The floor that cannot be told from gravity

A sealed box, no windows, and one occupant with every instrument physics can supply. The box is either standing on the surface of a planet or being towed through empty space by a rocket with a steady thrust. Everything inside behaves identically in the two cases: dropped objects fall with the same acceleration whatever they are made of, a spring balance reads the same weight, a pendulum keeps the same time. The claim of the equivalence principle is that the identity is exact, and that no experiment performed inside will ever separate the two.

That statement sounds like a remark about the limits of experiment. It is not. It is a constraint, and the things it forces are large: that light has to bend, that clocks at different heights have to disagree, and eventually that gravity is not a force at all but the shape of the space and time the experiments are done in.

The assumption underneath, which is measurable #

The principle rests on a coincidence that Newton’s mechanics permits and does not explain.

Mass appears twice in Newtonian physics and means different things each time. In $F = ma$ it is a body’s resistance to being accelerated — inertial mass. In $F = GMm/r^2$ it is the body’s coupling to gravity — gravitational mass, playing the same role that charge plays in the electrostatic force. There is no reason within Newtonian mechanics for the two to be equal. Charge and mass are not.

But if they are equal, gravity acquires a property no other force has: the acceleration it produces is the same for everything. Put the two together and $m$ cancels, leaving $a = GM/r^2$, with nothing about the falling body in it. A feather and a cannonball, a proton and a planet, all fall the same way.

That cancellation is what makes the sealed box ambiguous. Every other force can be identified from inside, because it acts differently on different things: an electric field accelerates a charged pellet and leaves a neutral one alone, so an occupant compares two pellets and knows. Gravity offers no such comparison.

So the principle is only as good as the coincidence, and the coincidence is a measurement. Eötvös tested it around 1900 with a torsion balance and found the two masses equal to a part in $10^8$; modern torsion and satellite experiments have reached a part in $10^{15}$. That number is the experimental floor under everything on this page, and it is a number rather than an axiom — a distinction worth keeping, since the whole structure would come apart if some material were found to fall a part in $10^{16}$ differently from another.

What it forces about light #

Here is the argument in its 1907 form, and its economy is the striking part.

Consider the accelerating box. A pulse of light crosses it from one wall to the other, taking $L/c$. In that time the box gains speed $a L/c$, and its far wall has moved up by $\tfrac12 a (L/c)^2$. So in the box’s own frame, the pulse arrives below the height it left at, by that amount, and the path traced is a parabola — the same parabola any projectile follows, for the same reason and with no gravity anywhere in the derivation.

Now invoke the principle. If the accelerating box and the box in a gravitational field are indistinguishable, and light bends in the first, then light bends in the second. There is no room for an exception: an exception would be an experiment that tells the two apart.

Two features of that derivation are worth pausing on, because they are what makes it more than a curiosity. The first is that no property of light appears anywhere in it. The drop is $\tfrac12 a (L/c)^2$: an acceleration, a length and the speed of light, with no frequency, no polarisation and no mass. Whatever light turns out to be made of, it bends by the same amount. The second is that the argument runs entirely inside the box. It never asks what the box is doing, only that whatever it is doing cannot be detected from within — so the conclusion is forced by the principle rather than added to it.

The number is minute. Over six metres at one gravity the sag is about $2\times10^{-15}$ m, which is roughly the diameter of an atomic nucleus, and no laboratory measurement of it has ever been made or is likely to be. The effect only becomes visible where the field is strong and the path is long — where starlight passes a massive body — and there, as it happens, this argument gets exactly half the right answer. That factor of two is the whole subject of the next rung, and it is the sharpest evidence available that gravity is geometry rather than a force.

The clock follows immediately #

The same argument run on frequency rather than direction gives something measurable in a building.

A photon climbing a height $h$ against gravity ought to lose energy $mgh$ if it had a mass $m$. It has none, but it has an energy $E = h\nu$ and therefore, by mass–energy equivalence, an effective inertia $E/c^2$. So the energy lost is $(E/c^2)gh$, and the fractional loss is $gh/c^2$ — independent of the photon.

The clock follows immediately, and it is worth putting a number to. A photon emitted at the foot of a 22.5-metre tower and received at the top loses a fraction $gh/c^2$ of its frequency — two and a half parts in $10^{15}$, which sounds unmeasurable and was measured in 1960. Nothing about gravitation was assumed to get it: the accelerating box gives the same answer by Doppler shift alone, and the equivalence principle says the two situations cannot be told apart.

A frequency is a clock. If the light arrives with a lower frequency than it left, and nothing has happened to it in flight, then the emitter’s clock is slow compared with the receiver’s. That is gravitational time dilation, derived from a sealed box and a conservation law, with no field equations anywhere. Pound and Rebka measured it in a Harvard lift shaft in 1959.

