Pound-Rebka Experiment

Of the three "classical tests" of general relativity, the gravitational redshift is the odd one out. Mercury's perihelion precession and the deflection of starlight probe the spatial curvature of spacetime — the part of the metric that the equivalence principle alone does not fix. The redshift does not: it follows from the equivalence principle alone. A clock deep in a gravitational well ticks slow, and light climbing out arrives reddened, whether or not Einstein's equations are exactly right. That makes it the most foundational test of all, and also the hardest to see — over any laboratory height the fractional shift $gh/c^2$ is only about $10^{-15}$ , a part in a thousand trillion, drowned a millionfold by the Doppler broadening of ordinary spectral lines. For half a century the effect could only be chased in starlight — the Sun, white dwarfs like Sirius B — where pressure shifts, convection, and line blending swamped any clean number. A bench measurement became possible only in 1958, when Rudolf Mössbauer discovered recoilless nuclear resonance: a $\gamma$ -ray from a nucleus locked in a crystal lattice hands its recoil to the whole crystal, so the line keeps the razor sharpness set by the nuclear lifetime. For the $14.4\,\mathrm{keV}$ transition of $^{57}\mathrm{Fe}$ this gives a fractional width near $3\times10^{-13}$ — still a hundred times the shift, but now within reach of a careful centroid measurement. In 1959 Robert Pound and Glen Rebka Jr. sent these $\gamma$ -rays up and down the $22.5\,\mathrm{m}$ tower of Harvard's Jefferson Physical Laboratory, mounting the source on a loudspeaker cone and finding the tiny Doppler velocity that cancelled the gravitational offset and restored resonance. They confirmed Einstein to $\sim10\%$ ; the 1965 Pound–Snider refinement reached $1\%$ . It remains the canonical proof that gravity bends time.

Key takeaways

The Pound–Rebka experiment (Harvard, 1959–1960) was the first clean laboratory confirmation of gravitational redshift, using $14.4\,\mathrm{keV}$ $\gamma$ -rays from $^{57}\mathrm{Fe}$ over a height $h = 22.5\,\mathrm{m}$ .
The predicted fractional shift is minuscule, $\Delta\nu/\nu = gh/c^2 \approx 2.46\times10^{-15}$ , far below the natural width of any ordinary spectral line.
The enabling trick was the Mössbauer effect: recoilless emission and absorption in a crystal lattice gives a resonance sharp enough to resolve the shift by Doppler compensation.
They moved the source at a known velocity to null the shift and reversed source and absorber to difference the up and down runs, cancelling systematics; agreement was $\sim 10\%$ , improved to $\sim 1\%$ by Pound–Snider (1965).
It probes only the time–time metric component $g_{00} = -(1+2\Phi/c^2)$ , fixed by the equivalence principle and the Newtonian potential — not spatial curvature or the PPN parameter $\gamma$ .
Key idea: a photon climbing a height $h$ against gravity loses energy as though it had gravitational mass $E/c^2$ , giving a fractional redshift $\Delta\nu/\nu = -gh/c^2$ ; the Mössbauer effect supplies a line narrow enough to see it, and a known Doppler velocity is tuned to null and thereby measure the shift.

The core insight

A photon climbing a height $h$ against gravity loses energy as though it had gravitational mass $E/c^2$ , producing a fractional redshift $\Delta\nu/\nu = -gh/c^2$ ; the Mössbauer effect supplies a resonance narrow enough to detect it, and a known Doppler velocity is tuned to cancel and thereby measure the shift.

Where this lives today

The number Pound and Rebka fought for over a tower is now measured across a single atom cloud. In 2010 a JILA group compared two aluminum-ion optical clocks separated by just $33\,\mathrm{cm}$ and resolved the blueshift of the lower — a height you produce by standing up. By 2022 a JILA experiment resolved the redshift within a single millimeter-tall lattice of $^{87}\mathrm{Sr}$ atoms, the sample's top ticking measurably faster than its bottom. Such clocks are turning the redshift from a test into a tool: "relativistic geodesy" would map the geoid by transporting optical clocks and reading their rate differences, sensing centimeter changes in effective height. The same principle flew in Gravity Probe A (1976), whose hydrogen maser confirmed the redshift to $70$ parts per million, and it beats inside every GPS and Galileo satellite, pre-corrected for its $\sim38\,\mu\mathrm{s/day}$ offset. The open frontier is to test whether the redshift is exactly $gh/c^2$ — a hunt for violations of local position invariance that would signal physics beyond general relativity.

