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Relativity and Fundamental Physics Sergei Kopeikin

University of -Columbia & Siberian State University of Geosystems and Technology

20th International Workshop on Laser October 9-14, 2016 1 Ranging - GFZ Potsdam - Outline

• Where does fundamental physics starts from? • Fundamental parameters • Zoo of alternative theories of • Discovery of gravitational waves • Experiments in the solar system - Time and clocks - Gravitomagnetic field - Laser ranging for testing big G • Summary

20th International Workshop on Laser October 9-14, 2016 2 Ranging - GFZ Potsdam - Germany

Modern theory of fundamental interactions relies heavily upon two main pillars both created by – special and general .

Special relativity is a cornerstone of elementary particle physics and the quantum field theory.

General relativity is a metric-based theory of gravitational field.

20th International Workshop on Laser October 9-14, 2016 3 Ranging - GFZ Potsdam - Germany • Understanding the nature of fundamental physical interaction is the ultimate goal of experimental physics.

• The most important but least understood is the gravitational interaction due to its weakness in the solar system – a primary experimental laboratory of gravitational for several hundred years.

• We study gravity by observing orbital/rotational motion of celestial bodies with light rays and radio waves.

• Physical motions of the bodies and propagation of light are described by solutions of equations of motion which, in their own turn, depend on the solutions of equations of a gravity field theory

• The mathematical model of motion fits to observational data to determine various fundamental parameters characterizing the structure of (NB: most of the fitting parameters are not fundamental though).

20th International Workshop on Laser October 9-14, 2016 4 Ranging - GFZ Potsdam - Germany Where does fundamental physics start from?

Physical fields = sections Volume of Lagrangian of fiber bundles integration

4 푺 = ∫ 푑 푥 −푔 퐿 푥, 휓푏, 휓푏;훼| 퐺, ℎ, 푐, 훽, 훾, …

The action Covariant coordinates derivatives Path integral the metric tensor Fundamental physical

20th International Workshop on Laser constants October 9-14, 2016 5 Ranging - GFZ Potsdam - Germany The principle of a stationary action

Future The path taken by the system ′ ′ has a stationary action (δS = 0) 휓푏(푥 ) under small changes 훿휓푐 푥 in the configuration of the system.

훿휓푏 훿휓푏 = 휓 푏 푥 − 휓푏(푥)

휓 푏(푥 ) Field equations/equations of motion 휓 (푥) 푏 훿( −푔퐿) = 0 훿휓푏

Past 20th International Workshop on Laser October 9-14, 2016 6 Ranging - GFZ Potsdam - Germany Lagrangian

퐿 = 퐿퐸(푔훼훽; 푔훼훽,휇) + 퐿푀(휙푏; 휙푏;훼) + 퐿퐼(휙푏, 휙푏;훼;휓푐, 휙푐;훼)

Lagrangian of Interaction of Einstein’s Lagrangian Matter Lagrangian matter fields

(PPN parameters) Depends only on gravity Depends on both matter variables – the metric and gravity variables through tensor and its first the covariant derivative derivatives The minimal coupling principle Gravitoelectric field The Gravitomagnetic field principle Gravitational waves

20th International Workshop on Laser October 9-14, 2016 7 Ranging - GFZ Potsdam - Germany Parametrized post-Newtonian (PPN) Formalism • A global barycentric coordinate system x   (,) ct x (BCRS)

• A metric tensor g  ( ct , x |  ,  ,  ,...) = gravitational field potentials: depends on 10 PPN parameters  - curvature of space ( = 1 in GR)  - non-linearity of gravity ( = 1 in GR)  - preferred location effects (= 0 in GR) 휶ퟏ, 휶ퟐ, 휶ퟑ - preferred frame effects (= 0 in GR) 휻ퟏ, 휻ퟐ, 휻ퟑ, 휻ퟒ - violation of the linear momentum conservation (=0 in GR) • Stress-energy tensor: a perfect fluid in most cases • Stress-energy tensor is conserved (“comma goes to semicolon” rule) • Test particles move along geodesics • Maxwell equations obey the principle of equivalence (“comma goes to semicolon” rule) 20th International Workshop on Laser October 9-14, 2016 8 Ranging - GFZ Potsdam - Germany

