The Equivalence Principle and Classical Tests
General relativity is admitted to this treatise on the same terms as every other theory in it: by its evidence, and evidence first. This chapter collects that evidence in its historical and logical order — the strict proportionality of inertial and gravitational mass, which is what makes gravity universal and first suggests that it is geometry rather than force; the anomalous perihelion advance of Mercury, measured half a century before the theory that explained it; the deflection of starlight and the gravitational redshift; the fourth test found by Shapiro in the radar age; and the engineering test performed daily by every satellite-navigation receiver. The geometric formulation these facts support is constructed in Geometric Formulation of Gravity, and the field equations they constrain in The Einstein Field Equations.
The organizing principle is the equivalence principle in its three strengths — weak, Einstein, strong — each tested by a different class of experiments and each licensing a different part of the geometric description [Will:2014]. The chapter states the principles precisely, reports the experiments with their numbers and uncertainties, and keeps prediction and postdiction carefully apart. The measurements themselves — apparatus, procedure, observations, error budgets — are set out in Experiment: The Classical Tests of General Relativity, which tests the phenomena stated here and the field equations of The Einstein Field Equations.
The universality of free fall
From Galileo to Newton
That all bodies fall alike is not obvious — everyday experience, where air resistance discriminates between a stone and a feather, argues against it — and it was established as an idealized law before it could be tested cleanly. Galileo's route, set out in the Discorsi, was to dilute free fall on an inclined plane until its acceleration could be timed, and to argue away the residual disagreement between bodies as the work of the medium; his logical argument is still the sharpest short statement of the law. Suppose the heavy body fell faster. Tying a light body to it should then retard it, as the light partner drags; but the two tied together are a heavier body still, and should fall faster than either. The supposition contradicts itself, and composition-dependent fall with it [Galilei:1638]. The kinematics of uniformly accelerated fall that this law makes universal is developed in Kinematics, and its direct measurement — Galileo's plane among the rest — in Experiment: Free Fall and Projectile Motion.
Newton made the statement quantitative, and he could, because his own dynamics is the first in which it is a statement at all: the inertial mass in the second law and the gravitational mass coupling a body to attraction are logically independent quantities (Newtonian Dynamics), and nothing in the theory forces them to be proportional. His instrument was the pendulum of Experiment: The Pendulum, with the two masses now kept carefully distinct: a bob of inertial mass \(m_{\mathrm{I}}\) and gravitational mass \(m_{\mathrm{G}}\) on a cord of length \(\ell\) obeys \(\ell\,\ddot{\vartheta} =-\left(m_{\mathrm{G}}/m_{\mathrm{I}}\right)g\sin\vartheta\), so that for small swings
Two pendulums of equal length therefore beat together if and only if the ratio \(m_{\mathrm{G}}/m_{\mathrm{I}}\) is common to their bobs, a fractional difference in the ratio appearing as half of itself in the period. Newton hung identical wooden boxes filled in turn with gold, silver, lead, glass, sand, salt, water, wood and wheat, swung them in pairs beside one another, and observed no difference in their beats; he concluded that the ratio is common to all these materials to about one part in a thousand [Newton:1687]. In the modern figure of merit — the Eötvös parameter \(\eta\) of Equation (42.2) below — Galileo's argument bounds nothing numerically, and Newton's pendulums give \(\abs{\eta}\lesssim10^{-3}\).
The Eötvös torsion balance
The next six orders of magnitude came not from refining the pendulum but from abandoning it for a null instrument. A pendulum comparison must difference two separately measured periods, each carrying the full value of \(g\) with all its local disturbances; a torsion balance measures the difference directly, as a torque that vanishes identically when universality holds. The instrument descends from the Cavendish balance of Experiment: The Cavendish Torsion Balance: a horizontal beam hung from a fine wire, with test bodies of different composition at its ends.
The signal is geometrically subtle, and worth stating because it explains the design. On the rotating Earth a suspended body is acted on by the gravitational attraction, proportional to \(m_{\mathrm{G}}\), and by the centrifugal acceleration, proportional to \(m_{\mathrm{I}}\); the horizontal component of the latter, about \(1.7\times 10^{-2}\,\mathrm{m}/\mathrm{s}^{2}\) at \(45^\circ\) of latitude, is what a composition dependence would fail to cancel. Two bodies with different ratios then pull the beam ends toward the equator with different horizontal forces, and the beam turns. The full geometry — the torque, its maximum near \(45^\circ\) of latitude, its reversal under a \(180^\circ\) turn of the beam in azimuth — is derived in Section 53.5, as Equation (53.8). The reversal is the crucial operation: Eötvös performed it by hand, differencing the two orientations so that instrumental torques cancel while a genuine composition dependence changes sign.
The campaign of Eötvös, Pekár and Fekete ran from 1906 to 1909 and was published, posthumously for Eötvös, only in 1922. It compared platinum against copper, water, asbestos and several other materials, and bounded \(\abs{\eta}\) at the level of a few parts in \(10^{9}\) [Eotvos:1922] — six orders of magnitude past Newton, with an instrument whose essential parts are a beam, a wire and a mirror.
Modern limits
Every later improvement has been an improvement in modulation — in making the sought signal vary in a known way while the instrument stays still. The Princeton experiment of Roll, Krotkov and Dicke took the Sun as attractor: a violation would then produce a torque that rotates with the Earth, modulated at the diurnal period with no hand ever touching the balance, and the bound reached \(\abs{\eta}\lesssim10^{-11}\) for aluminium against gold [Roll:1964]. The Moscow experiment of Braginsky and Panov refined the same scheme to \(10^{-12}\) [Braginsky:1972]. The Eöt-Wash balance in Seattle rotates continuously on a turntable, so the signal appears at the turntable frequency, far above the drift of the fibre, and bounds \(\abs{\eta}\) for beryllium against titanium at \(10^{-13}\) [Schlamminger:2008].