The physical picture

The cleanest way to feel the effect is to drop a photon down the tower. Give it energy $E = h_{\mathrm P}\nu$ and, borrowing Newtonian bookkeeping, an effective mass $E/c^2$ . Falling through height $h$ it gains kinetic energy $(E/c^2)gh$ and arrives blueshifted by $\Delta\nu/\nu = +gh/c^2$ ; sent the other way it must spend energy climbing and arrives reddened by the same fraction. The arithmetic that lifts an apple governs the climbing photon — nothing about curved space has entered yet. The rigorous version replaces gravity with acceleration. In a windowless rocket accelerating upward at $g$ , a signal leaving the floor takes time $h/c$ to reach the ceiling; in that interval the ceiling speeds up by $\Delta v = gh/c$ , so it receives the light while receding and sees a first-order Doppler redshift $\Delta v/c = gh/c^2$ . By the equivalence principle the static tower and the rocket are locally indistinguishable and must predict the same shift — which is exactly why this experiment tests the equivalence principle, not the field equations.

Figure

The same shift two ways: a γ climbing the static 22.5 m tower loses energy in Earth's field, and a γ crossing a rocket accelerating at a = g is redshifted by the receding ceiling — both give Δν/ν = −gh/c² = −2.46×10⁻¹⁵

The Mössbauer effect makes this measurable by abolishing recoil. A free $^{57}\mathrm{Fe}$ nucleus emitting a $14.4\,\mathrm{keV}$ $\gamma$ kicks back with $E_R = E_\gamma^2/(2Mc^2) \approx 1.9\times10^{-3}\,\mathrm{eV}$ — some four hundred thousand linewidths — shoving emission and absorption so far apart that no free source can be reabsorbed. Lock the nucleus in a rigid lattice, though, and a "recoilless fraction" of events hand their momentum to the whole crystal ( $M \to M_{\mathrm{crystal}} \to \infty$ ), so $E_R \to 0$ and the line snaps back to full sharpness. Even then the recoilless line cannot be split into two peaks: its fractional width $\sim 3\times10^{-13}$ is about a hundred times the shift, so Pound and Rebka never saw two lines — they swept the source velocity on a transducer, mapped the single resonance, and located its centroid to a fraction of a percent of its width. The figure below lays out the three energy scales that dictate the strategy: the recoil to remove, the linewidth that survives, and the shift hiding inside it.

Figure

Three energy scales on a log eV axis: free-nucleus recoil E_R ≈ 1.9×10⁻³ eV must be removed by the Mössbauer effect (about 4×10⁵ linewidths), leaving the recoilless linewidth Γ ≈ 4.7×10⁻⁹ eV, inside which the gravitational shift ΔE = E_γ·gh/c² ≈ 3.5×10⁻¹¹ eV sits about 130 times smaller

The core equations

The prediction is a single line, Newtonian in form: the fractional shift of an upward photon is minus the potential difference in units of $c^2$ , and everything the experiment measures is a determination of this number.