Example: PPN β and γ parameters as fundamental constants of Nature  3 , c , 4 S  R  ()()      gd x 164 R gravity matter scalar

1  1    (  )     '    "  2  ... 0 2

GM 1 U  G  r  0 U g  11    1 ij ij   2  1 c   2 22UU2  ' g  1     1 00 cc24 4(2 3)( 2)2 20th International Workshop on Laser October 9-14, 2016 9 Ranging - GFZ Potsdam - Germany Fundamental parameters

• Fundamental parameters stay invariant (= keep the same numerical value) under the change of computational algorithm, coordinates, gauge conditions

• Measured value converges to a unique limit as the number of observations (normal points) increase. • Examples: – c electrodynamics; – G, c ; – β, γ scalar-tensor theory; – Some of the post-Newtonian parameters or a gauge-invariant combination of the post-Newtonian parameters made up of the integrals of motion and/or adiabatic invariants.

20th International Workshop on Laser October 9-14, 2016 10 Ranging - GFZ Potsdam - Germany Zoo of alternative gravity theories • Alternative (“classic”) theories of gravity with short-range forces – Scalar-tensor – Vector-tensor MOND, TeVeS – Tensor-tensor (Milgrom, Bekenstein) – Non-symmetric connection (torsion) • Extra dimensions (Kaluza-Klein, etc.) • Gauge theories on a fiber bundles – Standard Model Extension (SME) • Super-gravity, M-theory • Strings, p-branes • Loop

• , dark energy The Bullet Cluster -- a harbor of dark matter 20th International Workshop on Laser October 9-14, 2016 11 Ranging - GFZ Potsdam - Germany Hierarchy of Relativistic Test Experiments

• Laboratory (torsion balance, atomic clocks, LHC,…) • Earth-Moon System (weak-field tests: GNSS, GPB, SLR, LLR) • Solar System (weak-field tests: deep-space spacecraft tracking, astrometry, VLBI, interplanetary ranging) • Binary/Double Pulsars (strong field tests: pulsar timing) • Gravitational Waves (strong-field tests: LIGO, VIRGO, PTA) • (strong-field tests: COBE, PLANCK, SKA,…)

20th International Workshop on Laser October 9-14, 2016 12 Ranging - GFZ Potsdam - Germany Gravitational Waves the evidence through pulsar timing

Hulse & Taylor binary pulsar PSR 1913 + 16 -- discovered in 1974 Rotational period P = 1/17 sec  mp

mc

Orbital period 푷풃~ 8 hr

20th International Workshop on Laser October 9-14, 2016 13 Ranging - GFZ Potsdam - Germany The principle of detection of gravitational waves

International network (LIGO, Virgo, free masses GEO, TAMA, AIGO) of suspended mass Michelson-type interferometers on earth’s surface detect signals from distant astrophysical sources

suspended test masses

20th International Workshop on Laser October 9-14, 2016 14 Ranging - GFZ Potsdam - Germany

20th International Workshop on Laser October 9-14, 2016 15 Ranging - GFZ Potsdam - Germany How small is 10-20 meter?

One meter, about 40 inches

10,000 Human hair, about 100 microns

100 Wavelength of light, about 1 micron

10,000 Atomic diameter, 10-10 meter

10,000 Nuclear diameter, 10-14 meter

10 Proton/neutron, 10-15 meter

1,000 Electron, < 10-18 meter

100 LIGO sensitivity, ~10-20 meter 20th International Workshop on Laser October 9-14, 2016 16 Ranging - GFZ Potsdam - Germany GW Template of a Coalescing BH system

Ringdown

Chirp signal

20th International Workshop on Laser October 9-14, 2016 17 Ranging - GFZ Potsdam - Germany signal GW150914

20th International Workshop on Laser October 9-14, 2016 18 Ranging - GFZ Potsdam - Germany 20th International Workshop on Laser October 9-14, 2016 19 Ranging - GFZ Potsdam - Germany Public outreach:

Guest | Feb 12, 2016 6:09 AM

“I feel that there is a very big blunder. The two diagrams show the recording of two sound signals from the collision of two black holes. However these sound signals can not propagate in vacuum.