The current limit belongs to orbit. The MICROSCOPE satellite compared the free fall of concentric titanium and platinum-alloy test masses around the Earth, drag-free and spin-modulated, where the driving acceleration is the full orbital \(g\) rather than a small horizontal residue of it; the first results bounded \(\abs{\eta}\) at \(10^{-14}\) [Touboul:2017] and the final analysis at parts in \(10^{15}\) [Touboul:2022]. The instruments, procedures and the full table of bounds are reported in Section 53.5 (Table 53.3).
Two bodies of different composition, released at the same place in the same external gravitational field, fall with accelerations that agree to within the precision of every experiment yet performed. Writing the fractional differential acceleration of a pair \((A,B)\) as the Eötvös parameter
every measurement is consistent with \(\eta_{AB}=0\): at the \(10^{-9}\) level in the terrestrial torsion balance of Eötvös, Pekár and Fekete [Eotvos:1922]; at \(10^{-11}\) with the Sun as attractor [Roll:1964]; at \(10^{-12}\) [Braginsky:1972] and \(10^{-13}\) [Schlamminger:2008] in later torsion balances; and at the level of parts in \(10^{15}\) for titanium and platinum test masses compared in orbital free fall [Touboul:2017] [Touboul:2022]. Rests on Postulate 19.8.
Derivation. Derives Phenomenon 42.1. Newtonian dynamics carries two logically independent masses: the inertial mass \(m_{\mathrm{I}}\) multiplying the acceleration in the second law, and the gravitational mass \(m_{\mathrm{G}}\) that couples the body to the field \(\vect{g}\). The equation of motion \(m_{\mathrm{I}}\vect{a}=m_{\mathrm{G}}\vect{g}\) gives
and both bodies sample the same \(\vect{g}\), so on substituting into Equation (42.2) the field cancels and
Hence \(\eta_{AB}\) vanishes for every pair of bodies if and only if the ratio \(m_{\mathrm{G}}/m_{\mathrm{I}}\) is one and the same number for all matter — in which case a choice of units sets it to unity and the two masses may be identified. Nothing in Newtonian mechanics requires this identification; it is precisely the experimental content of the null results above. Its consequence is that the trajectory of a freely falling body depends on nothing but its initial position and velocity, which is what permits gravitation to be described as a property of spacetime rather than as a force with a body-dependent coupling.
∎The equivalence principle in three strengths
Weak, Einstein and strong forms
Einstein's 1907 review took the null result of the pendulum and torsion experiments and elevated it to a principle: for an observer in free fall the gravitational field is locally absent, so a laboratory falling freely in a uniform field is indistinguishable, by any experiment performed within it, from a laboratory floating in empty space — and, equally, a laboratory at rest in the field is indistinguishable from one uniformly accelerated in the absence of gravity [Einstein:1907b]. The principle comes in three strengths, distinguished by which experiments are admitted to the comparison, and the distinction is not pedantry: each strength is tested by a different class of measurements and licenses a different part of the geometric description [Will:2014].
(i) Weak equivalence principle (universality of free fall): the trajectory of a freely falling test body — one small enough that tidal gradients and its own gravity are negligible — depends only on its initial position and velocity, not on its composition or internal structure. (ii) Einstein equivalence principle: the weak principle holds, and in addition the outcome of any local non-gravitational experiment performed in a freely falling laboratory is independent of the laboratory's velocity (local Lorentz invariance) and of where and when in the universe it is performed (local position invariance). (iii) Strong equivalence principle: the Einstein principle extended to bodies with non-negligible gravitational self-energy and to local gravitational experiments — self-gravitating bodies fall like test bodies, and the locally measured gravitational coupling is the same everywhere and everywhen [Will:2014].
Each strength buys a definite piece of the theory. The weak principle makes the family of free-fall trajectories a property of spacetime alone, which is what allows them to be declared the geodesics of a connection — the content of Postulate 43.1 in Geometric Formulation of Gravity. The Einstein principle forces all non-gravitational physics to couple to one and the same metric, which is what makes the theory a metric theory and puts the special relativity of Lorentz Transformations in every freely falling laboratory. The strong principle is the most restrictive: among the metric theories surviving the first two, it is satisfied by general relativity and violated by nearly every proposed alternative, scalar–tensor theories included, which is why its test — lunar laser ranging, below — discriminates where the others cannot [Will:2014].
The three strengths may not be as independent as their statements suggest. Schiff observed that the gravitational redshift, nominally a test of the Einstein principle, already follows from the weak principle together with energy conservation, and conjectured that any complete, self-consistent theory of gravity satisfying the weak principle necessarily satisfies the Einstein principle in full [Schiff:1960]. The conjecture remains unproved in generality, but every concrete framework in which it can be checked supports it; on this argument the composition tests of Phenomenon 42.1, at parts in \(10^{15}\), are also the sharpest available probes of the Einstein form, since a violation of local position invariance would feed back into composition-dependent binding energies and show as a nonzero \(\eta\) [Will:2014].
In the language of this chapter, then: the torsion balances and MICROSCOPE test the weak form; the redshift and clock comparisons of Section 42.3 test the Einstein form directly, and specifically its position invariance; and lunar laser ranging tests the strong form, through the Nordtvedt effect stated next [Nordtvedt:1968]. The quantitative bookkeeping in which all these tests are compared — the parametrized post-Newtonian expansion and its parameters \(\gamma\) and \(\beta\) — is taken up in Section 42.8.
Lunar laser ranging
Laboratory masses have no usable self-gravity: the gravitational binding energy of a metre-scale test body is a fraction of order \(10^{-27}\) of its rest energy, hopelessly below any balance. The strong form of the principle therefore needs astronomical test bodies, and the Earth–Moon pair falling toward the Sun is the cleanest available: the Earth stores about \(4.6\times10^{-10}\) of its rest energy as gravitational binding, the Moon about \(1.9\times10^{-11}\), so the two differ in the relevant quantity by half a part in \(10^{9}\) while being tracked with millimetre-level precision [Williams:2004] [Murphy:2012].