\frac{\Delta\nu}{\nu} = -\frac{gh}{c^2} \qquad (1)

where: $\Delta\nu = \nu_{\mathrm{received}}-\nu_{\mathrm{emitted}}$ — the frequency shift; $\nu$ — emitted $\gamma$ -ray frequency ( $\approx 3.5\times10^{18}\,\mathrm{Hz}$ for $14.4\,\mathrm{keV}$ ); $g$ — local gravitational acceleration, $9.81\,\mathrm{m/s^2}$ ; $h$ — vertical source–absorber separation, $22.5\,\mathrm{m}$ ; $c$ — speed of light. The minus sign: upward light loses energy; a downward photon is blueshifted. The relativistic pedigree of that formula is the time–time component of the weak-field metric — the only component the experiment touches, fixed entirely by the Newtonian potential $\Phi$ .

g_{00} = -\left(1 + \frac{2\Phi}{c^2}\right),\qquad \Phi = -\frac{GM_\oplus}{r} \qquad (2)

where: $g_{00}$ — time–time metric component; $\Phi$ — Newtonian potential (negative, deepening inward); $M_\oplus$ — Earth's mass; $G$ — Newton's constant. Reading it: clocks run at a rate $\sqrt{-g_{00}}\approx 1+\Phi/c^2$ , slower where $\Phi$ is more negative — that is, deeper in the well. Comparing the tick rates of two static clocks at different potentials gives the frequency ratio directly, with no reference to spatial curvature.

\frac{\nu_{\mathrm{rec}}}{\nu_{\mathrm{emit}}} = \frac{(\mathrm{d}\tau/\mathrm{d}t)_{\mathrm{emit}}}{(\mathrm{d}\tau/\mathrm{d}t)_{\mathrm{rec}}} = 1 + \frac{\Phi_{\mathrm{emit}} - \Phi_{\mathrm{rec}}}{c^2} \qquad (3)

where: $\nu_{\mathrm{rec}},\nu_{\mathrm{emit}}$ — received and emitted frequencies; $\Phi_{\mathrm{emit}},\Phi_{\mathrm{rec}}$ — potentials at emitter and receiver; $\tau$ — proper time; $t$ — global coordinate time. With the emitter low ( $\Phi_{\mathrm{emit}}<\Phi_{\mathrm{rec}}$ ) the ratio is below one — a redshift of magnitude $gh/c^2$ , recovering equation (1). The obstacle is nuclear: a free emitter recoils, and the recoil energy dwarfs the line it must hit.

E_R = \frac{E_\gamma^2}{2 M c^2} \qquad (4)

where: $E_R$ — recoil energy of a free nucleus; $E_\gamma$ — $\gamma$ -ray energy, $14.4\,\mathrm{keV}$ ; $M$ — nuclear mass, $\approx 57\,u$ . Numerically $E_R\approx 1.9\times10^{-3}\,\mathrm{eV}$ , which displaces emission from absorption by hundreds of thousands of linewidths — no resonance for free nuclei. What survives when the lattice swallows the recoil is a line whose width is set only by the nuclear lifetime.

\frac{\Gamma}{E_\gamma} = \frac{\hbar}{\tau E_\gamma} \qquad (5)

where: $\Gamma$ — natural energy width; $\tau$ — mean lifetime of the $14.4\,\mathrm{keV}$ state, $\approx 1.4\times10^{-7}\,\mathrm{s}$ ; $\hbar$ — reduced Planck constant. This gives $\Gamma\approx 4.7\times10^{-9}\,\mathrm{eV}$ , a fractional width $\approx 3\times10^{-13}$ — sharp, but still $\sim100$ times the shift. Finally, the readout: a source moving at velocity $v$ imprints a known Doppler shift, and reversing the beam doubles the gravitational signal while cancelling velocity-independent systematics.

\nu_{\mathrm{obs}}(v) = \nu_0\left(1 + \frac{v}{c}\right), \qquad \left.\frac{\Delta\nu}{\nu}\right|_{\mathrm{down}-\mathrm{up}} = \frac{2gh}{c^2} \qquad (6)

where: $v$ — transducer-driven source velocity; $\nu_0$ — rest-frame emission frequency; differencing downward and upward runs isolates $2gh/c^2$ . Setting the Doppler shift equal to the gravitational shift gives the null velocity $v = gh/c \approx 7.4\times10^{-7}\,\mathrm{m/s}$ .

