Gravitational Time Dilation from the Equivalence Principle

A clock on the floor runs slow. Not because it is moving — it is bolted down. Not because it is old, or cold, or badly made. It runs slow because it is lower, and there is no way to fix it. The strange part is not the effect. It is that you can derive it without general relativity, without a field equation, without curvature — and, in the sharpest version, without any theory of spacetime at all. All you need is $E = mc^2$ and the refusal to build a perpetual-motion machine. Three arguments arrive here from three directions:

Ride an accelerating rocket and fire a laser at the ceiling. The ceiling picks up speed during the flight, so it sees a Doppler redshift. Then say the magic words: a rocket and a planet are locally the same thing.
Give the photon an effective mass $h\nu/c^2$ and make it climb. It pays $\left(h\nu/c^2\right)gh$ out of its own energy and arrives redder.
Or derive nothing, and simply demand that you cannot get energy for free. Drop a mass, convert it to a photon at the bottom, send the photon up, convert it back. If the photon did not redshift by exactly $gh/c^2$ , the cycle prints energy forever.

All three give $\Delta\nu/\nu = -gh/c^2$ , and the third needs nothing but mass–energy equivalence and a conservation law. Gravitational time dilation is not a prediction of general relativity. It is a theorem of energy conservation, and any theory of gravity without it is already dead. At Earth's surface the number is $1.09\times10^{-16}$ per metre. Small. Not zero — and your phone corrects for it.

Key takeaways

Signature $(+,-,-,-)$ : $\mathrm{d}s^2 = c^2\mathrm{d}t^2 - \mathrm{d}x^2$ , timelike intervals positive. Flat Minkowski spacetime in an accelerated chart, plus one reinterpretation.
The heuristic: light climbing a height $h$ against $g$ is redshifted by $\boxed{\Delta\nu/\nu = -gh/c^2}$ ; equivalently the lower of two clocks runs slow by the same fraction. Generalise $gh \to \Delta\Phi$ : $\boxed{\Delta\nu/\nu = -\Delta\Phi/c^2}$ .
Owned result: three independent derivations, and their agreement is the evidence. The rocket-Doppler route (1907), the falling-photon route (1911), and the energy-conservation route — which assumes no relativity of spacetime at all. It needs $E = mc^2$ and the impossibility of perpetual motion, and it forces the answer.
Owned result: the metric, read off the clock rate. $\mathrm{d}\tau/\mathrm{d}t = 1 + \Phi/c^2 \Rightarrow \boxed{g_{00} = 1 + 2\Phi/c^2}$ . The factor of two is a trap: the clock rate is $\sqrt{g_{00}}$ , not $g_{00}$ . The error is second order — invisible in every table you would use to check it.
Owned result: it is the POTENTIAL, not the field. The slowest clock on Earth is at the centre, where $g = 0$ exactly and $\Phi$ is deepest: $|\Phi|/c^2 = 1.044\times10^{-9}$ , 1.5× the surface value. A theory keyed to field strength would make it the fastest. It is the slowest.
Owned result: the numbers. $g/c^2 = \mathbf{1.0911\times10^{-16}}$ per metre. Earth's surface: $|\Phi|/c^2 = \mathbf{6.961\times10^{-10}}$ below infinity, 22.0 ms per year. The Sun's surface: $\mathbf{2.1225\times10^{-6}}$ , a recession of $\mathbf{636}$ m/s.
The heuristic is a scaffold. Exact for the redshift; half-strength for light bending; blind to tides. Its reach ends where `aq-sr-equivalence-principle`'s tidal tensor begins.
Key idea: gravity does not slow clocks by acting on them. Clocks at different potentials simply do not agree about how much time has passed, and the disagreement is forced by energy conservation alone.

The core insight

The heuristic never asks what gravity does to a clock, because gravity does nothing to a clock. It asks a bookkeeping question: if a photon sent up does not lose exactly the energy a mass would lose on the same climb, can I close a cycle and come out ahead? The answer must be no, and that single "no" fixes the redshift to $gh/c^2$ with no room to negotiate. Everything else is scaffolding around a result energy conservation had already made inevitable.

Where this lives today

Your phone is running this equation right now. A GPS satellite clock at $r \approx 2.66\times10^7$ m sits higher in the potential and runs fast; its orbital speed makes it run slow. The two do not cancel — they do not come close. Canon `aq-sr-equivalence-principle` prints $+45.7$ µs/day gravitational against $-7.2$ kinematic, net $+38.5$ for a spherical Earth, noting that the canonical $+45.8/-7.2/+38.6$ uses the rotating geoid. Either way the gravitational term is six times the special-relativistic one, with the opposite sign, and an uncorrected receiver drifts roughly $11$ km per day. A system built by someone who believed "you only need SR near the Earth" would fail immediately, catastrophically, and in the wrong direction.

