Testing the Equivalence Principle on Earth and in space

Pisa, 17 Aprile 2002

Anna Nobili

Equivalence of inertial-to-gravitational mass

im inertial mass

,g ga pm m gravitational (active,

passive) massa pg g gm m m action/reaction principle the CM

of an isolated system moves with constant velocity in an inertial frame

gravitational mass (“gravitational charge” analog to “electric charge” ): conceptually unrelated to inertial mass

0g ia

ma

m Universality of Free Fall

(UFF)

3g g

iGM m

m r rr

law of gravitational

interaction

First experimental evidence of UFF (no awareness of equivalence inertial-to-gravitational mass)

Galileo (I)

Awareness of the “differential effect” of the medium:

“…..veduto, dico questo, cascai in opininione che se si levasse totalmente la resistenza del mezzo

tuttel le materie descenderebbero con eguali velocità ” •Mass dropping (limitations…)

•Inclined plane (reduced air drag effects…..)

•Pendula (the best solution….)

•Galileo: pendulum tests ( 1 part in 103 )

Galileo (II)

Galileo: “e finalmente ho preso due palle, una di piombo ed una di sughero, quella ben più di cento volte più grave di questa, e

ciascheduna di loro ho attaccata a due sottili spaghetti eguali, lunghi quattro o cinque braccia, legati ad alto; allontanata poi l'una

e l'altra palla dallo stato perpendicolare, gli ho dato l'andare nell'istesso momento, ed esse, scendendo per le circonferenze de' cerchi descritti da gli spaghi eguali, lor semidiametri, passate oltre al perpendicolo, son poi per le medesime strade ritornate indietro;

e reiterando ben cento volte per lor medesime le andate e le tornate, hanno sensatamente mostrato come la grave va talmente

sotto il passo della leggiera, che né in ben cento vibrazioni, né in mille, anticipa il tempo d'un minimo momento, ma camminano con passo egualissimo. Scorgesi anche l'operazione del mezzo, il quale,

arrecando qualche impedimento al moto, assai più diminuisce le vibrazioni del sughero che quelle del piombo, ma non però che le

renda più o meno frequenti; anzi quando gli archi passati dal sughero non fusser più che di cinque o sei gradi, e quei del piombo di cinquanta o sessanta, son eglin passati sotto i medesimi tempi.”

[Le Opere, Vol. VIII p. 128]

Newton: pendulum tests ( 1 part in 103 )But this was about 80 years after Galileo’s first

experiments…

Newton•Newton major contribution (beyond UFF

equivalence)

Principia, (Definition I) : “this quantity that I mean hereafter under the name of ...mass ...is known

by the weight ...for it is proportional to the weight as I have found by experiments on

pendulums, very accurately made...”

• Pendulum tests later improved to 1 part in 105 (Bessel….)

Eötvös•Eötvös (1888-1905/08s): 1 part in 108 by torsion balance tests.

It was indeed a very subtle idea that a deviation from the proportionality between inertial and

gravitational mass should show up as a rotation of the balance (truly differential instrument….)

Note: Eötvös was not aware of the work by Einstein just in the same years….he had a very sensitive instrument and used it!!!

Einstein (I)• Einstein (1907): “hypothesis of complete

physical equivalence”

In a freely falling system all masses fall equally fast, hence gravitational acceleration has no

local dynamical effects

Weak Equivalence Principle (WEP) An observer inside Einstein elevator will not be able to tell, before hitting the ground, whether he is moving with an acceleration g in empty space, far away from all masses, or else he is falling in the vicinity (height of fall<<Earth radius) of a body (the Earth) whose local gravitational acceleration is also g (and in the same direction).

With the WEP Einstein has moved from Newton’s concept of one global reference frame with gravitational forces and the UFF, to many free falling local (h<<Rsource body) frames without

gravitational forces

Einstein (II)The cancellation of gravity in a freely falling frame holds locally for each frame, but the direction of free fall is not the same in all of them (the gravitational field of a body, like Earth, is not uniform tidal forces between test particles whose centers of mass are not coincident, and also inside a body of non zero dimensions.

Einstein (III)

“ Strong (Einstein) Equivalence Principle” (assumes WEP): all gravitational effects are (globally) replaced by the metric of a curved, 4-dimension space-time… General Relativity is based on WEPWEP UFF: should bodies of different composition fall with different accelerations, the elevator and the test mass inside it would generally fall with different accelerations and the observer would be able to tell that he is close to the surface of the Earth and not in an accelerated frame in empty space. In this sense the Equivalence Principle expresses the very essence of General Relativity and as such it deserves to be tested as accurately as possible.

