Fundamental Physics:
High precision tests of Special
and General Relativity

Special Relativity

Basics of Special Relativity

Postulates: two of the following three postulates are needed in order to derive SR

1.   constancy of the speed of light in all frames of reference

2.   validity of the Relativity Principle: there is no way to distinguish between different frames

3.   a body which is in uniform linear force-free motion in one frame of reference is in a uniform linear force-free motion in all frames


An example of the Relativity Principle:
In the two frames in relative motion, the experimenters measure the same electric force between two charges

Basic effects of Special Relativity

  • Time dilation
  • Length contraction
  • Einstein's velocity addition theorem
  • Doppler effect
  • aberration
  • Thomas precession

Doppler effect



Aberration of light:
If an observer has a velocity component perpendicular to the line connecting it with a light source, the light reaching the observer from the source has an angle of incidence that points in the direction of the velocity component. This effect was discovered already in the 18th century.

Motivation for testing Special Relativity

There are three main reasons for performing new and improved tests:

  • Special Relativity is one basic pillar of modern physics. Each basic theory has to be tested as its best, because it is so fundamentally important.
  • Test of validity of Maxwell's equation: Since light is a consequence of Maxwell's equations, each test of SR is also a test of the validity of Maxwell's equations.
  • There are predictions from loop gravity (Pullin and Gambini 1999, Alfrao et al. 2000, Alfrao et al. 2001) and string theory (Ellis et al 1999, Ellis, Mavromatos and Nanopoulos 1999, Ellis, Mavromatos and Nanopoulos 1999a) giving modified Maxwell and Dirac equations that violate Lorentz invariance. Violations of Lorentz invariance also arise in extensions of the standard model (Colladay and Kostelecki 1997) and in non-commutative field theories (Carroll et al 2001).

Special Relativity Motivation

Tests of Special Relativity

There are two possibilities to test SR:

  • testing the postulates
  • testing the consequences

Testing the postulates: While it is conceptually clear how to test the first postulate, the other two can never be tested completely because they are statements about all physical phenomena or all physical objects. Nevertheless, searches for effects violating e.g. the relativity principle are being performed. If there is one physical phenomenon which violates the Relativity Principle, then SR must be wrong.

In order to have a guideline at hand how to make a complete test of SR, Robertson (1948) invented a test theory which singled out three tests which - given particular results - will prove the validity of SR. These three tests are

1.   tests of the isotropy of the velocity of light: In an arbitrarily moving frame the velocity of light is independent of the direction of propagation

2.   test of the independence of the velocity of light from the velocity of the source or of the laboratory

3.   test of the time dilation (Doppler effect)

The firsts test are called Michelson-Morley tests (Michelson and Morley 1887), the second Kennedy-Thorndike tests (Kennedy and Thorndike 1932) and the last Ives-Stilwell tests (Ives and Stilwell 1938), according to the persons who carried through the corresponding tests for the first time.

If the first two tests give a null result and the third the well known time-dilation effect, then SR has been tested to be valid - at least within the experimental accuracy.

Until the 1940s the constancy of the velocity (1. and 2. above) of light was tested with interferometers. Modern tests use optical resonators and lasers. Tests of the Doppler effect use spectroscopic techniques applied to fast atoms or ions

Testing the consequences: The consequencs of SR have been widely tested. Furthermore, the results of SR are used in many areas of physics and are even important in everyday's life. Most high-energy experiments at the big accelerators cannot be interpreted without SR. Furthermore, GPS, telecommunication, satellite ranging, etc. are heavily based on the validity of SR.

Isotropy of velocity of light:
The velocity is independent of propagation direction, no matter what the direction and velocity v of the laboratory (here shown in green) is.

