Equivalence principle argument predicting gravitational red- and blueshift

Einstein's 1911 argument that falling light is Doppler-shifted in a gravitational field

In the decade preceding Einstein's publication of the definitive version of his theory of general relativity, he anticipated several of the results of his final theory with heuristic arguments, not all of which were to prove to be correct.^{[note 1]}

To show that the equivalence principle implies that light is Doppler-shifted in a gravitational field, Einstein considered a light source ${\displaystyle S_{2}}$ separated along the z-axis by a distance ${\displaystyle h}$ above a receiver ${\displaystyle S_{1}}$ in a homogeneous gravitational field having a force per unit mass of 1 ${\displaystyle g.}$ A continuous beam of electromagnetic energy with frequency ${\displaystyle f_{2}}$ is emitted by ${\displaystyle S_{2}}$ towards ${\displaystyle S_{1}.}$ According to the equivalence principle, this system is equivalent to a gravitation-free system which moves with uniform acceleration ${\displaystyle g}$ in the direction of the positive z-axis, with ${\displaystyle S_{2}}$ separated by a constant distance ${\displaystyle h}$ from ${\displaystyle S_{1}.}$^{[p 2]}

In the accelerated system, light emitted from ${\displaystyle S_{2}}$ takes (to a first approximation) ${\displaystyle h/c}$ to arrive at ${\displaystyle S_{1}.}$ But in this time, the velocity of ${\displaystyle S_{1}}$ will have increased by ${\displaystyle v=gh/c}$ from its velocity when the light was emitted. The frequency of light arriving at ${\displaystyle S_{1}}$ will therefore not be the frequency ${\displaystyle f_{2},}$ but the greater frequency ${\displaystyle f_{1}}$ given by

$${\displaystyle f_{1}\approx f_{2}\left(1+{\frac {v}{c}}\right)=f_{2}\left(1+{\frac {gh}{c^{2}}}\right).}$$

According to the equivalence principle, the same relation holds for the non-accelerated system in a gravitational field, where we replace ${\displaystyle gh}$ by the gravitational potential difference ${\displaystyle \Phi }$ between ${\displaystyle S_{2}}$ and ${\displaystyle S_{1}}$ so that^{[p 2]}

$${\displaystyle f_{1}=f_{2}+{\frac {f_{2}}{c^{2}}}\Phi .}$$

Advent of general relativity

In 1916, Einstein used the framework of his newly completed general theory of relativity to update his earlier heuristic arguments predicting gravitational redshift to a more rigorous form.^{[p 5][note 2]} Gravitational redshift and two other predictions from his 1916 paper, the anomalous perihelion precession of Mercury's orbit and the gravitational deflection of light by the Sun, have become known as the "classical tests" of general relativity. The anomalous perihelion precession of Mercury had long been recognized as a problem in celestial mechanics since the 1859 calculations of Urbain Le Verrier. The observation of the deflection of light by the Sun in the 1919 Eddington expedition catapulted Einstein to worldwide fame. Gravitational redshift would prove to be by far the most difficult of the three classical tests to demonstrate.

There had been little rush by experimenters to test Einstein's earlier predictions of gravitational time dilation, since the predicted effect was almost immeasurably small. Einstein's predicted displacement for spectral lines of the Sun amounted to only two parts in a million, and would be easily masked by line broadening due to temperature and pressure, and by line asymmetry due to the fact the lines represent the superposition of absorption from many turbulent layers of the solar atmosphere.^[3] Several attempts to measure the effect were negative or inconclusive. The first generally accepted claim to have measured gravitational redshift was W.S. Adams's 1925 measurement of shifts in the spectral lines of the white dwarf star Sirius B.^[4] However, even Adams's measurements have since been brought into question for various reasons.^[5]

Mössbauer effect

In atomic spectroscopy, visible and ultraviolet photons resulting from electronic transitions of outer shell electrons, when emitted by gaseous atoms in an excited state, are readily absorbed by unexcited atoms of the same species. A corresponding absorbance of photons emitted by the nuclei of γ-emitters had never been observed because recoil of the nuclei resulted in so much loss of energy by the emitted photons that they no longer matched the absorbance spectra of the target nuclei. In 1958, Rudolf Mössbauer, who was analyzing the 129 keV transition of Iridium-191, discovered that by lowering the temperature of the emitter to 90K, he could achieve resonant absorbance. Indeed, the energy resolutions that he achieved were of unheard-of sharpness. He had discovered the phenomenon of recoilless γ-emission.^{[3][note 3]}

In 1959, several research groups, most notably Robert Pound and Glen Rebka in Harvard and a team led by John Paul Schiffer in Harwell (England), announced plans to exploit this recently discovered effect to perform terrestrial tests of gravitational redshift.

In February 1960, Schiffer and his team were the first to announce success in measuring the gravitational redshift, but with a rather high error of ±47%.^{[p 6]} It was to be Pound and Rebka's somewhat later contribution in April 1960, which used a stronger radiation source, longer path length, and several refinements to reduce systematic error, which was to be accepted as having provided a definitive measurement of the redshift.^{[p 4]}

Sources of error

After evaluating various γ-emitters for their study, Pound and Rebka chose to use ⁵⁷Fe because it does not require cryogenic cooling to exhibit recoil-free emission, has a relatively low internal conversion coefficient^{[note 4]} so that it is relatively free of X-ray emissions that would have been difficult to distinguish from the 14.4 keV transition,^{[note 5]} and its parent ⁵⁷Co has a usable half-life of 272 days.^[7]

Pound and Rebka found that a large source of systematic error resulted from temperature variations, which they attributed primarily to a second order relativistic Doppler effect due to lattice vibrations. A mere 1°C difference in temperature between emitter and absorber caused a shift about equal to the predicted effect of gravitational time dilation.^{[p 7][8]}