Where the model stops, and it stops at a measurable size #

The principle as usually quoted has a word in it that does the work quietly: locally. Take it away and the statement is false, and the way it is false is the beginning of the whole theory.

A real gravitational field is not uniform. It points toward a centre, so two objects released side by side in a falling box are falling along lines that converge, and they approach one another as they go. Nothing about choosing a frame removes that: it is a difference between the field at two places, and a difference does not vanish because the average has been subtracted off.

So “locally” means: over a region small enough that the residue is below the precision of the experiment being done. That makes it a quantity rather than a qualification. Two balls a metre apart in a ten-metre fall converge by 1.57 µm — undetectable in a school laboratory and enormous to an interferometer. Make the fall longer and the residue grows in proportion.

The residue has a name — the tidal field — and a form: it is the gradient of the gravitational field rather than the field itself. That is why it survives. A uniform field can be transformed away by falling with it; a gradient cannot, because there is no single acceleration that matches the field at every point at once.

What is left when the force has been removed #

Take the argument seriously and something odd follows about what gravity is.

Every other force is detected as a departure from free motion: a charge in an electric field deviates from the straight line an uncharged particle takes, and the deviation is the evidence. Gravity has no such control case. Everything deviates, identically, so nothing is left to deviate from. The natural conclusion is not that gravity is a very democratic force but that the straight lines were wrong.

What is left when the force has been removed is a geometry. On a spacetime diagram unaccelerated motion is a straight line and the light cone fixes which directions are available — and the proposal general relativity makes is that free fall is also a straight line, in a spacetime whose notion of straight has been altered. Gravity stops being a force acting on trajectories and becomes a statement about which trajectories are the unaccelerated ones.

This is the move that turns a principle into a theory. Free fall is not motion under a force; it is the closest thing to standing still that a curved geometry allows, and the paths taken are geodesics — the analogue of straight lines. Weight, the thing an occupant of a stationary box actually feels, is then not gravity at all. It is the floor pushing up, preventing the free fall that would otherwise happen: the same normal force that appears in every free-body diagram, and the only force acting on somebody standing on the ground.

That is worth restating, because it inverts the usual account. A person in orbit is weightless not because gravity is absent — at the height of a space station it is nearly ninety per cent of its surface value — but because nothing is interrupting their fall. A person standing on a floor is the one being accelerated.

The gradient becomes the field equations #

If gravity is not a force, then the thing a theory must supply is a rule for which lines are straight, and the tidal term is what that rule is about.

The observable content of gravitation, on this reading, is entirely in the relative acceleration of nearby freely falling particles — the convergence in the figures above, formalised as geodesic deviation. Every measurement of gravity is such a comparison, whether it looks like one or not: a spring balance compares the floor’s worldline with the free-fall worldline of the mass hanging on it.

The residue behaves as it does because of one power of $r$. A field falling as $1/r^2$ has a gradient falling as $1/r^3$, so tidal effects weaken faster with distance than the field producing them — which is why a small enough box far enough away is indistinguishable from one in free space, and why the equivalence principle is exactly local and not approximately global.

The cost of this reformulation is severe and worth naming. The gravitational field’s energy cannot be localised — there is no honest answer to how much gravitational energy sits in a given cubic metre, because the field can be transformed away at any point by falling. Total energy remains meaningful for an isolated system; a density does not. Anything expecting gravity to behave like the electromagnetic field, which carries an unambiguous energy density, has to give that up.

The residue, as an engineering quantity #

The convergence in the two tidal figures is not only a caveat about a principle. It is the limiting nuisance in every facility built to supply free fall, and in one case it is used as a free service.

A laboratory in orbit is a falling box, and its occupants call the result microgravity rather than zero gravity for exactly the reason this essay gives. Across a station a hundred metres long the tidal term comes to something like a hundred-thousandth of a gravity at the far ends — small, and larger than the disturbances from atmospheric drag or from somebody pushing off a wall. An experiment that needs the quietest available conditions is therefore placed as near the centre of mass as the layout allows, and the station’s centre of mass is a published quantity for that reason.

A drop tower is the same trade in a shorter time. A hundred metres of evacuated shaft gives four and a half seconds of fall at a residual acceleration of about a millionth of a gravity, which is a better environment than orbit provides and lasts for one twenty-thousandth as long.

And the same term, on a satellite that is not trying to be quiet, is a control system. An elongated body in orbit has more of itself further from the centre of the Earth at one end than the other, so the tidal field exerts a torque that swings its long axis toward the vertical and holds it there. Gravity-gradient stabilisation costs nothing, needs no fuel and no moving parts, and points a spacecraft at the planet indefinitely — a free attitude-control system whose entire mechanism is the piece of gravity no choice of frame removes.