Independence of the velocity of light from the velocity of the laboratory:
The actual value of c does not depend on v

Time dilation for a moving atom:
In the rest frame, the transition frequency of an atom is f. If it moves with a velocity v with respect to the laboratory, its fluorescence is reduced when it is irradiated simultaneously by two laser waves of frequencies f‘ and f‘‘ that propagate against and in the direction of the atom‘s motion, respectively. Special Relativity predicts that the two laser frequencies obey the relationship f‘ f‘‘ = f2 . Experiments that test this prediction are performed at the Max-Planck-Institut für Kernphysik in Heidelberg (Gwinner et al.)

A. Michelson

The relativistic Doppler effect on the frequencies of the atomic clocks on board of the GPS satellites must be taken into account in calculating the position of the observer (see Physics Today, May 2002).

Our tests

   We are performing laser tests of the first two postulates of SR. These tests use highly stable optical resonators. The measured quantity is the frequency of the standing wave in the resonator, which is related to the length of the cavity and the velocity of light.

   In the case of a violation of the first two postulates, the velocity of light would depend on the direction of the propagation of light and on the velocity of the laboratory. Usually this is described by the following ansatz where the velocity of light c depends on the orientation q and on the velocity v of the apparatus:




Special Relativity is valid, if A and B vanish. A hypothetical orientation and velocity dependence of the speed of light would imply a dependence of the resonance frequency of the resonator on its orientation and on its velocity. This is because the frequency of a resonator of length L is given by




where n is an integer mode number.
Therefore, the search for a variation of the resonator frequency correlated with the orientation and with the velocity amounts to a test of the first two postulates of SR.

One needs a rotating platform and a platform which moves at different velocities, respectively. For both it is possible to use the motion of the Earth: The rotation of the Earth gives a complete revolution in 24 hours and the motion of the Earth around its axis and around the sun gives a 24-hour and a 1-year period variation in the velocity, respectively. However, for the isotropy test it is favourable to use a mechanical rotation table, although this makes the experimental setup more complex.

The experiments require

  • highly stable optical resonators (while rotating)
  • methods for a precise determination of frequency variations


Past experiments: Earlier, we used crystalline optical resonators, COREs (Cryogenic Optical REsonator, see Fig. 1, and Seel et al 1997 ) kept at liquid Helium temperature. Under these conditions they don't show any discernible long-term variation in their length. The shorter-term instability was also excellent - the comparison of the frequencies of two independent resonators in two independent cryostats showed an instability of the order 5*10-16 for integration times of 1 hour, see Braxmaier et al (CPEM2000). The stability of COREs is therefore higher than the stability of Hydrogen masers.

A first experiments using COREs was performed in Konstanz, in the group founded by S. Schiller and later led by A. Peters (Fig. 3). They are described in (Braxmaier et al 2002), (Schiller, Braxmaier et al), and (H. Müller et al, Phys. Rev. Lett. 2003).

Later experiments were performed in Düsseldorf using a pair of COREs that were rotated (Antonini et al 2005, and Schiller et al). They improved the test of isotropy further, by approx. a factor of 8. (Figs. 4, 5)

Current experiments: Currently, we are developing a new Michelson-Morley experiment that will significantly improve on our previous ones. The concept for the experiment, presented at the 2005 Potsdam meeting on Special Relativity, includes:

1.   A monolithic glass block made from a special glass with ultralow thermal expansion coefficient (ULE), that contains to standing-wave cavities oriented at 90 degrees (Fig. 7).

2.   A very precise read-out of the cavity frequencies is performed using a sophisticated two-stage laser frequency locking technique.

3.   An actively rotated setup is used, with active tilt control.

4.   A special housing is used to minimize environmental disturbances (Fig. 8).


We are also continuing the development of a Kennedy-Thorndike-type experiment. For this purpose we are comparing the frequency of a CORE with an atomic reference. At present we are using a Hydrogen maser as the microwave atomic reference (with stabilization to GPS) and an optical frequency comb to link with the CORE (at 280 THz), see Fig.8.

Fig. 1 Cross sectional view and photograph of a sapphire CORE.
The mirror reflectivity is R = 99.997%

Fig. 2 Transmission spectrum of a longitudinal mode of a CORE as a function of laser frequency

Fig. 3 Setup of the Michelson-Morley experiment with stationary cryogenic resonators. This experiment (Müller et al, 2003) took data for more than one year and at the time provided the most precise test of the isotropy of velocity of light.

