## Archive for September, 2005

## Einstein’s quest for unification

Jan 5, 2005

The last 30 years of his life were spent on a fruitless search for a unified field theory, but as John Ellis explains, Einstein put this “holy grail” of modern physics on the theoretical map.

The definitive scientific biography of Einstein, *Subtle is the Lord…,* which was written by Abraham Pais in 1982, delivered an unequivocal verdict on Einstein’s quest for a unified field theory. Pais wrote that the time for unification had not come, and that Einstein’s work “led to no results of physical interest”. But a lot of water has flowed under the bridge of unification since then, allowing us to look back with perhaps more indulgence as we celebrate the centenary of Einstein’s 1905 papers.

Let us briefly recall the relevant physics that was known in the 1920s, when Einstein embarked on his quest. The only known subatomic particles were the proton and the electron: the neutron and the neutrino, for example, were not predicted or discovered until the 1930s. Most “fundamental” physicists were striving to understand quantum physics – an endeavour from which Einstein stood apart. The structure of the nucleus was regarded as an interesting but secondary problem, and the unification of forces was considered, in the words of Pais, a minor issue.

For Einstein and his few unification-minded colleagues the big issue was to unify general relativity – a theory of gravity – with Maxwell’s electrodynamics. Theodor Kaluza and Oskar Klein proposed starting from a 5D theory, which contained an extra “compactified” spatial dimension in addition to the three spatial and one temporal dimensions of everyday experience. Electromagnetism then emerged naturally from this extra dimension.

Perhaps more so than Pais, we now recognize these early theories as breakthroughs in unification because of their many echoes in the supergravity and string theories of the past 20 years. Einstein was an early enthusiast; as he wrote to Kaluza in April 1919, “The idea of achieving unification by means of a five-dimensional cylinder world would never have dawned on me…At first glance I like your idea enormously”. Kaluza published his idea in 1921, which Einstein pursued in his first unification paper with Jacob Grommer the following year. Indeed Einstein was to return to 5D theories every few years for the rest of his life.

However, even Einstein had to admit that his unification papers were not always ground breaking. For example, after some initial confusion he recognized that the two papers he wrote in 1927 were equivalent to the work of Klein. But he might have been happy to know that some of today’s particle physicists will search for Kaluza-Klein excitations using the Large Hadron Collider at CERN. Einstein had hoped to identify quantum fields with such higher components that only arose in the 5D theories.

Another recurring theme in Einstein’s quest for unification was to generalize the “metric” of relativity – the symmetric tensor that describes the curvature of space-time – so that it could also describe the electromagnetic field. He pursued many apparently blind alleys, such as asymmetric generalizations of the metric, and even postulated that there might be no tensor at all. As Einstein himself said in a letter to Klein in 1917, “this process of deepening the theory has no limits”.

Unfortunately, these ideas were unsuccessful. For example, in his first unification paper in 1925 the antisymmetric part of his tensor field was not suitable for describing all the components of the electric and magnetic fields. Indeed, none of Einstein’s unification attempts ever reproduced the free-field Maxwell equations. In Einstein’s defence, it should be mentioned that we now recognize that other types of antisymmetric tensor fields emerge naturally from string theory. However, this type of theory had not been invented in Einstein’s day.

A more basic problem with many of Einstein’s proposals was that they did not include the general theory of relativity itself. However, in his final years following 1945 he returned to a theory with a fundamental tensor that was not symmetric and would include both the metric and the electromagnetic tensor, which avoided some of these problems.

### No stone left unturned

It is difficult to accuse Einstein of leaving stones unturned – no matter how unpromising they might appear. For example, in the early 1940s he even toyed with the idea that nature might not be described by partial differential equations. Modern theorists can hardly be accused of excessive conservatism, but even they have not revived this startling speculation!

What is most impressive about Einstein’s quest for unification was his persistent indefatigability. He tried many different ideas, and often returned to earlier theoretical haunts, such as Kaluza-Klein theories, with something new to say. However, the truth is that he was adrift from many of the most important developments in physics at the time. For instance, he was famously sceptical – if not downright hostile – towards quantum physics, and he does not seem to have followed closely the discoveries of new particles and interactions. More surprisingly, perhaps, he seems to have missed out on some of the most far-reaching new theoretical ideas of that period, which now play key roles in modern approaches to unification.

