid �+ �+ �+ �+ �6 �6 �+ �+ + = 1 + � � + � + � + � + � �+ �+ 4 �+ + �+ 4 �6 + �6 7 �+ �+ 89:$# where � = ≃ 5.2×10B4C . This equation can be Recast to the following foRm ;# �+ �+ �+ �+ + + + = 1 + � �4 + �+ − �7 + + �Y �Z � � 4 4 where � = � 1 + �(� + � ) - equatorial radius, and � = � 1 + �(� + � ) - Y + 4 4 Z + + + polar radius. First Meeting of IAG JWG2.1, 5/15/2017 5 Hannover, Germany The residual gauge freedom is described by a PN coordinate transfromation �^_ = �_ + ��_ � where the gauge functions �_ obey the Laplace equation ∆�_ = 0 Solution of this equation are harmonic polynomials. In the case of PN ellipsoid they are Transformation of the PN ellipsoid equation to the new coordinates is form-invariant. It transforms the coefficients in the equation of the PN ellipsoid �4 → �4 + 2ℎ �+ → �+ + 2� �4 → �4 + 2� �+ → �+ − 4ℎ �+ �+ � → � − 8� + 6� 7 7 �+ �+ First Meeting of IAG JWG2.1, 5/15/2017 6 Hannover, Germany Gauge freedom and the choice of coordinates Arbitrary �4, �+, �4, �+, �7 �4 = �+ = �4 = �+ = 0, �7 ≠ 0 ∼ � ��� �� � � � � �� ~ 1 �� B4C �� �7 ≃ 10 The problem has four degrees of freedom corresponding to the three infinitesimal coordinate transformations having four free parameters. Hence, we can arbitrary fix four out of the five constants in the mathematical equation of the PN ellipsoid. The only non-vanishing parameter can be chosen to be �7 ≡ �. First Meeting of IAG JWG2.1, 5/15/2017 7 Hannover, Germany Non-homogeneous fluid with ellipsoidal distribution of density 1 Allows to keep the level surface � = � 1 + C �+ as a biaxial ellipsoid at least, in 1 PN approximation (perhaps in all Orders of PN approximations First Meeting of IAG JWG2.1, 5/15/2017 8 Hannover, Germany The normal gravity field (exterior solution) Coordinates Ellipsoidal Spherical Coordinate transformation First Meeting of IAG JWG2.1, 5/15/2017 10 Hannover, Germany Green’s Function in Elliptical Coordinates Legendre polynomials from Legendre functions from Spherical harmonics imaginary argument imaginary argument First Meeting of IAG JWG2.1, 5/15/2017 11 Hannover, Germany Scalar gravitational potential The second eccentricity First Meeting of IAG JWG2.1, 5/15/2017 12 Hannover, Germany Scalar multipole moments � – the first eccentricity } �+ − �+ � = �+ First Meeting of IAG JWG2.1, 5/15/2017 13 Hannover, Germany Vector gravitational potential First Meeting of IAG JWG2.1, 5/15/2017 14 Hannover, Germany Vector multipole moments First Meeting of IAG JWG2.1, 5/15/2017 15 Hannover, Germany The normal gravity field (i) First Meeting of IAG JWG2.1, 5/15/2017 16 Hannover, Germany The normal gravity field (ii) First Meeting of IAG JWG2.1, 5/15/2017 17 Hannover, Germany The Post-Newtonian Somigliana Formula First Meeting of IAG JWG2.1, 5/15/2017 18 Hannover, Germany The Kerr Metric and Multipoles First Meeting of IAG JWG2.1, 5/15/2017 19 Hannover, Germany First Meeting of IAG JWG2.1, 5/15/2017 20 Hannover, Germany.