Fig. 4 View of the open cryostat. The grey cylinder on the top is part of the pulse tube cooler. Vertical size is about 1.2 m.

Fig. 5 Detail view of the optical components inside the cryostat. An optical fiber (green) brings the laser light to the CORE (located inside the box).

Fig. 6 The monolithic double-cavity block, made from ULE glass. Dimensions are 8.4*8.4*3 cm. Cavity finesse is approx. 200,000.

Fig. 7 Thermal insulation chamber containing the actively rotated room-temperature Michelson-Morley-experiment..

Fig. 8 Frequency comparison between a CORE and a H-Maser

General Relativity

Basics of General Relativity

   General Relativity rests on the Einstein Equivalence Principle (EEP) which consist of three parts

1.   Universality of Free Fall (UFF), also called the Weak Equivalence Principle

2.   Local Lorentz Invariance (LLI) which implies the local validity of Special Relativity

3.   Local Position Invariance (LPI), which implies the universality of the gravitational red shift

UFF states that in a gravitational field all structureless point-like particles with the same initial position and initial velocity follow the same path. This is the condition for gravitation to be geometrizable.
LLI states that at a point and in a sufficient small neighbourhood (so that no gravitational gradient effects play any role) Special Relativity is valid. Therefore, tests of SR are also tests of the validity of GR. The consequence is that at each point in space-time there are a unit of length and of time which are related by the speed of light. (In more mathematical terms: At each space-time point there is a Minkowskian metric.)
LPI states that all local nongravitational experiments (an electromagnetic experiment, for example) are not influenced by the gravitatonal field. (This may be interpreted as a gravitational Relativity Principle: There is no way to distinguish with local experiments between various gravitational fields. - Of course, experiments and observations performed over larger scales, or extended objects, indicate the existence of a gravitational field in terms of tidal forces.) In other words: The outcome of any non-gravitational experiment is independent of where and when in the universe it is performed. The consequence of this postulate is that all length and time scales introduced at different points by means of the previous postulate, are the same.

   The consequence of the Einstein Equivalence Principle is that gravity has to be described by means of a space-time metric, that is, by means of the model of a Riemannian geometry.

   At this pont it is still open which equations determine the space-time metric. In the frame of GR this is done by the Einstein equations. A widely used formalism which describes tests confirming these equations is the PPN formalism.

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Basic effects and features

Light bending, red shift, perihelion shift, time delay, Lense-Thirring effect, gravitational waves, physics of binary systems, Black Holes, Cosmology


Tests of General Relativity

There are two classes of tests of GR:

1.   tests of the EEP

2.   tests of the validity of Einsteins equations

   Tests of the UFF started with Eötvös and today have reached an accuracy of 10-12. Satellite tests of the UFF are in preparation which aim at a accuracy of 10-18 (STEP).

  Tests of LLI (Special Relativity) have been described above.

   Tests of LPI can be implemented via tests of the universality of the gravitational red shift of clocks:





Here n(x1) and n(x0) are the frequencies of the clocks at position x1 and x0. The reason for the dependence of a clock's frequency on the gravitational potential U at the clock's location is that a clock is based on a fundamental physical interaction (the electromagnetic, the weak or the strong interaction). If one of these interactions "feels" the gravitational field, then the clock is sensitive to this field.

Consider now the possibility that gravity couples in a different way to the other three fundamental forces. This could be described by a clock dependent correction factor bclock.


 If two dissimilar clocks are moved together in a gravitational field from x 0 to x1, the ratio of the frequencies is related by




The LPI-violation effect one searches for is proportional to the difference in gravitational potential which the clocks experience between the two locations x 0 to x1. Therefore, space tests are best suited for that purpose. However, also on Earth such tests are possible if one takes the variation of the laboratory's position in the gravitational field of the sun into account. Due to the smallness of the gravitational potential, the universality has so far been tested to an accuracy of 10-4 only.

