Definitions

# Tests of general relativity

At its introduction in 1915, the general theory of relativity did not have a solid empirical foundation. It was known that it correctly accounted for the "anomalous" precession of the perihelion of Mercury and on philosophical grounds it was considered satisfying that it was able to unify Newton's law of universal gravitation with special relativity. That light appeared to bend in gravitational fields in line with the predictions of general relativity was found in 1919 but it was not until a program of precision tests was started in 1959 that the various predictions of general relativity were tested to any further degree of accuracy in the weak gravitational field limit, severely limiting possible deviations from the theory. Beginning in 1974, Hulse, Taylor and others have studied the behaviour of binary pulsars experiencing much stronger gravitational fields than found in our solar system. Both in the weak field limit (as in our solar system) and with the stronger fields present in systems of binary pulsars the predictions of general relativity have been extremely well tested locally.

On the largest spatial scales, such as galactic and cosmological scales, general relativity has not yet been subject to precision tests. Some have interpreted observations supporting the presence of dark matter and dark energy as a failure of general relativity at large distances, small accelerations, or small curvatures. The very strong gravitational fields that must be present close to black holes, especially those supermassive black holes which are thought to power active galactic nuclei and the more active quasars, belong to a field of intense active research. Observations of these quasars and active galactic nuclei are difficult, and the interpretation of the observations are heavily dependent upon astrophysical models 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 tests of general relativity, subsequently called the classical tests of general relativity, in 1916:

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

### Perihelion precession of Mercury

In Newtonian physics, under Standard assumptions in astrodynamics a two-body system consisting of 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, is fixed. There are a number of effects present in our solar system that cause the perihelions of the planets to precess, or rotate around the sun. These are mainly because of the presence of other planets, which perturb the orbits. Another effect is solar oblateness, which produces only a minor contribution. 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. A number of ad hoc and ultimately unsuccessful solutions had been proposed, but they tended to introduce more problems. In general relativity, this remaining precession, or change of orientation within its plane, is explained by 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 the orbiting object. Einstein showed that general relativity predicts exactly the observed amount of perihelion shift. This was a powerful factor motivating the adoption of general relativity.

Although earlier measurements of planetary orbits were made using conventional telescopes, the most accurate measurements are now made with radar. 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 to the following causes (the numbers quoted are the modern values):

Sources of the precession of perihelion for Mercury
Amount (arcsec/century) Cause
5025.6 Coordinate (due to the precession of the equinoxes)
531.4 Gravitational tugs of the other planets
0.0254 Oblateness of the Sun (quadrupole moment)
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. The periapsis shift has also been observed with radio telescope measurements of Binary pulsar systems, again confirming general relativity.

### Deflection of light by the Sun

Henry Cavendish in 1784 (in an unpublished manuscript) and Johann Georg von Soldner in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object. The same value as Soldner's was calculated by Einstein in 1911 based on the equivalence principle alone. However, Einstein noted in 1915 in the process of completing general relativity, that his (and thus Soldner's) 1911-result is only half of the correct value. So Einstein was the first to calculate the correct value for light bending.

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 . 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. When asked by his assistant what his reaction would have been if general relativity had not been confirmed by Eddington and Dyson in 1919, Einstein famously made the quip: "Then I would feel sorry for the dear Lord. The theory is correct anyway."

The early accuracy, however, was poor. Dyson et al. quoted an optimistically low uncertainty in their measurement, which is argued by some to have been plagued by systematic error and possibly confirmation bias, although modern reanalysis of the dataset suggests that Eddington's analysis was accurate. 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 Einstein ring is an example of the deflection of light from distant galaxies by more nearby objects.

### Gravitational redshift of light

Einstein predicted the gravitational redshift of light from the equivalence principle in 1907, but it is very difficult to measure astrophysically (see the discussion under Equivalence Principle below). Although it was measured by Adams in 1925, it was only conclusively tested when 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 and Schiff 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. Other important theoretical developments included the inception of alternative theories to general relativity, in particular, scalar-tensor theories such as the Brans-Dicke theory; the parameterized post-Newtonian formalism in which deviations from general relativity can be quantified; and 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.

The experiments testing gravitational lensing and light time delay limits 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.

### Gravitational lensing

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, some 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 the amount of deflection predicted by general relativity aspect 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).

### Light travel time delay testing

Irwin 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 gave agreement with general relativity at the 0.002% level.

### 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 attraction) should follow the same trajectories in a gravitational field, provided the same conditions are satisfied. This is called the Nordtvedt effect and is most precisely tested by the Lunar Laser Ranging Experiment. Since 1969, it has continuously measured the distance from several rangefinding stations on Earth to reflectors on the Moon to approximately centimeter accuracy. 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 gravitational constant be constant in time, and have the same value everywhere in the universe. There are many independent observations limiting the possible variation of Newton's gravitational constant, 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.

#### Gravitational redshift

The first of the classical tests discussed above, the gravitational redshift, is a simple consequence of the Einstein equivalence principle and was predicted 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, since the absence of gravitational redshift would have strongly contradicted relativity. The first observation of the gravitational redshift was the measurement of the shift in the spectral lines from the white dwarf star Sirius B by Adams in 1925. Although this measurement, as well as later measurements of the spectral shift on other white dwarf stars, agreed with the prediction of relativity, it could be argued that the shift could possibly stem from some other cause, and hence experimental verification using a known terrestrial source was preferable .