Relativistic geodesy has stopped being a thought experiment. Optical lattice clocks reach fractional uncertainties below $10^{-18}$ , which at $1.09\times10^{-16}$ per metre is a height resolution of about a centimetre. Two clocks and a fibre between them turn the redshift into an altimeter that measures the geoid directly, without surveying and without assuming anything about Earth's interior. The most precise clocks ever built are now instruments of geology.

And it constrains the frontier, though not in the way people expect: because the redshift is forced by energy conservation, a precision measurement tests the universality of the coupling rather than general relativity itself. The experiment that opened the account is the next article.

The physical picture

Put yourself in a windowless rocket in deep space, engine burning at $9.8\ \mathrm{m/s^2}$ . Your friend on the floor aims a laser at the ceiling, frequency fixed at $\nu$ . You are at the ceiling with a spectrometer. The light takes $t \approx h/c$ to reach you, and in that time the ceiling — you — accelerated. In the inertial frame where the rocket was at rest at the moment of emission, you are now moving away from that event at $v = gt = gh/c$ . You are a receding receiver, so you measure a Doppler redshift:

\frac{\nu_{\text{rec}}}{\nu_{\text{emit}}} = 1 - \frac{v}{c} = 1 - \frac{gh}{c^2}.

There is no gravity in that paragraph. It is a first-order Doppler shift in flat spacetime, true of a rocket or of a rising lift in a mine.

Now say the magic words. The equivalence principle asserts that a windowless box accelerating at $g$ and a windowless box on a planet of surface gravity $g$ cannot be told apart by any local experiment. So the redshift must happen on the planet too. That is the whole heuristic, and it is worth pausing on how outrageous it is. We derived a property of gravity by doing an experiment where there is no gravity, then asserting the two are the same. That assertion is a postulate — and every limit of the argument lives in that one step.

Figure

THREE ROADS TO ONE NUMBER — and the third one does not know what spacetime is. Left: Einstein's 1907 rocket. The ceiling accelerates away during the light's flight, so the receiver recedes at v = gh/c and sees a first-order Doppler redshift. Flat spacetime. No gravity anywhere in the calculation. Centre: Einstein's 1911 falling photon. Give the light an effective mass hν/c² and make it climb; it pays (hν/c²)gh out of its own energy account and arrives redder. Right: THE PUNCHLINE — the energy-conservation cycle. Drop a mass m from height h and it arrives with energy mc² + mgh. Convert ALL of it to a photon. Send the photon back up. Convert it back to the mass. If the photon did not lose exactly the fraction gh/c² on the way up, you would arrive at the top with MORE than mc² and the cycle would print energy forever. This argument uses NO relativity of spacetime, NO equivalence principle, NO metric — only E = mc² and the refusal to build a perpetuum mobile. It does not merely predict the redshift. It FORCES it. All three roads land on the same number, and it is the number in the box

The redshift is therefore not a delicate consequence of an exotic geometry. It is the cheapest prediction in gravitational physics, and it was available in 1907, eight years before the field equations existed.

The core equations

\boxed{\;\frac{\Delta\nu}{\nu} \;=\; -\,\frac{gh}{c^{2}}\;}\qquad\Longrightarrow\qquad \boxed{\;\frac{\Delta\nu}{\nu} \;=\; -\,\frac{\Delta\Phi}{c^{2}}\;}

where: $\Delta\nu = \nu_{\text{rec}} - \nu_{\text{emit}}$ (negative for a redshift), $h$ the height climbed, $\Delta\Phi = \Phi_{\text{rec}} - \Phi_{\text{emit}} > 0$ for light going up. Physical reading: the shift depends on the potential difference and on nothing else — not on the photon's energy, not on its polarisation. That universality is the equivalence principle's fingerprint, and it is what makes the effect a property of spacetime rather than of light.

\frac{\mathrm{d}\tau_{\text{low}}}{\mathrm{d}\tau_{\text{high}}} \;=\; 1 - \frac{gh}{c^{2}} \;=\; 1 + \frac{\Delta\Phi_{\text{low}\to\text{high}}}{c^{2}}\Big|_{\Delta\Phi<0}

where: $\mathrm{d}\tau$ is proper time on a static clock. Physical reading: redshift and time dilation are the same statement. A light wave is a running clock — its crests are ticks — so a slow clock and a reddened signal cannot be told apart.

