The term free fall cannot remove

The equivalence principle says that a laboratory in free fall is indistinguishable from one floating in empty space. Released objects hover; a thrown ball goes in a straight line; a beam of light crosses undeviated. Every experiment gives the answer it would give with no gravity anywhere — which is why the deflection of light by a mass can be computed from a lift and why a clock’s rate depends on where it is.

That is true, and it is true only locally, and the word “locally” is doing a great deal of work. It is not a hedge. It is a length, and given an instrument it can be computed.

What survives the fall #

Take two test masses released side by side inside a falling box, both a distance $r$ from the centre of a spherical mass $M$. Each falls along its own radius, and radii converge. The relative acceleration across the fall direction is

$$a_\perp = -\frac{GM}{r^3}\,L,$$

with $L$ their separation; along the fall direction, where the nearer mass is pulled harder, it is

$$a_\parallel = +\frac{2GM}{r^3}\,L.$$

Nothing about that is a small-angle approximation — the transverse case is similar triangles and the radial one is the derivative of the inverse square.

The combination $GM/r^3$ is the quantity that matters, and it is worth a name and a unit: it is a tidal parameter, with units of inverse time squared, and it is the only thing about the external world that enters. At the Earth’s surface it is $g/R_\oplus = 1.54 \times 10^{-6}$ s⁻².

Note what has happened to the mass and the distance. The gravitational field itself goes as $GM/r^2$ and can be removed entirely by falling. What cannot be removed goes as $GM/r^3$ — one power steeper, because it is a difference of fields — in exactly the way a dipole’s field falls one power faster than a monopole’s for the same reason.

The exponent can be measured off a drawing with no gravity in it at all. A field from a point source falls with slope −2 on logarithmic axes; a difference between two such fields, nearly cancelling, falls with slope −3. The tidal parameter is the second of those, which is why it dies away so much faster with distance than gravity itself does — and why the Moon raises larger tides on the Earth than the Sun, whose gravitational pull here is two hundred times greater.

Putting a number on “locally” #

Now suppose the laboratory carries an instrument that can detect a relative displacement $\delta$. Two masses a distance $L$ apart, released together, separate by $\tfrac12 (GM/r^3)L t^2$ after a time $t$. The principle survives, in the sense of being undetectable, as long as that is below $\delta$:

$$L_{\max} = \frac{2\delta}{(GM/r^3)\,t^2}.$$

Two things follow at once, and the second is the one usually left out.

The size depends on the instrument. With a nanometre interferometer at the Earth’s surface and a one-second experiment, $L_{\max}$ is 1.3 mm. With a millimetre ruler it is a kilometre. There is no absolute answer, and any statement of the form “the equivalence principle holds over such-and-such a distance” is incomplete without saying what is doing the looking.

The size falls as the square of the time. Waiting ten times as long shrinks the allowed box by a hundred. So the principle is local in time as well as in space, on the same footing, and with a worse exponent — patience is more expensive than room.

Written as a spacetime statement, the two arguments combine into one. Curvature has units of inverse length squared, and the four-dimensional region within which spacetime is flat to a stated tolerance is bounded in every direction — with the time direction measured in units of $c\,t$. The box being small in space and the experiment being short in time are the same condition applied to different components of one region.

The horizon that is not the dangerous one #

The tidal parameter at the horizon of a black hole is the most counter-intuitive number in the subject. Substituting $r = r_s = 2GM/c^2$,

$$\frac{GM}{r_s^3} = \frac{c^6}{8G^2M^2}.$$

It falls as $1/M^2$. A more massive black hole has a gentler horizon, and by a lot: at a stellar-mass hole the parameter is $5.15\times10^7$ s⁻², which would tear a human body apart well outside the horizon; at a billion-solar-mass one it is $5.15\times10^{-9}$ s⁻², a three-hundredth of the Earth’s surface value.