Derivation AThe redshift from the equivalence principle

The goal is to obtain the full result $\Delta\nu/\nu = gh/c^2$ using nothing but special relativity in a uniformly accelerating frame, proving this article's central claim: the gravitational redshift follows from the equivalence principle alone and needs no curved spacetime, no field equations, and no knowledge of what sources the gravity. We replace the static tower by a rocket of proper acceleration $g$ in flat empty spacetime, emit a photon from its floor toward a ceiling detector a height $h$ above, and track the Doppler shift the accelerating detector records.

Step 1. The light takes a transit time $\Delta t \approx h/c$ to cross floor to ceiling, to leading order in $v/c$ . Why: at these tiny velocities the photon travels essentially at $c$ across a fixed gap $h$ , so the naive light-travel time suffices.

Step 2. In that interval the ceiling, accelerating at $g$ , gains a velocity $\Delta v = g\,\Delta t = gh/c$ away from the emission event. Why: the detector is not inertial — by arrival it is moving away from where the light started, the seed of a redshift.

Step 3. The receding detector measures a first-order Doppler shift $\Delta\nu/\nu = -\Delta v/c = -gh/c^2$ . Why: a receiver fleeing the source sees fewer crests per second — a redshift, exactly as for the upward tower photon.

Step 4. By the equivalence principle a uniform field $g$ is locally indistinguishable from acceleration $g$ , so the static tower reproduces the rocket's shift, $\Delta\nu/\nu = -gh/c^2$ . Why: the shift arose from pure flat-spacetime kinematics, so the identical shift in a real field cannot be a probe of curvature.

Go deeper · Derivation B · 1 minThe redshift from the weak-field metric

Derivation BThe redshift from the weak-field metric

The aim is to reach the same result from the general-relativistic side, showing that equation (1) is the leading-order shadow of the time–time metric component $g_{00}=-(1+2\Phi/c^2)$ and nothing else. This pins down which piece of the theory the experiment interrogates: it counts wave crests between two static clocks at different potentials, converting the metric's clock-rate factor into a frequency ratio, and never once uses the spatial metric. That omission is why Pound–Rebka is blind to the PPN parameter $\gamma$ that governs light bending.

Step 1. A static clock at potential $\Phi$ ticks proper time $\mathrm{d}\tau = \sqrt{-g_{00}}\,\mathrm{d}t \approx (1+\Phi/c^2)\,\mathrm{d}t$ against global coordinate time $t$ . Why: for a static observer only $g_{00}$ enters the line element, so the clock rate is fixed entirely by the Newtonian potential to this order.

Step 2. The emitter radiates $N = \nu_{\mathrm{emit}}\,\Delta\tau_{\mathrm{emit}}$ crests, and in a stationary geometry the same $N$ arrive: $N = \nu_{\mathrm{rec}}\,\Delta\tau_{\mathrm{rec}}$ . Why: nothing creates or destroys crests along a static path, so the shift can only come from the two ends ticking at different rates.

Step 3. Both proper intervals map to one coordinate interval $\Delta t$ , so $\nu_{\mathrm{rec}}/\nu_{\mathrm{emit}} = \Delta\tau_{\mathrm{emit}}/\Delta\tau_{\mathrm{rec}} = (1+\Phi_{\mathrm{emit}}/c^2)/(1+\Phi_{\mathrm{rec}}/c^2)$ . Why: coordinate time is the common ledger for static observers, so the ratio cancels $\Delta t$ and leaves only the potentials.

Step 4. To first order $\nu_{\mathrm{rec}}/\nu_{\mathrm{emit}} \approx 1+(\Phi_{\mathrm{emit}}-\Phi_{\mathrm{rec}})/c^2$ ; with the emitter at the base, $\Phi_{\mathrm{emit}}-\Phi_{\mathrm{rec}} = -gh$ , giving $\Delta\nu/\nu = -gh/c^2$ . Why: only the potential difference $gh$ survives, matching the equivalence-principle answer exactly — the two roads meet. *As a check: the exact Schwarzschild frequency ratio $\sqrt{(1-2GM/r_2c^2)/(1-2GM/r_1c^2)}$ reduces to $1-gh/c^2$ for small $h = r_2 - r_1$ , confirming that the weak-field limit recovers equation (1).*

