Illustrated Physics
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Astrophysics

The binding energy that has to fall too

Every laboratory test of the equivalence principle compares bodies whose own gravity is a part in 10²⁵ of their mass, so none of them can ask whether gravitational binding energy falls like everything else. The Earth is bound by five parts in ten billion and the Moon by twenty times less, and if that difference fell differently the Moon's orbit would lean towards the Sun once a month — by a distance lasers have been measuring since 1969.
18 min read 5 figures Who is measuringMade into an instrument

Assumes: The fall that does not depend on what is falling · The floor that cannot be told from gravity

The fall that does not depend on what is falling followed three centuries of experiments to a bound of a part in 101510^{15} on any difference in how two materials fall. Every one of those experiments compared bodies that differ in what they are made of: platinum against titanium, beryllium against aluminium. The differences they probe are differences in nuclear binding, in electromagnetic binding, in the ratio of neutrons to protons — every contribution to mass that changes from one element to the next.

One contribution to mass does not change from one laboratory sample to the next, because it is too small to change in anything that fits in a laboratory. A body held together by its own gravity has less mass than its parts would have apart, by the gravitational binding energy divided by c2c^2, and mass is a form of energy says that the deficit has to be counted. Whether that part of a body’s mass falls like the rest of it is a separate question, called the strong equivalence principle, and it can be asked only of bodies big enough for their own gravity to matter.

A mass that is partly its own gravity

The gravitational self-energy of a body is the work needed to pull it apart into dust scattered to infinity. For a uniform sphere it is 35GM2/R\tfrac35 GM^2/R, so as a fraction of the body’s rest energy it is 35 GM/Rc2\tfrac35\,GM/Rc^2 — a number that depends only on how compact the body is. For anything whose density rises towards its centre, the coefficient is larger.

How much of a body's mass is its own gravity. The gravitational self-energy of seven bodies as a fraction of their mass-energy, |Ω|/Mc², on a logarithmic axis spanning twenty-seven decades. Each is integrated shell by shell over a density profile: uniform for the lead sphere and the Moon, an iron core inside a rock mantle for the Earth, and polytropes solved from the Lane–Emden equation for Jupiter, the Sun, a white dwarf and a neutron star, each of which reproduces its polytrope's 3/(5 − n) to a per cent. A 10 kg lead sphere: 7.5 × 10⁻²⁶; the Moon: 1.9 × 10⁻¹¹; the Earth: 4.5 × 10⁻¹⁰; Jupiter: 1.5 × 10⁻⁸; the Sun: 3.2 × 10⁻⁶; a white dwarf: 8.7 × 10⁻⁵; a neutron star: 1.3 × 10⁻¹. A laboratory mass is bound by a part in 10²⁵, which is why no composition test can ask whether binding energy falls like the rest of a body. The Earth and the Moon differ by 4.3 × 10⁻¹⁰, enough for an orbit to show it; a neutron star is bound by a tenth of itself, and the Newtonian estimate drawn for it is only that.
Fig. 1 Gravitational self-energy as a fraction of mass-energy for seven bodies, integrated over a density profile for each: uniform for a lead sphere and the Moon, an iron core in a rock mantle for the Earth, and polytropes for Jupiter, the Sun, a white dwarf and a neutron star. The range is twenty-five decades, from 7.5 × 10⁻²⁶ for the lead sphere to about a tenth for the neutron star.

The spread is enormous and it is the whole story of this subject. A ten-kilogram lead sphere is bound by its own gravity by 7.5×10−267.5\times10^{-26} of its mass. The Moon, nearly uniform, is bound by 1.9×10−111.9\times10^{-11}. The Earth, with an iron core twice as dense as its mantle, by 4.5×10−104.5\times10^{-10} in the two-layer model drawn — about 4.6×10−104.6\times10^{-10} in a detailed one. The Sun, modelled as a polytrope, by 3.2×10−63.2\times10^{-6}; a white dwarf by nearly 10−410^{-4}; and a neutron star by about a tenth of its entire mass, in a Newtonian estimate that relativity corrects in detail but not in size.