So an experimenter crossing the horizon of a large enough black hole notices nothing. There is no local measurement that identifies the crossing, because there is nothing locally to measure — the horizon is a global feature of the spacetime, not a place where anything is large. What has been lost is the ability to return, and no instrument inside a small box detects that.

How far a horizon is from the surface of an ordinary body settles the question of when any of this becomes dangerous. The Schwarzschild radius of a person is $10^{-25}$ m, of the Earth 8.9 mm, of the Sun about 3 km — so every one of them sits deep inside itself and none has a horizon at all. Where a horizon is outside the body, the tidal parameter at it is the one computed above, and for a large enough hole it is small: the horizon is not the dangerous place, and the danger scales the other way from the size.

A fall toward a stellar-mass hole takes a finite and unremarkable proper time to reach the horizon, which is exactly what the equivalence principle guarantees: locally, a fall is a fall. The coordinate time diverges, and that divergence is a statement about a bookkeeping choice made far away rather than about anything the falling observer undergoes. Two clocks, two answers, and only one of them is carried by anybody.

How the box grows if the experiment is patient #

The two arguments — small box, short experiment — can be traded against one another, and the trade is worth doing explicitly because it says which is the cheaper resource.

Fix the instrument and ask for the largest four-dimensional region within which nothing is detectable. The spatial extent is $L_{\max} \propto 1/t^2$, so the region is not a box of fixed shape being scaled: it is a wedge, wide and brief or narrow and long. Doubling the duration costs three quarters of the size.

The consequence for experiments is direct. A drop tower gives about five seconds of free fall and a laboratory a few metres across, so its residual tidal signal is around $10^{-6}$ m — micrometres, easily measured, and the reason drop-tower experiments are corrected for gradients rather than assumed free of them. A satellite gives years of free fall, which is why gravity-gradient satellites exist at all: their instrument is the tidal term, and the mission’s duration is what makes it enormous rather than what makes it negligible.

The general rule is that time is the expensive direction. Anyone who wants a longer experiment must either shrink the apparatus quadratically or move to a place with a smaller tidal parameter, and there are not many of those.

The instrument whose signal is the residue #

If the tidal term is what free fall leaves behind, an instrument that wants to measure the Earth rather than to escape it should be built to read exactly that — and several are.

The construction is the box argument used deliberately. Put two accelerometers a fixed distance apart in a freely falling satellite and difference their readings. Everything common to both — the orbital acceleration, the drag, the thruster firings — cancels, because both instruments experience it equally. What survives is the difference, and the difference is the gradient times the separation.

That is enormously more useful than measuring the acceleration itself. An accelerometer in orbit reads a number dominated by the orbit; a gradiometer reads a number dominated by whatever mass is directly underneath, because the gradient falls one power faster with distance than the field does and is therefore far more sensitive to nearby structure than to the planet as a whole. A satellite mapping the gradient sees a mountain range and an ocean trench where one mapping the field sees a smooth ellipsoid with small departures.

The same principle at laboratory scale uses falling atoms rather than falling instruments. Two clouds of cold atoms, released at different heights and interrogated by the same laser pulses, each measure the local acceleration by counting interference fringes; differencing the two gives the gradient with the laser’s own vibration common to both and therefore absent from the answer. The technique reaches a few parts in $10^9$ of $g$ per metre, and it is the same trick as the satellite’s — measure a difference so that everything shared cancels.

Which reverses this essay’s framing in a satisfying way. The tidal term was introduced as the thing that spoils an idealisation; for a whole class of instruments it is the entire signal, and the idealisation is the background they are built to subtract.

The signal that is nothing but this term #

Build a detector entirely out of freely falling masses and there is only one quantity left for it to report. Every uniform part of the field has been transformed away by the falling; what remains in the separations between the masses is the second derivative, and nothing else. That is what a gravitational-wave detector is.

