Tests der allgemeinen Relativitätstheorie

Tests of Einstein's general theory of relativity did not provide an experimental foundation for the theory until well after it was introduced in 1915. Physicists accepted the theory because it correctly accounted for the precession of the perihelion of Mercury, a phenomenon which had long baffled astronomers and physicists, and because it unified Newton's law of universal gravitation with special relativity in a conceptually simple way. (Einstein has been famously quoted as saying that if his theory was falsified, then he would have felt "sorry for the dear Lord.") Vorlage:Fact Despite Einstein's proposal of three classical tests, the theory was without strong experimental support until a program of precision tests was started in 1959. This program has systematically tested general relativity in weak gravitational fields and severely limited possible deviations from the theory. Since 1974, Hulse and Taylor have studied stronger gravitational fields in binary pulsars. In these regimes, on typical solar system scales, general relativity has been extremely well tested.

On the largest scales, such as galactic and cosmological scales, general relativity has not yet been subject to precision tests. Some have interpreted dark matter and dark energy as a failure of Einstein's theory at large distances, small accelerations, or small curvatures. Likewise, the very strong fields around black holes, especially supermassive black holes, which are thought to power quasars and less dramatic active galactic nuclei, are still objects of intense study. Observations of these objects are difficult, and the interpretation of these observations is heavily dependent upon astrophysics other than general relativity or competing fundamental theories of gravitation, but they are qualitatively consistent with the black hole concept as modeled in general relativity.

Classical tests

Einstein proposed three famous tests of general relativity - the classical tests - in 1916  :

1. the gravitational redshift of light
2. the deflection of light by the Sun
3. the precession of the perihelion of Mercury

In Newtonian physics, a lone object orbiting a spherical mass would trace out an ellipse with the spherical mass at a focus. The point of closest approach, called the perihelion in the solar system, is fixed. There are a number of solar system effects that cause the perihelion of a planet to precess, or rotate around the sun. These are mainly because of solar oblateness and the presence of other planets, which perturb orbits. The precession of the perihelion of Mercury was a longstanding problem in celestial mechanics. Careful observations of Mercury showed that the actual value of the precession disagreed with that calculated from Newton's theory by 43 seconds of arc per century, which was much larger than the experimental error at the time. A number of ad hoc and ultimately unsuccessful solutions had been proposed, but they tended to introduce more problems. In general relativity, this orbit will precess, or change orientation within its plane, due to gravitation being mediated by the curvature of spacetime. Since the orientation of an orbit is usually given by the position of its periapsis, this change of orientation is described as being a precession in the periapsis of an object. However, the problem was resolved by Einstein's theory , which predicted exactly the observed amount of perihelion shift. This was a powerful factor motivating the adoption of Einstein's theory.

The total observed precession of Mercury is 5600 arc-seconds per century with respect to the position of the vernal equinox of the Sun. This precession is due the following causes (the numbers quoted are the modern values):

Sources of the precession of perihelion for Mercury

Amount (arc s/century)	Cause
5025".6	Coordinate
531".4	Gravitational tugs of the other planets
< 0.1	Oblateness of the sun
42".98±0".04	General relativity
5600".0	Total
5599".7	Observed

Thus, the predictions of general relativity perfectly account for the missing precession (the remaining discrepancy is within observational error). All other planets experience perihelion shifts as well, but, since they are further away from the Sun and have lower speeds, their shifts are lower and harder to observe. For example, the perihelion shift of Earth's orbit due to general relativity effects is about 5 seconds of arc per century.

One of Eddington's photographs of the 1919 solar eclipse experiment, presented in his 1920 paper announcing its success.

The first observation of light deflection was performed by noting the change in position of stars as they passed near the Sun on the celestial sphere. The observations were performed by Sir Arthur Eddington and his collaborators during a total solar eclipse , so that the stars near the sun could be observed. Observations were made simultaneously in the city of Sobral, Ceará, Brazil and in the west coast of Africa Vorlage:Fact. The result was considered spectacular news and made the front page of most major newspapers. It made Einstein and his theory of general relativity world famous.