The Eötvös parameter , which quantifies the violation of equivalence for two bodies of composition A and B, inertial mass mi and gravitational mass mg, can be generalized into:

])/()/[(

])/()/[(2

k

k

BigkAig

BigkAigk mmmm

mmmm

Each k 0 would define a violation of Equivalence between the inertial and gravitational mass-energy of the k-th type. If the Equivalence Principle holds, all these separate Equivalence Principles must hold, corresponding to a very peculiar coupling of each field to gravity (true only for the gravitational interaction!!!).

Scientific relevance of the Equivalence Principle (II)

a/a post-Newtonian deviations from General Relativity (e.g. measured by *)

proportionality factor: 10-5 10-3

*= - 1 Eddington parameter  * 10-3 by post Newtonian or pulsar tests

a/a 10-12 10-13 by EP tests * 10-7 10-9 or 10-8 10-10

The superior probing power of Equivalence Principle tests is beyond question !!!!

fractional differential acceleration between bodies of different composition falling in the gravitational field of a source mass

Scientific relevance of the Equivalence Principle (III)

Nearly all attempts to extend the present framework of physics predict the existence of new interactions which are composition dependent and therefore violate the Equivalence Principle. Equivalence Principle tests are by far the most sensitive low energy probes of such new physics, which is consistent with the fact that General Relativity is based on the Equivalence Principle, assumed by Einstein (1907) as the

“HYPOTHESIS OF COMPLETE PHYSICAL EQUIVALENCE between a gravitational field and an accelerated reference frame”

Scientific relevance of the Equivalence Principle (IV)

•Dicke: Eötvös tests lack frequency modulation and zero check test for UFF in the field of the Sun (rotation of the balance with the Earth provides 24-h modulation and zero check..) 10-11 (Roll et al., Ann. Phys. 1964)

Where are we? The ground tests after Eötvös

•Braginsky & Panov (Sov. Phys. JPT 1972): 10-

12

•“Eöt-Wash” (PRL 1990, 1994, 1999….): 10-13

Fischbach et al. (PRL, 1986) …

Where are we? EP tests by LLR

Anderson & Williams: (CQG, 2001) … 10-13

Great leap forward in sensitivity possible only in space (low Earth orbit –i.e. radius of the Earth range , UFF in the field of the Earth, ….)

What next?

• Driving signal about 3 orders of magnitude larger

• Absence of weight in orbit ideal for small force experiments

• High frequency signal modulation (reduced “1/f ” noise) by s/c rotation [no motor, no noise of motor]

• suspensions can be weaker than on Earth by many million times

STEP (USA): 10-18 (10-17 from Phase A Studies in Europe)

(requires payload to be maintained close to absolute zero temperature) In the SMEX competitionSCOPE (France): 10-15 Phase B Study

GG (Italy): 10-17 Advanced Phase A Study

The costs

STEP : GG : SCOPE=1 : 20% : 13%

The international context

GREAT (balloon test): 10-15 , competitive with SCOPE, lower cost, repeatibility… (ASI-NASA study)

The mission concepts

STEP & SCOPE mission concept (1-axis sensitivity, while effect is in 2-D; slow spin 103 sec)

GG mission concept (sensitive in plane of

signal; fast spin 0.5 sec).

Test masses suspension (I)

STEP: magnetic (by means of premanent currents in superconducting coils at very low temperature

GG: mechanical SCOPE: electrostatic (+ gold wire for electrostatic discharging)

The nature of the test requires:

•Differential accelerometer with test masses very weakly coupled (small k increases sensitivity),

•High modulation frequency (i.e. fast rotation reduces 1/f electronic and mechanical noise)

(Both requirements better satisfied in space than at 1- g)

Test masses suspension (II)Whatever the kind of suspension:

s n super-critical rotation

21

1 ( / )eq

s n

r

22 2

2, n

s n eqs

r

self-centering (requires 2-D motion !!!)

sub-critical rotation,

s n eqr

nkm

test body CM

suspension point

rotation center

r

k

ms

offset by construction

Accelerometer design: the STEP case (I)

The test cylinders are continuously recentered on each other by a servo loop: the output of the loop will contain the signal of a putative EP violation and any other effect acting between the masses.