A ring of freely falling masses is stretched one way and squeezed the other by a passing wave, in a pattern that returns to itself twice per cycle. Each mass follows a geodesic and feels nothing at all; the signal is entirely in the changing separations between them. That is geodesic deviation with the curvature supplied by a wave instead of by a nearby mass — the same term this essay is about, detached from any body and sent across the universe on its own.

The strain a detector reports is $\Delta L/L$, which is the relative displacement per unit separation — the same quantity as the box argument, with the tidal parameter now a function of time. And the reason such a detector must be kilometres long is exactly the argument above run backwards: the displacement is proportional to $L$, so making the box larger is the only way to make an undetectable effect detectable.

The tides, which are the same term with a name #

The word “tidal” is not a metaphor here. Ocean tides are geodesic deviation applied to a body large enough for the effect to be conspicuous, and every number in the argument above appears in them.

The Moon’s tidal parameter at the Earth is $GM_{\text{Moon}}/d^3 = 2.5 \times 10^{-14}$ s⁻², and multiplied by the Earth’s radius that is a differential acceleration of $1.1 \times 10^{-6}$ m/s² between the near side and the centre — about a ten-millionth of $g$. That is enough to raise the equilibrium ocean by half a metre, because the ocean responds to the potential rather than to the acceleration and has a whole planet’s radius to accumulate over.

The Sun’s is instructive by comparison. It is 27 million times more massive than the Moon and 390 times further away, so its tidal parameter is smaller by $390^3/2.7\times10^7 = 2.2$ — the cube in the denominator beating the mass in the numerator. Solar tides are about 45% of lunar ones, and the two add at new and full moon and oppose at the quarters, which is the spring–neap cycle.

That the cube wins is the whole reason a nearby small body dominates a distant large one, and it is the same exponent that made the tidal parameter at a big black hole’s horizon small.

The heat a gradient makes #

The limitation about extended bodies is worth following, because it produces the most spectacular consequence the tidal term has in the solar system.

A body of finite size in a tidal field is stretched along the field’s axis and squeezed across it, and it settles into a slightly elongated shape. If nothing about that shape changes, nothing further happens: the body sits deformed, and there is no dissipation.

Change is what costs. If the tidal field’s direction or strength varies with respect to the body, the bulge has to be continually remade, the material is worked back and forth, and — because no real material is perfectly elastic — some of that work is lost as heat. Two things can produce the variation. The body can rotate with respect to the field, so that the bulge has to travel through it. Or the orbit can be eccentric, so that the field’s strength varies round the orbit even for a body that keeps one face toward its primary.

Io is the case where the second dominates. It keeps one face toward Jupiter, so there is no bulge travelling through it, and its orbit is measurably eccentric — kept so by a resonance with two of its neighbours, which nudge it into eccentricity as fast as the tides damp it out. The result is a continuously flexed body, dissipating enough to keep it molten: its heat output per unit area is more than an order of magnitude above the Earth’s, and it is the most volcanically active object in the solar system.

The chain of reasoning is worth appreciating for its length. A cube in a distance law produces a differential acceleration; the differential acceleration produces a deformation; a varying deformation produces dissipation; the variation is maintained by an orbital resonance with two other moons; and the output is a surface covered in sulphur volcanoes. Every step is the tidal term.

Why the Moon shows one face #

The other route to a varying bulge explains something everybody has noticed and few connect to this argument.

A satellite rotating faster than it orbits carries its tidal bulge through itself once per relative rotation, dissipating energy each time. Dissipation means the bulge lags — it is dragged slightly past the line to the primary by the rotation — and a lagging bulge feels a torque from the primary that opposes the rotation. So the rotation slows, until the relative rotation is zero and the bulge stops moving through the body.

That is tidal locking, and it is why the Moon keeps one face toward the Earth, why every large moon in the solar system does the same, and why Mercury is caught in a resonance rather than spinning freely. The timescale depends steeply on the distance — the torque involves the tidal parameter, so it carries the same cube — which is why close-in bodies lock quickly and distant ones do not.