The early accuracy, however, was poor and is described further in the article on predictive power. Dyson et al. quoted an optimistically low uncertainty in their measurement, which is thought to be plagued by systematic error and possibly confirmation bias. In 1801 J. Soldner had pointed out that Newtonian gravity predicts that starlight will bend around a massive object, but the predicted effect is only half the value predicted by general relativity as calculated by Einstein in his 1911 paper. The results of Soldner were revived by the anti-Semite Philipp Lenard (1921) in an attempt to discredit Einstein. Eddington had been aware in 1919 of the alternative predictions but had rejected eclipse data consistent with the Newtonian predictions. Considerable uncertainty remained in these measurements for almost fifty years, until observations started being made at radio frequencies. It was not until the late 1960s that it was definitively shown that the amount of deflection was the full value predicted by general relativity, and not half that number.

The gravitational redshift of light was predicted by Einstein from the equivalence principle in 1907, but it is very difficult to measure astrophysically. It was not conclusively tested until the Pound-Rebka experiment in 1959 measured the relative redshift of two sources situated at the top and bottom of Harvard University's Jefferson tower using an extremely sensitive phenomenon called the Mössbauer effect. The result was in excellent agreement with general relativity. This was one of the first precision experiments testing general relativity.

Modern tests

The modern era of testing general relativity was ushered in largely at the impetus of Dicke (1959, 1962) and Schiff (1960) who laid out a framework for testing general relativity. They emphasized the importance not only of the classical tests, but of null experiments, testing for effects which in principle could occur in a theory of gravitation, but do not occur in general relativity. Another important theoretical development were the new alternatives to general relativity theory – such as Brans-Dicke theory and other scalar-tensor theories – by the parameterized post-Newtonian formalism in which deviations from general relativity can be quantified; and by the framework of the equivalence principle.

Experimentally, new developments in space exploration, electronics and condensed matter physics have made precise experiments, such as the Pound-Rebka experiment, laser interferometry and lunar rangefinding possible.

Post-Newtonian tests of gravity

Early tests of general relativity were hampered by the lack of viable competitors to the theory: it was not clear what sorts of tests would distinguish it from its competitors. General relativity was the only known relativitistic theory of gravity compatible with special relativity and observations. Moreover, it is an extremely simple and elegant theory. This changed with the introduction of Brans-Dicke theory in 1960. This theory is arguably simpler, as it contains no dimensionful constants, and is compatible with a version of Mach's principle and Dirac's large numbers hypothesis, two philosophical ideas which have been influential in the history of relativity. Ultimately, this led to the development of the parameterized post-Newtonian formalism by Nordtvedt and Will, which parameterizes, in terms of ten adjustable parameters, all the possible departures from Newton's law of universal gravitation to first order in the velocity of moving objects (i.e. to first order in $v/c$, where v is the velocity of an object and c is the speed of light). This approximation allows the possible deviations from general relativity, for slowly moving objects in weak gravitational fields, to be systematically analyzed. Much effort has been put into constraining the post-Newtonian parameters, and deviations from general relativity are at present severely limited.

