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  1. Tests of Gravity Sternberg Astronomical Institute, Moscow 1986 Sergei Kopeikin Grishchuk Zeldovich

  2. Basic Levels of Experiments • Laboratory • Earth/Moon • Solar System • Binary Pulsars • Cosmology • Gravitational Detectors EFT Wokshop, Pittsburg, July 2007

  3. Laboratory Tests:theoretical motivations The Bullet Cluster • Alternative (“classic”) theories of gravity with short-range forces • Scalar-tensor • Vector-tensor TeVeS • Tensor-tensor (Milgrom, Bekenstein) • Non-symmetric connection (torsion) • Super-gravity, M-theory • Strings, p-branes • Loop quantum gravity • Extra dimensions, the hierarchy problem • Cosmological acceleration EFT Wokshop, Pittsburg, July 2007

  4. Laboratory Tests: experimental techniques • Principle of Equivalence • Torsion balance (Eötvös-type experiment) • Rotating torsion balance • Rotating source • Free-fall in lab • Free-fall in space • Newtonian 1/r² Law (a fifth force) • Torsion balance • Rotating pendulum • Torsion parallel-plate oscillator • “Spring board” resonance oscillator • Ultra-cold neutrons • Extra dimensions and the compactification scale • Large Hadron Collider EFT Wokshop, Pittsburg, July 2007

  5. Principle of Equivalence:torsion balance tests 2- limits on the strength of a Yukawa-type PE-violation coupled to baryon number. [Credit: Jens H Gundlach ] EFT Wokshop, Pittsburg, July 2007

  6. Principle of Equivalence: • Free-fall in Lab • Galileo Galilei • NIST Boulder • ZARM Bremen • Stratospheric balloons • Lunar feather-hammer test (David Scott – Apollo 15) • Free-fall in Space • SCOPE (French mission ) • STEP (NASA/ESA mission ) • GG (Italian mission A. Nobili’s lecture) EFT Wokshop, Pittsburg, July 2007

  7. Newtonian 1/r² Law 2- limits on 1/r² violations. [Credit: Jens H Gundlach 2005 New J. Phys.7 205 ] Eöt-Wash 1/r² test data with the rotating pendulum =1; =250 m Casimir force+1/r² law EFT Wokshop, Pittsburg, July 2007

  8. Local Lorentz Invariance [Credit: Clifford M. Will] The limitsassume a speed of Earth of 370 km/srelative to the mean rest frame of the universe. EFT Wokshop, Pittsburg, July 2007

  9. Gravitational Red Shift • Ground • Mössbauer effect (Pound-Rebka 1959) • Neutron interferometry (Colella-Overhauser-Werner 1975) • Atom interferometry • Clock metrology • Proving the Theory of Relativity in Your Minivan • Air • Häfele & Keating (1972) • Alley (1979) • Space • Gravity Probe A (Vessot-Levine 1976) • GPS (Relativity in the Global Positioning System) Mach-Zender Interferometer

  10. Global Positioning System • The combined effect of second order Doppler shift (equivalent to time dilation) and gravitational red shift phenomena cause the clock to run fast by 38 s per day. • The residual orbital eccentricity causes a sinusoidal variation over one revolution between the time readings of the satellite clock and the time registered by a similar clock on the ground. This effect has typically a peak-to-peak amplitude of 60 - 90 ns. • The Sagnac effect – for a receiver at rest on the equator is 133 ns, it may be larger for moving receivers. • At the sub-nanosecond level additional corrections apply, including the contribution from Earth’s oblateness, tidal effects, the Shapiro time delay, and other post Newtonian effects. EFT Wokshop, Pittsburg, July 2007

  11. Gravitational Red Shift [Credit: Clifford M. Will ] Selected tests of local position invariance via gravitational redshift experiments, showingbounds on which measures degree of deviation of redshift from the Einstein formula. In null redshift experiments, the bound is on the difference inbetween different kinds of clocks. EFT Wokshop, Pittsburg, July 2007

  12. The PPN Formalism: the postulates • A global coordinate frame • A metric tensor with 10 potentials and 10 parameters - curvature of space (= 1 in GR)  - non-linearity of gravity (=1 in GR)  - preferred location effects (=0 in GR) - preferred frame effects (=0 in GR) - violation of the linear momentum conservation (=0 in GR) • Stress-energy tensor: a perfect fluid • Stress-energy tensor is conserved (“comma goes to semicolon” rule) • Test particles move along geodesics • Maxwell equations are derived under assumption that the principle of equivalence is valid (“comma goes to semicolon” rule) EFT Wokshop, Pittsburg, July 2007

  13. The PPN Formalism: the difficulties • The structure of the metric tensor in arbitrary coordinates is known only in one (global) coordinate system • Gauge-invariance is not preserved • Oservables and gravitational variables are disentangled • PPN parameters are gauge-dependent • PPN formalism derives equations of motion of test point particles under assumption that the weak principle of equivalence is valid but it does not comply with the existence of the Nordtvedt effect • PPN is limited to the first post-Newtonian approximation • Remedy: • Damour & Esposito-Farese, Class. Quant. Grav., 9, 2093 (1992) • Kopeikin & Vlasov, Phys. Rep., 400, 209-318 (2004) EFT Wokshop, Pittsburg, July 2007