One consequence of the validity of LPI is that all fundamental constants, e.g. the fine structure constant


or the ratio of the electron and proton mass are time- and position-independent. In other words: any time- or position-dependent fundamental constant indicates a violation of LPI, thus a violation of EEP and therefore a violation of GR.

   Tests of the validity of Einstein's equations mainly consists of planatery and astrophysical observations. Such tests are the famous light bending at the sun, the perihelion shift of Mercury, Shapiro's time delay and the effects observed for binary systems like the Hulse-Taylor pulsar system. All these observations are in accordance with the Einstein equations at the 10-4 level.


New motivations for testing GR

There are again important reasons for testing GR:

  • GR as a fundamental theory always calls for more and more precise confirmation.
  • Theories attempting the unification of all forces predict a violation of the EEP.
  • String theory predicts an additional scalar gravitational field, which leads to a time- and position-variation of the fundamental constants, like the fine structure constant  





Furthermore, these theories also predict a deviation from the Newtonian gravitational 1/ r potential.

Our tests

We carry out two tests: the universality of the gravitational red shift, and a test of the Newtonian 1/r-potential at short distances.

Universality of gravitational red shift

The universality of the gravitational red shift can be tested using two oscillators of different physical nature. If the frequencies of these two oscillators depend on the various fundamental constants in a different way, then the ratio or the difference of these frequencies depend on time if these "constants" depend on the strength of the gravitational potential U.

   Such a situation can for example be realized by comparing the frequency defined by a resonator and the frequency of atomic or molecular transitions, see Storz et al 1998. Here the idea is the following: The frequency of a resonator mode n is inversely proportional to the length L of the resonator, n ~ 1/L. Since the length of the cavity scales with the Bohr radius and since this radius is inverse to the fine structure constant, the frequency of the cavity scales witha.
On the other hand, electronic transitions in atoms and molecules scale with
a2. Therefore, the ratio of frequencies depends on the fine structure constant. Consequently, if a depends on U, so does the ratio of frequencies. The results of our first test of the universality of the gravitational redshift for an electronic transition can be found in Braxmaier et al 2002.
Currently, we are setting up a new experiment in which the frequency of a CORE will be compared with the frequency of a hydrogen maser (Fig. 8). The comparison will be performed by means of an optical frequency comb based on a mode-locked femtosecond laser.

   The measurement is similar to our test of SR described above. Here the state of motion of the COREs do not play any role.

Testing the Newtonian 1/r-potential at short distances

  Non-Newtonian effects are usually described by an additional Yukawa-like gravitational potential of range l which adds to the usual Newtonian potential with a certain strength a.




    In 1999-2003 we developed an experiment to test whether non-Newtonian effects exist in the range 10-4-10-3 m. The experimental setup is shown in Fig. 9 on the right. The principle of this measurement is the following: a sensor (micromachined torsional oscillator, see lower photo), which can be modeled as a mass attached to a spring, is gravitationally excited by a source mass driven at the sensor's resonance frequency.

   The sensor motion is detected by a He-Ne laser beam, which is reflected by the oscillator head and impinges on a position sensitive photodiode. In order to detect the small gravitational signal, the elimination of external disturbances is crucial. For this reason sensor and exciting mass (toothed wheel) are mounted on vibration isolation systems, which are composed of metall disks separated by small elastomer pieces. A further background effect to be taken into account is the electrostatic attraction caused by residual charges on the sensor and on the exciting mass. A thin grounded metallic plate is therefore inserted between the two and acts as electrostatic shield (see figure above).

   The experiment is described in Haiberger 2005. Unfortunately, we did not succeed in removing all systematics, which remained about a factor 5 higher than the gravitational force.

Future experiments:

It is expected that precise tests of constancy of speed of light can be performed on satellites – see “Optical Clocks in Space”.




Fig. 9 a.  Experimental setup

Fig. 9 b.  Silicon torsional oscillator