Go deeper · Derivation C · 1 minWhy recoilless emission is essential

Derivation CWhy recoilless emission is essential

This derivation establishes the nuclear-physics precondition for the whole measurement: a free $^{57}\mathrm{Fe}$ nucleus recoils so violently on emitting its $14.4\,\mathrm{keV}$ $\gamma$ that emission and absorption frequencies no longer overlap, and only the Mössbauer effect — recoil taken up by the entire crystal — restores a line sharp enough to carry a $10^{-15}$ signal. We compute the free recoil, compare it to the natural linewidth, and show quantitatively why binding the nucleus in a lattice rescues the resonance.

Step 1. Momentum conservation gives the recoiling nucleus $p = E_\gamma/c$ , equal and opposite to the photon's momentum. Why: the nucleus and photon are the only two bodies, so the photon's recoil is dumped entirely into the emitter.

Step 2. The kinetic energy of that recoil is $E_R = p^2/2M = E_\gamma^2/2Mc^2$ , non-relativistic since $E_R \ll Mc^2$ . Why: at keV energies the nucleus recoils far below relativistic speeds, so the Newtonian kinetic-energy formula is exact to many digits.

Step 3. Numerically $E_R \approx (1.44\times10^4)^2/[2(57\times931.5\times10^6)]\,\mathrm{eV} \approx 1.9\times10^{-3}\,\mathrm{eV}$ , versus $\Gamma\approx 4.7\times10^{-9}\,\mathrm{eV}$ — a mismatch of $\sim4\times10^5$ linewidths. Why: emission drops by $E_R$ and absorption needs an extra $E_R$ , so the lines miss by $\sim2E_R$ and no free nucleus can reabsorb another's photon.

Step 4. Bind the nucleus in a rigid lattice: a "recoilless fraction" of events transfer momentum to the whole crystal, so $M\to M_{\mathrm{crystal}}\to\infty$ and $E_R\to 0$ , recovering full sharpness. Why: with the recoil quenched, emission and absorption lines coincide and the only remaining width is the lifetime-limited $\Gamma$ — narrow enough that a $10^{-15}$ shift is a measurable fraction of it. *As a numerical check: using $E_\gamma=14.4\,\mathrm{keV}$ , $M=57\,u$ , and $\tau=1.4\times10^{-7}\,\mathrm{s}$ gives $E_R = 1.95\times10^{-3}\,\mathrm{eV}$ and $E_R/\Gamma = 4.2\times10^5$ , confirming the mismatch quoted above.*

Limiting cases

Case	Result	Meaning
Predicted shift, full tower	$\dfrac{\Delta\nu}{\nu}=\dfrac{gh}{c^2}=\dfrac{(9.81)(22.5)}{(2.998\times10^8)^2}$	Numerically $\approx 2.46\times10^{-15}$ over the $22.5\,\mathrm{m}$ height — the number to be measured.
Equivalent null velocity	$v = c\,\dfrac{\Delta\nu}{\nu}=\dfrac{gh}{c}\approx 7.4\times10^{-7}\,\mathrm{m/s}$	Sub-micron-per-second source speed that cancels the shift; supplied by slow transducer modulation.
Gravitational energy shift	$\Delta E = E_\gamma\,\dfrac{gh}{c^2}\approx 3.5\times10^{-11}\,\mathrm{eV}$	About $1/130$ of the recoilless linewidth — hence centroid location, not line splitting.
Schwarzschild (exact) form	$\dfrac{\nu_2}{\nu_1}=\sqrt{\dfrac{1-2GM/(r_1c^2)}{1-2GM/(r_2c^2)}}$	For weak fields and small $h=r_2-r_1$ this reduces exactly to $1-gh/c^2$ ; as $r_1\to 2GM/c^2$ it diverges — the black-hole limit the lab formula cannot reach.

Common trapPound–Rebka "tested spacetime curvature and the field equations."