  14. Solar System Tests: Classic • Advance of Perihelion • Bending of Light • Shapiro Time Delay EFT Wokshop, Pittsburg, July 2007

  15. Advance of Perihelion p Q: To what extent does the orbital motion of the Sun contribute to ? EFT Wokshop, Pittsburg, July 2007

  16. Bending of Light Traditionally the bending of light is computed in a static-field approximation. Q: What physics is behind the static approximation?  EFT Wokshop, Pittsburg, July 2007

  17. (PRL, 26, 1132, 1971) The Shapiro Time Delay Eikonal Equation: A plane-wave eikonal (static gravity field):

  18. Limits on the parameter  [Credit: Clifford M. Will ] EFT Wokshop, Pittsburg, July 2007

  19. Solar System Tests: Advanced • Gravimagnetic Field Measurement • LAGEOS • Gravity Probe B • Cassini • The Speed of Gravity • The Pioneer Anomaly EFT Wokshop, Pittsburg, July 2007

  20. LAGEOS(Ciufolini, PRL, 56, 278, 1986) Measured with 15% error budget by Ciufolini & Pavlis, Nature 2004 J2 perturbation is totally suppressed with k = 0.545 EFT Wokshop, Pittsburg, July 2007

  21. Gravity Probe B Residual noise: GP-B Gyro #1 Polhode Motion (torque-free Euler-Poinsot precession) => Mission begins => Mission ends EFT Wokshop, Pittsburg, July 2007

  22. Cassini Measurement of Gravimagnetic Field (Kopeikin et al., Phys. Lett. A 2007) Mass current due to the orbital motion of the Sun Bertotti-Iess-Tortora, Nature, 2004 -1=(2.1±2.3) EFT Wokshop, Pittsburg, July 2007

  23. Propagation of light in time-dependent gravitational field: light and gravity null cones Observer Future gravity null cone Star’s world line Observer Future gravity null cone Future gravity null cone Future gravity null cone Light null cone Future gravity null cone Light null cone Observer’s world line Planet’s world line

  24. The null-cone bi-characteristic interaction of gravity and light in general relativity Any of the Petrov-type gravity field obeys the principle of causality, so that even the slowly evolving "Coulomb component" of planet’s gravity field can not transfer information about the planetary position with the speed faster than the speed of light (Kopeikin, ApJ Lett., 556, 1, 2001). EFT Wokshop, Pittsburg, July 2007

  25. The speed-of-gravity VLBI experiment with Jupiter (Fomalont & Kopeikin, Astrophys. J., 598, 704, 2003) Position of Jupiter taken from the JPL ephemerides (radio/optics) undeflected position of the quasar 5 1 Position of Jupiter as determined from the gravitational deflection of light from the quasar 4 2 3 Measured with 20% of accuracy, thus, proving that the null cone is a bi-characteristic hypersurface (speed of gravity = speed of light) 10 microarcseconds= the width of a typical strand of a human hair from a distance of 650 miles.

  26. The Pioneer Anomaly The anomaly is seen in radio Doppler and ranging data, yielding information on the velocity and distance of the spacecraft. When all known forces acting on the spacecraft are taken into consideration, a very small but unexplained force remains. It causes a constant sunward acceleration of (8.74 ± 1.33) × 10−10 m/s2 for both Pioneer spacecrafts.

  27. Lunar Laser Ranging: Retroreflector’s Positions on the Moon

  28. Credit: T. Murphy (UCSD) Lunar Laser Ranging: Technology EFT Wokshop, Pittsburg, July 2007

  29. LLR and the Strong Principle of Equivalence Inertial mass Gravitational mass The Nordtvedt effect: 4(-1)-(-1)=-0.0007±0.0010 Moon Earth Moon Earth To the Sun To the Sun

  30. Gauge Freedom in the Earth-Moon-Sun System Sun Moon Earth Boundary of the local Earth-Moon reference frame

  31. Example of the gauge modes: • TT-TCB transformation of time scales • Lorentz contraction of the local coordinates • Einstein contraction of the local coordinates • Relativistic Precession (de Sitter, Lense-Thirring, Thomas)

  32. Effect of the Lorentz and Einstein contractions Magnitude of the contractions is about 1 meter! Ellipticity of the Earth’s orbit leads to its annual variation of about 2 millimeters. The Lorentz contraction Earth The Einstein contraction

  33. The gauge modes in EIH equations of a three-body problem: • “Newtonian-like” transformation of the Einstein-Infeld-Hoffman (EIH) force • This suppresses all gauge modes in the coordinate transformation from the global to local frame but they all appear in the geocentric EIH equations as spurious relativistic forces EFT Wokshop, Pittsburg, July 2007

  34. Are the gauge modes observable? • Einstein: no – they do not present in observational data • LLR team (Murphy, Nordtvedt, Turyshev, PRL 2007) • yes – the “gravitomagnetic” modes are observable • Kopeikin, S., PRL.,98, 229001 (2007) The LLR technique involves processing data with two sets of mathematical equations, one related to the motion of the moon around the earth, and the other related to the propagation of the laser beam from earth to the moon. These equations can be written in different ways based on "gauge freedom“, the idea that arbitrary coordinates can be used to describe gravitational physics. The gauge freedom of the LLR technique shows that the manipulation of the mathematical equations is causing JPL scientists to derive results that are not apparent in the data itself.