The tracking is lunar laser ranging: timing pulses against the retroreflector arrays left on the Moon by the Apollo 11, 14 and 15 crews and the two Lunokhod rovers, a programme running continuously since 1969, with the APOLLO instrument now returning normal points at the millimetre level [Murphy:2012]. The apparatus and procedure are reported in Section 53.6. Two relativistic quantities are extracted from the decades-long range series. The first is the Nordtvedt signal of Phenomenon 42.4 below, whose null result is the strong principle's test. The second is a bound on any secular variation of the gravitational constant: \(\dot{G}/G\) comes out consistent with zero to within about \(10^{-12}\,/\mathrm{yr}\) [Williams:2004]. That number answers a question older than the data: Dirac's large-numbers hypothesis, which read the coincidence of the great dimensionless ratios of physics as a law and required \(G\) to fall with cosmic age at about that rate [Dirac:1938], is excluded by it — a connection developed with the constants of nature in Measurement, SI Units, and the Theory of Errors.
The Earth and the Moon store different fractions of their mass-energy as gravitational binding energy, so if the gravitational mass of a body depended on that fraction the two would fall toward the Sun at different rates and the lunar orbit would be displaced along the Earth–Sun line, with a signature periodic at the synodic month [Nordtvedt:1968]. Lunar laser ranging to the retroreflectors left on the surface, with residuals at the centimetre level, sees no such displacement: the Nordtvedt parameter is consistent with zero at the level of parts in \(10^{4}\) [Williams:2004]. Self-gravitating bodies therefore fall like test bodies, which is the observational content of the strong form of the equivalence principle [Will:2014]. Rests on Phenomenon 42.1 and Theorem 29.51.
Derivation. Derives Phenomenon 42.4. Let the gravitational-to-inertial mass ratio of a body depend on its fractional gravitational self-energy,
where \(U<0\) is the body's Newtonian self-energy and the Nordtvedt parameter \(\eta_{\mathrm{N}}\) vanishes exactly when self-gravitating bodies obey Phenomenon 42.1 [Nordtvedt:1968]. For a uniform sphere \(\abs{U}=3GM^{2}/\left(5R\right)\) (Equation (49.23)), which gives \(\abs{U}/(Mc^{2})=4.2\times 10^{-10}\) for the Earth and \(1.9\times 10^{-11}\) for the Moon, both computed here; interior models concentrate the Earth's mass and raise its value to \(4.6\times 10^{-10}\) [Williams:2004]. The two ratios Equation (42.3) therefore differ by about \(4.4\times 10^{-10}\,\eta_{\mathrm{N}}\), and in the Sun's field at the Earth's orbit, \(GM_{\odot}/r_{\mathrm{ES}}^{2} =5.93\times 10^{-3}\,\mathrm{m}/\mathrm{s}^{2}\), the Moon falls sunward relative to the Earth with the differential acceleration
directed along the Earth–Sun line (toward the Sun for \(\eta_{\mathrm{N}}>0\), since the Earth, with the larger binding fraction, then couples the more weakly).
What matters is how the lunar orbit responds. Work in the frame rotating about the Earth with the Moon's mean motion \(n=2.662\times 10^{-6}\,/\mathrm{s}\), in which the unperturbed Moon rests on a circular orbit of radius \(r_{0}\) with \(n^{2}=\mu/r_{0}^{3}\), \(\mu\) the gravitational parameter of the pair. Linearizing the rotating-frame equation of motion Equation (29.66) about this orbit — gravity contributing \(+2n^{2}\xi\) radially and \(-n^{2}\zeta\) along track, the centrifugal term \(+n^{2}\xi\) and \(+n^{2}\zeta\), the Coriolis term coupling the two — gives, for a radial displacement \(\xi\) and an along-track displacement \(\zeta\),
The forcing rotates at the synodic frequency \(D=n-n_{\odot}=2.463\times 10^{-6}\,/\mathrm{s}\) (synodic month \(29.53\) days), because the frame rotates at \(n\) while the Earth–Sun direction advances at the annual rate \(n_{\odot}=1.991\times 10^{-7}\,/\mathrm{s}\). The ordinary solar tidal terms are omitted: they are independent of \(\eta_{\mathrm{N}}\) and already carried by the Newtonian lunar ephemeris.
Substituting the forced solution \(\xi=X\cos Dt\), \(\zeta=Z\sin Dt\) into Equation (42.4), the along-track equation gives \(Z=\left(\delta a-2nDX\right)/D^{2}\), and eliminating \(Z\) from the radial equation leaves \(X\left(n^{2}-D^{2}\right)=\delta a\left(1+2n/D\right)\), that is
The denominator is the physics: \(n-D=n_{\odot}\), so the forcing lies close to the orbit's own frequency — they differ only by the annual rate — and the response is amplified by more than an order of magnitude over the naive estimate \(\delta a/n^{2}\). Numerically the response coefficient \(\left(1+2n/D\right)/\left(n^{2}-D^{2}\right)\) is \(3.1\times 10^{12}\,\mathrm{s}^{2}\), so
computed here from the linear response alone. The full lunar theory does better: the classical solar perturbations of the orbit feed the displacement back at the same frequency and amplify it by roughly a further \(1.6\), to about \(13\,\mathrm{m}\) per unit \(\eta_{\mathrm{N}}\) — a figure quoted from the detailed computations rather than derived here [Nordtvedt:1968] [Williams:2004]. The signature is a range oscillation \(X\cos Dt\), largest along the Earth–Sun line and periodic at the synodic month, which is what separates it from the Newtonian terms in the fit.