\frac{\mathrm{d}\tau}{\mathrm{d}t} \;=\; 1 + \frac{\Phi}{c^{2}} \qquad\Longrightarrow\qquad \boxed{\;g_{00} \;=\; 1 + \frac{2\Phi}{c^{2}}\;}

where: $t$ is the coordinate time of a reference clock at $\Phi = 0$ (infinitely far away), and $\mathrm{d}\tau = \sqrt{g_{00}}\,\mathrm{d}t$ for a static clock. Physical reading: the heuristic has just measured a metric component — the first brick of general relativity, and `aq-sr-special-to-general-relativity` builds on it. The factor of two is real and it is a trap: the tick rate is $\sqrt{g_{00}} \approx 1 + \Phi/c^2$ , while $g_{00}$ is $1 + 2\Phi/c^2$ . Confuse them and every clock-rate prediction is wrong by exactly two. For a point mass, $\Phi(r) = -GM/r$ and the same law reads $\Delta\nu/\nu = -(GM/c^2)(1/r_{\text{emit}} - 1/r_{\text{rec}})$ . That extension is a guess, not a derivation — nothing in the rocket argument says a real, non-uniform field obeys the uniform-field law with $gh \to \Delta\Phi$ . The rocket only ever knew a uniform field. That the guess is right, to leading order, is a fact about general relativity and not a fact the heuristic proved.

Where the number comes from

Derivation AThe rocket, the Doppler shift, and the one step that is a postulate

The goal is to get a gravitational prediction out of a calculation in which no gravity appears — and to be scrupulous about where the gravity is smuggled in, because that smuggling point is the whole boundary of the heuristic's validity. Everything up to the final step is flat-spacetime special relativity with a receding receiver. The final step is the equivalence principle: an assumption, not a consequence.

Step 1. Work in the momentarily comoving inertial frame (MCIF) at the instant of emission. At $t = 0$ the rocket is at rest in it, and a photon of proper frequency $\nu$ leaves the floor heading up. Why: the clock postulate licenses this — an ideal clock's rate depends on instantaneous velocity, not acceleration, so at $t=0$ the emitter is an ordinary inertial emitter. `aq-sr-proper-acceleration-comoving-frame` owns the MCIF.

Step 2. The light crosses the rocket in $t \approx h/c$ ; corrections are of relative order $gh/c^2 \sim 10^{-16}$ . Why: a first-order correction to the flight time contributes only at second order. That is the licence — not sloppiness.

Step 3. In that time the ceiling, rigidly attached to the engine, has picked up $v = gt = gh/c$ in the MCIF, directed away from the emission event. Why: this is the entire physical content. The receiver is receding. Everything downstream is bookkeeping on that one fact.

Step 4. A receding receiver sees the relativistic Doppler shift

\frac{\nu_{\text{rec}}}{\nu_{\text{emit}}} = \sqrt{\frac{1-\beta}{1+\beta}} \;=\; 1 - \beta + \tfrac12\beta^2 - \cdots \;=\; 1 - \frac{gh}{c^{2}} + O\!\left(\!\left(\tfrac{gh}{c^2}\right)^{\!2}\right).

Why: the first-order term is the whole answer at any measurable precision — the second-order piece is $\sim 10^{-32}$ at Earth's surface. But note what this means: the heuristic delivers a first-order result and cannot, even in principle, produce the exact clock-rate law. `aq-sr-rindler-metric` derives the exact law $\Delta\tau_2/\Delta\tau_1 = x_2/x_1$ and owns it.

Step 5. Now invoke the equivalence principle: replace the accelerating rocket by one standing still on a planet of surface gravity $g$ . By assumption no local experiment distinguishes them, so the redshift is identical:

\boxed{\;\frac{\Delta\nu}{\nu} \;=\; -\frac{gh}{c^{2}}\;}

Why: this step is a postulate and it is the only place gravity enters. Steps 1–4 contain no gravity at all. The equivalence principle buys the right to reinterpret them — so every limitation of the equivalence principle is inherited here. `aq-sr-equivalence-principle` prices them exactly: a $1$ m laboratory is an inertial frame for $0.81$ s and no longer. *Verified in verify_aq-sr-gravitational-time-dilation-heuristic.py: the exact relativistic Doppler factor is expanded symbolically in sympy and its first-order term confirmed to be $-gh/c^2$ ; the neglected second-order term is evaluated at Earth's surface and shown to be below $10^{-31}$ ; and $g/c^2 = 1.0911\times10^{-16}\ \mathrm{m^{-1}}$ is recomputed at fifty digits and grepped out of this file.*

Go deeper · Derivation B · 2 minEnergy conservation forces the redshift — and it never mentions spacetime

Derivation BEnergy conservation forces the redshift — and it never mentions spacetime

Both Einstein arguments lean on relativity — one on the Doppler effect and the equivalence principle, the other on giving a photon an effective gravitational mass, which is uncomfortable. The goal here is to remove both crutches and show the redshift is already forced by two facts that predate general relativity entirely: mass–energy equivalence, and the impossibility of a perpetual-motion machine. This is the strongest form of the result, and the reason no theory of gravity has ever been proposed without it.