It did neither. The redshift depends only on $g_{00}=-(1+2\Phi/c^2)$ , fixed by the equivalence principle and the Newtonian potential; spatial curvature and the PPN parameter $\gamma$ — which govern light deflection and the Shapiro delay — do not enter, and a theory could pass Pound–Rebka perfectly yet fail the bending-of-light test. A second, subtler trap: the natural linewidth of $^{57}\mathrm{Fe}$ is about a hundred times larger than the shift, so the experiment never resolved two lines. It worked by locating the centroid of a single overlapping resonance to a fraction of a percent of its width, using the Doppler null and up-minus-down reversal to beat down systematics — measuring a tiny displacement of a line's center, not splitting it in two.

Worked examples

1 · The gravitational shift and Doppler null for the Harvard tower (fully worked)

Compute the predicted fractional redshift over the $22.5\,\mathrm{m}$ tower, the source velocity needed to cancel it, and the gravitational energy shift, and compare the shift to the natural linewidth. Use $g=9.81\,\mathrm{m/s^2}$ , $c=2.998\times10^8\,\mathrm{m/s}$ , $E_\gamma=14.4\,\mathrm{keV}$ .

Step 1. Numerator: $gh = 9.81 \times 22.5 = 220.7\,\mathrm{m^2/s^2}$ . Denominator: $c^2 = (2.998\times10^8)^2 = 8.99\times10^{16}\,\mathrm{m^2/s^2}$ .

Step 2. Fractional shift: $\dfrac{\Delta\nu}{\nu} = \dfrac{220.7}{8.99\times10^{16}} = \mathbf{2.46\times10^{-15}}$ . This is the redshift for an upward photon; a downward photon is blueshifted by the same amount.

Step 3. Null velocity: to cancel it the source must approach the absorber at $v = c\,\Delta\nu/\nu = gh/c = 220.7/(2.998\times10^8) = \mathbf{7.4\times10^{-7}\,m/s}$ — less than a micron per second, supplied by a loudspeaker cone.

Step 4. Energy shift: $\Delta E = E_\gamma\,\Delta\nu/\nu = 1.44\times10^4 \times 2.46\times10^{-15} = \mathbf{3.5\times10^{-11}\,eV}$ .

Step 5. Compare to the line: with $\Gamma/E_\gamma \approx 3\times10^{-13}$ , the shift is $\Delta E/\Gamma \approx (3.5\times10^{-11})/(4.7\times10^{-9}) \approx \mathbf{1/130}$ — about $0.8\%$ of the linewidth, invisible as a splitting but measurable as a centroid displacement, exactly what the experiment did.

2 · Free-nucleus recoil versus the recoilless line (fully worked)

Show quantitatively why a free $^{57}\mathrm{Fe}$ source cannot be resonantly reabsorbed, and why the Mössbauer effect is indispensable. Take $E_\gamma=14.4\,\mathrm{keV}$ , $Mc^2 = 57\times931.5\,\mathrm{MeV}$ , $\tau = 1.4\times10^{-7}\,\mathrm{s}$ .

Step 1. Recoil energy: $E_R = \dfrac{E_\gamma^2}{2Mc^2} = \dfrac{(1.44\times10^4)^2}{2(5.31\times10^{10})}\,\mathrm{eV} = \mathbf{1.95\times10^{-3}\,eV}$ .

Step 2. Natural width: $\Gamma = \hbar/\tau = (6.58\times10^{-16}\,\mathrm{eV\,s})/(1.4\times10^{-7}\,\mathrm{s}) = \mathbf{4.7\times10^{-9}\,eV}$ .

Step 3. Mismatch: emission is pushed down by $E_R$ and absorption needs an extra $E_R$ , so the lines are separated by $\sim 2E_R$ , a gap of $E_R/\Gamma \approx \mathbf{4\times10^{5}}$ linewidths. Overlap is utterly negligible — a free source is a dead source.

Step 4. With the lattice absorbing the recoil, $E_R\to 0$ , the lines coincide, and only $\Gamma$ survives — inside which the $\Delta E \approx 3.5\times10^{-11}\,\mathrm{eV}$ shift becomes accessible. The Mössbauer effect buys a factor $\sim10^{5}$ in line position and is the sole reason the measurement is possible.