Ranging residuals at the level of millimetres to centimetres against a \(13\,\mathrm{m}\) coefficient then bound the parameter at parts in \(10^{4}\): the joint fit of the ranging data gives \(\eta_{\mathrm{N}}=\left(4.4\pm4.5\right)\times10^{-4}\) [Williams:2004]. In the parametrized post-Newtonian bookkeeping of Section 42.8 the parameter is, for a metric theory with no preferred-frame or preferred-location effects, the combination
a result imported from the post-Newtonian analysis of massive bodies [Nordtvedt:1968] [Will:2014]. General relativity has \(\beta=\gamma=1\) and predicts \(\eta_{\mathrm{N}}=0\) identically — the null result above — while any theory in which the effective coupling depends on the local gravitational potential predicts \(\eta_{\mathrm{N}}\neq0\) at some level.
∎Gravitational redshift
The prediction
The redshift was the first new phenomenon Einstein extracted from the equivalence principle, in the 1907 review, and it is the one prediction of this chapter that needs no field equation whatever: it follows from the principle together with kinematics, as the derivation below shows, and Einstein reworked it in 1911 with the optics made explicit — a periodic signal climbing a potential difference \(\Delta\Phi\) arrives with its frequency shifted by \(-\Delta\Phi/c^{2}\) [Einstein:1907b] [Einstein:1911]. Over a laboratory height \(H\) in a uniform field the fraction is \(gH/c^{2}\); from the surface of the Sun to the Earth it is \(GM_{\odot}/\left(c^{2}R_{\odot}\right)=2.12\times 10^{-6}\), computed here from the solar mass parameter and radius.
Because no field equation enters, a redshift measurement tests the equivalence principle itself — in the classification of Definition 42.2, the local position invariance of the Einstein form — and not the dynamics of gravity. A theory that failed it would not need different field equations but a different foundation; conversely, any metric theory passes it, which is why the classical redshift experiments, for all their beauty, do not discriminate between general relativity and its metric rivals [Schiff:1960] [Will:2014].
The obvious place to look first was the Fraunhofer spectrum of the Sun, and the search there stayed inconclusive for half a century: the solar lines are shifted by pressure and by the Doppler effect of convective granulation by amounts comparable to the sought \(2.12\times 10^{-6}\), and the early claims scattered on both sides of the prediction [Will:2014]. The clean confirmation came indoors, from an effect discovered in 1958 that turned a nucleus into a frequency standard of quality \(10^{13}\).
Pound–Rebka and Pound–Snider
Recoil-free gamma emission — the Mössbauer effect — pins the emitting nucleus to its crystal lattice, so that the recoil momentum is taken up by the crystal as a whole and the emitted line keeps its natural width, a fractional width of about \(3\times10^{-13}\) for the \(14.4\,\mathrm{keV}\) transition of \(^{57}\)Fe [Moessbauer:1958]. That is what made a shift of order \(10^{-15}\) measurable at all. Pound and Rebka sent these gamma rays up and down the \(22.5\,\mathrm{m}\) tower of the Jefferson Physical Laboratory at Harvard, detected the shift as an asymmetry in resonant absorption, and confirmed the prediction to about ten per cent in 1960 [Pound:1960]; the 1965 repetition with Snider reached one per cent [Pound:1965].
The decisive systematic is thermal: the second-order Doppler shift of nuclei vibrating in their lattices amounts to \(-2.4\times 10^{-15}\,/\mathrm{K}\) of temperature difference between source and absorber — comparable to the entire gravitational effect per kelvin — and controlling it is the limiting art of the experiment. The apparatus, the reversal procedure that cancels almost everything else, and the derivation of that temperature coefficient (Equation (53.4)) are set out in Section 53.3.
A signal emitted by a source at rest low in a static gravitational field and received by an observer at rest higher up arrives with its frequency lowered. Over a height difference \(H\) small enough that the field strength \(g\) may be treated as uniform,
For the \(22.5\,\mathrm{m}\) tower of the Harvard experiment this is a fractional shift of \(2.5\times10^{-15}\), resolvable only because the recoil-free emission and absorption of gamma rays by nuclei bound in a lattice [Moessbauer:1958] supplies a line of comparable relative width. The measurement confirmed Equation (42.7) to about ten per cent [Pound:1960] and, in the 1965 repetition, to about one per cent [Pound:1965]. A hydrogen maser carried on a suborbital rocket verified it at the \(7\times10^{-5}\) level [Vessot:1980], and optical clocks now resolve it across a height difference of less than one metre [Chou:2010]. Rests on Definition 42.2.
Derivation. Derives Phenomenon 42.5. By the Einstein equivalence principle a laboratory held static in a uniform field \(g\) is locally indistinguishable from a laboratory in flat spacetime accelerating with proper acceleration \(g\), and the second is the easier to analyse [Einstein:1907b] [Einstein:1911]. Work in the momentarily comoving inertial frame in which the emitter is at rest at the instant of emission. Light leaves the floor and reaches the ceiling a height \(H\) above it after a time \(t=H/c\), up to corrections of higher order in \(gH/c^{2}\). During that time the receiver, which is accelerating, has acquired the velocity
directed away from the source. The receiver is therefore receding, and to first order in \(v/c\) the Doppler formula of Lorentz Transformations gives
which is Equation (42.7). Two features of the argument deserve notice. It uses only the equivalence principle, kinematics and the conservation of energy — no field equation enters — so a redshift measurement tests the equivalence principle rather than the dynamics of the gravitational field; and the sign is fixed by the geometry, an upward-travelling signal always being received redshifted.
∎Clocks on rockets and in orbit
Everything since Pound and Snider has been a better clock at a bigger height difference. Gravity Probe A flew a hydrogen maser on a suborbital trajectory to \(10^{7}\,\mathrm{m}\) altitude in 1976, comparing it continuously against a ground maser through a Doppler-cancelling radio link, and verified the combined gravitational and kinematic frequency shift to \(7\times10^{-5}\) [Vessot:1980] — still the sharpest dedicated test of Equation (42.7) on a terrestrial potential difference. At the opposite extreme of scale, optical clocks of fractional instability below \(10^{-17}\) resolve the shift across laboratory height differences: first \(0.33\,\mathrm{m}\) between two aluminium-ion clocks [Chou:2010], and by 2022 across a single millimetre-scale sample of strontium atoms, mapping Equation (42.7) within one atomic cloud [Bothwell:2022].