The polytropes are worth a word, because they are what makes the stellar numbers more than guesses. A star in the balance between pressure and its own gravity whose pressure goes as a power of its density has a structure fixed by one equation, the Lane–Emden equation, and its self-energy comes out as exactly 3/(5−n)3/(5-n) times GM2/RGM^2/R for index nn. The integrations behind the figure reproduce that coefficient to a per cent for each index, which is the check that the density profiles are the ones they claim to be.

No laboratory test can see a violation that scales with self-energy, however good the instrument. If binding energy fell at a rate differing by a fraction η from the rest of mass, two laboratory masses would fall differently by η times a number of order 10−2510^{-25}, and the best torsion balance in the world, sensitive to two parts in 101310^{13}, would bound η only below about 101210^{12} — which is to say, not at all.

Two bodies already falling together

The Earth and the Moon are a pair of test masses that nobody had to build. They fall together towards the Sun once a year, separated by 384,000 kilometres, and their self-energies differ by 4.3×10−104.3\times10^{-10} of their mass.

Suppose the Earth’s binding energy fell towards the Sun a little less readily than the rest of its mass. The Earth as a whole would then accelerate towards the Sun slightly less than the Moon does, by η times 4.3×10−104.3\times10^{-10} times the Sun’s pull at that distance. The Sun pulls at 5.93 millimetres per second squared, so the Moon would be pushed sunward relative to the Earth by 2.6×10−122.6\times10^{-12} metres per second squared for every unit of η.

That push is a tiny force of fixed direction in space — always towards the Sun — acting on an orbit that turns. Kenneth Nordtvedt worked out what it does in 1968, which is why the effect bears his name: the Moon’s orbit is not shifted as a whole but pulled towards the Sun on one side and away from it on the other, so that the Earth–Moon distance oscillates once per synodic month, the 29.53 days from one new moon to the next.

The month a binding energy would bend. The Moon's orbit seen with the Sun held off to the right, drawn as it would be if the Earth's binding energy fell towards the Sun more weakly than the rest of it. The Earth is bound by 4.5 × 10⁻¹⁰ of its mass-energy and the Moon by 1.9 × 10⁻¹¹, so with the Sun pulling at 5.93 mm/s² the Moon is pushed sunward relative to the Earth by 2.6 × 10⁻¹² m/s² for every unit of η. That push turns once a synodic month relative to the orbit, 29.53 days, and Hill's equations about a circular orbit — integrated from the forced solution and held on it to 4 × 10⁻¹¹ over twelve months — give a radial displacement of 8.0 m times η times the cosine of the lunar phase: outward at new moon, inward at full. The complete lunar theory, with the Sun's tide on the orbit included, gives 13.1 m. The displacement is drawn about 7 × 10⁶ times larger than it would be at η = 1.
Fig. 2 The Moon’s orbit with the Sun held to the right, as it would be if the Earth’s binding energy fell more weakly than the rest of it. The orbit moves out towards the Sun at new moon and in at full moon, by 8.0 m per unit η in Hill’s equations about a circular orbit and 13.1 m in the complete lunar theory. The displacement is drawn about seven million times too large.

The pattern has a phase, and the phase is part of the prediction. At new moon the Moon lies between the Earth and the Sun and is pushed further out; at full moon it lies on the far side and is pulled in. A signal with exactly that period and that phase is what the measurement looks for, and nothing about the lunar orbit’s other monthly wobbles is obliged to share both.

It is worth being clear which way the sign runs, because the literature writes it both ways. What is physically unambiguous is this: if the Earth’s binding energy falls less readily than the rest of its mass, the Earth lags behind the Moon in their shared fall towards the Sun, and the Moon’s orbit is displaced sunward. The opposite violation displaces it away.