One of the most important tests is gravitational lensing. It has been observed in distant astrophysical sources, but these are poorly controlled and it is uncertain how they constrain general relativity. The most precise tests are analogous to Eddington's 1919 experiment: they measure the deflection of radiation from a distant source by the sun. The sources that can be most precisely analyzed are distant radio sources. In particular, quasars are very strong radio sources. The directional resolution of any telescope is in principle limited by diffraction; for radio telescopes this is also the practical limit. An important improvement in obtaining positional high accuracies (from milli-arcsecond to micro-arcsecond) was obtained by combining radio telescopes across the Earth. The technique is called very long baseline interferometry (VLBI). With this technique radio observations couple the phase information of the radio signal observed in telescopes separated over large distances. Recently, these telescopes have measured the deflection of radio waves by the Sun to extremely high precision, confirming this aspect of Einstein's theory to the 0.04% level. At this level of precision systematic effects have to be carefully taken into account to determine the precise location of the telescopes on Earth. Some important effects are the Earth's nutation, rotation, atmospheric refraction, tectonic displacement and tidal waves. Another important effect is refraction of the radio waves by the solar corona. Fortunately, this effect has a characteristic spectrum, whereas gravitational distortion is independent of wavelength. Thus, careful analysis, using measurements at several frequencies, can subtract this source of error.

The entire sky is slightly distorted due to the gravitational deflection of light caused by the Sun (the anti-Sun direction excepted). This effect has been observed by the European Space Agency astrometric satellite Hipparcos. It measured the positions of about 10⁵ stars. During the full mission about 3.5 × 10⁶ relative positions have been determined, each to an accuracy of typically 3 milliarcseconds (the accuracy for an 8–9 magnitude star). Since the gravitation deflection perpendicular to the Earth-Sun direction is already 4.07 mas, corrections are needed for practically all stars. Without systematic effects, the error in an individual observation of 3 milliarcseconds, could be reduced by the square root of the number of positions, leading to a precision of 0.0016 mas. Systematic effects, however, limit the accuracy of the determination to 0.3% (Froeschlé, 1997).

I. Shapiro proposed another test, beyond the classical tests, which could be performed within the solar system. It is sometimes called the fourth "classical" test of general relativity. He predicted a relativistic time delay (Shapiro delay) in the round-trip travel time for radar signals reflecting off other planets . The curvature of the path of a photon passing near the Sun is too small to have an observable delaying effect, but general relativity predicts a time delay which becomes progressively larger when the photon passes nearer to the Sun due to the time dilation in the gravitational potential of the sun. Observing radar reflections from Mercury and Venus just before and after it will be eclipsed by the Sun gives agreement with general relativity theory at the 5% level . More recently, the Cassini probe has undertaken a similar experiment which gives perfect agreement with general relativity at the 0.002% level.

These experiments all test the same post-Newtonian parameter, the so-called Eddington parameter γ, which is a straightforward parameterization of the amount of deflection of light by a gravitational source. It is equal to one for general relativity, and takes different values in other theories (such as Brans-Dicke theory). It is the best constrained of the ten post-Newtonian parameters, but there are other experiments designed to constrain the others. Precise observations of the perihelion shift of Mercury constrain other parameters, as do tests of the strong equivalence principle.

The equivalence principle

The equivalence principle, in its simplest form, asserts that the trajectories of falling bodies in a gravitational field should be independent of their mass and internal structure, provided they are small enough not to disturb the environment or be affected by tidal forces. This idea has been tested to incredible precision by Eötvös torsion balance experiments, which look for a differential acceleration between two test masses. Constraints on this, and on the existence of a composition-dependent fifth force or gravitational Yukawa interaction are very strong, and are discussed under fifth force and weak equivalence principle.

A version of the equivalence principle, called the strong equivalence principle, asserts that self-gravitation falling bodies, such as stars, planets or black holes (which are all held together by their gravitational attaction) should follow the same trajectories in a gravitational field, provided the same conditions are satisfied. This is called the Nordtvedt effect (Nordvedt, 1968) and is most precisely tested by the Lunar Laser Ranging Experiment. It has continuously, since 1969, measured the distance from several rangefinding stations on Earth to reflectors on the Moon to approximately centimeter accuracy (Williams, 2004). These have provided a strong constraint on several of the other post-Newtonian parameters.

Another part of the strong equivalence principle is the requirement that Newton's constant be constant in time, and not varying cosmologically. There are many independent constraints on the variation of Newton's constant (Uzan, 2003), but one of the best comes from lunar rangefinding which suggests that the gravitational constant does not change by more than one part in 10¹¹ per year. The constancy of the other constants is discussed in the Einstein equivalence principle section of the equivalence principle article.