Two further measurements belong to this family though they are reported elsewhere. The caesium clocks Hafele and Keating flew around the world in 1972 measured the redshift summed with the kinematic time dilation of Phenomenon 38.13, eastward and westward [Hafele:1972a] [Hafele:1972b]; the flights are treated in full in Section 41.5. And the same shift has now been seen in an astronomical spectrum under a potential four orders of magnitude deeper than the Sun's: the star S2, at its 2018 closest approach to the Galactic-centre black hole, showed the predicted combined redshift [Abuter:2018], a measurement belonging to Experiment: Black-Hole Observations. The engineering counterpart of all of these — the satellite-navigation clock budget — closes this chapter in Section 42.7.
The perihelion of Mercury
Le Verrier's anomaly
The oldest evidence in this chapter predates the theory by fifty-six years, and it was found by accounting. Le Verrier, having predicted Neptune from the misbehaviour of Uranus, applied the same perturbation theory — the secular expansion of Central Forces and Statics — to Mercury, using the accumulated record of transits across the solar disc. The observed advance of Mercury's perihelion, referred to the moving equinox, is about 5600 seconds of arc per century; the precession of the equinoxes accounts for some 5025 of them, and the Newtonian perturbations of Venus, the Earth, Mars, Jupiter and Saturn for another 531. In 1859 Le Verrier announced that the books do not balance: a residual advance remained, near 38 seconds of arc per century in his reduction [LeVerrier:1859], moved to the modern 43 by Newcomb's re-reduction with improved planetary masses [Newcomb:1895]. The full budget, with the numbers above, stands as Table 53.1 in Section 53.1.
Every Newtonian repair was tried and failed. An intramercurial planet — Vulcan, hunted for decades — would have been seen in transit; a sufficient solar oblateness contradicts the observed figure of the Sun; a modified force-law exponent tuned to Mercury disturbs the Moon [Will:2014]. In the terms of Epistemology and the Scientific Method this is the textbook case of an anomaly held open: a small, stubborn, well-measured discrepancy, resistant to parametric patching, waiting half a century for a theory with no free parameter to claim it.
After every Newtonian perturbation by the other planets has been computed and subtracted, the perihelion of Mercury is observed to advance in the direction of the orbital motion at a residual rate of about 43 seconds of arc per century — a discrepancy isolated by Le Verrier in 1859 and left unexplained for half a century [LeVerrier:1859]. General relativity accounts for it with no adjustable parameter: a bound orbit of semi-major axis \(a\) and eccentricity \(e\) about a mass \(M\) has its perihelion advanced by
per revolution, to lowest order in \(GM/(c^{2}a)\) [Einstein:1915a]. Inserting Mercury's orbital elements and the Sun's mass parameter, and multiplying by the 415 revolutions Mercury completes in a century, reproduces the observed residual. Rests on Postulate 43.1 and Equation (53.2).
Derivation. Derives Phenomenon 42.6. This is carried out in Section 53.1, in the derivation following Phenomenon 53.1. In outline: the timelike geodesics of the Schwarzschild geometry (Schwarzschild Geometry and Black Holes) give, for the inverse radius \(u=1/r\) in the orbital plane, the orbit equation Equation (53.2), which is the Newtonian equation of Central Forces and Statics plus the single correction term \(3GMu^{2}/c^{2}\), smaller than the Newtonian terms by a factor of order \(v^{2}/c^{2}\sim10^{-7}\) for Mercury. Treating it as a perturbation of the Keplerian ellipse, only its \(\cos\varphi\) harmonic is resonant with the orbit and accumulates; the ellipse's apsis line turns by \(\Delta\varphi=6\pi G^{2}M^{2}/(c^{2}h^{2})\) per revolution, and eliminating the angular momentum \(h\) through \(h^{2}=GMa\left(1-e^{2}\right)\) gives Equation (42.8). Numerically, Mercury's elements yield \(5.02\times 10^{-7}\) radian per revolution, or 42.98 seconds of arc per century — the residual of Table 53.1.
∎Einstein's 1915 computation
Einstein computed the advance in the 18 November 1915 communication to the Prussian Academy, from the vacuum field equations in the approximation that was all he had — the exact Schwarzschild solution was still two months away — and obtained the observed 43 seconds of arc per century with nothing to adjust [Einstein:1915a]. The same paper carries the doubled light deflection of Section 42.5, so the one communication contains both the postdiction that validated the field equations and the prediction that would test them four years later.
It must be said plainly, as Section 53.1 says it, that this is a postdiction: the number was known, and Einstein knew what he had to match. It is equally important that this costs the result almost nothing. Equation (42.8) contains no parameter that could have been tuned — the field equations were fixed by general covariance and the Newtonian limit, and the orbit followed — whereas every rival explanation on offer was parametric, and each, tuned to Mercury, then contradicted some other observation. A parameter-free postdiction carries the evidential weight of a prediction; the epistemology is Epistemology and the Scientific Method's. The same formula has since been confirmed as a genuine prediction far outside the solar system, in the Schwarzschild precession of the star S2 about the Galactic-centre mass [Abuter:2020], reported with its instrument in Experiment: Black-Hole Observations.
The deflection of light
Half and full values
If the equivalence principle holds, light must fall: a horizontal beam crossing a laboratory that accelerates upward arrives lower on the far wall, so the same must happen in the laboratory held static in a gravitational field. The principle alone even fixes the rate, because whatever falls, falls universally. Treat the ray as a trajectory traversed at speed \(c\) in the Newtonian field of a mass \(M\), passing at impact parameter \(b\), and integrate the transverse acceleration \(GMb/\left(b^{2}+x^{2}\right)^{3/2}\) along the undeflected straight line \(x=ct\):
For a ray grazing the Sun, \(b=R_{\odot}\), this is \(4.24\times 10^{-6}\) in radian measure, or 0.87 seconds of arc, computed here. Einstein published exactly this value in 1911, from the equivalence principle dressed as optics — the potential-dependent clock rate of Equation (42.7) makes the coordinate speed of light vary with height, and Huygens' principle turns the gradient into a bending of the wavefront — which is the same physics as the falling corpuscle and gives the same number [Einstein:1911].