Experimental verification of gravitational redshift using terrestrial sources 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 line width. 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, where 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 not designed as a test of fundamental physics, it must account for the gravitational redshift in its timing system, and physicists have analyzed timing data from the GPS to confirm other tests. 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 that was later built into subsequent satellites. It showed the predicted shift of 38 microseconds per day. This rate of discrepancy is sufficient to substantially impair function of GPS within hours if not accounted for. 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 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.

### Frame dragging tests

The Gravity Probe B satellite, launched in 2004 and operating until 2005 was attempting to detect frame dragging (the Lense-Thirring effect). At the moment (as of December 2007) final results of analyzing the measurements from the satellite have not been announced. This is believed to be mainly due to the high noise levels and difficulties in modelling the noise accurately so that a useful signal can be found. A "definitive final result is anticipated by May 2008 Mission Update December 2007,'' Probe B website, accessed 12/12/2007

## Strong field tests

Pulsars are rapidly rotating neutron stars which emit regular radio pulses as they rotate. As such they act as clocks which allow very precise monitoring of their orbital motions. Observations of pulsars in orbit around other stars 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 (a pair of neutron stars in which one is detected as a pulsar) has an observed precession of over 4o of arc per year. This precession has been used to compute the masses of the components.

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. These gravitational waves are predicted to travel at the speed of light. For example, planets orbiting the Sun constantly lose energy via gravitational radiation, but this effect is so small that it is unlikely it will be observed in the near future (Earth radiates about 300 Watts (see gravitational waves) of gravitational radiation). Gravitational waves have been indirectly detected from the Hulse-Taylor binary. Precise timing of the pulses show that the stars orbit only approximately according to Kepler's Laws, – over time they gradually spiral towards each other, demonstrating an energy loss in close agreement with the predicted energy radiated by gravitational waves. Thus, although the waves have not been directly measured, their effect seems necessary to explain the orbits. For this work Hulse and Taylor won the Nobel prize.

A "double pulsar" discovered in 2003, J0737−3039, has a perihelion precession of 16.90o per year; unlike the Hulse-Taylor binary, both neutron stars are detected as pulsars, allowing precision timing of both members of the system. Due to this, the tight orbit, the fact that the system is almost edge-on, and the very low transverse velocity of the system as seen from Earth, J0737−3039 provides by far the best system for strong-field tests of general relativity known so far. Several distinct relativistic effects are observed, including orbital decay as in the Hulse-Taylor system. After observing the system for two and a half years, four independent tests of general relativity were possible, the most precise (the Shapiro delay) confirming the general relativity prediction within 0.05%.

## Gravitational waves

The laser interferometer gravitational-wave observatory (LIGO) is currently the most sensitive experiment designed to detect gravitational waves. So far (as of December 2007) no detections have been reported, but a second observatory, dubbed "Advanced LIGO" will have an event rate at 100 times that of the initial design, to a possible several events per year (each "event" a black hole or neutron star binary in the final stages of merging). Advanced LIGO is planned to begin construction in 2009, with funding expected to be received in 2008; it is expected to go online in 2011 (see official LIGO website, external links below). Also, the planned laser interferometer space antenna (LISA) will have enough sensitivity to directly detect gravitational waves from numerous binary systems in the Milky Way; LISA is planned for launch some time near the year 2015. Gravitational waves have so far not been detected directly, but if they exist as predicted they will certainly be detected by LISA and probably by Advanced LIGO. Finding or falsifying the existence of gravitational waves as predicted by general relativity is of course a critical test of the validity of the theory.

## Cosmological tests

Tests of general relativity on the largest scales are not nearly so stringent as solar system tests. 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 is perhaps due to the effect of living on a brane, or due to other corrections to the Einstein field equations.

## References

### Other research papers

• 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).
• 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.
• Shapiro, S. S.; Davis, J. L.;Lebach, D. E.; Gregory J.S. "Measurement of the solar gravitational deflection of radio waves using geodetic very-long-baseline interferometry data, 1979–1999". Physical Review Letters 92 (121101): 121101. Retrieved on 2007-01-18.
• M. Froeschlé, F. Mignard and F. Arenou, " Determination of the PPN parameter γ with the Hipparcos data" Hipparcos Venice '97, ESA-SP-402 (1997).
• Will, Clifford M. Was Einstein Right? Testing Relativity at the Centenary. arXiv eprint server. Retrieved on August 8., 2005.

### Textbooks

• S. M. Carroll, Spacetime and geometry: an introduction to general relativity, Addison-Wesley, 2003 An introductory general relativity textbook.
• A. S. Eddington, Space, Time and Gravitation, Cambridge University Press, 1987 (originally published 1920).
• A. Gefter, "Putting Einstein to the Test", Sky and Telescope July 2005, p.38. A popular discussion of tests of general relativity.
• H. Ohanian and R. Ruffini, Gravitation and Spacetime, 2nd Edition Norton, New York, 1994, ISBN 0-393-96501-5. A general relativity textbook.
• 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.