A push that turns once a month

The displacement is not simply the push divided by some stiffness, because the orbit is not a spring. Near a circular orbit, small disturbances obey Hill’s equations, the same equations that govern two spacecraft flying in formation, and they contain two things a spring does not have: a Coriolis coupling between radial and along-track motion, of the kind the forces that are not there introduces, and a natural radial oscillation at exactly the orbital frequency.

The sunward push, seen from the rotating orbit, has a radial part and an along-track part, and both turn at the synodic rate. Solving for the response that turns with them gives a radial displacement equal to the push times (1+2n/ω)(1 + 2n/\omega) divided by (n2−ω2)(n^2 - \omega^2), where nn is the orbital frequency and ω\omega the rate at which the push turns. The figure’s value, 8.0 metres per unit η, is that formula; it was also checked by integrating Hill’s equations from the forced solution for twelve synodic months and finding the motion still on it to four parts in 101110^{11}.

The complete lunar theory gives 13.1 metres, and the difference is not a flaw in the arithmetic but a real effect of the Sun. The Sun’s own tide already distorts the Moon’s orbit by thousands of kilometres — an oscillation called the variation, known since Tycho — and a perturbation added on top of a strongly perturbed orbit is amplified further. Hill’s equations about a circle get the scale and not the number, and the essay’s bounds use the number.

Why the orbit magnifies it

Why the Moon magnifies a push that turns once a month. The radial response of a circular orbit to a small push whose direction turns at a rate Ω relative to the orbit, as a multiple of the push divided by n squared, where n is the orbital frequency, on a logarithmic axis. The closed form is (1 + 2n/Ω) divided by (1 − Ω squared over n squared), and Hill's equations integrated at 0.6, 0.925 and 1.5 of the orbital frequency stay on it. It has two factors: a resonance, because a push turning at nearly the orbital rate keeps pushing in step with the orbit's own radial oscillation, and a factor from the along-track part of the push, which the Coriolis term turns into more radial motion. A binding-energy push turns once a synodic month, at 0.9252 of the orbital frequency, where the response is 22.0 times the push divided by n squared — 3.16 of that from the Coriolis term and 6.94 from resonance. A push fixed relative to the stars would turn at exactly the orbital frequency and not settle at all.
Fig. 3 The radial response of a circular orbit to a small push turning at a given rate, in units of the push divided by the orbital frequency squared. At the synodic month, 0.925 of the orbital frequency, the response is 22 — a factor 6.94 from being close to resonance and 3.16 from the Coriolis term turning the along-track push into radial motion.

The response has two factors and both favour the Moon. The first is resonance. A push turning at 0.925 of the orbital frequency stays nearly in step with the orbit’s own radial oscillation for many months before drifting out of phase, and the response is amplified by 1/(1−ω2/n2)1/(1-\omega^2/n^2), a factor of 6.94 — the same denominator the frequency that gets an answer finds for any oscillator driven near its natural frequency.

The second is the along-track part of the push. Pushing a body along its orbit changes its speed, which changes the radius it can hold, and the Coriolis coupling turns an along-track push into additional radial motion — the same coupling that makes a pushed parcel in the deflection that closes on itself move sideways. That contributes the factor 1+2n/ω=3.161 + 2n/\omega = 3.16.

Together they make the Moon a sensitive instrument for exactly this force, and it is a coincidence of the Solar System rather than a design: the synodic month is close to the sidereal month because a year is long compared with a month. A force fixed among the stars rather than pointing at the Sun would turn at exactly the orbital frequency, and the response would not settle at all — it would grow the orbit’s eccentricity steadily, which is a different signature, and the orbit’s measured eccentricity would bound it separately.

Ranging to a millimetre, and what stops a millimetre being enough

The distance to the Moon has been measured by timing laser pulses since the first retroreflector was left on the surface by Apollo 11 in July 1969. Apollo 14 and 15 left two more, and two Soviet rovers carried French-built reflectors. A pulse sent from a telescope spreads to several kilometres across by the time it reaches the Moon, a handful of photons per pulse make the return trip, and the round-trip time gives the distance — the same measurement the shift a mirror gives twice makes of a frequency rather than a delay.