The same torque acts on the primary. The Earth’s own tidal bulge is dragged ahead by its rotation, which is faster than the Moon’s orbit, so it pulls the Moon forward and is pulled back in return: the Earth’s day is lengthening by a couple of milliseconds a century, and the Moon is receding by a few centimetres a year. Both numbers are measured — the first from ancient eclipse records, the second by bouncing lasers off reflectors left on the surface — and both are the residue of free fall, integrated over four billion years.

Where the model stops #

The whole account is Newtonian. The tidal parameter $GM/r^3$ is a derivative of Newton’s law, and the relativistic version replaces it with a component of the Riemann tensor. Near a horizon the two differ by factors of order one, so every number quoted for a black hole above is right in magnitude and not in its last digit. The structure — that what survives free fall is a second derivative of the potential and falls one power faster than the field — is the same in both theories.

The source is a static spherical mass. For anything else the tidal effect is not a single number: it is a tensor with five independent components, and its pattern differs. Near a rotating mass there is an additional frame-dragging term with no Newtonian counterpart.

The masses are test masses. They are assumed to have no gravity of their own and no size. A real extended body responds to the tidal field with internal stresses and deforms, and the deformation feeds back — which is the whole of tidal heating and is not in the geodesic deviation equation.

It also assumes the two masses fall the same way, which is the equivalence principle’s other half and is the thing actually being tested by the experiments that test it. If inertial and gravitational mass differed by one part in $10^{15}$ between two materials, two released masses of different composition would separate at a rate that has nothing to do with the gradient — and separating that signal from the tidal one is the whole design problem of a torsion balance and of the satellite experiments that have superseded it.

The expansion is to first order in the separation. The deviation equation keeps only the leading term in $L$, so for a box comparable with $r$ it is wrong. That is exactly the regime where “locally” has failed anyway, so the two limitations coincide.

The observers are freely falling and not held. A laboratory bolted to the ground is not in free fall at all, and the field it measures is $g$ rather than the gradient of $g$. Every number here belongs to an experiment that has released its apparatus, which is why the interesting instruments — drop towers, atom interferometers, satellites — are the ones that let go.

What is being differentiated is the inverse-square law itself. Everything here is the derivative of that law with respect to position — the difference between the field here and the field a short distance away — and the tidal term is therefore not an additional force at all. It is what is left of a force after the part that is the same everywhere has been transformed away, which is why free fall removes the first term and cannot touch the second.

What the pictures cannot show #

The box figures plot a size and treat the box as a cube. A real limit is direction-dependent: the radial extent allowed is half the transverse one, because the radial deviation is twice as large. Every number quoted here is the transverse one, and a box that is undetectably flat across is detectably curved along.

The tidal figure exaggerates the convergence by a stated factor, and it must — the real approach over a ten-metre fall is 1.6 micrometres across a metre of separation, which at true scale is invisible.

And no figure here shows curvature. What is drawn throughout is the relative motion of pairs of test particles, which is the observable consequence of curvature, and the geometric object itself has twenty independent components in four dimensions and is not drawable at all.

Where the ladder goes next #

The rung below establishes that a floor pushing up and gravity pulling down cannot be told apart. This one asks how far that indistinguishability extends, and finds that the answer is a length and a time, computable from the curvature and from whatever is doing the looking.

The rungs above are where the tidal term stops being a correction and becomes the subject. A wave is a tidal field that propagates. A body pulled apart by one is a question about material strength against curvature. And the mathematical rung is the geodesic deviation equation itself, in which the tidal parameter becomes the Riemann tensor and the whole of general relativity’s field content is the statement that this tensor is what mass produces.

The habit worth carrying out of it is the one the numbers here enforce. When a principle is stated with a qualifier — locally, approximately, to first order — the qualifier is a quantity. Find what it is proportional to, ask what instrument would violate it, and the qualifier stops being an apology and becomes a prediction.