The completed theory of November 1915 doubled it, to \(4GM/\left(c^{2}b\right)=1.75\) seconds of arc at the limb [Einstein:1915a]. The second half has a precise pedigree: the equivalence principle constrains only the time–time part of the metric, but in the full theory the mass curves space as well, and the two contributions to the bending of a null ray are equal. In the parametrized post-Newtonian bookkeeping the deflection is \(\left(1+\gamma\right)/2\) times the full value, with \(\gamma\) the space-curvature parameter: \(\gamma=0\) reproduces Equation (42.9), and general relativity's \(\gamma=1\) doubles it [Will:2014]. The factor of two is therefore exactly what separates a metric theory from Newtonian gravity dressed in geometry, and it is what the eclipse expeditions were designed to resolve.
The 1919 eclipse expeditions
The measurement asked for stars photographed close to the eclipsed Sun, compared with the same field photographed months away from it. The Greenwich expeditions of 29 May 1919 — Sobral, in northern Brazil, and Príncipe, in the Gulf of Guinea — were placed under an eclipse of unusual length with the Sun standing in the Hyades, and their announced result favoured the full deflection: \(1.98\pm0.12\) seconds of arc at the limb from the Sobral \(10\,\mathrm{cm}\) lens and \(1.61\pm0.30\) from Príncipe, bracketing the predicted 1.75 and both far above 0.87 [Dyson:1920]. The apparatus, the plate reduction, and the celebrated difficulty — the Sobral astrographic plates, set aside for a coelostat mirror that had changed figure in the heat — are reported in Section 53.2 (Table 53.2), together with the modern re-measurement of those same plates, which returned values consistent with the full deflection and showed that the rejection had not biased the conclusion [Harvey:1979].
The announcement of November 1919 made general relativity a public fact, and the treatise's assessment of its evidential weight is the experiment chapter's: the two accepted determinations were designed to separate hypotheses differing by a factor of two, and they did; the precision claimed was modest, and what settles the matter today is not photography at all.
Starlight passing close to the Sun is deflected toward it, so that the apparent positions of stars seen near the eclipsed limb are displaced radially outward. For a ray of impact parameter \(b\) passing a mass \(M\) with \(b\gg GM/c^{2}\), the total deflection is
which for a ray grazing the solar limb amounts to \(8.49\times10^{-6}\) in radian measure, or 1.75 seconds of arc. This is exactly twice the value that the equivalence principle alone yields [Einstein:1911], the second half coming from the curvature of space [Einstein:1915a]. The eclipse expeditions of 1919 measured a deflection favouring the full value [Dyson:1920]; radio interferometry on very long baselines now confirms it at the \(10^{-4}\) level [Will:2014]. The same bending, exerted by galaxies and clusters, produces the multiple images of gravitational lenses, of which the twin quasar 0957+561 was the first identified [Walsh:1979]. Rests on Postulate 43.1 and Equation (53.2).
Derivation. Derives Phenomenon 42.7. The full computation is carried out in Section 53.2, in the derivation following Phenomenon 53.2. In outline: for a null geodesic of the Schwarzschild geometry the orbit equation Equation (53.2) loses its Newtonian constant term — that term came from the rest mass — leaving \(\dd^{2}u/\dd\varphi^{2}+u=3GMu^{2}/c^{2}\); perturbing the straight line \(u_{0}=\sin\varphi/b\) and following the asymptotes where \(u=0\) shows the ray's two asymptotic directions closing by \(4GM/\left(c^{2}b\right)\), which is Equation (42.10), and numerically 1.751 seconds of arc at the solar limb.
The origin of the factor of two over Equation (42.9) is cleanest in the isotropic form of the weak-field metric used for the Shapiro derivation in Section 53.4: the potential enters twice, once in the time–time coefficient and once in the spatial one, and the two retard the coordinate speed of light equally. The equivalence principle fixes only the first — it is the clock-rate effect of Equation (42.7) — and by itself reproduces the half value Equation (42.9); the spatial curvature, absent from any account of gravity as a mere universal acceleration, contributes the equal second half. The parametrized statement is that the deflection carries the factor \(\left(1+\gamma\right)/2\), and the 1919 result is the measurement that first required \(\gamma\neq0\) [Will:2014].
∎Radio interferometry and gravitational lensing
The modern measurement is radio astrometry. Very-long-baseline interferometry locates compact quasars to fractions of a milliarcsecond, and the Sun's deflection field is not a limb-grazing curiosity but a whole-sky distortion: even a source at right angles to the Sun is displaced by about four milliarcseconds, so global astrometry must carry Equation (42.10) everywhere it points [Will:2014]. Dedicated campaigns following quasars as the Sun passes them measure the space-curvature parameter directly: \(\gamma-1\) consistent with zero at the level of a few parts in \(10^{4}\) [Lebach:1995] [Fomalont:2009], four orders of magnitude past the eclipse plates.
The same bending, scaled up to galaxy and cluster masses, is gravitational lensing. The twin quasar 0957+561 — two images, identical spectra, one source — was identified in 1979 as the first gravitational lens [Walsh:1979], and lensing has since become an astrophysical instrument in its own right: it weighs the dark matter of The Dark Sector: Evidence Without Explanation, and it magnifies and multiplies the strong-field sources of Experiment: Black-Hole Observations. In this chapter it stands as the demonstration that Equation (42.10) operates identically from \(b\sim R_{\odot}\) to megaparsec impact parameters.