Step 1. Build the cycle. Take a mass $m$ at the top of a tower of height $h$ . Its total energy there is $mc^2$ . Drop it. At the bottom it has gained kinetic energy $mgh$ , so its total energy is

E_{\text{bottom}} \;=\; mc^{2} + mgh \;=\; mc^{2}\left(1 + \frac{gh}{c^{2}}\right).

Why: pure Newtonian mechanics plus $E = mc^2$ . Nothing about the gravitational field has been assumed — only that energy and mass are the same quantity.

Step 2. At the bottom, convert all of it into a single photon: $h\nu_{\text{bottom}} = mc^{2}(1 + gh/c^{2})$ . Why: an idealisation, but a legal one — annihilation converts rest energy to photons, and nothing forbids capturing the kinetic energy too. The argument needs the conversion only to be possible in principle; an impossibility proof does not care how inconvenient the apparatus is.

Step 3. Send the photon back up, where it arrives with frequency $\nu_{\text{top}}$ , and convert it back into the mass $m$ . Closure demands $h\nu_{\text{top}} = mc^{2}$ exactly — no more, no less. More, and running the cycle again makes energy from nothing. Less, and running it backwards does. Why: this is the whole argument. We have not computed the redshift; we have cornered it. The cycle returns the apparatus to its initial state, so the first law leaves no freedom. The cycle is Einstein's own — Ann. Phys. 35, 899–901 (1911), §2, where he runs it in the other direction to prove that energy has gravitational mass $E/c^2$ : "By the energy principle, we must therefore have $E\gamma h/c^2 = M'\gamma h - M\gamma h$ , or $M' - M = E/c^2$ ." Bondi's and Feynman's versions are later restatements, not the source.

Step 4. Divide.

\frac{\nu_{\text{top}}}{\nu_{\text{bottom}}} \;=\; \frac{1}{1 + gh/c^{2}} \;=\; 1 - \frac{gh}{c^{2}} + O\!\left(\!\left(\tfrac{gh}{c^{2}}\right)^{\!2}\right) \qquad\Longrightarrow\qquad \boxed{\;\frac{\Delta\nu}{\nu} \;=\; -\,\frac{gh}{c^{2}}\;}

Why: the same number, from an argument that never mentioned a rocket, a metric or a coordinate system. No relativity of spacetime was used anywhere. Light must pay exactly what mass pays to climb the same tower — otherwise two freely convertible currencies disagree about the price of a height, and that is an arbitrage.

Step 5. Read the consequence, which is larger than the result. Gravitational redshift is not a test of general relativity the way perihelion precession is. It is a consistency requirement on any theory of gravity that accepts $E = mc^2$ . What the precision experiments test is the universality: that every clock, of every construction, shifts by the same $\Delta\Phi/c^2$ . Why: which is why the redshift is called a test of the equivalence principle rather than of general relativity — and the terminology is exactly right. `aq-sr-gravitational-redshift` runs the experiment; `aq-sr-rindler-metric` owns the analysis of what Pound and Rebka could and could not have seen. *Verified in verify_aq-sr-gravitational-time-dilation-heuristic.py: the cycle is closed symbolically in sympy — the ratio $\nu_{\text{top}}/\nu_{\text{bottom}}$ is computed from energy balance alone and shown to equal $1/(1+gh/c^2)$ identically; its series expansion is confirmed to be $1 - gh/c^2 + (gh/c^2)^2 - \cdots$ ; and the "no free energy" condition is checked by asserting that any other redshift factor leaves a non-zero energy surplus per cycle.*

Go deeper · Derivation C · 2 minFrom the clock rate to a metric component, and the factor of two that eats people

Derivation CFrom the clock rate to a metric component, and the factor of two that eats people

The heuristic's real prize is not the redshift — it is that the redshift, read backwards, hands you a piece of the spacetime metric before a single field equation is written. The goal is to extract $g_{00}$ honestly, say exactly what has and has not been determined, and expose a factor-of-two error that is easy to make, endemic, and impossible to catch numerically.

Step 1. Let $t$ be the proper time of a static clock infinitely far from all masses, where $\Phi = 0$ . A static clock at potential $\Phi$ ticks at the rate the heuristic gives:

\frac{\mathrm{d}\tau}{\mathrm{d}t} \;=\; 1 + \frac{\Phi}{c^{2}} \qquad (\Phi < 0 \Rightarrow \text{slow}).

Why: apply $\Delta\nu/\nu = -\Delta\Phi/c^2$ between the clock and infinity, and identify the frequency ratio with the proper-time ratio by crest counting — an argument `aq-sr-rindler-metric` owns.

Step 2. For a static clock ( $\mathrm{d}x^i = 0$ ) the interval is $\mathrm{d}s^2 = g_{00}c^2\mathrm{d}t^2 > 0$ and proper time is $\mathrm{d}\tau = \sqrt{\mathrm{d}s^2}/c$ , so

\frac{\mathrm{d}\tau}{\mathrm{d}t} \;=\; \sqrt{g_{00}} .

