Equivalence Principle

Transcription

1 July 16, 2015

2 Universality of free fall (Galileo) Aristoteles view: the manner in which a body falls, does depend on its weight (at least plausible, if one does not abstract from air resistance etc.) Galileo s experiments on an inclined plane: the speed of a body is independent of its weight; all bodies, regardless of their constitution, fall with the same acceleration This behaviour is fairly unique to gravitation; usually: the larger the acting force, the larger the acceleration.

3 Weak (WEP) The two different definitions of mass (i.e. inertial, gravitational) are equivalent; bodies of different constitution feel the same acceleration. in classical physics, it is not entirely clear why this is the case. Early experiments were already performed by Newton, Bessel; much more accurate ones between by Eötvös. in GR, gravity is explained geometrically: matter deforms space and time and all bodies are following the straightest lines in this distorted geometry no need for different mass-definitions like Newton: (a) force acting on a body depends on its gravitational mass (b) but the reaction on this force depends on the bodys inertial mass in GR, all bodies feel the same acceleration because their motion is determined by the very same space-time around them. WEP test traditionally interpreted in Newtonian terms as specified by the Eötvös is parameter a1 a2 η = 2 a 1 + a = 2 (mg /m i) 1 (m g /m i ) 2 (1) 2 (m g /m i ) 1 + (m g /m i ) 2

4 The Eötvös - experiment,, measurement using torsion balance two test-masses of different composition with equal weight equal gravitational mass a net torque will show up if the equivalence principle is violated in term of the Eötvös-parameter a1 a2 η = 2 a 1 + a (2)

5 : Dicke s Experiment The experiment is based on measuring the effect of the gravitational field on two masses of different material in a torsional pendulum. GMm (E) G R 2 = m(e) I v 2 R The forces acting on both masses are: F (j) + GMm(j) G R 2 = m(j) I v 2 R and the total torque applied is: L = we assumed that : m (1) I = m (2) I = m. v 2 = GM R F (j) = GMm(j) I R 2 ( ) (E) mg m I [ ( ) (E) mg m I ) [ (F (1) F (2) l = GM m (1) R 2 G ( mg m I m(2) G ) (j) ] ] l here

6 The Eötvös - experiment Operating principle of the Eötvös torsion balance. 1 This idealized balance consists of two test bodies attached to a rigid, massless frame that is supported by a perfectly flexible torsion fiber. F 1 and F 2 denote the external forces on the test bodies. The torque about the fiber axis is ( F1 ) F 2 r 12 T z = F 1 + F (3) 2 The signal is the change in T z when the instrument is rotated about the fiber axis so that the component of r 12 along the direction of F 1 F 2 changes sign. 1 T A Wagner et al 2012 Class. Quantum Grav

7 The Eötvös - experiment

8 List of Experiments Figure:

9 in Einstein s Theory The gravitational field has only a relative existence... Because for an observer freely falling from the roof of a house - at least in his immediate surroundings - there exists no gravitational field. (Einstein) Because of the equivalence between gravitational and inertial mass, a freely falling observer won t feel his own weight, nor any effect of gravity gravity can be (nearly) transformed away (at least locally) locally, a gravitational field and a uniform accelerated frame of reference are equivalent in any and every local Lorentz frame, anywhere and anytime in the universe, all the (non-gravitational) laws of physics must take on their familiar special-relativistic forms.

10 Strong The existence of local inertial frames in a gravitational field forms the basis of Einsteins Strong (SEP): SEP: Locally and at any point of spacetime physics is that of special relativity (Lorentz invariant) and is not affected by the presence of a gravitational field. We can quantify the notion of locally: the spatial size D of the local inertial frame is much smaller that the typical lengthscale L of the gravitational field, R αβγδ 1/L 2. Hence, in a region D << L the field is almost homogeneous and can be cancelled by inertial forces. The SEP implies the WEP: the behavior of freely falling test-bodies in a gravitational field is locally (i.e. in a local inertial frame) indistinguishable from that of a free body. Since the latter is universal, also freely falling bodies should behave in a universal way.