How then the sound of the collision of the black holes came to Earth?!!! “ 20th International Workshop on Laser October 9-14, 2016 20 Ranging - GFZ Potsdam - Germany Relativity in Global Positioning System • The combined effect of the second order Doppler shift (equivalent to ) and gravitational red shift phenomena causes the GPS 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 – 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, irregularity of the Earth’s rotation, tidal effects, the Shapiro time delay, and other post Newtonian effects (ISSI Workshop 2015, ) • GREAT GR tests experiment (ZARM, SYRTE, ISLR) in progress from May 1, 2016 – the goal is to improve on the GP-A limit 1×10−4 in measuring the gravitational red shift down to an uncertainty around (3−4)×10−5 after one year of integration of Galileo-201 data. ACES time transfer experiment (U. Schreiber et al, this workshop) 20th International Workshop on Laser October 9-14, 2016 21 Ranging - GFZ Potsdam - Germany Time Scales in Fundamental Physics Kopeikin, PRD, 86, 064004 (2012) “Celestial ephemerides in an expanding universe”

Universal Time (UT) Pulsar Time (PT) Inertia Tensor 2 3 UT = 퐺 푀 Spin PT = 푘 c4Spin

Ephemeris Time (ET) Binary Pulsar Time (BPT) 1 푅3 1 푅3 ET = BPT = 2휋 퐺푀 2휋 퐺푀

Atomic Time (TAI) “Time is merely an illusion” - A. Einstein 3 2 ℎ 휀0 TAI = 4 푚e푒 “Make time an observable” - U. Schreiber

Einstein’s Time (optical cavity) 퐿 T = 푐 Kopeikin, Gen. Rel. Grav., 47, 5 (2015) “Optical cavity resonator in an expanding universe” L 20th International Workshop on Laser October 9-14, 2016 22 Ranging - GFZ Potsdam - Germany Optical cavity as a probe of the local Hubble expansion S. Schiller et al., Univ. Düsseldorf (submitted)

Resonator optical frequency variation 횫풇풓풆풔 corrected for 풇풎풂풔풆풓 drift. Blue points are the measured values of 풇풓풆풔 The bars indicate the range twice the standard deviation. Red line: time-linear fit, exhibiting a drift rate 퐷푟푒푠−푚푎푠푒푟 = 5.1 × 10−21/푠. Blue shaded area: 2휎 uncertainty range of the time-linear fit. Zero ordinate value is defined as the mean of the data points. 20th International Workshop on Laser October 9-14, 2016 23 Ranging - GFZ Potsdam - Germany Optical clocks for relativistic . http://www.geoq.uni-hannover.de/a03.html

푊2 휏2 h

푊1 휏1

푊2 − 푊1 Δ휏 = − 푡 푐2

LOD, precession/nutation

Optical clocks for TAI realization. Fateev & Kopeikin: Measur. Tech., 58, 647 (2015) 푡 푊푖 1 1 휏 = 1 − 푡 − 훀(t) × 푹 2 + (1 + 푘 − ℎ)푈 (푡) 푑푡 푖 푐2 푐2 2 푖 푡푖푑푒 푡0 20th International Workshop on Laser October 9-14, 2016 24 Ranging - GFZ Potsdam - Germany Relativistic Geodesy: Altai Mountain Experiment Kopeikin et al. Gravitation and Cosmology, 22, 234 (2016) Stationary: cesium clock with instability ∼ ퟏퟎ−ퟏퟓ Transportable: hydrogen clock with instability ∼ ퟏퟎ−ퟏퟒ Route: Novosibirsk -> Shebalino -> Seminsky Pass -> Novosibirsk (Height difference 850 m) Time transfer : “Common View” GLONASS/GPS

Shebalino Seminsky Pass

ns ns

훿푓 = 9.5×10−14 ± 1.5×10−17 푓 0 GNSS Δℎ =859 m

Time, s Time, s 훿푓 = 7.9×10−14 ± 7.3×10−15 푓 0 clock 20th International Workshop on Laser Δℎ = 725 ± 64 m October 9-14, 2016 Ranging - GFZ Potsdam - Germany 25 Solar System Tests • Advance of Perihelion • Bending of Light • Shapiro Time Delay • Gravitomagnetic Field Measurement

– The field induced by rotational mass current • LAGEOS/LARES • Gravity Probe B

– The field induced by translational mass current • Cassini • VLBI Planetary Time Delay