The precision has moved from about 25 centimetres in the first years to a few centimetres by the 1980s and to about a millimetre per normal point since 2006, when the APOLLO instrument began ranging with a 3.5-metre telescope in New Mexico. Against a 13.1-metre coefficient, a millimetre corresponds to η of about 10−410^{-4} in a single measurement, and there are tens of thousands of them.

What a decade of ranging can see, before the systematics. Two sets of 6000 synthetic range residuals, with per-point errors of 20 mm and 2 mm, generated with no binding-energy signal and folded by lunar phase. Each is fitted by least squares for a constant and a cosine and sine of the phase. At 20 mm the synodic amplitude comes out at 0.075 ± 0.368 mm, the σ times the square root of 2/N that the fit is required to reproduce, which at 13.1 m per unit η bounds η below 5.6 × 10⁻⁵ at two standard errors; at 2 mm the synodic amplitude comes out at 0.000 ± 0.037 mm, the σ times the square root of 2/N that the fit is required to reproduce, which at 13.1 m per unit η bounds η below 5.6 × 10⁻⁶ at two standard errors. The dashed curves are the signal η = 10⁻³ would produce on the upper set and η = 10⁻⁴ on the lower. Published limits from lunar ranging sit near 10⁻⁴, well above what statistics alone would allow, because the real residuals carry thermal expansion of the reflectors, atmospheric delay, the Moon's interior and a synodic term in the Sun's own perturbation of the orbit, all of which the fit has to separate — and because the phases where the signal peaks, new and full moon, are the ones hardest to range at.
Fig. 4 Two sets of six thousand synthetic range residuals with no signal in them, at 20 mm and 2 mm per point, folded by lunar phase and fitted for a monthly cosine. The fitted amplitudes are 0.075 ± 0.368 mm and 0.000 ± 0.037 mm, bounding η below 5.6 × 10⁻⁵ and 5.6 × 10⁻⁶ at two standard errors. The dashed curves are the signal η = 10⁻³ would give on the upper set and η = 10⁻⁴ on the lower.

The figure asks how far statistics alone would take a fit. Six thousand points at 20 millimetres each bound the monthly amplitude to 0.37 mm and η below 5.6×10−55.6\times10^{-5}; the same number at 2 millimetres, ten times better. The uncertainty on the amplitude is the per-point error times 2/N\sqrt{2/N}, the law any average of independent noise obeys, and it is the ceiling of what ranging could achieve if nothing but noise stood between the data and the answer.

Published limits from lunar ranging sit near 10−410^{-4}, an order of magnitude or more above that ceiling, and the reason is instructive. The residuals are not noise around a known orbit. They contain the thermal expansion of the reflectors and the telescopes, the delay of the pulse through the atmosphere, the tides raised in the Earth and in the Moon, the Moon’s liquid core and the dissipation in it, and — worst of all — a monthly term in the Sun’s own perturbation of the orbit that has exactly the period of the signal. The fit has to determine all of these together, and every parameter that correlates with the monthly cosine inflates its uncertainty.

There is also a practical problem that is almost a joke. The signal is largest at new moon and at full moon. At new moon the Moon is close to the Sun in the sky and daylight swamps the returning photons; at full moon the sunlit surface around the reflectors floods the detector. The phases where the effect peaks are exactly the phases hardest to observe, and a fit to data concentrated around the quarters is fitting a cosine near its zeros.

The composition of the two bodies is a separate complication. The Earth has an iron core and the Moon does not, so a difference in how iron and rock fall — the weak principle — would produce the same monthly signal. Lunar ranging measures the total, and separating the strong part requires a laboratory test with materials chosen to mimic the Earth’s core and the Moon’s mantle, which torsion balances have done to a level well below what the ranging can resolve. The two principles are tested together and disentangled by an experiment that has nothing to do with the Moon.