For example, Einstein recognized Hermann Weyl’s seminal 1918 work on scale transformations in four dimensions, even paying it the backhanded compliment that “apart from the agreement with reality, it is at any rate a grandiose achievement of the mind”. Weyl’s ideas led to the discovery in the late 1920s of local phase transformations, which laid the foundations for the gauge theories of the weak and electromagnetic interactions in the 1950s and beyond. However, Einstein was never involved personally in these far-reaching developments.

He also seems to have been affected by frequent mood swings during his quest for unification. On several occasions he switched rapidly from unwarranted optimism about the prospects of a new idea to complete rejection. More alarmingly, his mood often swung in the full glare of publicity. For many years a new scientific paper by Einstein was a major public event, with hundreds of journalists hanging on the utterances of the great man. The closest present-day parallel would be Stephen Hawking and his recent comments on black holes and quantum mechanics.

### Einstein’s legacy

Why were Einstein’s papers on unification not more successful? It is surely insufficient simply to say that only young theorists have brilliant new ideas. The many distractions of fame in his later years should also not get all the blame. Einstein himself wrote in his early years that “formal points of view…fail almost always as heuristic aids”. But later he seems to have abandoned this insight in his quest for unification, and instead was seduced more by mathematical novelty than by physical intuition.

It could be, however, that Einstein was simply ahead of his time, since even if he had been following contemporary physics more closely, the information available before his death was probably insufficient to make significant progress in unification. For example, the unification of the weak and electromagnetic interactions in the 1960s required many unforeseen experimental discoveries as well as new theoretical ideas. Even now, the unification of gravity with the other interactions – which was Einstein’s true dream – still eludes us.

Following Einstein, most theoretical physicists assign a central role to geometrical ideas. Most of the particle-physics community believes, for example, that string theory provides the appropriate framework for realizing Einstein’s dream. Here, fascinating generalizations of Kaluza and Klein’s hidden dimensions, such as “Calabi-Yau manifolds”, are able to dispose of the several extra dimensions required by the theory. However, not all general relativists are convinced, and there is absolutely no experimental evidence for string theory. Are we also in danger of being seduced by formal beauty?

Although some of the unification ideas pursued by Einstein are now recognizable in developments such as string theory, this is not to say that Einstein’s work actually inspired these modern unification attempts. It seems to me that the real significance of Einstein’s quest for unification lies in its quixotic ambition. Einstein, more than any of his contemporaries, put unification on the theoretical map and established it as a respectable intellectual objective. Even if we do not have all the necessary theoretical tools or experimental information, unification is the “holy grail” towards which our efforts should be directed.

### About the author

John Ellis is in the Theory Division at CERN, Geneva, Switzerland.

## Relativity at the centenary

Jan 5, 2005

Gravitational physics has become a truly experimental science as tests of the special and general theories of relativity reach new levels of precision

When I was a first-term graduate student in the late 1960s, it was said that the field of general relativity was “a theorist’s paradise and an experimentalist’s purgatory”. There were some experiments – Irwin Shapiro, for instance, had just measured the effects of general relativity on radio waves as they passed the Sun – but the field was dominated by theory and by theorists. This seemed to reflect Einstein’s own attitudes: although he had a keen insight into the workings of the physical world, he felt that the bottom line was the theory. As he once famously said, when asked how he would have reacted if an experiment had contradicted the theory, “I would have felt sorry for the dear Lord. The theory is correct”.

Since that time the field has been completely transformed. Today, at the centenary of Einstein’s *annus mirabilis,* experiment has become a central component of gravitational physics. I know of no better way to illustrate this than to cite a paper by the LIGO Scientific Collaboration that was published in *Physical Review D* last year (see Abbott *et al.* in further reading). This was one of the papers reporting results from the first science run of the Laser Interferometer Gravitational-wave Observatory (LIGO), but with 374 authors from 41 institutions in 8 countries it is reminiscent of particle physics, not general relativity.

Box 1: Special and general relativity).

Put in simple terms, the equivalence principle states that gravity and acceleration are equivalent. Embellished over the years, this idea is now called the Einstein equivalence principle and encompasses three separate principles: the weak equivalence principle, and the principles of local Lorentz and local position invariance.

The weak equivalence principle states that test bodies fall with the same acceleration independent of their internal structure or composition: in other words *gravitational mass* (the *m* in *F* = *GMm*/*r*^{2}, where *F* is the gravitational attraction between two masses a distance *r* apart and *G* is the Newtonian gravitational constant) and *inertial mass* (the *m* in *F* = *ma,* where a is the acceleration caused by any force *F*) are the same. There is also a strong version of the equivalence principle that goes beyond the weak version by stating that *gravitational energy* will fall with the same acceleration as ordinary matter and other types of energy in a gravitational field (see Box 2: Self-energy and the strong equivalence principle).