20th International Workshop on Laser October 9-14, 2016 26 Ranging - GFZ Potsdam - Germany LAGEOS/LARES: spin-orbital interaction (Ciufolini, PRL, 56, 278, 1986)

 2S LT ae3(1 2 ) 3/ 2

 -1 LT 31 mas yr

−1 Ω 퐿−푇(푡표푡푎푙) = 48 mas yr

Gravitomagnetic field measured with 10% error budget: Ciufolini & Pavlis, Nature 2004

Ω 퐿−푇 J2 perturbation is totally suppressed with k = 0.545

20th International Workshop on Laser October 9-14, 2016 27 Ranging - GFZ Potsdam - Germany A test of general relativity using the LARES and LAGEOS satellites and a GRACE Earth gravity model. European Phys. J. C (2016) 76:120 (I. Ciufolini, A. Paolozzi, E. Pavlis, R. Koenig, J. Ries, et al)

흁 = ퟎ. ퟗퟗퟒ ± ퟎ. ퟎퟎퟐ ± ퟎ. ퟎퟓ 퐞퐱퐩퐞퐫퐢퐦퐞퐧퐭

흁 = ퟏ (general relativity)

20th International Workshop on Laser October 9-14, 2016 28 Ranging - GFZ Potsdam - Germany Gravity Probe B: spin-spin interaction Leonard I. Schiff (1960) – with R. Cannon and W. Fairbank dS   S d

  SLTT    

1 GM  rv -39.2 mas/yr S   23 2 cr

11GS s3 n n s  LT  1   1 23 24cr

Ω =-37.2±7.2 mas/yr 퐿푇 T vA  Residual noise: GP-B Gyro #1 Polhode Motion (torque-free Euler-Poinsot precession) From website: http://einstein.stanford.edu/highlights/hl_polhode_story.html

Mission Mission begins => => ends

20th International Workshop on Laser October 9-14, 2016 29 Ranging - GFZ Potsdam - Germany Gravitational Time Delay by a moving body

24GM2GM GM v h h ij h  00ij 0 i  |x z ()|t | x  z ()| t | x  z ()| t cg

: x xN (t ) x0  c k ( t  t 0 ) massive body: z ( t )  z 0  v ( t  t 0 )

GM 1 |xz (s ) |k  x  z (s ) (tt , )  2 1 kv  ln 11 11 10 3  ccg |x0 z (ss 0 ) |  k  x 0  z ( 0 )

vvvv22    z(s ) z ( t )  | x  z ( t ) |  O  z ( s )  z ( t )  | x  z ( t ) |  O   1 1 1 122  0 0 0 0   cg c g  c g  c g  Look like a retarded time 1 1 s1 t 1 |xz 1  ( t 1 ) | s0 t 0  |xz 0  ( t 0 ) | cg cg 20th International Workshop on Laser October 9-14, 2016 30 Ranging - GFZ Potsdam - Germany Gravitomagnetic Field in the Cassini Experiment Kopeikin et al., Phys. Lett. A, 367, 276 (2007) Gravitomagnetic Doppler shift 풗1 due to the orbital motion of the Sun

1 (풗⊙ ⋅ 휶퐵) 푐푔

Bertotti-Iess-Tortora, Nature, 2004 5 풗0  1  (2.1  2.3)  10 However, the gravitomagnetic field contribution has been never analyzed 20th International Workshop on Laser October 9-14, 2016 31 Ranging - GFZ Potsdam - Germany The speed-of-gravity experiment (2002)

Edward B. Fomalont Sergei M. Kopeikin

ApJ Lett, 556, 1-5 (2001)

ApJ, 598, 704-711(2003) Albuquerque 2002

VLBA support team: NRAO and MPIfR (Bonn)

October 9-14, 2016 20th International Workshop on Laser 32 Ranging - GFZ Potsdam - Germany The Jovian 2002 experiment Positions of Jupiter determined from the Shapiro time delay by Jupiter

Positions of Jupiter taken from JPL ephemerides

10 microarcseconds = the width of a typical The retardation effect was measured with 20% of strand of a human hair from a distance of accuracy, thus, proving that the 650 miles!!! does not exceed the with 20% of accuracy. 20th International Workshop on Laser October 9-14, 2016 Ranging - GFZ Potsdam - Germany 33 Light Deflection Experiment with Saturn and Cassini spacecraft Fomalont, Kopeikin et al., Proc. IAU Symp. 261, 291-295 (2009)