What a zero here says about gravity itself

The strong principle sounds like a refinement of the weak one, and the connection to the rest of gravitational physics is more surprising than that. General relativity satisfies it exactly, and nearly every alternative theory does not. In the parametrised post-Newtonian framework, which describes the weak-field predictions of a whole family of metric theories with a handful of numbers, the Nordtvedt parameter is

η=4β−γ−3,\eta = 4\beta - \gamma - 3,

where γ measures how much space curvature a unit mass produces — the parameter that sets the factor of two in the bend Newton got half right — and β measures how nonlinear gravity is: how much gravitational field is produced by gravitational energy itself. General relativity has β = γ = 1 and so η = 0. Brans–Dicke theory, the simplest theory with a scalar field alongside the metric, has η equal to 1/(2+ω)1/(2+\omega) for its coupling constant ω.

So the Moon’s orbit is doing something more than checking a principle. Radio timing of the Cassini spacecraft pinned γ to 1 within a few parts in 10510^5. With γ fixed, a bound on η is a bound on β, and lunar ranging is a measurement of whether gravitational energy gravitates the way general relativity says — a question about the nonlinearity of the field equations, answered by timing light to a mirror. The shift of Mercury’s perihelion, the subject of the orbit that does not come back to itself, depends on the combination 2β−γ+22\beta - \gamma + 2… and the two measurements together constrain β from two directions.

The physical picture behind the equation is worth keeping. The Earth’s binding energy is negative gravitational field energy stored in the Earth’s own field. If it fell differently from the rest of the Earth’s mass, gravitational energy would have a different ratio of gravitational to inertial mass from every other kind of energy, which is the same as saying it sources gravity differently. The weak principle is about how everything falls; the strong one is about gravity’s own energy, and that is why it probes the structure of the theory rather than its coupling to matter.

A star that is a tenth binding energy

The Earth is bound by five parts in ten billion. If a violation of the strong principle grew faster than linearly with self-energy, or appeared only where gravity is strong, the Moon would never see it. A body that is a tenth binding energy would.

The self-energy a test can put to work. Three tests placed by the difference in gravitational self-energy between the two bodies they compare, and by the bound each sets on a difference in how those bodies fall. The diagonal lines are constant η, the ratio of the two. The bounds are rounded published values and the self-energy differences are the integrated ones. Torsion balance, two laboratory masses: 7.5 × 10⁻²⁶ of self-energy difference, falls alike to 2.0 × 10⁻¹³, so η below 2.7 × 10¹²; lunar ranging, the Earth against the Moon: 4.3 × 10⁻¹⁰ of self-energy difference, falls alike to 5.0 × 10⁻¹⁴, so η below 1.2 × 10⁻⁴; triple pulsar, neutron star against white dwarf: 1.3 × 10⁻¹ of self-energy difference, falls alike to 2.6 × 10⁻⁶, so η below 2.0 × 10⁻⁵. The torsion balance is the best differential accelerometer of the three and says nothing about binding energy at all. Lunar ranging moves to the right by 16 decades by using bodies that are bound; the pulsar moves another 8 by using a body that is a tenth binding energy, and it reaches a smaller η with a far worse bound on the acceleration difference — the same move from numerator to denominator that the composition tests made.
Fig. 5 Three tests placed by the self-energy difference between the bodies they compare and the bound each sets on a difference in how they fall, with lines of constant η. The torsion balance bounds η only below 2.7 × 10¹²; lunar ranging below 1.2 × 10⁻⁴; a neutron star and a white dwarf falling together round a second white dwarf below 2.0 × 10⁻⁵, from a bound on the fall eight decades worse than the Moon’s.

The system is PSR J0337+1715, found in 2014: a millisecond pulsar and a white dwarf in a 1.6-day orbit, both circling a second white dwarf every 327 days. The inner pair is a neutron star and a white dwarf falling together towards the outer star, exactly as the Earth and the Moon fall towards the Sun, and the pulsar is a clock whose ticks can be timed to a few microseconds over years. In 2018 the timing bounded any difference in how the two inner bodies fall towards the outer one to 2.6 parts in a million.

