Binary spinning black hole Hamiltonian in canonical center-of-mass and rest-frame coordinates through higher post-Newtonian order

The recently constructed Hamiltonians for spinless binary black holes through third post-Newtonian order and for spinning ones through formal second post-Newtonian order, where the spins are counted of zero post-Newtonian order, are transformed into fully canonical center-of-mass and rest-frame variables. The mixture terms in the Hamiltonians between center-of-mass and rest-frame variables are in accordance with the relation between the total linear momentum and the center-of-mass velocity as demanded by global Lorentz invariance. The various generating functions for the center-of-mass and rest-frame canonical variables are explicitly given in terms of the single-particle canonical variables. The no-interaction theorem does not apply because the world-line condition of Lorentz covariant position variables is not imposed.Comment: 18 pages, no figure

Prospects in the orbital and rotational dynamics of the Moon with the advent of sub-centimeter lunar laser ranging

Lunar Laser Ranging (LLR) measurements are crucial for advanced exploration of the laws of fundamental gravitational physics and geophysics. Current LLR technology allows us to measure distances to the Moon with a precision approaching 1 millimeter. As NASA pursues the vision of taking humans back to the Moon, new, more precise laser ranging applications will be demanded, including continuous tracking from more sites on Earth, placing new CCR arrays on the Moon, and possibly installing other devices such as transponders, etc. Successful achievement of this goal strongly demands further significant improvement of the theoretical model of the orbital and rotational dynamics of the Earth-Moon system. This model should inevitably be based on the theory of general relativity, fully incorporate the relevant geophysical processes, lunar librations, tides, and should rely upon the most recent standards and recommendations of the IAU for data analysis. This paper discusses methods and problems in developing such a mathematical model. The model will take into account all the classical and relativistic effects in the orbital and rotational motion of the Moon and Earth at the sub-centimeter level. The new model will allow us to navigate a spacecraft precisely to a location on the Moon. It will also greatly improve our understanding of the structure of the lunar interior and the nature of the physical interaction at the core-mantle interface layer. The new theory and upcoming millimeter LLR will give us the means to perform one of the most precise fundamental tests of general relativity in the solar system.Comment: 26 pages, submitted to Proc. of ASTROCON-IV conference (Princeton Univ., NJ, 2007

Post-Newtonian Theory for Precision Doppler Measurements of Binary Star Orbits

The determination of velocities of stars from precise Doppler measurements is described here using relativistic theory of astronomical reference frames so as to determine the Keplerian and post-Keplerian parameters of binary systems. We apply successive Lorentz transformations and the relativistic equation of light propagation to establish the exact treatment of Doppler effect in binary systems both in special and general relativity theories. As a result, the Doppler shift is a sum of (1) linear in $c^{-1}$ terms, which include the ordinary Doppler effect and its variation due to the secular radial acceleration of the binary with respect to observer; (2) terms proportional to $c^{-2}$, which include the contributions from the quadratic Doppler effect caused by the relative motion of binary star with respect to the Solar system, motion of the particle emitting light and diurnal rotational motion of observer, orbital motion of the star around the binary's barycenter, and orbital motion of the Earth; and (3) terms proportional to $c^{-2}$, which include the contributions from redshifts due to gravitational fields of the star, star's companion, Galaxy, Solar system, and the Earth. After parameterization of the binary's orbit we find that the presence of periodically changing terms in the Doppler schift enables us disentangling different terms and measuring, along with the well known Keplerian parameters of the binary, four additional post-Keplerian parameters, including the inclination angle of the binary's orbit, $i$. We briefly discuss feasibility of practical implementation of these theoretical results, which crucially depends on further progress in the technique of precision Doppler measurements.Comment: Minor changes, 1 Figure included, submitted to Astrophys.

Horava-Lifshitz gravity: tighter constraints for the Kehagias-Sfetsos solution from new solar system data

We analytically work out the perturbation induced by the Kehagias-Sfetsos (KS) space-time solution of the Horava-Lifshitz (HL) modified gravity at long distances on the two-body range for a pair of test particles A and B orbiting the same mass M. We apply our results to the most recently obtained range-residuals \delta\rho for some planets of the solar system (Mercury, Mars, Saturn) ranged from the Earth to effectively constrain the dimensionsless KS parameter \psi_0 for the Sun. We obtain \psi_0 >= 7.2 x 10^-10 (Mercury), \psi_0 >= 9 x 10^-12 (Mars), \psi_0 >= 1.7 x 10^-12 (Saturn). Such lower bounds are tighter than other ones existing in literature by several orders of magnitude. We also preliminarily obtain \psi_0 >= 8 x 10^-10 for the system constituted by the S2 star orbiting the Supermassive Black Hole (SBH) in the center of the Galaxy.Comment: LaTex2e, 15 pages, 1 table, 3 figures, 31 references. Version matching the one at press in International Journal of Modern Physics D (IJMPD

Earth-Moon Lagrangian points as a testbed for general relativity and effective field theories of gravity