Why: the square root is not decoration. In $(+,-,-,-)$ a static worldline has $\mathrm{d}s^2 > 0$ — timelike positive — and proper time is its square root. This is the only place the signature enters, and it enters decisively.

Step 3. Equate and square:

\sqrt{g_{00}} = 1 + \frac{\Phi}{c^{2}} \qquad\Longrightarrow\qquad g_{00} = \left(1 + \frac{\Phi}{c^{2}}\right)^{2} = 1 + \frac{2\Phi}{c^{2}} + \frac{\Phi^{2}}{c^{4}} \;\simeq\; \boxed{\;1 + \frac{2\Phi}{c^{2}}\;}

Why: there is the two. The tick rate carries $\Phi/c^2$ , the metric component $2\Phi/c^2$ . The discarded $\Phi^2/c^4$ is exactly the order at which the heuristic has no authority anyway. At Earth's surface the difference between $\sqrt{g_{00}}$ and $1 + \Phi/c^2$ is $\mathbf{2.42\times10^{-19}}$ — far below anything measurable. That is precisely why the error is dangerous: substituting numbers will never catch it. Only the algebra will.

Step 4. Scope it ruthlessly. We have determined one component of the metric, to first order, and nothing else. $g_{ij}$ is completely undetermined — a clock at rest never samples the spatial geometry. Why: and that blindness has a famous price. Einstein used exactly this in 1911 to predict the deflection of starlight grazing the Sun, and printed $\mathbf{0.83''}$ (Ann. Phys. 35, 904). General relativity doubles the effect, to $\mathbf{1.75''}$ : the missing half is the spatial curvature in $g_{ij}$ , which no clock-rate argument can see. (The doubling is in the physics, not in his arithmetic — see the cheat-sheet, and do not compute $2\times0.83$ .)

Step 5. Bank it. $g_{00} = 1 + 2\Phi/c^2$ is the Newtonian limit of general relativity's time-time component, forced by nothing more than energy conservation; any metric theory of gravity must reproduce it. `aq-sr-special-to-general-relativity` takes this line and, with the tidal tensor of `aq-sr-equivalence-principle`, walks it to $\nabla^2\Phi = 4\pi G\rho$ . Why: the heuristic's job was never to be the truth. It was to leave behind one true equation the real theory would have to honour — and this is it. *Verified in verify_aq-sr-gravitational-time-dilation-heuristic.py: sympy expands $\sqrt{1 + 2\Phi/c^2}$ and confirms the leading term is $1 + \Phi/c^2$ ; the difference $\sqrt{g_{00}} - (1 + \Phi/c^2)$ is evaluated at fifty digits at Earth's surface and asserted to equal $-2.42\times10^{-19}$ ; and the script asserts that the article nowhere states the tick rate to be $g_{00}$ .*

Figure

IT IS THE POTENTIAL, NOT THE FIELD — and the clean test case sits at the centre of the Earth. The bars are |Φ|/c², the fractional rate by which a static clock runs slow relative to a clock at rest infinitely far away. Read the ladder from the bottom: a GPS satellite, high in the well, is the fastest clock in the system. Earth's surface is slower. And the SLOWEST clock anywhere on Earth is at the CENTRE — where the gravitational field g is EXACTLY ZERO, by symmetry, and where an astronaut would float weightless. A theory in which clocks respond to the local field strength would predict the centre clock to tick at the same rate as a clock in deep space. It does not. It runs slow by 1.04 × 10⁻⁹, one and a half times the surface value, because the POTENTIAL is what it responds to and the potential is deepest exactly where the field vanishes. The whole 22.5 m of the Pound–Rebka tower is the hairline at the top of the Earth-surface bar: 2.46 × 10⁻¹⁵, five and a half orders of magnitude below the bar it sits on

Limiting cases

Case	Result	Meaning
$c \to \infty$	$\Delta\nu/\nu \to 0$	the Newtonian limit. Light climbs for free and the perpetual-motion cycle closes trivially, because a photon carries no mass
$g \to 0$	$\Delta\nu/\nu \to 0$ , $g_{00} \to 1$	flat spacetime, no potential, Minkowski
$\|\Phi\|/c^2 \ll 1$	$\mathrm{d}\tau/\mathrm{d}t = 1 + \Phi/c^2$	the whole domain of the heuristic. At Earth's surface $\|\Phi\|/c^2 = 6.96\times10^{-10}$ — deep inside it
$g$ finite, $\Phi$ deep	clock still slow	the potential governs, not the field. At Earth's centre $g = 0$ and the clock is the slowest on the planet
real $1/r$ field, big lab	heuristic fails	tides appear; no accelerated frame mimics them. `aq-sr-tidal-forces-equivalence-limits` prices the failure
light bending	heuristic gives half strength	$0.83''$ against GR's $1.75''$ . The spatial metric $g_{ij}$ , which no clock can see, supplies the other half