As a bound on an acceleration difference that is feeble — eight decades worse than lunar ranging’s. As a bound on η it is six times better, because the neutron star’s self-energy is a tenth of its mass rather than half a billionth. The pulsar moved right on the figure rather than down, which is exactly the move the fall that does not depend on what is falling traced from the torsion balance to the satellite: when a bound is a ratio, a bigger denominator beats a better numerator.

It also asks a question the Moon cannot. The η in the figure is defined by assuming the violation grows in proportion to self-energy. A theory in which the violation is negligible below some threshold of compactness and large above it would pass every lunar test and fail the pulsar’s, and several scalar-tensor theories with that behaviour were proposed. The triple system rules out the strongest of them.

Where the parameterisation stops being honest

η assumes a violation proportional to self-energy. It is a one-parameter description of an effect that could depend on self-energy in any way at all, and quoting a single η across bodies whose compactness differs by eight decades assumes a linearity nothing guarantees. The strong-field bound and the weak-field bound are best read as two separate results.

Self-energy is a Newtonian quantity. In general relativity gravitational energy cannot be assigned a local density, and the self-energy of a body is well defined only in the weak-field limit where it is computed here. For the Earth, the Moon and the Sun that is no restriction; for a neutron star the figure’s tenth is an estimate, and the triple-system analysis uses the star’s relativistic binding energy, which differs in detail but not in order of magnitude.

The lunar coefficient comes from a theory, not from the drawing. Hill’s equations about a circular orbit give 8.0 metres per unit η; the 13.1 metres used for the bounds comes from the complete theory of the Moon’s motion with the Sun’s tide included. The figure computes the first and quotes the second.

The composition signal has to be removed by another experiment. Lunar ranging measures a total difference between the Earth and the Moon, and the strong-principle bound depends on a laboratory bound on the weak principle for Earth-like and Moon-like materials. A future disagreement between the two would be ambiguous until one of them improved.

The bounds quoted are rounded. Published values from lunar ranging have moved within a factor of a few of 10−410^{-4} as the data lengthened and the models changed; the figure places the test at the round number and draws no confidence interval.

What the drawings cannot carry

The orbit figure shows a displacement seven million times too large and a circular orbit that the Moon does not have. The real orbit is an ellipse perturbed by the Sun by thousands of kilometres, precessing, tilted and slowly receding, and the Nordtvedt term is a centimetre-sized pattern hidden inside all of that. No drawing at a readable scale can show both at once, and the difficulty of the measurement is precisely that they coexist.

The fit figure shows noise around nothing. Real residuals carry structure from every model imperfection, correlated from one night to the next and from one year to the next, and what makes a fifty-year dataset valuable is not the number of points but the range of solar and lunar geometries they sample. A scatter plot of independent errors is the one thing ranging data is not.

Still open: whether the principle holds where gravity is strong

Lunar ranging tests the strong principle where gravity is weak, and the triple system extends it to a neutron star. The deepest version of the question concerns bodies that are nothing but gravitational energy. A black hole’s mass is entirely field; in general relativity it falls exactly like everything else, while in many scalar-tensor theories it cannot carry the scalar charge that a neutron star carries, so the two would fall differently and a binary of the two would radiate energy in a way general relativity forbids.

That radiation is the next place to look. Binary pulsars with white-dwarf companions already constrain it through the rate at which their orbits shrink, the measurement the orbit that has to shrink follows, and gravitational-wave observations of merging black holes and neutron stars probe it where the fields are strongest of all. Whether the principle survives there is a question about which theory of gravity is right, asked in the only regime where the candidates stop agreeing.

The habit worth carrying away is to ask which part of a body a test can actually weigh. A null result bounds only the kinds of energy in which the compared bodies differ, and the most precise instrument ever built says nothing about any contribution to mass that it cannot make its two samples disagree about.

