# Presentation: 559745

## Testing General Relativity in Fermilab:

- Sergei Kopeikin
- University of Missouri-Columbia

***   ******A Bridge between the Particles Physics and Relativistic Gravity***** **

## Photograph by Paul Ehrenfest. Image Source: AIP Emilio Segrè Visual Archives . 

*Photograph by Paul Ehrenfest. Image Source: *_*[AIP Emilio ](http://photos.aip.org/)*__*[Segrè](http://photos.aip.org/)*__*[ Visual Archives](http://photos.aip.org/)*_* .*

- The U.S. program in particle physics is at a crossroads. The continuing vitality of the program requires new, decisive, and forward-looking actions. In addition, sustained leadership requires a willingness to take the risks that always accompany leadership on the scientific frontier. Thus, the committee recommends the thoughtful pursuit of a high-risk, high-reward strategy.
**Committee on Elementary Particle Physics in the 21st Century, National Research Council (2006)**
**Revealing the Hidden Nature of Space  and Time: Charting the Course for  Elementary Particle Physics **

## It seems natural to neglect gravity in particle physics

## Tests of Gravity at Macroscopic Distances 

- Laboratory
- Earth/Moon (Lunar Laser Ranging)
- Solar System (VLBI, Doppler/radio ranging, GPS)
- Binary Pulsars
- Black Hole in the Milky Way
- Cosmic Microwave Background
- Gravitational-wave detectors (bars, interferometers)
**Gravity regime tested in the solar system:**
- weak field (U << c2)
- slow motion (v << c)
**Gravity regime tested in binary pulsars:**
- strong field (U ≤ c2)
- slow motion (v << c)
- radiation-reaction force                          2.5 post-Newtonian approximation

## Why Do We Need to Measure Gravity at Microscopic Scale?

**Over the past 50 years accelerators have explored the energy range from 1 MeV in nuclear reactions up to about 1000 GeV at the Tevatron. We have a remarkably accurate theory to predict and explain what we see at present**

## Cosmological Evidence for Vacuum Energy

## Revolution in Gravity

- Cris Quigg: Fermi National Accelerator Laboratory

## Newton’s Law in n dimensional space

## Testing 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

-  parameterization of the presumable violation of the Newtonian 1/r2 law.

## Gravity Field in Fermilab 

- Weak    (U << c2)
- Ultra-relativistic (v  c)
- Post-Newtonian
- _Possible experiments:_
- Post-Newtonian gravity force produced by magnetic stresses and mechanical strains
- Gravity force at ultra-relativistic velocities (testing standard theory extension, other possible long-range relativistic forces between particles)

## The Tevatron Characteristics

**The Tevatron Characteristics**

**6**

## Local Inertial Reference Frame

*x1 = X*

*x3 = Z*

*x2 = Y*

## The metric tensor in the local frame

- Acceleration of gravity 9.81 m/s2

- Luni-solar tidal force

- Angular velocity of
- the Earth’s rotation

## Forces exerted on proton in the beam

- Protons in the beam falls down to Earth
- with acceleration of gravity twice as g=9.81 m/s2

## Tevatron Magnets

***Two main types of magnets: dipole and quadrupole. Dipoles are able to bend a particle beam. Quadrupoles focus the particle beams. ***
***The strength of the magnetic field is determined by the amount of electric current flowing through the coils. The 8-Tesla magnetic field requires current of 12,000 amperes. ***
***The SC magnets rely on a niobium-titanium compound. At ~10 K, the NTC becomes super-conducting and carries electric currents without resistance. ***
***Fermilab operates its 4.4 Tesla magnets. ***
***The magnets are bathed in 4.2-Kelvin liquid helium******.****** ***

**Tevatron Magnet Cross Section**

## Magnetic Field Structure in Tevatron Magnet

## Gravity by the Magnetic Stress

- General relativity predicts that gravity should be produced by stress as well as by mass-energy of matter
- Magnetic field is a matter field with energy and stresses (Maxwell’s tensor)
- Post-Newtonian gravity field of a magnet crucially depends on the magnetic (and mechanical) stress besides the part generated by mass-energy

**“****Believe it or not, gravity and magnetism are two totally different and **
**  ****really  unrelated phenomena.” **
**                                                           ****WikiAnswers ****http://wiki.answers.com/**** **

## The Gravity by the Magnetic Stress Why is it important to study?

- _Astrophysics:_ the maximum mass of neutron (and quark) stars depends on stress-produced by magnetic field
- _Gravitational Physics:_ validity of the principle of equivalence depends on how the stress contributes to the inertial and gravitational masses of a self-gravitating body (motion of binary pulsars, the effacing principle _[gr-qc/0612017](http://adsabs.harvard.edu/abs/2006gr.qc....12017K)__)_

## Gravity by the Magnetic Stress.

- Current circulating in magnet’s coil  generates the magnetic force
- The magnetic force leads to mechanical deformations that cause stresses in the material of the magnet that are comparable with the magnetic stress
- Strain-stress tensor calculation:
- 	L. Landau & E. Lifshitz “Electrodynamics of Continuum Media**”**

## Magnetic Strain-Stress Tensor

## Gravitational Field of the Magnet

## Gravitational Field of a Magnetized Sphere

**Gravitational Field of a Magnetized Sphere**

## Gravitational Field of the Magnetized Sphere

## Gravitational Force of the Magnetized Sphere

## Numerical Estimate and Measurability

- 

- 

- 

## Torsion balance in University of Washington 

- The free resonant period of our balances is about 2 mHz.
- The torsional spring constant is about 0.03 in cgs units and the angular displacement sensitivity is about               radians. A typical level arm of
- our torsion balances is about 2.5 cm. So the force sensitivity is about
-          cgs units or            Newtons.
- The quality factor of our torsion oscillator is about 5000. But if the signal has a definite period (as is the case in all of our experiments so far) one can integrate much for longer longer times (and we do so).
- To resolve              radians of angular deflection (given our noise and other disturbances) we need days of running time.

