04/22/23 1

Testing General Relativity in

Fermilab:

Sergei Kopeikin

University of Missouri-Columbia

A Bridge between the Particles Physics and Relativistic Gravity

04/22/23 2

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

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

04/22/23 3

It seems natural to neglect gravity in particle physics

04/22/23 4

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

5 5( ~ / )v c

04/22/23 5

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

04/22/23 6

Cosmological Evidence for Vacuum Energy

04/22/23 7

Revolution in GravityCris Quigg: Fermi National Accelerator Laboratory

04/22/23 8

Newton’s Law in n dimensional space

04/22/23 9

Testing Newtonian 1/r² Law 2- limits on 1/r² violations.[Credit: Jens H Gundlach 2005 New J. Phys. 7 205 ]

/1 212

21 22

1

1 ...2

rGm mV er

Gm m r rr

Eöt-Wash 1/r² test data with therotating pendulum

=1; =250 m

Casimir force+1/r² law

parameterization of the presumable violation of the Newtonian 1/r² law.

04/22/23 10

Gravity Field in Fermilab

• Weak (U << c²)• Ultra-relativistic (v c)• Post-Newtonian

Possible experiments:1. Post-Newtonian gravity force produced by magnetic

stresses and mechanical strains 2. Gravity force at ultra-relativistic velocities (testing

standard theory extension, other possible long-range relativistic forces between particles)

04/22/23 11

04/22/23 12

The Tevatron Characteristics

6

04/22/23 13

Local Inertial Reference Frame

x¹ = X

x³ = Z

x² = Y

04/22/23 14

The metric tensor in the local frame

300 0 02 2

30 02

30 02 2

2 11

1 1

2 11

k k pk k p

k p k pi ikp kpi

k k pij ij k k p

g a x R x x O xc c

g x R x x O xc c

g a x R x x O xc c

Acceleration of gravity 9.81 m/s² Luni-solar tidal force

Angular velocity of the Earth’s rotation

04/22/23 15

Forces exerted on proton in the beam

2 001 1 ...2 2

12 ...

iij jk j k i

i k i

i ii i i

du eu u F ud m

g ggdv ec v v F udt x x x m

dv ea a v v E v Bdt m c

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

04/22/23 16

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

04/22/23 17

Magnetic Field Structure in Tevatron Magnet

04/22/23 18

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/

04/22/23 19

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)

04/22/23 20

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”

04/22/23 21

Magnetic Strain-Stress Tensor

1 2

,

1 tensor of deformations2

magnetic permeability

the stre16

jiij j i

ik ij ij kk ij

ik j jk i kij ij

ij T

uuux x

a u a u

H H HFFu

(0)

(0) 21 2

(0) 20

ss tensor

the free energy of the magnetic8

2 inside the magnet8 8

1 outside th4 2

ik i k

ij ij i j ij

ij ij i j ij

H HF F

a aH H H

H H H

e magnet

04/22/23 22

Gravitational Field of the Magnet

00 2 2

2(0) 21 2

2 20 0

2 1 4

3 ; inside the magnet8 8

; outside the mag8 8

kk

kk kk

kk

g h

h Gc c

a aH H

H H

21 22

20

2

net

2 34 inside the magnet2

4 outside the magnet

a aG Hc

HG

c

04/22/23 23

Let us take a uniformly magnetized sphere with a radius as an example:

- permanent magnetization vector

R

M

0

3

;

21 ; = permanent magnetic field inside3 3

1( ) 3

in in

out

xn mr

H B

RH n m n mr

MM

M M

M dipole magnetic field outside

Gravitational Field of a Magnetized Sphere

x

0out outB H

04/22/23 24

Gravitational Field of the Magnetized SphereBy making use of the boundary conditions