20th International Workshop on Laser October 9-14, 2016 34 Ranging - GFZ Potsdam - Germany Does LLR measure the gravitomagnetic field? Kopeikin, PRL, 98, 22, 229001 (2007) Kopeikin & Yi, Cel. Mech. Dyn. Astr., 108, 245-263 (2010)

푀퐶

푽퐶 푹퐵퐶

푽퐵 Gravitomagnetic field Gravitoelectric field 푀퐵

20th International Workshop on Laser October 9-14, 2016 Ranging - GFZ Potsdam - Germany 35 Ranging Time Delay

푅12 퐺푀퐵 푅1퐵 + 푅2퐵 + 푅12 푡2 − 푡1 = + 2 3 ln 푐 푐 푅1퐵 + 푅2퐵 − 푅12 퐵

2 2 퐺푀퐵 푅1퐵 − 푅2퐵 − 푅12 + 흀푩 3 (푅1퐵 + 푅2퐵) 푐 2푅1퐵푅2퐵푅12 퐵

퐺푀퐵 풗퐵 ⋅ 푹1퐵 풗퐵 ⋅ 푹2퐵 + 휶푩 4 − 푐 푅1퐵 푅2퐵 퐵

20th International Workshop on Laser October 9-14, 2016 36 Ranging - GFZ Potsdam - Germany Deep space experiment to measure G M.R. Feldman, J.D. Anderson, G. Schubert, V. Trimble, S. Kopeikin, C. Laemmerzahl

CQG 33 (2016) 125013 arXiv:1605.02126 [gr-qc] • Measure G with relative uncertainty surpassing 10 parts per million – National Science Foundation solicitation NSF 16-520 • Perform in isolated environment with minute and accountable number of forces – Relative vacuum of space would work • Lifetime on the order of years to test reality of a periodic signature

20th International Workshop on Laser October 9-14, 2016 37 Ranging - GFZ Potsdam - Germany How to produce lifetime of years? • Gravity train mechanism – Originally a thought experiment • Drill hole through center of Earth to other side • Unrealistically approximating Earth as uniform solid, observer inside the hole experiences simple harmonic motion along diameter of tunnel • Period of oscillation:

R3 푅6 2 T = 2p 퐺 = 4휋 2 MG 푀푇 – Using this mechanism but with much smaller object of known mass and radius, can produce an experiment on the order of years Gif courtesy: By Gotant6884 • G determinations result if one can accurately (http://www.gnu.org/copyleft/fdl.html) measure period of oscillator

20th International Workshop on Laser October 9-14, 2016 38 Ranging - GFZ Potsdam - Germany Deep space experiment to measure G

CCR

Laser Laser Massive Sphere Ranging distance: 5-6 miles

DG DR DM DT Q Uncertainty Experiment = 3 + + 2 G R M T R 7.3e-09 Avogadro Project

M 5.0e-09 Avogadro Project 63 ppb uncertainty: potential three orders of T 1.8e-08 Femtosecond laser magnitude improvement v.s. previous G G 6.3e-08 experiments

20th International Workshop on Laser October 9-14, 2016 39 Ranging - GFZ Potsdam - Germany Summary • Solar system tests continue to be competitive with pulsar timing and gravitational wave detectors in testing fundamental gravitational physics

• SLR continues to improve the accuracy in testing gravitomagnetic field with LARES/LAGEOS

• Light-ray deflection experiments are sensitive to the time-dependent component of the gravitational field of moving planets and Sun. Interplanetary laser ranging may improve testing of the “speed-of-gravity” effect by 10-100 times

• Relativistic geodesy with optical clocks opens a new window to a cm-precise normal height system on the global scale.

• Laser ranging systems for spacecraft in deep space are invaluable for future tests of general relativity and determination of fundamantal constants like big G.

• Much better theoretical model of the orbital/rotational motion of the Moon is required for providing an unambiguous testing relativistic theory of gravity. 20th International Workshop on Laser October 9-14, 2016 40 Ranging - GFZ Potsdam - Germany