The first of the classical tests discussed above, the gravitational redshift, is a simple consequence of the Einstein equivalence principle and was discovered by Einstein in 1907. As such, it is not a test of general relativity in the same way as the post-Newtonian tests, because any theory of gravity obeying the equivalence principle should also incorporate the gravitational redshift. Nonetheless, confirming the existence of the effect was an important substantiation of relativistic gravity. Experimental verification of this principle took several decades, because it is difficult to find clocks (to measure time dilation) or sources of electromagnetic radiation (to measure redshift) with a frequency that is known well enough that the effect can be accurately measured.

It was confirmed experimentally for the first time in 1960 using measurements of the change in wavelength of gamma-ray photons generated with the Mössbauer effect, which generates radiation with a very narrow linewidth. The experiment, performed by Pound and Rebka and later improved by Pound and Snyder, is called the Pound-Rebka experiment. The accuracy of the gamma-ray measurements was typically 1%. The blueshift of a falling photon can be found by assuming it has an equivalent mass based on its frequency $E=hf$ (where h is Planck's constant) along with $E=mc^{2}$, a result of special relativity. Such simple derivations ignore the fact that in general relativity the experiment compares clock rates, rather than energies. In other words, the "higher energy" of the photon after it falls can be equivalently ascribed to the slower running of clocks deeper in the gravitational potential well. To fully validate general relativity, it is important to also show that the rate of arrival of the photons is greater than the rate at which they are emitted. A very accurate gravitational redshift experiment, which deals with this issue, was performed in 1976 (Vessot, 1980). A hydrogen maser clock on a rocket was launched to a height of 10,000 km, and its rate compared with an identical clock on the ground. It tested the gravitational redshift to 0.007%.

Although the Global Positioning System (GPS) is neither designed nor operated as a test of fundamental physics, it must account for the gravitational redshift in its timing system. When the first satellite was launched, some engineers resisted the prediction that a noticeable gravitational time dilation would occur, so the first satellite was launched without the clock adjustment built into subsequent satellites. It showed the predicted shift of 38 microseconds per day. If general relativity suddenly stopped working tomorrow, the GPS control center in Colorado would know within hours; the relativistic correction to the timing is large enough to make GPS useless if it is not allowed for. Also, while it is true that GPS is not operated by the Defense Department as a test of general relativity, physicists have analyzed timing data from the GPS to confirm other tests. An excellent account of the role played by general relativity in the design of GPS can be found in Ashby 2003.

Other precision tests of general relativity, not discussed here, are the Gravity Probe A satellite, launched in 1976, which showed gravity and velocity affect the ability to synchronize the rates of clocks orbiting a central mass; the Gravity Probe B satellite, launched in 2004, is currently attempting to detect frame dragging (Lense-Thirring effect); the Hafele-Keating experiment, which used atomic clocks in circumnavigating aircraft to test general relativity and special relativity together; and the forthcoming Satellite Test of the Equivalence Principle.

Strong field tests

Observations of binary pulsars have all demonstrated substantial periapsis precessions that cannot be accounted for classically but can be accounted for by using general relativity. For example, the Hulse-Taylor binary pulsar PSR B1913+16, has an observed precession of over 4^o of arc per year. This precession has been used to compute the masses of the components. A binary pulsar discovered in 2003, J0737-3039, has a perihelion precession of 16.88^o.