We first analyse the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid. This scheme allows us to take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L4 and L5 both in the classical regime and in the context of effective field theories of gravity. A vehicle initially placed at L4 or L5 will not remain near the respective points. In particular, in the classical case the vehicle moves on a trajectory about the libration points for at least 700 days before escaping away. We show that this is true also if the modified long-distance Newtonian potential of effective gravity is employed. We also evaluate the impulse required to cancel out the perturbing force due to the Sun in order to force the spacecraft to stay precisely at L4 or L5. It turns out that this value is slightly modified with respect to the corresponding Newtonian one. In the second part of the paper, we first evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L4 and L5, the corrections of coordinates are of order a few millimeters and describe a tiny departure from the equilateral triangle. After that, we set up a scheme where the theory which is quantum corrected has as its classical counterpart the Einstein theory, instead of the Newtonian one. In other words, we deal with a theory involving quantum corrections to Einstein gravity, rather than to Newtonian gravity. By virtue of the effective-gravity correction to the long-distance form of the potential among two point masses, all terms involving the ratio between the gravitational radius of the primary and its separation from the planetoid get modified. Within this framework, for the Lagrangian points of stable equilibrium, we find quantum corrections of order two millimeters, whereas for Lagrangian points of unstable equilibrium we find quantum corrections below a millimeter. In the latter case, for the point L1, general relativity corrects Newtonian theory by 7.61 meters, comparable, as an order of magnitude, with the lunar geodesic precession of about 3 meters per orbit. The latter is a cumulative effect accurately measured at the centimeter level through the lunar laser ranging positioning technique. Thus, it is possible to study a new laser ranging test of general relativity to measure the 7.61-meter correction to the L1 Lagrangian point, an observable never used before in the Sun-Earth-Moon system. Performing such an experiment requires controlling the propulsion to precisely reach L1, an instrumental accuracy comparable to the measurement of the lunar geodesic precession, understanding systematic effects resulting from thermal radiation and multi-body gravitational perturbations. This will then be the basis to consider a second-generation experiment to study deviations of effective field theories of gravity from general relativity in the Sun-Earth-Moon system

Uniqueness of collinear solutions for the relativistic three-body problem

Continuing work initiated in an earlier publication [Yamada, Asada, Phys. Rev. D 82, 104019 (2010)], we investigate collinear solutions to the general relativistic three-body problem. We prove the uniqueness of the configuration for given system parameters (the masses and the end-to-end length). First, we show that the equation determining the distance ratio among the three masses, which has been obtained as a seventh-order polynomial in the previous paper, has at most three positive roots, which apparently provide three cases of the distance ratio. It is found, however, that, even for such cases, there exists one physically reasonable root and only one, because the remaining two positive roots do not satisfy the slow motion assumption in the post-Newtonian approximation and are thus discarded. This means that, especially for the restricted three-body problem, exactly three positions of a third body are true even at the post-Newtonian order. They are relativistic counterparts of the Newtonian Lagrange points L1, L2 and L3. We show also that, for the same masses and full length, the angular velocity of the post-Newtonian collinear configuration is smaller than that for the Newtonian case. Provided that the masses and angular rate are fixed, the relativistic end-to-end length is shorter than the Newtonian one.Comment: 18 pages, 1 figure; typos corrected, text improved; accepted by PR

Note on the generalized Hansen and Laplace coefficients

Recently, Breiter et al (2004) reported the computation of Hansen coefficients $X_k^{\gamma,m}$ for non integer values of $\gamma$. In fact, the Hansen coefficients are closely related to the Laplace $b_{s}^{(m)}$, and generalized Laplace coefficients $b_{s,r}^{(m)}$ (Laskar and Robutel, 1995) that do not require $s,r$ to be integers. In particular, the coefficients X_0^{\g,m} have very simple expressions in terms of the usual Laplace coefficients b_{\g+2}^{(m)}, and all their properties derive easily from the known properties of the Laplace coefficients.Comment: 9/11/200

Augmented kludge waveforms for detecting extreme-mass-ratio inspirals

The extreme-mass-ratio inspirals (EMRIs) of stellar-mass compact objects into massive black holes are an important class of source for the future space-based gravitational-wave detector LISA. Detecting signals from EMRIs will require waveform models that are both accurate and computationally efficient. In this paper, we present the latest implementation of an augmented analytic kludge (AAK) model, publicly available at github.com/alvincjk/EMRI_Kludge_Suite as part of an EMRI waveform software suite. This version of the AAK model has improved accuracy compared to its predecessors, with two-month waveform overlaps against a more accurate fiducial model exceeding 0.97 for a generic range of sources; it also generates waveforms 5-15 times faster than the fiducial model. The AAK model is well suited for scoping out data analysis issues in the upcoming round of mock LISA data challenges. A simple analytic argument shows that it might even be viable for detecting EMRIs with LISA through a semi-coherent template bank method, while the use of the original analytic kludge in the same approach will result in around 90% fewer detections.Comment: Published versio

Gravitational bending of light by planetary multipoles and its measurement with microarcsecond astronomical interferometers

General relativistic deflection of light by mass, dipole, and quadrupole moments of gravitational field of a moving massive planet in the Solar system is derived. All terms of order 1 microarcsecond are taken into account, parametrized, and classified in accordance with their physical origin. We calculate the instantaneous patterns of the light-ray deflections caused by the monopole, the dipole and the quadrupole moments, and derive equations describing apparent motion of the deflected position of the star in the sky plane as the impact parameter of the light ray with respect to the planet changes due to its orbital motion. The present paper gives the physical interpretation of the observed light-ray deflections and discusses the observational capabilities of the near-future optical (SIM) and radio (SKA) interferometers for detecting the Doppler modulation of the radial deflection, and the dipolar and quadrupolar light-ray bendings by the Jupiter and the Saturn.Comment: 33 pages, 10 figures, accepted to Phys. Rev.