Physics of the portrait

The redshift is the oldest of Einstein's predictions and among the last nailed down in the lab. That it fell to a Harvard tower rather than an astronomer owes everything to one physicist's genius for turning a nuclear curiosity into a precision instrument.

Origins · 1 minRobert Pound and how this was discovered

Robert Pound

1919–2010 · Harvard precision tests of general relativity by the Mössbauer method

Robert Vivian Pound was a Canadian-American physicist who, remarkably, never earned a doctorate yet rose to a full professorship at Harvard and the National Academy of Sciences. Trained at the wartime MIT Radiation Laboratory, he was a virtuoso of radio-frequency technique — with Purcell and Torrey he co-discovered nuclear magnetic resonance in 1946, and the Pound–Drever–Hall locking scheme he helped father still stabilizes lasers and gravitational-wave detectors. When Mössbauer's 1958 result crossed his desk, Pound saw what few did: a line finally sharp enough — a part in $10^{12}$ — to weigh a photon's climb up a building. With his student Glen Rebka Jr. he built the 1959 experiment, threading a $^{57}\mathrm{Fe}$ source and absorber through a $22.5\,\mathrm{m}$ helium-filled tube, temperature-matching both ends to suppress the thermal (second-order Doppler) shift and modulating the source on a transducer to sweep the resonance. Pound's genius was less new theory than the relentless engineering of systematics — knowing that at a part in $10^{15}$ the enemy is not statistics but every temperature gradient, vibration, and stray field in the building.

History

1911

— Einstein derives the gh/c² gravitational redshift from the equivalence principle

1958

— Mössbauer discovers recoilless nuclear resonance absorption of γ-rays

1959

— Pound and Rebka build the 22.5 m Jefferson tower experiment at Harvard

1960

— Pound and Rebka confirm the gravitational shift to about 10 percent

1965

— Pound and Snider refine the agreement with general relativity to about 1 percent

Going further

Texts. Misner, Thorne & Wheeler, Gravitation (1973), §38.5 and Box 38.1 · Wald, General Relativity (1984), §6.2 · Carroll, Spacetime and Geometry (2004), §5.6 · Hartle, Gravity: An Introduction to Einstein's General Relativity (2003), §6.2 · Schutz, A First Course in General Relativity (2009), §5.1 · Will, Theory and Experiment in Gravitational Physics (2018), §2.4.

Primary sources. Pound & Rebka, "Apparent Weight of Photons," Phys. Rev. Lett. 4, 337 (1960) · Pound & Snider, "Effect of Gravity on Gamma Radiation," Phys. Rev. 140, B788 (1965) · Mössbauer, Z. Phys. 151, 124 (1958). The key numerical benchmarks for this article are: $gh/c^2\approx 2.46\times10^{-15}$ ( $g=9.81\,\mathrm{m/s^2}$ , $h=22.5\,\mathrm{m}$ , $c=2.998\times10^8\,\mathrm{m/s}$ ); null velocity $v=gh/c\approx 7.4\times10^{-7}\,\mathrm{m/s}$ ; free-nucleus recoil $E_R=E_\gamma^2/(2Mc^2)\approx 1.9\times10^{-3}\,\mathrm{eV}$ ( $E_\gamma=14.4\,\mathrm{keV}$ , $M=57\,u$ ); recoilless linewidth $\Gamma=\hbar/\tau\approx 4.7\times10^{-9}\,\mathrm{eV}$ ( $\tau=1.4\times10^{-7}\,\mathrm{s}$ ); and the net GPS clock offset $\approx +38\,\mu\mathrm{s/day}$ . Experimental results: agreement to $\sim10\%$ (Pound–Rebka, 1960) and $\sim1\%$ (Pound–Snider, 1965).

Prerequisites. Gravitational Redshift · Newtonian Limit of General Relativity · Weak-Field and Linearized Gravity. Leads to. Parametrized Post-Newtonian (PPN) Formalism · Shapiro Time Delay · Deflection of Light by the Sun.