Similarly to the way in which atoms and molecules emit electromagnetic radiation, a gravitating mass that is in quadrupole type or higher order vibration, or is asymmetric and in rotation, can emit gravitational waves. Two mutually orbiting bodies can also do so. These gravitational waves are predicted to travel at the speed of light. In general relativity, a perfectly spherical star (in vacuum) that expands or contracts while remaining perfectly spherical cannot emit any gravitational waves (similar to lack of e/m radiation from pulsating charge), as Birkhoff's theorem says that the geometry remains the same exterior to the star. More generally, a rotating system will only emit gravitational waves if it lacks the axial symmetry with respect to the axis of rotation. For example, planets orbiting the Sun constantly lose their energy via gravitational radiation, but this effect is so small that it is unlikely it will be observed in the near future. For example, Earth radiates about 200 watt of gravitational radiation. Gravitational waves originating from a binary system of a neutron star and a pulsar outside our solar system have been indirectly detected, for which Hulse and Taylor won the Nobel prize. The stars orbit only approximately according to Kepler's Laws, – over time they gradually spiral towards each other, demonstrating an energy loss in agreement with general relativity. Thus, although the waves have not been detected, their effect is necessary to explain the orbits.

The laser interferometer gravitational-wave observatory (LIGO) is an experiment designed to detect gravitational waves. It may lack the sensitivity to detect gravitational waves of astrophysical origin, but an equipment overhaul, dubbed "Advanced LIGO" will have an event rate at 100 times that of the initial design. The upgrade is planned for 2007. Also, the planned laser interferometer space antenna (LISA) is expected to directly detect gravitational waves, which will launch some time near the year 2015.

Cosmological tests

Tests of general relativity on the largest scales are not nearly so stringent as solar system tests. These are discussed in Peebles 2003. Some cosmological tests include searches for primordial gravity waves generated during cosmic inflation, which may be detected in the cosmic microwave background polarization or by a proposed space-based gravity wave interferometer called Big Bang Observer. Other tests at high redshift are constraints on other theories of gravity, and the variation of the gravitational constant since big bang nucleosynthesis (it varied by no more than 40% since then).

Some physicists think dark energy (energy density of virtual particles) is an indication of a failure of general relativity on the large scales, perhaps due to the effect of living on a brane (Dvali, 2000), or due to other corrections to the Einstein field equations.

Other physicists think dark matter is an indication of a failure of general relativity on galactic scales, and that the observed flat galactic rotation curves are due to a theory of Modified Newtonian dynamics or a relativistic variant, such as the TeVeS theory of Bekenstein. Some physicists point to the Pioneer anomaly as evidence of the failure of general relativity. This is not the view of most cosmologists, however, who think that the rotation curves are best explained by cold dark matter.

