Tests of general relativity

Tests of general relativity are the direct consequences for experimental verification of Albert Einstein's theory of general relativity, the most accurate theory of gravity we have to date. Three types of experiments, called the classical tests, were proposed soon after publication of the Einstein field equations in 1916. These include:

1. Gravitational redshift or Einstein shift (clocks in a gravitational potential well observed from a stationary clock at a distant point [i.e. farther from the source of gravity] appear to tick slower),
2. Deflection of light (when light passes near a mass concentration such as the Sun, its path is slightly bent, also called gravitational lensing),
3. Perihelion shifts of planets (the deviation from Kepler-orbits of a test mass around a massive object (for example, a planet in orbit about the Sun).

Other tests include:

1. Pound-Rebka-Snider experiment, which confirmed the existence of the gravitational red-shifting of light.
2. Time delay in radar propagation near the Sun
3. The existence of gravitational waves, still unconfirmed by direct detection, e.g. LIGO experiment
4. Haefele-Keating Experiment, which used atomic clocks in aircraft to test GR and SR together.
5. GPS, which was found to be inaccurate unless the effects of GR were taken into account.
6. Gravity Probe A satellite launched 1976, that showed gravity and velocity affect the ability to synchronize the rates of clocks orbiting a central mass
7. Gravity Probe B satellite launched 2004, to detect "frame dragging" (Lense-Thirring effect). Still in the data collection stage.
8. Tests of the strong equivalence principle
9. Searches for a fifth force
10. Strong gravitational field tests, such as pulsar timing

Gravitational redshift

The gravitational redshift is a simple consequence of the Einstein equivalence principle and was found by Albert Einstein eight years before the full theory.

Experimental verification of this principle requires either a carefully controlled nuclear resonance or high precision clocks. It was confirmed experimentally for the first time only in 1960 using measurements of the change in wavelength of gamma-ray photons, the Pound-Rebka experiment (Pound, R.V., Rebka, G.A., 1960, Phys. Rev. Lett., 4, 337), later improved by Pound and Snider. This famous experiment is generally called the Pound-Rebka-Snider experiment. The accuracy of the gamma-ray measurements was typically 1%. The experimenters entitled their paper Apparent Weight of Photons, not a "validation of GR" or the like because it did not firmly differentiate general relativity from other semi-Newtonian theories. The blueshift of a falling photon can be found by assuming it has an equivalent mass based on its frequency $E=hf\,$ (due to Planck) along with $E=mc^{2}$, a result of special relativity. Such simple derivations ignore the fact that in GR the experiment compares clock rates, more than energies. In other words, the "higher energy" of the photon after it falls is properly ascribed to the slower running of clocks deeper in the gravitational potential well! To fully validate GR, one would have to show that the rate of arrival of the photons is also greater than the rate at which they are emitted. The clock experiments reported below deal with that issue.

A very accurate gravitational redshift experiment was performed in 1976 (Vessot, R.F.C,. Levine, M.W., Mattison, E.M., et al., 1980, Phys. Rev. Lett. 45, 2081-2084). A hydrogen maser clock on a rocket was launched to a height of 10000 km, and its rate compared with an identical clock on the ground. It tested the gravitational redshift to 0.02%.

For a full discussion see http://relativity.livingreviews.org/open?pubNo=lrr-2001-4&page=node3.html

Gravitational deflection of light

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Starlight bending around a massive object showing the star's apparent location (right) and its actual location (left).

The first observation of light deflection was performed by noting the change in position of stars near the Sun (F.W.Dyson, A.S. Eddington, C. Davidson, 1920, Philos. Trans. Royal Soc. London, Vol.~220A, p. 291-333). It took place during a total solar eclipse (so that stars near the Sun could be observed) in 1919 and was observed on an island near Brazil and near the west coast of Africa. The result was considered spectacular news and made the front page of most international journals. It made Einstein and his theory of general relativity world famous. The early accuracy, however, was poor (20% at best) and remained poor for about 40 years, until methods were found to accurately measure stellar positions in the sky.

Newton's theory of gravitation also predicts that starlight will bend around a massive object, but the predicted effect is only half the value predicted by general relativity (which gives a bending effect much closer to the observed value). The bending of starlight is important in modern astronomy, one of the important phenomena in this field being gravitational lensing where a massive object (usually a galaxy) distorts the light emitted from a distant object (usually a star), so that the observer sees distorted images of the distant object. In some cases, the distant object appears to take the form of a ring (a so-called Einstein ring).

Using radio interferometry

Radio interferometry observations (using stars that emit in the radio range) during the 1960s are able to provide accurate (relative) positions of radio sources. The sources used are quasars, some of which are strong radio sources in the sky. The quasars 3C273 and 3C279 have a small angular separation. Each year around October 8 they pass near the Sun, whereby the quasar 3C279 is eclipsed by the Sun. During its approach to the Sun the bending of light near the Sun can be verified to 1.5%. (Fomalont, E.B., and Sramek, R.A., 1976, Phys. Rev. Lett., 236, 1475-1478.)