11 and the Laws of Physics The principle of equivalence has great power. One can generalise all the spacial relativistic laws of physics to curved space-time, and the curvature not needs to be small. Apart from singularities the EP acts as atoll to mesh all the non-gravitational laws of physics with gravity. EXAMPLE This special relativistic rule, i.e. T µν,ν = 0 holds also true in presence of gravitation; it is valid in a freely falling frame of reference In a freely falling system, the connection coefficients (Christoffel-symbols) vanish, i.e. Γ µ νκ = 0 at the origin of the freely falling system In such a system at that point it is T µν ;ν = 0 Laws of physics are independent of the coordinate system, so in curved spacetime we have also T µν ;ν = 0. MORAL The laws of physics, written in astract geometric form, differ in no way whatsoever between curved space-time and flat space-time The laws of physics, written in component form, change on passage from flat space-time to curved space-time by a mere replacement of all commas by semicolons.

12 Tests of the Weak (WEP) One elementary equivalence principle is the kind Newton had in mind when he stated that the property of a body called mass is proportional to the weight, and is known as the weak equivalence principle (WEP). An alternative statement of WEP is that the trajectory of a freely falling test body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition. In the simplest case of dropping two different bodies in a gravitational field, WEP states that the bodies fall with the same acceleration (this is often termed the Universality of Free Fall (UFF)). The Einstein equivalence principle (EEP) is a more powerful and far-reaching concept; it states that: 1. WEP is valid. 2. The outcome of any local non-gravitational experiment is independent of the velocity of the freely-falling reference frame in which it is performed - local Lorentz invariance (LLI). 3. The outcome of any local non-gravitational experiment is independent of where and when in the universe it is performed - Local Position Invariance (LPI).

13 Test of WEP Figure: Selected tests of the weak equivalence principle, showing bounds on η, which measures fractional difference in acceleration of different materials or bodies. The free-fall and Eötvös experiments were originally performed to search for a fifth force (green region, representing many experiments). The blue band shows evolving bounds on η for gravitating bodies from lunar laser ranging (LLR). [C.M. Will]

14 Lunar Laser Ranging (LLR) By using laser shootings to the Moon, one can accurately measure the distance from Earth and to detect possible shifts (initially suggested by Newton). During the 1st lunar landing ( Apollo 11) a reflector was deposited on the Moon. In total 5 reflectors, deposited by robotized Soviet missions and by US astronauts ( ). If the distance Earth-Moon is known with 2cm accuracy, one can compare the measured orbit with the theoretical orbit. In this way they have been able to validate the EP with a relative accuracy better than After 2006, a new station of laser telemetry in construction in New Mexico (Ranging Laser APOLLO) will provide Earth-Moon distances with an accuracy of a few millimetres. The integration of the signal on several years will enable a relative precision of for the test of the principle of equivalence.

15 Tests of Local Lorentz Invariance (LLI) A simple and useful way of interpreting some of these modern experiments, called the c 2 -formalism, is to suppose that the electromagnetic interactions suffer a slight violation of Lorentz invariance, through a change in the speed of electromagnetic radiation in other words, c 1 Such a violation necessarily selects a preferred universal rest frame, presumably that of the cosmic background radiation, through which we are moving at about 370 km s 1. Such a Lorentz-non-invariant electromagnetic interaction would cause shifts in the energy levels of atoms & nuclei that depend on the orientation of the quantization axis of the state relative to our universal velocity vector, and on the quantum numbers of the state. The presence or absence of such energy shifts can be examined by measuring the energy of one such state relative to another state that is either unaffected or is affected differently by the supposed violation. One way is to look for a shifting of the energy levels of states that are ordinarily equally spaced, such as the Zeeman-split 2J + 1 ground states of a nucleus of total spin J in a magnetic field; another is to compare the levels of a complex nucleus with the atomic hyperfine levels of a hydrogen maser clock. The magnitude of these clock anisotropies turns out to be proportional to δ = c 2 1.

16 Test of LLI Figure: Selected tests of local Lorentz invariance showing the bounds on the parameter δ, which measures the degree of violation of Lorentz invariance in electromagnetism. The Michelson-Morley, Joos, Brillet-Hall and cavity experiments test the isotropy of the round-trip speed of light. The centrifuge, two-photon absorption (TPA) and JPL experiments test the isotropy of light speed using one-way propagation. The most precise experiments test isotropy of atomic energy levels. The limits assume a speed of Earth of 370 km s 1 relative to the mean rest frame of the universe.