Common trapThe photon does not lose energy climbing. It has no potential energy to lose

The "falling photon" road is a scaffold, and it collapses if you lean on it. A photon has no rest mass and no gravitational potential energy; $h\nu/c^2$ is not a mass sitting in a well. Push the picture and you find yourself asking where the energy went — and there is no answer, because nothing happened to the photon at all. The correct statement is about the ends of the journey, not the journey. The source emits $N$ crests in its own proper time; exactly $N$ arrive — none are created or destroyed in a static field — and the receiver counts them against a faster clock, so it reports a lower frequency. Nothing was done to the light. Something is different about the two people looking at it. `aq-sr-rindler-metric` owns the crest-counting argument. Why keep the bad picture? It gets the right number, it got there first, and it makes the energy-conservation argument visible. Use it to see; do not use it to think.

Common trapThree defects in this article's own base, and one of them is a sign that changes the physics

(1) The base never declares a signature. $g_{00} = 1 + 2\Phi/c^2$ is a $(+,-,-,-)$ statement; in $(-,+,+,+)$ the same physics reads $g_{00} = -(1 + 2\Phi/c^2)$ . A metric component without a signature is not a number — it is half of one.

(2) The base puts the Rindler horizon at $x = -c^2/g$ , canon `aq-sr-rindler-metric` at $x = 0$ . Both are right — different charts (Møller's $\xi$ runs from the observer, canon's $x$ from the horizon). Mixing them silently is how a sign gets lost. This article measures height above the emitter.

(3) The base's GPS card prints " $+45$ , $-7$ , net $+38$ " µs/day. Canonical: SR $-7.2$ , GR $+45.8$ , net $+38.6$ (Ashby, Living Rev. Rel. 6 (2003) 1). The commonly-seen $+45.9$ does not sum: $45.9 - 7.2 = 38.7 \neq 38.6$ .

Worked examples

1 · How slow is the clock in your basement?

Step 1 — identify. Two identical caesium clocks, one on the floor, one at ceiling height $h = 3.0$ m. The fractional rate difference is $gh/c^2$ .

Step 2 — apply.

\frac{\Delta\nu}{\nu} = \frac{(9.80665)(3.0)}{8.98755\times10^{16}} = \mathbf{3.273\times10^{-16}} .

Step 3 — make it human. Over one year ( $3.156\times10^7$ s) the floor clock falls behind by $3.273\times10^{-16} \times 3.156\times10^{7}\ \mathrm{s} = \mathbf{10.3\ nanoseconds\ per\ year}$ .

Step 4 — is it measurable? Easily. An optical lattice clock reaches a fractional uncertainty near $10^{-18}$ — three hundred times smaller than the effect over a single metre. Your basement clock is not slow in principle; it is slow in a laboratory, on a Tuesday.

Step 5 — the boundary number. $g/c^2 = 1.0911\times10^{-16}$ per metre is the whole of gravitational time dilation near Earth's surface. Everything else is multiplication.

2 · The slowest clock on Earth is at its centre, where gravity is zero

Step 1 — identify the trap. Ask where on Earth a clock runs slowest and the reflex answer is "wherever gravity is strongest." That is wrong, and the cleanest place to see it wrong is the centre, where by Newton's shell theorem the field is exactly zero.

Step 2 — the potential at the centre. For a uniform sphere the interior potential is $\Phi(r) = -\dfrac{GM}{2R}\left(3 - \dfrac{r^2}{R^2}\right)$ , so at $r = 0$ ,

\Phi_{\text{centre}} = -\frac{3GM}{2R} = \frac{3}{2}\,\Phi_{\text{surface}} .

Step 3 — the numbers. With $GM_\oplus = 3.986\times10^{14}\ \mathrm{m^3/s^2}$ , $R_\oplus = 6.371\times10^6$ m,

\frac{|\Phi_{\text{surface}}|}{c^{2}} = \mathbf{6.961\times10^{-10}}, \qquad \frac{|\Phi_{\text{centre}}|}{c^{2}} = \mathbf{1.044\times10^{-9}} .

Step 4 — the verdict. The centre clock runs slow by $1.044\times10^{-9}$ — half again as slow as one at the surface — while an observer beside it floats weightless, accelerometer reading exactly zero. Over a year it loses 33 ms against infinity, to the surface clock's 22.0 ms.