References

• N. Ashby, "Relativity in the Global Positioning System", Living Reviews in Relativity (2003).
• B. Bertotti, L. Iess and P. Tortora, "A test of general relativity using radio links with the Cassini spacecraft", Nature 425, 374 (2003).
• C. Brans and R. H. Dicke, "Mach's principle and a relativistic theory of gravitation", Phys. Rev. 124, 925-35 (1961).
• S. M. Carroll, Spacetime and geometry: an introduction to general relativity, Addison-Wesley, 2003 [1]. An introductory general relativity textbook.
• R. H. Dicke, "New Research on Old Gravitation," Science 129, 3349 (1959).
• R. H. Dicke, "Mach's Principle and Equivalence," in Evidence for gravitational theories: proceedings of course 20 of the International School of Physics "Enrico Fermi", ed C. Møller (Academic Press, New York, 1962).
• G. Dvali, G. Gabadadze and M. Porrati, "4-D gravity on a brane in 5-D Minkowski space", Phys. Lett. B485, 208–14 (2000).
• A. S. Eddington, Space, Time and Gravitation, Cambridge University Press, 1987 (originally published 1920).
• A. Einstein, "Über das Relativitätsprinzip und die aus demselben gezogene Folgerungen," Jahrbuch der Radioaktivitaet und Elektronik 4 (1907); translated "On the relativity principle and the conclusions drawn from it," in The collected papers of Albert Einstein. Vol. 2 : The Swiss years: writings, 1900–1909 (Princeton University Press, Princeton, NJ, 1989), Anna Beck translator. Einstein proposes the gravitational redshift of light in this paper, discussed online at The Genesis of General Relativity.
• A. Einstein, "Über den Einfluß der Schwerkraft auf die Ausbreitung des Lichtes," Annalen der Physik 35 (1911); translated "On the Influence of Gravitation on the Propagation of Light" in The collected papers of Albert Einstein. Vol. 3 : The Swiss years: writings, 1909–1911 (Princeton University Press, Princeton, NJ, 1994), Anna Beck translator, and in The Principle of Relativity, (Dover, 1924), pp 99–108, W. Perrett and G. B. Jeffery translators, ISBN 0-486-60081-5. The deflection of light by the sun is predicted from the principle of equivalence. Einstein's result is half the full value found using the general theory of relativity.
• M. Froeschlé, F. Mignard and F. Arenou, "Determination of the PPN parameter γ with the Hipparcos data" Hipparcos Venice '97, ESA-SP-402 (1997).
• A. Gefter, "Putting Einstein to the Test", Sky and Telescope July 2005, p.38. A popular discussion of tests of general relativity.
• P. Lenard, "Über die Ablenkung eines Lichtstrahls von seiner geradlinigen Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie", Physik. Zeitschr. 19, 156–63.
• K. Nordtvedt, "Equivalence principle for massive bodies. II. Theory", Phys. Rev. 169, 1017–25 (1968). Introduces the parameterized post-Newtonian formalism.
• K. Nordtvedt, Testing relativity with laser ranging to the moon", Phys. Rev." 170 1186–7 (1968).
• H. Ohanian and R. Ruffini, Gravitation and Spacetime, 2nd Edition Norton, New York, 1994, ISBN 0-393-96501-5. A general relativity textbook.
• P. J. E. Peebles, "Testing general relativity on the scales of cosmology", 2004 arXiv:astro-ph/0410285. Discusses tests of general relativity on the largest scales.
• L. I. Schiff, "On experimental tests of the general theory of relativity", Am. J. Phys. 28, 340–3.
• S. S. Shapiro, J. L. Davis, D. E. Lebach and J. S. Gregory, "Measurement of the solar gravitational deflection of radio waves using geodetic very-long-baseline interferometry data, 1979–1999", Phys. Rev. Lett. 92, 121101 (2004).
• J. P. Uzan, "The fundamental constants and their variation: Observational status and theoretical motivations," Rev. Mod. Phys. 75, 403 (2003). [2] This technical article reviews the best constraints on the variation of the fundamental constants.
• R. F. C. Vessot, M. W. Levine, E. M. Mattison, E. L. Blomberg, T. E. Hoffman, G. U. Nystrom, B. F. Farrel, R. Decher, P. B. Eby, C. R. Baugher, J. W. Watts, D. L. Teuber and F. D. Wills, "Test of Relativistic Gravitation with a Space-Borne Hydrogen Maser", Phys. Rev. Lett. 45, 2081-2084 (1980).
• C. M. Will, Theory and experiment in gravitational physics, Cambridge University Press, Cambridge (1993). A standard technical reference.
• C. M. Will, Was Einstein Right?: Putting General Relativity to the Test, Basic Books (1993). This is a popular account of tests of general relativity.
• C. M. Will, The Confrontation between General Relativity and Experiment, Living Reviews in Relativity (2001). An online, technical review, covering much of the material in Theory and experiment in gravitational physics. It is less comprehensive but more up to date.
• J. G. Williams, S. G. Turyshev and D. H. Boggs, "Progress in lunar laser ranging tests of relativistic gravity", Phys. Rev. Lett. 93 261101 (2004).

External links

• the USENET Relativity FAQ experiments page
• Mathpages.com article on Mercury's perihelion shift (for amount of observed GR shift).
• Article on computing the classical precession of Mercury (for classical precession value with respect to the stars).