(0) ., ( ) 0, and the matching conditions on the surface of the sphere

( ) ( ),

yields

in out

in outin out

r R r R

const

R Rr r

2 2 2321 2 2 2 0

2 2

6 20

2 4

monopole radial monopole radia

the gravitational field of the magnetized sphere

2 3( ') ' 1 14( )' 36 3 90 3

( )54

M

inV

out

G a a G Rx d xx G R r m nx x c c

G M M G Rx

r c r

M M

M

5 220

2 3

l quadrupole

3 2

0 1 22

5 1115 6 3

2 2 354

G R R m nc r r

RM a ac

M

M

04/22/23 25

Gravitational Force of the Magnetized Sphere

5 20

2 2 4

5 2 6 20 0

2 2 4 2 5

radial quadrupole

in the equatorial plane5

2 on the axis of symmetry5 9

The

G M M G RF n n

r c rG M M G R G RF n n n

r c r c r

M

M M

gravity field due to the magnetic/mechanical stress has two components: monopole quadrupole

04/22/23 26

Numerical Estimate and Measurability

For 10 , 100 cm, and 200 cm the characteristic magnitude

of the gravity force per unit mass (contribution of term to the relativistic mass of the magnet)

22 17 2 10 cm/s2

in

in

H T R r

M

F G R H Rm rc

2ec

which is measurable with the technique of the modern weak-force-meters (V.B. Braginsky, PNAS, , 3677-80, 2007).

The experiment must make use of a DC magnetic field, which is slowly turned on and

104

off at the frequency of the detector (~ 0.001 Hz). One has to watch whether gravity due to the oscillating magnetic/mechanical stress energy produces a change in the amplitude and phase of the oscillator.

04/22/23 27

Dr. Eric Adelberger: (a letter from 1/14/2008)

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

91 10

91 10

1110 1610

04/22/23 28

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-qc 0712.4119)

04/22/23 29

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)

04/22/23 30

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 asmeasured from thegravitational deflectionof light by Jupiter

1

2

3

54

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

04/22/23 31

Relativity in high-energy accelerators

-7

-7

2

900-GeV protons: v – c 586 km/h; 1-v/c 5.43 10

980-GeV protons: v – c 495 km/h; 1-v/c 4.58 10Bunch of protons makes 45 000 turns/second: 2.83 MHz

Acceleration of protons in Tevatron: 8ca

13 2

2 213 2

.48 10 m/s

Acceleration of a test particle slightly above the horizon of a BH

of one solar mass: 8.48 10 m/s - the same !!BHg

c car

Fermilab’s Tevatron:

CERN’s Large Hadron Collider-8100.93v/c-1 km/h; 10 c– v:protons TeV-7

04/22/23 32

The Minkowski geometry of the gravity force

measurement at ultra-relativistic speed

World line of the detectorWorld line of the proton bunch

space

time

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

04/22/23 33

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)

04/22/23 34

Einstein-Maxwell Equations; ;

(4)

1 14 4

( ) ( ) [ ( )]

the electromagnetic field tensor

the electromagnetic stress-energy tensor

f

p

F A A

T F F g F F

T m d u u x z

(4)

; ; ; ;

the stress-energy tensor of particles

the electric current

Maxw

( ) [ ( )]

4

ell Equations

Einstein Equ

0

ations

J e d u x z

F J F F Fc

du emc F ud c

R

4

1 82 p f

Gg R T Tc

04/22/23 35

The RN metric is a black hole that is electrically charged but non-rotating (q<<m). A Reissner-Nordström black hole has two separate event horizons; the more charge the black hole carries, the closer are its event horizons.