The principle of local Lorentz invariance states that the outcome of any local non-gravitational experiment carried out in a freely falling reference frame is independent of the velocity of that frame, while the principle of local position invariance holds that the outcome of any local non-gravitational experiment is also independent of where and when in the universe it is performed. In this context “local” means confined to a suitably small region of space and time, while “freely falling” means falling freely under gravity with no other forces acting.

Although Einstein used it to derive general relativity, his equivalence principle implies only that gravitation must be described by a “metric theory” – a theory in which matter responds to the geometry of space-time and nothing else. However, general relativity is not the only metric theory of gravity, and other examples include the “scalar-tensor” theory developed by Carl Brans and Robert Dicke at Princeton University in 1961, building on earlier work by Markus Fierz and Pascual Jordan.

When it comes to testing metric theories of gravity, we need to distinguish between the weak-field limit, which is valid in the solar system (see figure 2 and Box 3: Testing metric theories in the solar system), and the strong-field regime that is needed to describe regions where gravity is extremely strong, such as in the vicinity of a black hole or a neutron star. If we are being really ambitious, we might also try to describe situations where gravity is strong and quantum effects are important, such as during the Big Bang, but that is a separate story (see “Welcome to quantum gravity”).

In non-metric theories matter can respond to something other than the geometry of space-time, and this can lead to violations of one or more pieces of the Einstein equivalence principle. For instance, in the string theories that seek to unify gravity with the other three forces of nature, the equivalence principle is violated because matter can respond to additional long-range fields. Searching for violations of the Einstein equivalence principle is therefore a good way to search for new physics beyond the standard metric theories of gravity.

### In the balance

To test the weak equivalence principle one compares the accelerations of two bodies with different compositions in an external gravitational field. Such experiments are often called Eötvös experiments after Baron von Eötvös, the Hungarian physicist whose pioneering experiments with torsion balances provided a foundation for Einstein’s ideas on general relativity.

4 and were obtained with Very Long Baseline Radio Interferometry (VLBI; see Shapiro

*et al.*in further reading). A recent measurement of the Shapiro time delay by the Cassini spacecraft, which was on its way to Saturn, was accurate to 1 part in 10

^{5}(see Bertotti

*et al.*in further reading).” href=”http://images.iop.org/objects/phw/world/18/1/5/PWrel4_01-05.jpg”>

*“Breaking Lorentz symmetry”*). In earlier times, such thinking would have been called “crackpot”, but these new ideas are well rooted in attempts to find a quantum theory of gravity and, ultimately, a unified theory of the four fundamental forces of nature.

Various string theories, for instance, allow for the possibility of long-range fields that are linked to the average matter distribution of the universe. If these fields couple weakly to local matter, they could lead to effects that can be observed in experiments. In particular, we know from observations that the Earth moves through the cosmic background radiation at a speed of 350 km s^{-1}. With the right kind of long-range field, this motion could produce an effective interaction that has a preferred direction associated with it. If this long-range field were then to couple weakly to, say, electromagnetism, then the electromagnetic fields in atoms could be changed by an amount that depends on the orientation of the atom relative to our direction of motion through the universe.

During the late 1980s researchers at Seattle, Harvard and NIST looked for these effects by checking if atomic transition frequencies change over the course of a year as their orientation changes relative to our cosmic velocity. Exploiting the then newly developed techniques of atom trapping and cooling, the researchers found no effects down to a few parts per 10^{26}.

These “clock anisotropy” experiments are latter-day versions of the classic Michelson-Morley experiments of 1887. In the Michelson-Morley experiment the “clocks” being compared were defined by the propagation of light along each of the two perpendicular arms of an interferometer. Einstein took the null result of these experiments for granted in his 1905 paper on special relativity, although he never referred to them by name.

Looking to the future, the discreteness of space-time at the Planck scale that is found in some quantum theories of gravity could also lead to effective violations of Lorentz invariance. However, a wide range of experiments, including tests of CPT (charge-parity-time) symmetry in particle-physics experiments and careful observations of gamma rays and synchrotron radiation from astrophysical sources, have ruled these out to a high-level of precision.

### Does space-time do the twist?

A central prediction of general relativity is that moving matter generates a gravitational field that is analogous to the magnetic field generated by a moving charge. Thus, a rotating body produces a “gravitomagnetic” field that drags space-time around with it, and this “frame-dragging” may play an important role in the dynamics of matter spiralling into supermassive black holes in quasars and other active galaxies. Frame-dragging might also be partly responsible for the collimated relativistic jets seen in such systems.