Step 5 — the boundary number. This is the sharpest demonstration that gravitational time dilation is about $\Phi$ and not $g$ . A force is a gradient; a clock rate is not. (The uniform-density model overstates Earth's central potential — the real Earth is centrally condensed — but the ordering is model-independent: $|\Phi_{\text{centre}}| > |\Phi_{\text{surface}}|$ for any density profile, while $g_{\text{centre}} = 0$ for any spherically symmetric one.)

Origins · 2 minGravitational potential and how this was discovered

Gravitational potential

1687–1916 · Newton's inverse square · Laplace's equation · Einstein turns a potential into a clock rate

The quantity. The gravitational potential began as a bookkeeping convenience. Newton's law gives a force; Lagrange and Laplace noticed in the 1770s–80s that the force is the gradient of one scalar, $\mathbf{g} = -\nabla\Phi$ , obeying one beautifully simple equation, $\nabla^2\Phi = 4\pi G\rho$ . Three components became one function, and gravitation became computable. For a century and a half the potential was regarded as pure mathematics — convenient, but not there. The reinterpretation. Then Einstein, in 1907, did something without precedent. He showed that $\Phi$ — the mathematical convenience — is what clocks respond to, while $g$ — the obvious thing, the thing you feel — is what they ignore. A clock at Earth's centre feels nothing and runs slowest on the planet. The potential is not an antiderivative of the field; the field is a derivative of the thing that is real. General relativity completes the promotion: $\Phi$ becomes a component of the metric, $g_{00} = 1 + 2\Phi/c^2$ , and the gravitational force becomes a Christoffel symbol — a coordinate artefact free fall removes. The quantity nobody could see turned out to be the geometry; the quantity everybody could feel turned out to be a fiction. What survives. The additive constant in $\Phi$ is still arbitrary and still cancels from everything measurable — only $\Delta\Phi$ appears in the redshift, exactly as in Newtonian energy conservation. What changed is not the mathematics. It is that the potential difference now has a stopwatch attached.

History

1687

— Newton's Principia: gravity as an inverse-square force. No potential, no clocks.

1782

— Laplace writes the potential equation for gravity in vacuum; the potential becomes the central object of celestial mechanics, and is regarded as a mathematical convenience.

1905

— Einstein, "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?", Annalen der Physik 18, 639: E = mc². Without it the energy argument cannot even be stated.

1907

— Einstein, in his review for the Jahrbuch der Radioaktivität und Elektronik 4, 411–462, postulates the complete equivalence of a uniformly accelerated frame and a homogeneous gravitational field, and derives from it both gravitational time dilation and the redshift — eight years before the field equations.

1911

— Einstein, "Über den Einfluss der Schwerkraft auf die Ausbreitung des Lichtes", Annalen der Physik 35, 898–908. Section 2 runs the drop-convert-lift-convert cycle explicitly and concludes from the energy principle alone that energy has gravitational mass E/c². Section 3 gives the redshift, and the paper predicts a starlight deflection of 0.83 arcseconds.

1915

— The field equations double the deflection to 1.75 arcseconds: general relativity's spatial curvature supplies a second contribution equal to the first, which no clock-rate argument could have seen.

1960

— Pound and Rebka measure the shift over the 74-foot Jefferson tower at Harvard, confirming the 1907 number; `aq-sr-gravitational-redshift` owns the experiment.

1976

— Vessot and Levine fly a hydrogen maser and confirm the redshift to 7 parts in 100,000.

2022

— Optical lattice clocks resolve the redshift across a millimetre of atomic sample. The prediction has become infrastructure.

Going further

Texts. Einstein, Jahrbuch der Radioaktivität und Elektronik 4, 411–462 (1907) · Einstein, Ann. Phys. 35, 898–908 (1911), §2 for the energy cycle · Taylor & Wheeler, Spacetime Physics, 2nd ed., Project F · Rindler, Relativity, 2nd ed., ch. 1 · Hartle, Gravity, ch. 6 · MTW, Gravitation, §7.2–7.4 · Ashby, Living Rev. Rel. 6 (2003) 1.

Provenance. Signature $(+,-,-,-)$ . Every equation and number is recomputed in verify_aq-sr-gravitational-time-dilation-heuristic.py, which reads this file, regexes the printed values out and asserts each at fifty digits, with artefact guards on $g$ and $c$ . Base defects repaired: the base never declares a signature (declared here); its horizon convention ( $x = -c^2/g$ ) silently conflicts with canon's ( $x = 0$ ), and this article states which chart it uses; its GPS triple (" $+45$ , $-7$ , $+38$ ") is replaced by canon's.

Scope. The equivalence principle, the tidal tensor and the expiry time are owned by `aq-sr-equivalence-principle` — recalled by name, never re-derived. The exact uniform-field clock law $\Delta\tau_2/\Delta\tau_1 = x_2/x_1$ and the crest-counting argument are owned by `aq-sr-rindler-metric`; this article derives only the first-order heuristic. Pound–Rebka is `aq-sr-gravitational-redshift`'s. The Schwarzschild metric, the field equations and the exact deflection belong to the general-relativity track — forward-linked, not annexed.