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

12 2

2 2 2 2 2 2 22 2

2 21 1 sinm q m qds dt dr r d dr r r r

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

24

10 3 2 1/2

216

0

2

34

2

5 1.67 10 g 0.938 GeV

4.80 10

Proton rest mass

Proton electric charge

Classical proton

(g cm /s )

radiu = =1.54 10 cm

Notice that in th

1.24 10 cm

1

e

.3 1 c

s

8 0 m

m

e

ermc

2 2 2 beam , while .pm Nm q N e

04/22/23 36

Electromagnetic-Gravitational Field Model in a post-Minkowskian Approximation

1. Gravity field potentials

2. Lorentz – de Donder gauge3. Linearized field equations

2( )

12

g Gh O G

h h

, ,0 0A

4

4 16 GA J Tc c

=

04/22/23 37

Electromagnetic Lienard-Wiechert Potentials

Retarded time s=s(t,x)

04/22/23 38

Electromagnetic Stress-Energy Tensor

Reissner-Nordstrom (velocity-induced

2

f

(

ield)

2

)4

00 11 33 22

8

=

124 2

magnet magnet magnet magnet

magnet beam mag

magnet magnet beam be

beam

net be

m

am

R

a

qT u k k

BT T

T T

T

T T

T

T F

r

B

k

acceleration-induced field

synchrotron-induced field

2( ) ( )

3

2 2

2

4

4

R

R

q a k a k u k k kr

q a a a k k kr

04/22/23 39

Gravitational Lienard-Wiechert Potentials

2

2

( ) ( )

4 (Reissner-Nordstom)

4 4

( , )

velocity induced fieldR R

beam magnet coupling field magnet

acceleration i

vif bmcf aif sif

nduced

mu u q k kr r

x s

m a k a k qF u k

in progress

in progressfield

synchrotron induced field

Gravitational mass of the particle

Inertial mass of the particle

04/22/23 40

The frame used for calculation of the gravity force

inertial mass rigiditydampingof detector of "spring"factor

m grav Lorentz noiseM x H x K x F F F

r nr

dr

Z

Y

X

O

R

0s

Gravity force detector(torsion balance)

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

04/22/23 41

Christoffel4-acceleration 4-velocity 4-veloci

the fo

ty external force symbols

exte

per unit rce of gr

r

mavity

s

l

as

na

probe mass (detecto

1

r)

Measur

2a

2

d x dx dx+ = Fd d d M

Equation of motion of a

ement is executed at the local (probe-mass-centered)

reference frame, where ( ,0,0,0), and the proper

time of the probe mass, , is 'practically equivalent'

to the coordinate time t in the labor

dx cd

2 2atory, ( / ). t O v c

04/22/23 42

Equations of Motion of a Mechanical Detector

inertial mass rigiditydampingof detector of "spring"factor

2

*

The resonant frequency:

The amplitude relaxation time:

The quality f t

/

/

c

2

a

m grav Lore

m

m m

ntz noise

K M

M H

M x H x K x F F F

*

or: 2

m m mm

m

MQ

H

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

04/22/23 43

Gravitational versus Lorentz Force

gravitationalmass of de

2 00

tector

residual chargeon the dete r

0

ct

0

o

1 12

1

i igravity i

i iLorentz i

h hF Mcx c t

A AF Qcx c t

Tensor force

Vector force

04/22/23 44

Gravitational versus Lorentz Force

2 2

3 22 2

3 2

22

32 2

2 4 21

1 1

4 4 1 2 ..

( )1

1 1

.1

grav

ret

re

Lor

t

entz

nF GMm r n

r n r n

nGMm Gq

n nnF Qqr n c

r n

r

3

( ) - a unit vector from the retarded position, ( ),of the particle to the ( )

point of observation, . Distance ( ) .

retn

x z sn z sx z s

x r x z s

04/22/23 45

Gravity Force in ultra-relativistic approximation

3 32

2

22

53 5

2 2 211

3 1 21

(

1

)1

1 1L

g

oren

a

z

r v

t

GMm nF nrnr n

GMq n r nnr

n nnF Qqr n cr n

n

04/22/23 46

Motion of the particle in the beam

20

0 0

0

0

0

0

0

0

0

( ) 1 cos sin

( ) sin cos

( ) cos sin

sin co

ˆ ˆ ˆ1 cos sin c

ˆ

os sin

21

1

ˆ

ˆ ˆ

ˆ ˆ

ˆ

sˆ

d d

d

d

d

d

i z

i

R s s s

s s z

i z

s

s s

j

r r R i

r

s

s jr z r

c cc

r

r

z

s

r

2

0 0cos sin 2 1 cosd d

s sr r

04/22/23 47

The Dipole Approximation

0 0 0

0

2

The 'observer' will be affected by the field that is emitted from a small section of the trajectory around the origin of angular length . Hence,