Accelerometer design: the STEP case (II)

The accelerometer is differential, but reduces the 2 degrees of freedom of the motion to 1 super-critical rotation in 1-D not allowed, fast spin not allowed, active self centering needed (requires changing supercurrents in the suspension coils…)

SQUID read out (Common Mode & Differential Mode)

Accelerometer design: the SCOPE case

Each test cylinder is suspended independently, has its own capacitance read out and its own servo loop to keep itself “motionless”. A small signal is extracted as the difference of two large ones not a differential instrument

As in STEP, 1-D motion forbids fast spin and

makes passive self centering impossible

(active centering by off-setting the servo loops); balancing the restoring

force of the gold wire requires large active

forces continuosly applied to the test

masses

“Free“ test cylinders not possible…

Test cylinders cannot be free falling: must be suspended because of inevitable tidal effects if initial conditions differ by r (release error)

3

33equivalent tider GM

a rr d

STEP & SCOPE test cylinders centered actively (physical equilibrium centring not sufficient); GG test cylinders centered by physical equilibrium):

, , , ,

, ,

110 42

0.4

tide STEP signal STEP tide SCOPE signal

tide GG signal GG

a a a a SCOPE

a a

Blaser (CQG, 2001): “In the clever concept of GG, the CM of the two masses confined by super-critical rotation actually exactly follow the orbits of figure 1”

Accelerometer design: the GG case (I)

GG instrument differential: test

bodies coupled by the capacitance

read-out & by the suspensions (masses free to oscillate) like in an ordinary beam

balance (effective and well tested) + 2

Hz spin

Note: Best sensitivity on Earth achieved with differential instrument & frequency modulation (rotating torsion balance)

Accelerometer design: the GG case (II)

2 differential accelerometeres centered on CM of spacecraft: internal acceleormeter for EP testing

(different composition), external one for systematic checks (equal composition)

GG: the s/c around the accelerometer (I)

• 97.5 inclination

• 520 km altitude

• 0 eccentricity

Sun-synchronous rather than equatorial…..

Launch at good price with high latitude DNEPR russian rocket (former SS-18 missile, from Baikonour cosmodrome)

300 kg mass

spin axis (passive) stabilization

1m

GG: the s/c around the accelerometer (II)

Largest disturbance due to residual air drag: 108 times smaller than 1g…but 108 times larger than target signal….. Partial drag compensation (FEEP thrusters) & partial CMR (Common Mode Rejection)

Equatorial orbit easier (no regression of the nodes, stays perpendicular to s/c spin axis…) but sun-synchronous orbit much less expensive to reach (from high latitude russian launchers….)

• High frequency modulation of expected signal (reduction of "1/f" electronic and mechanical noise; satellite designed for rapid spin, easy attitude stabilization).

• Mechanical suspensions ensure electric grounding, hence eliminate charging effects passively (low stiffness ensured by weightlessness high sensitivity)

• Instrument inherently differential: test bodies coupled by the read out and by very weak suspensions like in an ordinary beam balance for higher sensitivity to differential forces

GG: main novelties and advantages (I)

• Capacitance read-out at room temperature: sensitive, reliable, well known

• Fast spin makes many perturbing effects DC (constraints on payload and s/c released)

• Ground testing of FULL payload prototype possible !!!!

• 2-axis sensitivity (in the plane perp. to symmetry axis): maintains 2 DOF of signal; allows supercritical rotation ( self-centering + high frequency modulation); improves sensitivity by factor 2 w.r.t having only one sensitive axis

GG: main novelties and advantages (II)

It would be mis-interpreted as a violation signal because, like it, it is not detected by the accelerometer made of test bodies of equal composition (density) used for checking of systemtics

EP tests in LEO: the most subtle disturbance (I)

Radiometer effect: residual gas pressure p and temperature gradient dT/ds (due to exposure of the s/c to a radiation source) cause an acceleration on the test body of density (different for bodies of different composition):

2rep dT

aT ds

In LEO, exposure to infrared radiation from the Earth causes a differential radiometer effect with the same signature as an EP violation signal

STEP: kills the radiometer effect by very low pressure (thanks to very low temperature) and by keeping temperature gradients across the test cylinders small)

Note: Thermal noise depends on T/m (m=mass of test body), which is about the same in STEP and GG (GG has lower temperature but more massive test bodies)

EP tests in LEO: the most subtle disturbance (II)

SCOPE (design similar to STEP, but room temperature): unless it is thermally isolated better than LISA, the radiometer effect is 20 times larger than its target signal !!