Builds on. The Equivalence Principle · Relativistic Uniform Acceleration · Proper Acceleration and the Comoving Frame. Leads to. Gravitational Redshift and the Pound–Rebka Experiment · The Rindler Metric · Tidal Forces and the Limits of the Equivalence Principle · From Special to General Relativity.

If the photon does not lose energy climbing, why does the "falling photon" derivation work?

Because it is a correct energy accounting dressed up as an incorrect mechanical story. What it really encodes is that light and matter must agree about the price of a height — $h\nu$ and $mc^2$ climb the same tower for the same fractional cost, or you have an arbitrage between two convertible currencies. That is exactly what the perpetual-motion argument states, and it is true. The mechanical picture, of a parcel of mass $h\nu/c^2$ toiling upward and paying out of its energy account, is a mnemonic — and the reason to distrust it is that it invites the question "where did the energy go?", which has no answer. The photon that arrives is the photon that left, unchanged. Only the clock it is measured against has changed.

Does the redshift depend on the local gravitational field strength g, or on the potential Φ?

On the potential, and the distinction is not academic — it is the whole character of the effect. Two clocks at the same potential tick alike even if the local $g$ differs, and a clock at Earth's centre, where $g$ is exactly zero, is the slowest on the planet. The uniform-field formula $gh/c^2$ merely looks like a field-strength law, because in a uniform field $\Delta\Phi = gh$ and there is nothing to distinguish. Real fields separate them at once. If clocks answered to $g$ you could null the effect inside a hollow shell, where the field vanishes identically — and you cannot: the potential inside a shell is constant, deep and non-zero, and clocks there run slow at exactly the shell's surface rate.

The heuristic is "first order". First order in what, and does that mean the answer is approximate?

First order in $\Phi/c^2$ , and the answer is approximate — but not where people assume. In a genuinely uniform field the linear law is not an approximation at all: `aq-sr-rindler-metric` derives the exact ratio $\Delta\tau_2/\Delta\tau_1 = x_2/x_1$ and shows it equals $1 + g_1h/c^2$ exactly, provided $g_1$ is the proper acceleration at the lower clock. What is approximate is the leap at the end: from "uniform field, shift $gh/c^2$ " to "any field, shift $\Delta\Phi/c^2$ ". Nothing in the rocket licenses it — the rocket only ever knew a uniform field. That it turns out right, to leading order, is a fact about general relativity; the exact Schwarzschild answer $\sqrt{1 - 2GM/rc^2}$ has higher-order terms the heuristic cannot guess.

Why is the metric component 1 + 2Φ/c² but the clock rate only 1 + Φ/c²?

Because proper time is the square root of the interval, and the square root eats the two. For a static clock $\mathrm{d}s^2 = g_{00}c^2\mathrm{d}t^2$ , so $\mathrm{d}\tau/\mathrm{d}t = \sqrt{g_{00}} = \sqrt{1 + 2\Phi/c^2} \approx 1 + \Phi/c^2$ . The trap is that the mistake is numerically invisible: at Earth's surface $\sqrt{g_{00}}$ and $1 + \Phi/c^2$ differ by $2.4\times10^{-19}$ , a hundred million times below the effect itself. You cannot catch this error by substituting numbers — only by looking at the algebra. Which is a good general lesson about which errors survive a numerical check.

Is gravitational time dilation a prediction of general relativity or not?

It is a prediction of any theory of gravity consistent with $E = mc^2$ — which is why the energy-conservation derivation matters: it shows the redshift is not evidence for general relativity in particular. Newtonian gravity plus mass–energy equivalence already gives $\Delta\Phi/c^2$ . What the precision experiments test is the universality — whether every clock, whatever its internal physics, shifts by the same fraction — because that is what makes the effect geometric rather than a quirk of one oscillator. General relativity's distinctive content lies elsewhere: the tidal tensor, the spatial curvature that doubles the light bending, the field equations. The redshift is its entrance fee, not its signature.

The rocket argument uses the Doppler effect, which is about velocity. Isn't the clock's acceleration the real cause?

No, and there is an experiment. The clock postulate says an ideal clock's rate depends on its instantaneous velocity and not at all on its acceleration, and it has been tested at $\sim10^{18}\,g$ — muons in the CERN ring, whose dilated lifetime tracked $\gamma$ with no acceleration-dependent term whatever (`aq-sr-equivalence-principle` prints the numbers). What the rocket argument uses is not the emitter's acceleration but the receiver's velocity at reception — a first-order Doppler shift between two inertial states of motion. The acceleration's only job is to arrange for the receiver to be moving when the light arrives; once it has, it steps out of the calculation.