The Shapiro time delay
Radar ranging to the inner planets
The fourth classical test was proposed, not by Einstein, but by radar. Shapiro observed in 1964 that a pulse whose path passes close to the Sun takes measurably longer to return than the flat-space geometry predicts — an excess of some \(200\,\mu\mathrm{s}\) for an Earth–Venus round trip at superior conjunction, growing logarithmically as the ray approaches the limb [Shapiro:1964]. The logarithmic conjunction profile is the effect's signature: it is not degenerate with an ephemeris error, which varies on the orbital timescale rather than with the impact parameter. The first measurement followed within four years, radar echoes from Mercury and Venus timed at Haystack and Arecibo agreeing with the prediction at the level of tens of per cent [Shapiro:1968]; the transmitters, the coded pulse trains and the conjunction-by-conjunction procedure are reported in Section 53.4.
Viking and Cassini
Replacing the passive planetary echo with a spacecraft transponder sharpened the test by orders of magnitude. Ranging to the Viking landers on Mars through several superior conjunctions confirmed the delay to about one part in \(10^{3}\) [Reasenberg:1979]. The Cassini spacecraft, tracked on its cruise to Saturn in 2002, ran its radio link simultaneously at multiple widely separated frequencies, using the known dispersion of the solar-corona plasma to solve the corona away — the systematic that had limited every predecessor — and delivered \(\gamma-1=\left(2.1\pm2.3\right)\times10^{-5}\) [Bertotti:2003], the tightest solar-system constraint on any metric theory of gravity. The same delay, imposed by a companion star on the timing of a pulsar, is now a standard tool for weighing neutron stars [Demorest:2010]; that use belongs to Compact Stars and Relativistic Astrophysics.
A radar pulse sent from Earth to an inner planet and reflected back takes measurably longer when its path passes close to the Sun than when the same geometric distance is traversed far from it. For a round trip between points at radii \(r_{1}\) and \(r_{2}\) from a mass \(M\) past a closest approach \(b\), with \(b\ll r_{1},r_{2}\), the excess is
of order \(200\,\mu\mathrm{s}\) for an Earth–Venus circuit at superior conjunction. Proposed as a fourth test in 1964 [Shapiro:1964] and first measured against Mercury and Venus [Shapiro:1968], the delay was confirmed to one part in \(10^{3}\) by ranging to the Viking landers [Reasenberg:1979]. The Cassini radio link measured the post-Newtonian parameter \(\gamma\), which enters Equation (42.11) through an overall factor \(\left(1+\gamma\right)/2\), as \(\gamma-1=\left(2.1\pm2.3\right)\times10^{-5}\) [Bertotti:2003]. Rests on Postulate 43.1 and Proposition 45.14.
Derivation. Derives Phenomenon 42.8. Carried out in Section 53.4, in the derivation following Phenomenon 53.4. In outline: write the weak-field Schwarzschild metric in isotropic coordinates, whose spatial part is conformally flat, so that the unperturbed ray is a coordinate straight line and the whole effect sits in the coordinate speed of light, \(c\,\dd t\simeq\dd l\left(1+2GM/c^{2}r\right)\) — one factor of \(GM/c^{2}r\) from time dilation and one from spatial curvature, the same equal split as in the deflection. Integrating the excess along the straight path with \(r=\sqrt{b^{2}+l^{2}}\) and doubling for the return trip gives Equation (42.11); for an Earth–Venus circuit grazing the limb the logarithm is \(11.8\) and \(\Delta t=2.3\times 10^{-4}\,\mathrm{s}\). The characteristic weakness of the dependence on \(b\) — halving the impact parameter adds only \(2.7\times 10^{-5}\,\mathrm{s}\) — is why the measurement needs a well-sampled conjunction rather than a single well-placed shot.
∎Relativity in the Global Positioning System
Satellite navigation is a clock experiment run as infrastructure. A receiver fixes its position by differencing the arrival times of signals from satellites whose clocks are synchronized to a common timescale; light covers \(30\,\mathrm{cm}\) per nanosecond, so metre-level positioning demands nanosecond-level timekeeping, held continuously, by caesium and rubidium standards in orbit [Ashby:2003]. At that precision both relativistic clock effects of this chapter are not corrections but the largest terms in the error budget: the orbiting clocks sit higher in the Earth's potential than the geoid — the redshift of Phenomenon 42.5 makes them run fast — and they move at orbital speed — the time dilation of Phenomenon 38.13 makes them run slow. The budget below is computed here, and the design of the system confirms it daily.
The atomic clocks carried by the satellites of the Global Positioning System, in nearly circular orbits of radius \(r=2.656\times 10^{7}\,\mathrm{m}\), run fast with respect to clocks on the geoid by about \(38.5\,\mu\mathrm{s}\) per day. The offset is not a small correction: it is designed into the hardware, the flight oscillators being deliberately set low in frequency before launch, with the residual terms modelled in every receiver. Left unaccounted for it would introduce a ranging error \(c\,\Delta t\) growing by about \(11\,\mathrm{km}\) per day, so that positions would become useless within minutes [Ashby:2003]. Rests on Phenomena 38.13 and 42.5.
Derivation. Derives Phenomenon 42.9. To first order in \(\Phi/c^{2}\) and \(v^{2}/c^{2}\), a clock at Newtonian potential \(\Phi\) moving with speed \(v\) in a static weak field accumulates proper time at the rate
the second term being the gravitational redshift of Phenomenon 42.5 written with \(\Phi=gH\), and the third the time dilation of Lorentz Transformations. The two have opposite signs: height makes a clock run fast, motion makes it run slow.
