Pound–Rebka experiment

Consider an electron bound to an atom in an excited state. As the electron undergoes a transition from the excited state to a lower energy state it will emit a photon with a frequency corresponding to the difference in energy between the excited state and the lower energy state. The reverse process will also occur: if the electron is in the lower energy state then it can undergo a transition to the excited state by absorbing a photon at the resonant frequency for this transition. In practice the photon frequency is not required to be at exactly the resonant frequency, but must be in a narrow range of frequencies centred on the resonant frequency: a photon with a frequency outside this region cannot excite the electron to a higher energy state.

Now consider two copies of this electron-atom system, one in the excited state (the emitter), the other in the lower energy state (the receiver). If the two systems are stationary relative to one another and the space between them is flat (i.e. we neglect gravitational fields) then the photon emitted by the emitter can be absorbed by the electron in the receiver. However, if the two systems are in a gravitational field then the photon may undergo gravitational redshift as it travels from the first system to the second, causing the photon frequency observed by the receiver to be different to the frequency observed by the emitter when it was originally emitted. Another possible source of redshift is the Doppler effect: if the two systems are not stationary relative to one another then the photon frequency will be modified by the relative speed between them.

In the Pound–Rebka experiment, the emitter was placed at the top of a tower with the receiver at the bottom. General relativity predicts that the gravitational field of the Earth will cause a photon emitted downwards (towards the Earth) to be blueshifted (i.e. its frequency will increase) according to the formula:

${\displaystyle f_{r}={\sqrt {\frac {1-{\dfrac {2GM}{(R+h)c^{2}}}}{1-{\dfrac {2GM}{Rc^{2}}}}}}f_{e},}$

where ${\displaystyle f_{r}}$ and ${\displaystyle f_{e}}$ are the frequencies of the receiver and emitter, h is the distance between the receiver and emitter, M is the Earth's mass, R is the radius of the Earth, G is Newton's constant and c is the speed of light. To counteract the effect of gravitational blueshift, the emitter was moved upwards (away from the receiver) causing the photon frequency to be redshifted, according to the Doppler shift formula:

${\displaystyle f_{r}={\sqrt {\frac {1-v/c}{1+v/c}}}f_{e},}$

where ${\displaystyle v}$ is the relative speed between the emitter and receiver. Pound and Rebka varied the relative speed ${\displaystyle v}$ so that the Doppler redshift exactly cancelled the gravitational blueshift:

${\displaystyle {\sqrt {{\frac {1-v/c}{1+v/c}}\cdot {\frac {1-{\dfrac {2GM}{(R+h)c^{2}}}}{1-{\dfrac {2GM}{Rc^{2}}}}}}}=1.}$

In the case of the Pound–Rebka experiment ${\displaystyle h\ll R}$; the hight of the tower is tiny compared to the radius of the earth, and the gravitational field can be approximated as constant. Therefore, the Newtonian equation can be used:

${\displaystyle v\approx {\frac {gh}{c}}}$ = 7.5×10⁻⁷ m/s

The energy associated with gravitational redshift over a distance of 22.5 meters is very small. The fractional change in energy is given by δE/E, is equal to gh/c² = 2.5×10⁻¹⁵. As such, short wavelength high energy photons are required to detect such minute differences. The 14 keV gamma rays emitted by iron-57 when it transitions to its base state proved to be sufficient for this experiment.

Normally, when an atom emits or absorbs a photon, it also moves (recoils) a little, which takes away some energy from the photon due to the principle of conservation of momentum.

The Doppler shift required to compensate for this recoil effect would be much larger (about 5 orders of magnitude) than the Doppler shift required to offset the gravitational redshift. But in 1958 Rudolf Mössbauer reported that all atoms in a solid lattice absorb the recoil energy when a single atom in the lattice emits a gamma ray. Therefore, the emitting atom will move very little (just as a cannon will not produce a large recoil when it is braced, e.g. with sandbags). This allowed Pound and Rebka to set up their experiment as a variation of Mössbauer spectroscopy.

The test was carried out at Harvard University's Jefferson laboratory. A solid sample containing iron (⁵⁷Fe) emitting gamma rays was placed in the center of a loudspeaker cone which was placed near the roof of the building. Another sample containing ⁵⁷Fe was placed in the basement. The distance between this source and absorber was 22.5 meters (73.8 ft). The gamma rays traveled through a Mylar bag filled with helium to minimize scattering of the gamma rays. A scintillation counter was placed below the receiving ⁵⁷Fe sample to detect the gamma rays that were not absorbed by the receiving sample. By vibrating the speaker cone the gamma ray source moved with varying speed, thus creating varying Doppler shifts. When the Doppler shift canceled out the gravitational blueshift, the receiving sample absorbed gamma rays and the number of gamma rays detected by the scintillation counter dropped accordingly. The variation in absorption could be correlated with the phase of the speaker vibration, hence with the speed of the emitting sample and therefore the Doppler shift. To compensate for possible systematic errors, Pound and Rebka varied the speaker frequency between 10 Hz and 50 Hz, interchanged the source and absorber-detector, and used different speakers (ferroelectric and moving coil magnetic transducer).[2] The reason for exchanging the positions of the absorber and the detector is doubling the effect. Pound subtracted two experimental results:

1. the frequency shift with the source at the top of the tower
2. the frequency shift with the source at the bottom of the tower

The frequency shift for the two cases has the same magnitude but opposing signs. When subtracting the results, Pound and Rebka obtained a result twice as big as for the one-way experiment.

The result confirmed that the predictions of general relativity were borne out at the 10% level.[3] This was later improved to better than the 1% level by Pound and Snider.[4][5]

Another test, Gravity Probe A, involving a space-borne hydrogen maser increased the accuracy of the measurement to about 10⁻⁴ (0.01%).[6]

• Physical Review focus story.
• The Pound-Rebka experiment.