Ask the tutor: Ask me to walk through why the same $gh/c^2$ that Pound and Rebka nulled over a tower is the exact correction quietly running inside every GPS satellite.

Is the redshift a real loss of photon energy, or just an illusion of clocks?

Both pictures are correct and must agree. In the photon picture the upward $\gamma$ does work against gravity, spending energy $E\,gh/c^2$ and arriving at lower frequency. In the clock picture the frequency never changes; instead the emitter's clock, deep in the well, ticks slow, so the higher receiver counts fewer crests per (faster) second. These are the same fact seen through energy conservation versus proper time — general relativity insists they are one statement, and Pound–Rebka confirms it to $1\%$ .

Why does this test the equivalence principle rather than "all of general relativity"?

Because the derivation of $gh/c^2$ uses only special relativity in an accelerating frame plus the assertion that acceleration and gravity are locally identical — no curvature, no field equations, no knowledge of Earth's mass distribution. Formally the shift is set entirely by $g_{00}$ , which every metric theory respecting the equivalence principle predicts identically at leading order. The genuinely curvature-dependent effects — light bending, Shapiro delay, perihelion precession — live in the spatial metric and the PPN parameter $\gamma$ , to which Pound–Rebka is completely insensitive. So a positive result confirms the theory's deepest assumption while leaving its distinctive field-equation content untested.

If the linewidth is a hundred times the shift, how could they measure it?

The same way a surveyor reads a needle to a fraction of a tick mark. The recoilless resonance is a smooth, symmetric dip in transmission versus source velocity, and its center can be located far more precisely than its width, given enough counts and controlled systematics. Pound and Rebka swept the source velocity through resonance and found the small offset that centered it — that offset is the shift. They also reversed the beam, sending $\gamma$ -rays both up and down and differencing: the gravitational term doubles to $2gh/c^2$ while direction-blind thermal and vibration shifts cancel. Centroid location plus differencing pulls a $10^{-15}$ signal out of a $10^{-13}$ line.

What was the biggest systematic error, and how did they defeat it?

The second-order (transverse) Doppler shift — a thermal effect. Nuclei in a warm crystal vibrate, and time dilation of that random motion shifts the frequency by $-\langle v^2\rangle/2c^2$ , of order $10^{-15}$ per kelvin of temperature difference between source and absorber — comparable to the entire signal. A one-degree imbalance between the tower's ends would mimic the effect. Pound and Rebka, and far more carefully Pound and Snider in 1965, matched the two temperatures precisely and used beam reversal to expose any residual. Taming this thermal shift, not counting statistics, was the real experimental battle.

How does this connect to GPS actually working?

Directly and quantitatively. A GPS satellite clock sits higher in Earth's potential, so by $\Delta\Phi/c^2$ it runs fast by $\approx +46\,\mu\mathrm{s/day}$ ; orbital speed slows it by $\approx -7\,\mu\mathrm{s/day}$ , for a net $+38\,\mu\mathrm{s/day}$ . Since positions are fixed by timing signals to nanoseconds, an uncorrected $38\,\mu\mathrm{s}$ drift would smear positions by $\sim11\,\mathrm{km}$ within a day, so the satellite oscillators are frequency-offset before launch to compensate. Every phone finding itself on a map is running a continuous re-test of the physics Pound and Rebka pioneered.

Could a future experiment ever falsify the exact gh/c² prediction?

That is the point of pushing on. General relativity predicts the redshift is exactly the potential difference over $c^2$ , coefficient one, independent of the clock's composition — local position invariance. Theories with a scalar field or an equivalence-principle violation would give a coefficient slightly off one, or a composition-dependent shift. Optical-clock comparisons already constrain any anomaly to better than a part in $10^{4}$ , and clock-in-space and clock-transport experiments aim to tighten this by orders of magnitude. A confirmed deviation would signpost physics beyond Einstein — which is why an experiment "done in 1960" is still very much alive.