For a circular orbit of radius \(r\) about a mass \(M\) the Newtonian relations \(\Phi=-GM/r\) and \(v^{2}=GM/r\) hold simultaneously, so the two effects combine into one term,
while a clock at rest on the ground at radius \(R\) has \(\dd\tau/\dd t=1-GM/(Rc^{2})\), the Earth's rotation and its equatorial bulge being absorbed into the definition of the geoid. The fractional rate at which the orbiting clock gains is therefore
With \(GM=3.986\times 10^{14}\,\mathrm{m}^{3}/\mathrm{s}^{2}\) for the Earth, \(R=6.371\times 10^{6}\,\mathrm{m}\) and \(r=2.656\times 10^{7}\,\mathrm{m}\), the bracket is \(1.005\times10^{-7}\,\mathrm{m}^{-1}\) and \(GM/c^{2}=4.435\times 10^{-3}\,\mathrm{m}\), so \(\delta=4.46\times10^{-10}\). Over one day of \(86400\,\mathrm{s}\) the accumulated gain is \(3.85\times 10^{-5}\,\mathrm{s}\). Separating the two contributions, \(GM/(Rc^{2})-GM/(rc^{2})\) gives \(45.7\,\mu\mathrm{s}\) per day of gravitational blueshift and \(GM/(2rc^{2})\) gives \(7.2\,\mu\mathrm{s}\) per day of velocity dilation, whose difference is the quoted \(38.5\,\mu\mathrm{s}\). Multiplying by \(c\) turns the daily offset into \(11.5\,\mathrm{km}\) of range.
∎The system absorbs the constant rate \(\delta\) in the factory: the fundamental frequency of \(10.23\,\mathrm{MHz}\) is set before launch to \(10.22999999543\,\mathrm{MHz}\), a fractional offset of \(-4.4647\times 10^{-10}\) [Ashby:2003], agreeing with the \(\delta\) computed here to a fraction of a per cent — the residue reflecting the rounded orbital radius used above. Two smaller terms remain for the receiver. An orbit of eccentricity \(e\) trades height against speed around each revolution, producing a periodic residual of amplitude \(2e\sqrt{GMa}/c^{2}\) [Ashby:2003] — about \(23\,\mathrm{ns}\) at \(e=0.01\), computed here — which every receiver applies from the broadcast elements. And a receiver fixed to the rotating Earth is not in an inertial frame: consistent synchronization requires the Sagnac correction of Section 67.6.4,
with \(A_{z}\) the equatorial-plane projection of the area swept by the position vector of the signal, counted positive eastward [Ashby:2003]. For a receiver on the equator watching a satellite near the horizon this reaches about \(133\,\mathrm{ns}\), computed here — \(40\,\mathrm{m}\) of pseudorange, far above the system's accuracy goals.
What the system tests, in the classification of Definition 42.2, is the Einstein principle's position dependence of clock rates summed with special-relativistic time dilation, continuously and publicly; the field equations proper it does not touch, the next order in \(GM/\left(rc^{2}\right)\) entering at \(10^{-19}\) and accumulating femtoseconds per day, far below the flight clocks' stability. Nor is it a precision test — its residuals answer to engineering tolerances, orders of magnitude short of Gravity Probe A [Vessot:1980] — but it is the test that never stops running: every position fix is a check that Equation (42.12), with both signs, is physics [Ashby:2003].
Beyond the classical tests
The tests of this chapter are compared, across theories, in one common language: the parametrized post-Newtonian (PPN) expansion, which writes the metric of any metric theory, to first order beyond Newton, in terms of a small set of dimensionless parameters and lets each experiment bound a combination of them [Will:2014]. Two parameters carry nearly everything measured here: \(\gamma\), the space curvature per unit mass, and \(\beta\), the nonlinearity of superposition. General relativity has \(\gamma=\beta=1\). The deflection and the Shapiro delay each measure \(\left(1+\gamma\right)/2\), and Cassini's \(\gamma-1=\left(2.1\pm2.3\right)\times10^{-5}\) is the sharpest entry [Bertotti:2003]; the perihelion advance measures \(\left(2+2\gamma-\beta\right)/3\); and the Nordtvedt combination Equation (42.6) closes the system, lunar laser ranging bounding it at parts in \(10^{4}\) [Williams:2004]. Taken together the solar-system tests pin \(\gamma\) and \(\beta\) to unity at the level of parts in \(10^{4}\) or better [Will:2014].
One sector of the weak field is not touched by any test above: the gravitational effect of rotation. A spinning mass drags inertial frames around with it, precessing an orbiting gyroscope about the spin axis [Lense:1918] [Thirring:1918], on top of the larger geodetic precession that any orbiting gyroscope suffers from the curvature of the static field [Schiff:1960a]. Gravity Probe B measured both against a guide star — the geodetic rate to about \(0.3\,\mathrm{\%}\), the frame-dragging rate to about twenty per cent [Everitt:2011] — and lunar laser ranging sees the geodetic precession of the lunar orbit independently [Williams:2004]. The rates, the apparatus and the honest accounting of Gravity Probe B's error budget are given in Section 53.6.
Everything above is weak-field: the dimensionless potential \(GM/\left(c^{2}r\right)\) never exceeds a few parts in \(10^{6}\) in any solar-system test. The evidence beyond that regime opens with the binary pulsar PSR B1913+16, discovered in 1974 [Hulse:1975]: a clock in an eight-hour relativistic orbit, whose periastron advances four degrees per year where Mercury manages 43 seconds of arc per century, and whose orbit shrinks by energy lost to gravitational radiation at the rate the quadrupole formula of Gravitational-Wave Theory predicts — confirmed within a few per cent by 1982 [Taylor:1982] and to a fraction of a per cent over four decades of timing [Weisberg:2016]. The radiation itself was detected directly in 2015, two black holes of thirty-odd solar masses observed in their final orbits by laser interferometry [Abbott:2016]; that story is Experiment: Gravitational Waves's. The strongest fields of all — orbits, spectra and images at a few gravitational radii — are the subject of Experiment: Black-Hole Observations.
The order of this part follows the logic these tests establish. The universality of free fall licenses geometry (Geometric Formulation of Gravity); the classical tests constrain the field equations that the geometry must obey (The Einstein Field Equations); and the chapters after them follow the equations into the strong field, confident at the level of parts in \(10^{5}\) where the confidence was earned.