17 Test of Local Position Invariance (LPI) The principle of LPI, the 3rd part of EEP, can be tested by the gravitational redshift experiment Despite the fact that Einstein regarded this as a crucial test of GR, we now realize that it does not distinguish between GR and any other metric theory of gravity, but is only a test of EEP. The gravitational redshift experiment measures the frequency or wavelength shift Z ν/ν = λ/λ between two identical frequency standards (clocks) placed at rest at different heights in a static gravitational field. If the frequency of a given type of atomic clock is the same when measured in a local, freely falling frame (Lorentz frame), then the comparison of frequencies of two clocks at rest at different locations boils down to a comparison of the velocities of two local Lorentz frames The frequency shift is then a consequence of the first-order Doppler shift between the frames. The structure of the clock plays no role whatsoever. The result is a shift Z = U/c 2 where U is the difference in the Newtonian gravitational potential between the receiver and the emitter. If LPI is not valid, then it turns out that the shift can be written Z = (1 + α) U/c 2 (4)

18 Test of LPI (Pound-Rebka-Snider) The first successful, high-precision redshift measurement was the series of Pound-Rebka-Snider experiments of that measured the frequency shift of gamma-ray photons (14keV) from 57 Fe as they ascended or descended the Jefferson Physical Laboratory tower at Harvard University. The high accuracy achieved one percent was obtained by making use of the Mössbauer effect to produce a narrow resonance line whose shift could be accurately determined. Other experiments since 1960 measured the shift of spectral lines in the Sun s gravitational field and the change in rate of atomic clocks transported aloft on aircraft, rockets and satellites.

19 Test of LPI Figure: Selected tests of local position invariance via gravitational redshift experiments, showing bounds on α, which measures degree of deviation of redshift from the formula ν/ν = U/c 2. In null redshift experiments, the bound is on the difference in α between different kinds of clocks.

20 Test of LPI - Gravity Probe-A The most precise standard redshift test to date was the VessotLevine rocket experiment known as Gravity Probe-A (GPA) that took place in June A space probe was launched vertically, at less than escape velocity, to an altitude of 10,000 km, after which it fell back to earth and crashed down in the Atlantic Ocean. The probe carried a hydrogen maser clock which was used to control the frequency of a radio signal. The radio signal was received on the ground, the nonrelativistic Doppler shift was subtracted out, and the residual blueshift was interpreted as the gravitational effect effect on time, matching the relativistic prediction to an accuracy of 0.01%.

21 Test of LPI - Gravity Probe-A The experiment took advantage of the masers frequency stability by monitoring the frequency shift as a function of altitude. A sophisticated data acquisition scheme accurately eliminated all effects of the first-order Doppler shift due to the rockets motion, while tracking data were used to determine the payloads location and the velocity (to evaluate the potential difference U, and the special relativistic time dilation). Since the rocket is in motion relative to the earth for most of the flight the frequency of the maser on the rocket compared to that on the earth is given by: ν rocket = ( gµνuµ u ν ) 1/2 ν earth g 00 earth rocket (5) in the weak-field limit ( g µνu µ u ν ) 1/2 rocket g00 rocket ν 2 rocket = 1 + 2U rocket ν 2 rocket (6) Analysis of the data yielded a limit α <

22 MICROSCOPE: The Mission French space mission with participation of CNES, ESA, DLR (ZARM & PTB ) Mission goal: Test of with an accuracy of η = Mission overview: Payload: Micro-satellite of CNES Myriade series Dragfree satellite Sunsynchronous orbit Altitude about 800 km Mission lifetime of 1 year Two highprecision capacitive differential accelerometers Science sensor: Ti and Pt test mass Reference sensor: two Pt test masses

23 STEP - Satellite Test of the STEP is a joint European-U.S. space program to investigate the Equivalence of inertia and passive gravitational mass. The STEP experiment is conceptually a modern version of Galileo s Free-Fall Experiment. STEP will advance the sensitivity of EP tests by 5 or 6 orders of magnitude, into regions where the principle may break down. A violation of EP at any level would have significant consequences for modern gravitational theory. In STEP, the masses are in free fall in an orbit around the Earth, and if there is a violation of the EP they tend to follow slightly different orbits. The orbiting masses fall all the way around the Earth and never strike the ground, so that any small difference in the rate of fall can build a large displacement. SQUID magnetometer circuits are used to measure displacements as small as cm.

24 STEP - Satellite Test of the

25 Triangular Torsional Balance - Dicke