The wave that is nothing but the residue #

The sharpest illustration is a signal that consists of the tidal term and of nothing else.

A gravitational wave passing through a region produces no net acceleration of anything — there is no uniform part to it, so there is no frame it could be transformed away in even locally. What it produces is precisely a relative acceleration between neighbouring freely falling bodies: it stretches the separation along one axis while squeezing it along the perpendicular one, and reverses twice a cycle.

So a detector for such a wave is the figure at the top of this essay’s tidal section, built at kilometre scale. Four mirrors are suspended so that they are as nearly in free fall as an object attached to the Earth can be, and the instrument measures the changing distance between them. Nothing else is being measured: the whole signal is the convergence of nearby geodesics.

The size is the reason it took a century. The strain a detectable source produces is about one part in $10^{21}$, so over an arm four kilometres long the separation changes by a ten-thousandth of the diameter of a proton. That the measurement is possible at all rests on the same reciprocal relation that governs every interferometer — and the first detection, in 2015, was of two black holes merging a thousand million light-years away.

What is worth carrying from it is what the instrument is. It is not a device that responds to gravity; it is a device that responds to the difference of gravity between two places, because that difference is the only part of gravity there is anything to respond to.

What the picture cannot show #

The hero figure draws a beam of light sagging inside a box, and the sag is exaggerated by a factor of about $6\times10^{14}$. That is stated on the canvas, and it should be read as a warning rather than a courtesy: at true scale the drawing is a straight line, and every intuition built on the picture’s curvature is an intuition about the drawing.

The convergence figures are worse in the same way. The balls are drawn approaching by a fifth of the frame; they approach by a distance smaller than a wavelength of light. The exaggeration is what makes the geometry legible and it is also what makes it feel like a large effect, which it is not — except where the gradient is enormous, which is the regime a horizon supplies.

Neither figure shows the thing that is actually being claimed, because the claim is about spacetime and the figures draw space. The parabola in the hero is a path in a plane; the object general relativity says is straight is a worldline in four dimensions, and its straightness is a statement involving the time axis at least as much as the spatial ones. On the scale of a laboratory the time direction is stretched by $c$, which is why the trajectory looks so nearly straight in spacetime while looking so obviously curved in space.

What it would take to break it #

A principle worth stating is one an experiment could refuse, and this one can be refused in three distinct ways, each of which is being looked for.

The first is the original coincidence: some material falling at a different rate from another. That is the weak equivalence principle, and the torsion balances have pushed it to a part in $10^{15}$ without finding a departure. The second is local Lorentz invariance — whether the outcome of a non-gravitational experiment in a falling box depends on how fast the box is moving or which way it is pointing. The third is local position invariance: whether it depends on when and where the box is, which would show up as a drift in the fine-structure constant or as a redshift that does not match $gh/c^2$.

The three together are the strong equivalence principle, and the reason for separating them is that theories fail them separately. A theory in which the gravitational constant varies slowly with time keeps the first two and breaks the third. It is a rather better structure than a single slogan, because each part names an experiment rather than a conviction.

The history, in the place where it earns its space #

Einstein called it der glücklichste Gedanke meines Lebens — the happiest thought of his life — and dated it to 1907, while sitting in the patent office in Bern: a man in free fall does not feel his own weight. It took eight further years to reach the field equations, and the intervening work was mostly the mathematics of curvature, which he did not know and had to be taught.

What deserves attention is how much was extracted before any of that machinery arrived. The bending of light, the gravitational redshift and the impossibility of a local gravitational force were all in place by 1911, from the sealed box alone. They are consequences of a symmetry rather than of a specific theory of gravitation, which is why they survived the theory’s own development and why alternative theories that keep the equivalence principle keep them too.

The prediction that did not survive unchanged is the one that made the reputation. In 1911 Einstein published a deflection of 0.87 arcseconds for starlight grazing the Sun — the sealed-box answer. The complete theory doubled it. Had the 1914 eclipse expedition to Crimea not been interrupted by a war, the measurement would probably have been made against the wrong prediction.

The ladder from here #

The next rungs on this anchor: the redshift as a fully-fledged clock effect, with the two relativities pulling in opposite directions in a satellite; the deflection of light and the factor of two that separates a falling photon from a geodesic; geodesic deviation written properly, which is the tidal figure above turned into an equation; the strong equivalence principle, which extends the claim to gravitational binding energy itself and is tested by watching whether the Earth and the Moon fall toward the Sun at the same rate; and the frames in which the principle fails on purpose — a rotating one, where the centrifugal and Coriolis terms are exactly the fictitious forces the principle is about.

The neighbouring ladder is special relativity, whose clocks and slices are what every local frame here is built out of. The principle’s whole content is that special relativity holds in a small enough falling box; everything gravitational is the failure of “small enough” at the size of the experiment.