The positional accuracy 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-arcsec to micro-arcsec) was obtained by combining radio telescopes across the Earth. The technique is called VLBI, Very Long Baseline Interferometry. With this technique radio observations couple the phase information of the radio signal observed in telescopes separated over large distances. With these accuracies the Einstein light deflection can be determined to an accuracy of 0.2% (D.S. Robertson & W.E. Carter, 1984, Nature, 310, p.572-574; and D.S. Robertson, W.E. Carter & W.H.Dillinger, 1991, Nature, 349, p.768-770).

At this level of precision all sorts of systematic effects have to be taken into account to determine the precise location of the telescopes on Earth. Such effects are important for the following reasons: Earth nutation (which has an error in the annual term of 2 milli-arcsec (mas)), Earth rotation, atmospheric refraction, tectonic displacement, tidal waves in the ocean, etc. An astronomical limitation is the refraction of radio waves around the Sun, in the so called solar corona, extending to several Solar radii. Fortunately, gravitational reflection is achromatic (it doesn't depend on wavelength) while the Solar corona bends electromagnetic radiation in the radio depending on wavelength. This chromatic effect can be used to eliminate the refraction in the solar corona, but uncertainties remain.

Observations with the Hipparchos satellite

In principle we observe almost all sky slightly distorted due to the gravitational deflection of light caused by the Sun (the anti-Sun direction excepted). This effect has been observed. The ESA astrometric satellite Hipparchos has 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 mas (1 mas= 0.001 arcsec; this accuracy is for a 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 mas, 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 0.1% (Froeschl\'e, M.\, Mignard, F., Arenou F., 1997, Hipparchos Venice, ESA-SP-402, "Determination of the PPN parameter γ with the Hipparchos data").

Orbital Precession

The two previous effects, the gravitational redshift and the deflection of light, are derived from the study of null geodesics, the paths of photons. Trajectories of nonrelativistic objects in Einstein's theory of gravitation also differ from those expected on the basis of Newtonian theory.

In Newtonian physics, a lone object orbiting a spherical mass would trace out an ellipse with the spherical mass at a focus, and this ellipse would maintain its orientation in space. 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.

Often, there are classical reasons why an orbit will precess. The presense of more than two objects in a gravitational system (such as the solar system) or the primary object being oblate will both cause precession in Newtonian physics. In these cases, the orbital precession due to general relativity will be in addition to the classical precession, and will manifest itself as a discrepancy between the observed amount of precession and the classically predicted amount.

Perihelion shift of Mercury

The total observed precession of Mercury is approximately 5600 arc-seconds per century with respect to the position of the Vernal Equinox of the Sun. This precession is due the following causes:

Sources of the precession of perhielion for Mercury

Type	Amount (arc-seconds/century)	Cause
Coordinate	~5030	Movement of the Vernal Equinox with respect to the stars due to precession of the Earth's axis of rotataion
Perturbational	532	Gravitational tugs of the other planets
Quadrapole	< 0.1	Oblateness of the Sun
Geodesic	43.1 +/- 0.5	General Relativity

Discounting the coordinate precession leaves a precession of 575 arc-seconds/century for the precession of Mercury with respect to the stars. This figure is often cited on-line and in the literature. (This is often reported as being 570 arc-seconds per century, which is the 19th Century value used when the discrepancy with theory was first noticed.)

For Mercury, the predicted amount of the added precession due to the goedesic precession of GR is 42.98 arc-seconds per century. This is in agreement with the observations of the discrepancy between the Newtonian prediction and observation.

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 GR effects is about 5 seconds of arc per century.

The Binary pulsar

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 Huse-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.

Sources

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).

Gravitation and Spacetime 2nd Edition by Hans Ohanian and Remo Ruffini, 1994, ISBN 0-393-96501-5, p.266

Putting Einstein to the Test by Amanda Gefter, Sky and Telescope, July 2005, p.38

Time-delay in radar propagation

The previous three tests are called the three classical tests of General Relativity. Much later, in 1964, Shapiro proposed another test to be performed within the solar system. It is generally called the fourth "classical" test of General Relativity. Shapiro predicted a relativistic time delay in the round-trip travel time for radar signals reflecting off other planets (Shapiro,I.I., 1964, Phys. Rev. Lett. 13, p.~789-791, "Fourth test of general relativity").

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 progessively larger when the photon passes nearer to the Sun. Observing radar reflections from a planet just before and after it will be eclipsed by the Sun shows this effect.

Existence of gravitational waves

Main article: Gravitational radiation 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, as Birkhoff's theorem (relativity) says that the geometry remains the same exterior to the star (the metric is time-independent. 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 tiny that it is unlikely it will be observed in near future. Gravitational waves originating from orbiting neutron stars outside our Solar System have been indirectly detected as described in the 1993 Nobel Prize lecture of Russell Alan Hulse and Joseph_Hooton_Taylor_Jr.. The stars only orbit 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 per se have not been detected, their effect is necessary to explain the orbits. Several detection experiments are currently underway, but no direct evidence of gravitational waves have been found so far.

References

• C. M. Will, Theory and experiment in gravitational physics, Cambridge University Press, Cambridge (1993).
• S. Carroll, Spacetime and geometry: an introduction to general relativity, Addison-Wesley [1].

External links

• the USENET Relativity FAQ experiments page