**Torsion balance in University of Washington **

## Long range forces and spontaneous violation of the Lorentz symmetry 

- One may expect that the Lorentz invariance for gravity field is spontaneously broken.
- Theories with spontaneous local Lorentz and diffeomorphism violation contain massless
- Nambu-Goldstone modes, which arise as field excitations in the minimum of the
- symmetry-breaking potential. If the shape of the potential also allows excitations above
- the minimum, then an alternative gravitational Higgs mechanism can occur in which
- massive modes involving the metric appear (see discussion of a class of bumblebee models by A. Kostelecky et al. _[gr](http://arxiv.org/abs/0712.4119)__[-qc 0712.4119](http://arxiv.org/abs/0712.4119)_)

## Local Lorentz Invariance

- [Credit: Clifford M. Will]

***The limits***** *****assume a speed of the solar system of***** 370 km/s  *****relative to the mean rest frame of the universe that is considered as a preferred frame. ***
***However, a genuine test of the Lorentz invariance of gravity must not rely upon this assumption (requires an experiment in a variable gravitational field)***

## The speed-of-gravity VLBI experiment with Jupiter(Kopeikin, ApJL, 556, 1, 2001;   Fomalont & Kopeikin, ApJ., 598, 704, 2003)

**The speed-of-gravity VLBI experiment with Jupiter**(Kopeikin, ApJL, 556, 1, 2001;   Fomalont & Kopeikin, ApJ., 598, 704, 2003)

- Position of  Jupiter from  JPL ephemerides (radio/optics)

- Position of Jupiter as
- measured from the
- gravitational deflection
- of light by Jupiter

- 1

- 2

- 3

- 5

- 4

**10 microarcseconds** = the width of a typical strand of a human hair from a distance of 650 *miles !*

- Measured with 20% of accuracy, thus, proving that the fundamental speed in General Relativity (the speed of gravity) equals the speed of light.

-  gravity-unperturbed position of quasar

## Relativity in high-energy accelerators

**Fermilab’s Tevatron:**
**CERN’s Large Hadron Collider**

## The Minkowski geometry of the gravity force measurement at ultra-relativistic speed 

- World line of the detector

- World line of the
- proton bunch

- space

- time

- In general relativity ultra-
- relativistic gravity force behaves
- similar to the synchrotron EM radiation

## What makes it plausible to measure the ultra-relativistic force of gravity in Tevatron?

- The bunch consists of N=3×1011  protons
- Ultra-relativistic speed = large Lorentz factor
- 	 =1000
- Synchrotron character of the force = beaming factor gives additional Lorentz factors
- Spectral density of the gravity force grows as a power law as frequency decreases
- The gravity force is a sequence of pulses (45000 “pushes” per second   36 bunches =1,620,000)

## Einstein-Maxwell Equations

## The Reissner-Nordström Metric (an exact solution of Einstein-Maxwell equations)

- If q>>m, the two horizons disappear and the singularity becomes a naked one. Many physicists believe that such a situation can't arise: there is a principle of "cosmic censorship", which prevents naked singularities from ever forming. Theories with super-symmetry usually guarantee that such "super-extremal" black holes can't exist.

**The Reissner-Nordstr****ö****m Metric ****(an exact solution of Einstein-Maxwell equations)**

- Main features of a Reissner-Nordstrom black hole. Credit: N. Rumiano

## Electromagnetic-Gravitational Field Model in a post-Minkowskian Approximation

- Gravity field potentials
- Lorentz – de Donder gauge
- Linearized field equations

- =

## Electromagnetic Lienard-Wiechert Potentials

**Electromagnetic Lienard-Wiechert Potentials**

- Retarded time s=s(t,x)

## Electromagnetic Stress-Energy Tensor

## Gravitational Lienard-Wiechert Potentials

- Gravitational mass of the particle

- Inertial mass of the particle

## The frame used for calculation of the gravity force 

*Z*

*Y*

*X*

- O

**

- Gravity force detector
- (torsion balance)

- X-Z plane is the plane of the Tevatron ring.
- Y axis is a local vertical.

- 

## Slide 41

## Equations of Motion of a Mechanical Detector

- Braginsky V.B. and Manukin, A.B., “Measurement of Weak Forces in
- 			               Physics Experiments”, Univ. Chicago Press, 1977
- Braginsky, V.B., Caves, C.M. & K.S. Thorne, “Laboratory experiments to test relativistic
- 				                gravity”, PRD, 15, 2047-2068, 1977
- Braginsky, V.B., “Experiments with Probe Masses”, PNAS, 104, 3677-80, 2007

## Gravitational versus Lorentz Force

- Tensor force

- Vector force

## Gravitational versus Lorentz Force

## Gravity Force in ultra-relativistic approximation

## Motion of the particle in the beam

## The Dipole Approximation

- 

## The ultra-relativistic approximation

## Fourier  Transform of the Gravity Force

- Fourier frequency

## Fourier Transform in The Ultra-relativistic Approximation

## Numerical estimate of the gravity force

- 

- 

- 

## Respond of the detector to the gravity force 

- 

## Laser Interferometer for Gravity Physics at Tevatron?

- 

- 

- L

-  = 999.5 m
- d = 0.07 m
- L = 16.7 m
- Current technology of LIGO coordinate meters allows to measure position with an error

- +d

- Probe mass

- Proton’s beam

- Probe mass