1 1 1sin ; 1- cos ; sin2

s 2 /

We dem

s s s

0

and 1, and approximate

cos sin the - dipole approximationd

d

r

r r s

04/22/23 48

The ultra-relativistic approximation

3 20 0

0 0 0

2 2 3 3

2

22 2 202

33

0

sin ; sin ; cos 1 ; 6 2

1 + ;2 6

11 1 ; 2

It is convenient to introduce the critical frequency:

and p

3 32

a

2

d

c

ct

n r

s ss s s

ss

s r

c

2/3 2 23 ; rameters: 4

1c

a b a

04/22/23 49

Fourier Transform of the Gravity Force

( )1

( )

( )( ) Exp

( ) 1 ( ) ( )

1 ( ) 1 ( ) ( )

( ) ( )( ) Exp( ) 1 ( ) ( )

pqp

d

p dqp

fF GMm i t dt

r n

t r r dt n dc

f r rF GMm i dcr n

s

s s s

s s s s s

s ss ss s s

Fourier frequency

04/22/23 50

Fourier Transform in The Ultra-relativistic Approximation

2 2/3 1 1/30

2 1 3( )2 13

2

1

Introduce a new variable: 2

2( ) Exp3

Further simplification comes from re-parameterization:

2( )

qq

pqqp

d

q

d

a a

fF GMm a i b d

c r b

b

F GM

s u s u

u uu uu

u

mc r

12

32

2 3( )

2 2 2

ˆExp

1 31

qp

qp

fib d

04/22/23 51

Numerical estimate of the gravity force

mass 2mass 19 2

0charge mass2 2 3

charge 23 202 3/2

0

40 0

44.4 10 cm/sec

6.8 10 cm/sec( )

at the fundamental beam-revolution frequency 2 2.3 10 rad/sec,

grav pgrav

grav gravgrav

F GNma

M d

F FNrGN eaM Mc d d

d

3

12

160

10 cm - minimal distance between the beam and detector, = 1 km - radius of Tevatron,

=10 - the Lorentz factor,

10 - number of protons in bunch,

1.54 10 cm - radius of proton.

N

r

04/22/23 52

Respond of the detector to the gravity force

0

Equation of motion of the detector ( coordinate):

Solution of the inhomogeneous equation (Laplace-transform techn( ) ( ) ( ) ( )

1( ) ( ) Exp Sin

ique

2

):m x

tm

x mm t

Mx t H x t Kx t F t

Hx t F t t d

x

M M

0

Gravity force is pulse-like ( " ") and periodic: ( ) ( ), 2where - a period of one revolution of a proton's bunch. Hence, after

revolutions ( " ") of the n

bu

x xa hammer blow F T F

T n

n hammer blows

2 /2

0

0 /2

Sin2 2(

ch, the respond of the detector is approximated as follows (here we neglect the damping factor

) ( )Sin

):

Maximal

dis

(

pla

)

c

mT

x mm T

m

t

x t F t d t nT

H

M

2

0max

0

ement of the probe ma( )

ss: m

m

Fx

M

04/22/23 53

Laser Interferometer for Gravity Physics at Tevatron?

16 18(10 10 ) cm/L Hz

L = 999.5 md = 0.07 mL = 16.7 m

Current technology of LIGO coordinate meters allows to measure position with an error

+d

Probe mass

Proton’s beam

Probe mass

16 18

3

13

15

cm (10 10 )

For the detector frequency 10 Hz

4.4 10 cm

3.2 10 cm

m

LHz

x

L

04/22/23 54