The Gravity Probe B satellite is currently measuring this effect near the Earth. Launched on 20 April 2004, its goal is to measure the precessions of four gyroscopes relative to a telescope trained on a nearby guide star called IM Pegasi over the course of a year (until the liquid helium that is used to cool the experiment runs out). The gyroscopes are spheres that are perfect to a few parts in 10 million and are coated with a thin layer of superconducting niobium. When the spheres rotate, the superconducting films develop magnetic moments that are precisely parallel to their spin axes. This means that any precession of the spins can be measured by monitoring changes in the magnetic flux through superconducting current loops fixed in the spacecraft.

General relativity predicts that frame-dragging will lead to a precession of 41 milliarcseconds per year, and the Gravity Probe B team hopes to measure this with an accuracy of 1%. The experiment will also measure the “geodetic” precession caused by the ordinary curvature of space around the Earth. General relativity predicts a value of 6.6 arcseconds per year for this effect. Gravity Probe B has been designed so that these precessions are perpendicular to one another, and the first results from the mission are expected in early 2006 (see figure 3).

Meanwhile, last October Ignazio Ciufolini of the University of Lecce in Italy and Erricos Pavlis of the University of Maryland used techniques in which laser beams were reflected from satellites to make a measurement of frame-dragging on the orbit of a satellite. Their result agreed with general relativity, with errors at the level of 10% (see *Physics World* November 2004 p7).

### The binary pulsar

In 1974 Russell Hulse and Joseph Taylor, then at the University of Massachusetts, discovered a binary pulsar called PSR 1913+16 that was to play a crucial role in tests of general relativity. Pulsars emit pulses of radio waves at very regular intervals and are thought to be rotating neutron stars. PSR 1913+16 was special because it was a pulsar that was in orbit around another compact object.

By carefully measuring small changes in the rate of the pulsar “clock”, Hulse and Taylor were able to determine both non-relativistic and relativistic orbital parameters with extraordinary precision. In particular they were able to measure three relativistic effects: the rate of advance of the periastron (the analogue of the perihelion in a binary system); the combined effects of time-dilation and gravitational redshift on the observed rate of the pulsar; and the rate of decrease of the orbital period.

If we assume that general relativity is correct and make the reasonable assumption that both objects are neutron stars, then all three relativistic effects depend on the two unknown stellar masses. Since we have, in effect, three simultaneous equations and just two unknowns, we can determine the mass of both objects with an uncertainty of less than 0.05%, and *also* test the predictions of general relativity. If we assume that the orbital period of the system is decreasing due to the emission of gravitational waves, then theory and experiment agree to within 0.2%. Hulse and Taylor shared the 1993 Nobel Prize for Physics for this work.

Binary pulsars can also be used to distinguish between different theories of gravity because they have very strong internal gravity (see Stairs in further reading). Indeed, several tenths of the rest-mass energy of a neutron star is contained in the gravitational forces that hold the star together, while the orbital energy only accounts for 10^{-6} of the total mass energy of the system. In the Brans-Dicke theory this internal self-gravity leads to the prediction that binary pulsars should emit both dipole and quadrupole gravitational radiation, whereas general relativity strictly forbids the dipole contribution. The emission of dipole radiation would have a characteristic effect on the orbital period of the system, but such an effect has not been seen. Several recently discovered binary-pulsar systems may allow new tests of general relativity.

### Gravitational waves

One of the outstanding challenges in physics today is to detect gravitational waves, and new gravitational-wave observatories in the US, Europe and Japan hope to achieve this, possibly before the end of the decade. In addition to exploring various astrophysical phenomena, these observatories might also be able to carry out new tests of fundamental gravitational physics (see *Physics World* January 2005 p37, print version only).

General relativity makes three predictions about gravitational radiation that can be tested: gravitational waves have only two polarization states, whereas other theories can predict as many as six; gravitational waves travel at the speed of light, while other theories may predict different speeds; and the emission of gravitational waves acts back on the source that is emitting them in a characteristic manner.

For example, as is described above, scalar-tensor theories and general relativity make different predictions for the nature of the gravitational waves emitted by binary pulsars, and it may be possible to detect these differences. Moreover, if gravitational waves with long wavelengths travel more slowly than those with shorter wavelengths, then it might be possible to observe this behaviour – which is generally associated with massive (as opposed to massless) elementary particles – in the gravitational radiation from binary systems.