Part 4 of 4

This essay is one argument about Equivalence principle. The others:

Three centuries of finding nothing. The upper limit on η against the year it was set, on a logarithmic scale. Newton (1687) reached 1e-3; Bessel (1832) reached 2e-5; Eötvös (1922) reached 5e-9; Dicke (1964) reached 1e-11; Braginsky (1972) reached 1e-12; Eöt-Wash (2008) reached 2e-13; MICROSCOPE (2022) reached 1e-15. That is 12 orders of magnitude in 335 years, and every one of those measurements returned zero. A sequence of null results is not a sequence of failures. Each one is a statement that a principle assumed by every theory of gravity holds to a new level, and each new level excludes a class of theories that would have shown a departure there — a long-range force coupling to something other than mass-energy, a scalar partner to the graviton, a violation arising at some energy scale. The measurement is worth making again precisely because it has always come out the same way, which is what makes any departure decisive. The fall that does not depend on what is falling Part 3 — Everything falls at the same rate, and the statement has been tested for three hundred years by people looking for the exception. Twelve orders of magnitude have been added to the limit and every measurement has returned zero. The last three orders came not from a better instrument but from finding something bigger to fall towards.

Where this goes next

Three centuries of finding nothing. The upper limit on η against the year it was set, on a logarithmic scale. Newton (1687) reached 1e-3; Bessel (1832) reached 2e-5; Eötvös (1922) reached 5e-9; Dicke (1964) reached 1e-11; Braginsky (1972) reached 1e-12; Eöt-Wash (2008) reached 2e-13; MICROSCOPE (2022) reached 1e-15. That is 12 orders of magnitude in 335 years, and every one of those measurements returned zero. A sequence of null results is not a sequence of failures. Each one is a statement that a principle assumed by every theory of gravity holds to a new level, and each new level excludes a class of theories that would have shown a departure there — a long-range force coupling to something other than mass-energy, a scalar partner to the graviton, a violation arising at some energy scale. The measurement is worth making again precisely because it has always come out the same way, which is what makes any departure decisive. The fall that does not depend on what is falling Everything falls at the same rate, and the statement has been tested for three hundred years by people looking for the exception. Twelve orders of magnitude have been added to the limit and every measurement has returned zero. The last three orders came not from a better instrument but from finding something bigger to fall towards. Light crossing an accelerating box. A pulse crosses a box 6 m wide while the box accelerates at 9.81 m/s². The crossing takes 2·10⁻⁸ s, in which the far wall gains 1.96·10⁻⁷ m/s, so the pulse lands 1.96·10⁻¹⁵ m below the height it left at — and the path is a parabola. An observer sealed inside cannot tell that from a beam of light bending in a gravitational field, and the equivalence principle says there is nothing to tell. The sag is drawn 5.6·10¹⁴ times its true size. The floor that cannot be told from gravity Seal a laboratory, take away the windows, and no experiment inside it can distinguish standing in a gravitational field from accelerating through empty space. That is not a philosophical remark — it forces light to bend, forces clocks to disagree, and has a size at which it stops being true. The line that slopes the wrong way. Mean temperature against total energy, for a self-gravitating sphere at nine radii. The points fall on a straight line of negative slope: taking energy away makes the body hotter. The heat capacity read off the drawing is -4.10e+34 J/K against the -4.10e+34 J/K the theorem gives, and the sign is the whole content. The dashed line is an ordinary gas in a rigid box, whose temperature rises when energy is added, as everything one can put a thermometer in does. The ball of gas that heats up as it cools Take energy away from a self-gravitating cloud and its temperature rises. Its heat capacity is negative, which nothing else stable does, and the consequence is that a star radiating into cold space is not cooling down — it is running up, and its whole life is a slow fall it cannot stop.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

Binding energyEquivalence principleGravitational massLunar laser rangingNull experimentPost-newtonian parametersResonanceSelf-gravityStrong equivalence principle