GG: kills the radiometer effect at room temperature because the test cylinders spin (2 Hz) around their symmetry axis [natural choice] thus averaging out temperature gradients

Nobili et al. PRD, 63, 101101, 2001; submitted 2001

Unstable whirl motions: slow if Q factor high; can be damped

Nobili et al, CQG 16, 1463 (1999)

Whirl damping in GG (I)The 6-body GG system with realistic noise: damping of whirl motions

GG Phase A Report, ASI, 1998 http://eotvos.dm.unipi.it/nobili/ggweb/phaseA

Whirl damping in GG (II)

The 6-body GG system with realistic noise. The plot shows the relative displacement between the inner and the outer test mass(differential displacement) obtained after applying the active control of their whirl motions

GG Phase A Report, ASI, 1998

Result obtained

under worst case assumptions:

• launch close to solar maximum

• maximum drag value along orbit

• Q as measured in the lab

GG: error budget consistent with EP testing to 10-17

(GG Phase A Study Report, ASI Nov 1998; 2nd Edition Jan. 2000; GG Advanced Pahse A Study, 2002)

Ground test of proposed space apparatus

The GGG prototype

(“GG on the Ground”)

Design of GGG apparatus mounted inside vacuum chamber (1 m diameter)

•First generation experiment completed (LABEN laboratories, Florence)

GGG prototype: overview GGG apparatus inside vacuum chamber in the basement of LABEN lab (Florence)

Test cylinders (rotors) 10 kg each.

3 laminar suspensions (stiff along vertical, soft in horizontal plane, suspend and couple the

test cylinders like in an ordinary beam balance

(vertical beam in this case)

EP signal differential best sensitivity with differential apparatus (so far, torsion balance) test bodies must be coupled

GGG prototype: test cylinders coupled by the suspensions

4 capacitance plates halfway in between the test cylinders form 2 capacitance bridges sensitive to differential displacements in the horizontal plane (they sense common mode effects too, but only to second order)

GGG prototype: test cylinders coupled by read out too

Central laminar suspension carrying whole weight. It connects the midpoint of the coupling arm (bottom) to the suspension shaft (top).

Inclination of coupling arm is adjusted (in 2D): corse 5 gr; fine 0.5 gr; driving signal

comes from main sensors as a constant

offset in rotating frame — once capacitance

bridge has been electrically balanced)

GGG prototype: the central suspension

GGG prototype: verticality of shaft

Verticality of suspension shaft is adjusted with differential micrometric screws (0.07 m resolution fine) and with PZT (much finer and from remote). Driving signal comes from main sensors (as an offset fixed in non rotating frame)(3 Micrometric screws at 120 and 3 PZTs at 120)

GGG prototype: the read out (I)

Capacitance plates forming 2 capacitance bridges to measure differential displacements of test cylinders in the horizontal plane (main sensors). Sensitivity depends on good centering peek frame for high vacuum built in 1 piece and then cut to ensure symmetry….; test cylinders measured accurately..

Test cylinders (10 kg ach) being precisely measured at Laben

Result data needed as input in the design of the mounting structure of the capacitance sensors (to be located halfway between the test cylinders for best sensitivity to differential effects)

GGG prototype: the read out (II)

Sensitivity obtained (on bench): 5 picometer in 1 sec integration time (ok for GG target in 100 sec)

GGG prototype: the read out (III)

Led for optical transfer of measurement data from rotating bridges to computer outside vacuum chamberRotating electronics takes care of: demodulating signal of bridges, digitizing and formatting it for computer acquisition

GGG prototype: the read out (II)

Calibration of read out capacitance bridges performed by applying a relative displacement of test cylinders of known amount (with micrometric screw) Calibration factors (FebJune 2001) of bridges in X and Y sensitive axes:

X 1 Volt= 29.87 m

Y 1 Volt=30.46 m

GGG prototype: calibration of read out

Led (non rotating)used to get a voltage signal once per spin period of the rotor (reference signal)Microprocessor outside the vacuum chamber will process the reference signal, combine it with measurement data of the bridges, and format it for computer acquisition as binary file

C program reads binary file of all data

(bridges+reference signal) in a run and transforms it in

a text file ready to be analysed

GGG prototype: the reference signal (I)

Reference signal gives rotation rate and phase of the rotor allows us to transform data from reference frame rotating with test cylinders to the laboratory (non rotating frame) where the expected signal is a constant displacement vector (for EP violation in the field of the Earth) or slowly following the daily motion of the Sun (for EP violation in the field of the Sun