Although the collision of two compact objects to form a black hole is too complex to allow precision tests of general relativity, analysis of the gravitational waves produced in the collision will reveal information about the masses and spins of the compact objects themselves, and also about the mass and angular momentum of the final black hole. Such observations will therefore reflect dynamical, strong-field general relativity in its full glory.

Making firm predictions for this situation involves solving Einstein’s equations in a regime where weak-field methods fail, and therefore requires large-scale numerical computations. This challenging task has been taken up by many “numerical relativity” groups around the world. The discovery and study of the formation of a black hole through gravitational waves would provide a stunning test of general relativity.

### Relativity and beyond

Einstein’s special and general theories of relativity altered the course of science. They were triumphs of the imagination and of theory, with experiment playing a secondary role. In the past four decades we have witnessed a second triumph for Einstein, with general relativity passing increasingly precise experimental tests with flying colours. But the work is not done. Tests of strong-field gravity in the vicinity of black holes and neutron stars need to be carried out. Gamma-ray, X-ray and gravitational-wave astronomy will all play a critical role in probing this largely unexplored aspect of the theory.

General relativity is now the “standard model” of gravity. But as in particle physics, there may be a world beyond the standard model. Quantum gravity, strings and branes may lead to testable effects beyond general relativity. Experimentalists will continue to search for such effects using laboratory experiments, particle accelerators, instruments in space and cosmological observations. At the centenary of relativity it could well be said that experimentalists have joined the theorists in relativistic paradise.

### Box 1: Special and general relativity

When Einstein introduced the concept of “relativity” in 1905 – the notion that there is no absolute motion in the universe, only relative motion – he overthrew ideas that had been in place since the time of Newton some 200 years before. In addition to *E = mc*^{2}, special relativity predicted various novel effects that occurred when bodies moved at close to the speed of light: time slowed down (an effect known as time-dilation) and lengths became shorter (Fitzgerald contraction). With the general theory Einstein then went on to show that we do not reside in the flat (Euclidean) space and uniform time of everyday experience, but in curved space-time instead.

Special relativity helped us to understand the microworld of elementary particles and interactions, while general relativity revolutionized our view of the universe by predicting astrophysical phenomena as bizarre as the Big Bang, neutron stars, black holes and gravitational waves.

The theory of relativity is a single, all-encompassing theory of space-time, gravity and mechanics, although special relativity and general relativity are often viewed as being independent. Special relativity is actually an approximation to curved space-time that is valid in sufficiently small regions called “local freely falling frames”, much as small regions on the surface of an apple are approximately flat, even though the overall surface is curved.

Einstein’s great insight was to realize that gravity and acceleration are equivalent in free fall, and he then went on to show that the laws of physics, such as the equations of electromagnetism, should have built-in local Lorentz and local position invariance.

In special relativity the “distance” between two points in space–time is given by the line element, d*s,* which is defined as d*s*^{2} = –*c*^{2}d*t*^{2} + d*x*^{2} + d*y*^{2} + d*z*^{2}, where *t* is time and *c* is the speed of light in a vacuum. In the curved space-time of general relativity d*s* is defined as d*s*^{2} = *g*_{μν}d*x*^{μ}d*x*^{ν}, where *x*^{1}, *x*^{2} and *x*^{3} are the three spatial dimensions, *x*^{0} = *ct,* and *g*μν, which is called the metric, is a function in space–time. The right-hand side of the equation must be summed over all values of μ and ν between 0 and 3.

General relativity provides a set of field equations that allow us to calculate the space-time metric (i.e. the amount of curvature) from a given distribution of matter – something that is not defined by the equivalence principle. Einstein’s aim was to find the simplest field equations that made this possible. The result was a set of 10 equations, symbolized by the seductively simple equation *G*_{μν} = 8π*GT*_{μν}/*c*^{4}, where *G*_{μν} is Einstein’s curvature tensor, which can be obtained directly from *g*_{μν} and its derivatives, and *T*_{μν} is the stress-energy tensor of normal matter. Sweating the details hidden in this equation has kept generations of relativists occupied.

In the past it was customary to speak of the three classical tests proposed by Einstein: the deflection of light by a massive body; the advance of the perihelion of Mercury; and the gravitational redshift of light (although this is actually a test of the Einstein equivalence principle rather than general relativity itself). Many new tests have been developed since Einstein’s time: in 1964 Irwin Shapiro, then at the Massachusetts Institute of Technology, predicted a delay in the propagation of light past a massive body; and in 1968 Kenneth Nordtvedt Jr of Montana State University showed that theories other than general relativity do not necessarily obey the equivalence principle in certain situations. One of the most striking predictions of general relativity is the black hole: when a massive star collapses under its own gravity it can warp space-time to such an extent that nothing, not even light, can escape. There is now convincing observational evidence for these objects.