  35. Binary Pulsar Tests • Equations of Motion • Orbital Parametrization • Timing Formula • Post-Keplerian Formalism • Gravitational Radiation • Geodetic Precession • Three-dimensional test of gravity • Extreme Gravity: probing black hole physics EFT Wokshop, Pittsburg, July 2007

  36. Deriving the Equations of Motion Lagrangian-based theory of gravity Field equations: tensor, vector, scalar Boundary and initial conditions: External problem - global frame Boundary and initial conditions: Internal problem - local frame(s) External solution of the field equations: metric tensor + other fields in entire space Internal solution of the field equations: metric tensor + other fields in a local domain; external and internal multipole moments Matching of external and internal solutions Coordinate transformations between the global and local frames External multipole moments in terms of external gravitational potentials Laws of motion: external Laws of transformation of the internal and external moments Laws of motion: internal; Fixing the origin of the local frame Equations of motion: external Equations of motion: internal Effacing principle: equations of motion of spherical and non-rotating bodies depend only on their relativistic masses – bodies’ moments of inertia does not affect the equations

  37. Lorentz-Droste, 1917 Einstein-Infeld-Hoffman, 1938 Petrova, 1940 Fock, 1955 (see Havas, 1989, 1993 for interesting historic details) Equations of Motionin a binary system Carmeli, 1964 Ohta, Okamura, Kiida, Kimura, 1974 Damour-Deruelle, 1982 Kopeikin, 1985 Schaefer, 1985 … Grishchuk-Kopeikin, 1983 Damour, 1983 Kopeikin, PhD 1986 EFT Wokshop, Pittsburg, July 2007

  38. Orbital Parameterization(Klioner & Kopeikin, ApJ, 427, 951, 1994) f To observer • Osculating Elements • Blandford-Teukolsky • Epstein-Haugan • Brumberg • Damour-Deruelle EFT Wokshop, Pittsburg, July 2007

  39. Timing Model Pulsar’s rotational frequency derivative Pulsar’s rotational frequency Pulse’s number Emission time Roemer delay Time of arrival Proper motion delay Parallax delay Einstein delay Shapiro delay Bending Delay Plasma delay Atomic (proper) time EFT Wokshop, Pittsburg, July 2007

  40. Keplerian Parameters • Projected semi-major axis: • Eccentricity: • Orbital Period: • Longitude of periastron: • Julian date of periastron: • Keplerian parameters => Mass function: EFT Wokshop, Pittsburg, July 2007

  41. Post-Keplerian Parameters s EFT Wokshop, Pittsburg, July 2007

  42. Credit: Esposito-Farese Four binary pulsars tests EFT Wokshop, Pittsburg, July 2007

  43. A test of general relativity from the three-dimensional orbital geometry of a binary pulsar(van Straten, Bailes, Britton, Kulkarni, et al. Nature 412, 158, 2001) PSR J0437-4715 Shapiro delay in the pulsar PSRJ 1909-3744 timing signal due to the gravitational field of its companion. EFT Wokshop, Pittsburg, July 2007

  44. Geodetic precession in PSR 1913+16 1.21 deg yr -1 Credit: M. Kramer & D. Lorimer Pulsar’s Spin Axis Orbital Spin Axis To observer

  45. Extreme Gravity: detecting black hole with pulsar timing (Wex & Kopeikin, ApJ, 1999) • Timing of a binary pulsar allows us to measure the quadrupolar-field and spin-orbit-coupling perturbations caused by the presence of the pulsar’s companion • Since these perturbations have different orbital-phase dependence, one can measure the quadrupole and the spin of the companion • Black hole physics predicts a unique relationship between the spin and the quadrupole because of the “no-hair theorem” • Comparision of the mesured value of spin against the quadrupole allows us to see if the companion is a black hole and explore the black hole physics EFT Wokshop, Pittsburg, July 2007

  46. Finite Size Effects in the PN Equations of Motion: gravitational wave detector science • Reference frames in N-body problem • Definition of body’s spherical symmetry • The effacing principle EFT Wokshop, Pittsburg, July 2007

  47. Reference Frames in N-body Problem: global and local frames R L

  48. Matching of Local and Global Frames (u, w) Global coordinates (t, x) Matching Domain

  49. Coordinate Transformations between Local and Global Frames EFT Wokshop, Pittsburg, July 2007

  50. The Law of Motion of the Origin of the Local Frame in the Global Frame External Grav. Potentials Inertial Forces EFT Wokshop, Pittsburg, July 2007