Narrow band of raw data (in rotating frame) is filtered around spin

frequency, and then transformed to non

rotating frame cleaner results (advantage of signal modulation)

GGG prototype: the reference signal (II)

Reference signal also sent to another computer (as analogic signal via NI card and Labview acquisition) for independent analysis of system rotation, to check that combination of the reference signal with data coming from the capacitance bridges is performed correctly by the microprocessor

Check proved very important in data analysis

of first generation GGG measurements

GGG prototype: the reference signal (III)

GGG prototype: damping of whirl motions (I)

Passive oil damping

On-off control from remote

GGG prototype: damping of whirl motions (II)

Active damping (for better accuracy) with small capacitance sensors/actuators (2 bridges each)

Electronics for sensing bridges same as electronics of main sensors; ad hoc electronics for actuating bridges

GGG: results from first generation prototype (I)

Relative displacements (X,Y) between the centers of mass of the test cylinders at zero spin rate, 8 sec natural periods of differential oscillations. The amplitudes of these oscillations are slowly decreasing with time; data sets taken at subsequent times under no changes in the system yield Q1590. Residual air pressure during this measurement is of 210-5 mbar.

Q at natural period of

differential oscillation of

the test cylinders with

respect to one another

GGG: results from first generation prototype (II)

Q measurements as function of the residual air pressure. Each point refers to a separate run. For pressures lower than10-3 mbar Q reaches about 1590 and is then independent of pressure since it is the maximum value allowed by mechanical losses in the laminar suspensions.

Q at natural period of

differential oscillation of

the test cylinders with respect o one

another

GGG: results from first generation prototype (III)Measurement of quality factor Q of the system (at 2.5 Hz spin): with damping system turned off, raw data show no appreciable growth in 3 hr run at 2.5 Hz

In a cleaner run, undamped whirl

motion at 0.74 Hz shows growth

compatible with Q 5000

(Q 2000 measured for the suspension

alone at larger amplitudes)

(GG target in space requires Q=20000)

Proof that whirls are no limitation and experimental value of Q (measured with full system running) close to requirement !!!

GGG: results from first generation prototype (IV)Measurement of gyroscopic effect (test cylinders spin in laboratory subject to the diurnal rotation of the Earth) Effect

constant in North-

South direction; changes sign with sense of

spin; increases

lineraly with spin

rate

GGG: results from first generation prototype (V)Stability around displacement vector due to gyroscopic effect: 1.5 m in 1 hr (at 2.5 Hz), with differential natural period of 3.5 sec

After scaling for the much stiffer suspensions on

Earth as compared to

space in absence of

weight: 540 sec vs 3.5 sec

differential period) this is

only a factor 100 away from what is requested for

GG in space

GGG: results from first generation prototype (VI)Measured effects around gyroscopic vector due to seismic tilts, as checked with ISA acclereometer (by V. Iafolla) operating nearby

Time interval corresponding to GGG run

GGG: 2nd generation prototype

Improved electronics with hardwired phase information on each data point (ring with holes …). Accuracy of reference system transformation crucial ….Weaker laminar (cardanic) suspensions: reduction of seismic tilt effects, better rejection of common mode effects (i.e. horizontal seismic noise), longer differential periods higher sensitivity … (need to gain factor 100)Bi-axial tiltmeter to provide driving signal for active control of tilts with PZT: crucial to improve sensitivity and for long term measurements (testing for EP violation in the field of the Sun)

GGG: 2nd generation prototype

construction drawing of GGG second generation prototype: general view

GGG: 2nd generation prototype (II)

GGG: 2nd generation prototype (III)

Improved electronics with hardwired phase information on each data point (ring with holes …). Accuracy of reference system transformation crucial ….

Ad hoc testing apparatus

GGG: 2nd generation prototype (IV)

GGG: 2nd generation prototype (V)

Calibration (open chamber): after 2 days is constant by 0.4 % in one channel; by 3 % in the other)

GGG: 2nd generation prototype (VI)

Q measurements: 2002 data at 12 sec natural period; 2001 data at 8 sec natural period

GGG: 2nd generation prototype (VII)

Best Q value at 12 sec natural period: 1500

GGG: 2nd generation prototype (VIII)

2( ) (0) (0)t Q tA t A e A e

Q=20000 @ 2 Hz needed in space experiment

Q=16540 measured @ 0.90625 Hz

work in progress….

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