One of the outstanding problems in physics is to unify general relativity, which is our best theory of gravity, with the quantum field theories that describe the three other fundamental forces. Although this challenge defeated Einstein, it should not surprise us that all the leading candidates for a unified theory – string theory, branes and loop quantum gravity – are all fundamentally geometrical.

### Box 2: Self-energy and the strong equivalence principle

Special relativity and *E = mc*^{2} tell us that energy and mass are essentially the same. The mass of a proton and an electron is greater than that of a hydrogen atom because energy must be supplied to break the electromagnetic bond in the atom. The weak equivalence principle asserts that this difference will change both the gravitational mass and the inertial mass by the same amount. This means that all forms of energy at microscopic scales – electromagnetic, strong and weak – respond to gravity in the same way. But what about large bodies like the Earth and Sun, or even extreme gravitational bodies like black holes, which also have measurable gravitational binding energy? The strong equivalence principle goes beyond the weak version by stating that gravitational energy falls with the same acceleration as ordinary matter and other forms of energy in a gravitational field. Although the gravitational self-energy contained in the gravitational forces that hold the Earth together only changes its total mass energy by less than 1 part in a billion, lunar laser-ranging experiments (see main text) can achieve a precision of 1 part in 10^{13} and can therefore test the strong equivalence principle. General relativity obeys the strong equivalence principle, whereas the Brans–Dicke theory and many other alternative theories do not.

### Box 3: Testing metric theories in the solar system

General relativity is one of several “metric” theories in which gravity arises from the geometry of space-time and nothing else. If we want to distinguish between different metric theories in the weak-field limit, it is customary to use a formalism that dates back to Arthur Eddington’s 1922 textbook on general relativity and was later extended by Kenneth Nordtvedt Jr and the present author. This parametrized post-Newtonian (PPN) formalism contains 10 parameters that characterize how the predictions of the different metric theories differ from those of Newtonian gravity, and therefore from each other, for various phenomena that can be measured in the solar system.

Six of these parameters are shown in the table below. For instance, γ is related to the amount of spatial curvature generated by mass and determines the size of classic relativistic effects such as the deflection of light by mass, while β is related to the degree of nonlinearity in the gravitational field. Another four parameters – ξ, α_{1}, α_{2} and α_{3} – determine if gravity itself violates a form of local position invariance or local Lorentz invariance (such as *G* depending on our velocity through the universe).

In the PPN formalism the deflection of light and the Shapiro time delay are both proportional to (1 + γ)/2. The “1/2” corresponds to the so-called Newtonian deflection (i.e. the deflection that a body moving at the speed of light would experience according to Newtonian gravity). This result was derived over two centuries ago by Henry Cavendish, who never published it, and then discovered again by Johann von Soldner in 1803, who did publish it. The “γ/2” comes directly from the warping of space near the massive body.

The PPN parameters can have different values in the different metric theories of gravity. In general relativity, for instance, γ and β are exactly equal to one and the other eight parameters all vanish. Four decades of experiments have placed bounds on the PPN parameters that are consistent with general relativity (see figure 2).

### About the author

Clifford M Will is in the McDonnell Center for the Space Sciences and the Department of Physics, Washington University in St Louis, Missouri, US.

### Further reading

B Abbott *et al.* 2004 Analysis of LIGO data for gravitational waves from binary neutron stars *Phys. Rev.* D **69** 122001

B Bertotti, L Iess and P Tortora 2003 A test of general relativity using radio links with the Cassini spacecraft *Nature* **425** 374

S S Shapiro *et al.* 2004 Measurement of the solar gravitational deflection of radio waves using geodetic very-long-baseline interferometry data, 1979-1999 *Phys. Rev. Lett.* **92** 121101

I H Stairs 2003 Testing general relativity with pulsar timing *Living Reviews in Relativity* www.livingreviews.org/lrr-2003-5

C M Will 2001 The confrontation between general relativity and experiment *Living Reviews in Relativity* www.livingreviews.org/lrr-2001-4

J G Williams, S Turyshev and T W Murphy Jr 2004 Improving LLR tests of gravitational theory *Int. J. Mod. Phys.* **D** 13 567