They also found frequency offsets between the lines of different combinations of source and absorber stemming from the sensitivity of the nuclear transition to an atom's physical and chemical environment. They therefore needed to adopt methodology which would allow them to distinguish these offsets from their measurement of gravitational redshift. Extreme care was also needed in sample preparation, otherwise inhomogeneities would limit the sharpness of the lines.^{[p 4]}

Experimental setup

Pound and Rebka's experimental arrangement

The experiment was carried out in a tower at Harvard University's Jefferson laboratory that was, for the most part, vibrationally isolated from the rest of the building. An iron disk containing radioactive ⁵⁷Co diffused into its surface was placed in the center of a ferroelectric or a moving coil magnetic transducer (speaker coil) which was placed near the roof of the building. A 38 cm diameter absorber consisting of thin square foils of iron enriched to a level of 32% ⁵⁷Fe (as opposed to a 2% natural abundance), which were pasted side by side in a flat pattern on a Mylar sheet, was placed in the basement. The distance between the source and absorber was 22.5 meters (74 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 absorber to detect the gamma rays that passed through.^[7]

The vibrating speaker coil imposed a continuously varying Doppler shift on the gamma ray source. Superimposed on the sinusoidal motions of the transducer was the slow (typically about 0.01 mm/s) constant motion of a slave hydraulic cylinder driven by a small diameter master cylinder controlled by a synchronous motor. The hydraulic cylinder motion was reversed multiple times during each data run after a constant integral number of transducer vibrations. Every several days, the position of the source and absorber would be reversed so that half the data runs would be of blueshift, and half would be of redshift.^[7]

Three thermocouples mounted on the source in a spiral pattern and three on the absorber were connected to Wheatstone bridges to measure the temperature differences between the source and absorber. The recorded temperature differences were used to correct the data before analysis.^[7]

Among the other steps used to compensate for possible systematic errors, Pound and Rebka varied the speaker frequency between 10 Hz and 50 Hz and tested different transducers (ferroelectric transducers versus moving coil magnetic speaker coils).^[9]

A Mössbauer monitor near the source (not illustrated) checked for possible distortions of the source signal resulting from the cylinder/transducer assembly being regularly inverted from facing downwards to facing upwards.^[7]

Modulation technique to detect small shifts

Animation: Modulation Technique To Detect Small Shifts. Counts were accumulated in four channels corresponding to the two maximum velocity (positive or negative) quarter-cycles of sinusoidal modulation in the two directions of motion of the hydraulic piston.

Although the 14.4 keV recoilless emission line of ⁵⁷Fe had a half-width of 1.13×10⁻¹², the anticipated gravitational frequency shift was only 2.5×10⁻¹⁵. Measurement of this minute amount of frequency shift, 500 times smaller than the half-width, required a sophisticated protocol for data acquisition and data analysis. The best way to measure a small shift is often by "slope detection", measuring the resonance not at its peak, but rather comparing the absorption curve near its points of maximum slope (inflection points) on either side of the peak.^[7]

The speaker coil typically operated at about 74 Hz with a maximum velocity amplitude corresponding to the maximum change of absorption with velocity of the resonance curve for a given combination of source and absorber (typically around 0.10 mm/s).^{[note 6]} Counts that were received in the quarter cycles of the oscillation period centered around the velocity maxima were accumulated in two separate registers. Likewise, counts received with the hydraulic cylinder in reverse motion were accumulated in another two separate registers, for a total of four registers of accumulated counts.^[7]

Animation: Precise determination of the center of the absorbance line

The combined motions of the vibrating transducer and hydraulic cylinder allowed the incoming photons to be collected in four channels representing source motions of +0.11 mm/s, +0.09 mm/s, −0.11 mm/s, and −0.09 mm/s. They collectively operated at a 50% duty cycle, so that out of, say, 80 million incoming photons, 10 million would fit into the time slots of each of the four recording channels. From these counts, the velocity corresponding to the absorbance maximum could be calculated.^[7]

The accuracy of determination of the line center depended on (1) the sharpness of the line, (2) the depth of the absorbance maximum, and (3) the total number of counts. They typically achieved a fractional absorbance maximum depth of about 0.3^[7] and recorded about 1×10¹⁰ γ-rays, of which most will have been recoilless.^{[10][note 7]}

Results

Each data run yielded eleven numbers, i.e. four absorber register counts, four monitor register counts, and three average temperature differences. The register counts were generally recorded after twelve full back-and-forth cycles of the hydraulic piston, where each reversal of piston motion occurred after 22,000 periods of source vibration.^[7]

The source and absorber units were interchanged every several days to allow comparison between the results with the γ-rays rising versus the γ-rays falling. Combining data from runs having gravitational frequency shift of equal but opposite sign enabled the fixed frequency shift between a given source/target combination to be eliminated by subtraction.^[7]

In their 1960 paper, Pound and Rebka presented data from the first four days of counting. Six runs with the source at the bottom, after temperature correction gave a weighted average fractional frequency shift between source and absorber of −(19.7±0.8)×10⁻¹⁵. Eight runs with the source at the top, after temperature correction gave a weighted average fractional frequency shift of −(15.5±0.8)×10⁻¹⁵.^{[p 4]}

The frequency shifts, up and down, were both negative because the magnitude of the inherent frequency difference of the source/absorber combination considerably exceeded the magnitude of the expected gravitational redshifts/blueshifts. Taking half the sum of the weighted averages yielded the inherent frequency difference of the source/absorber combination, −(17.6±0.6)×10⁻¹⁵. Taking half the difference of the weighted averages yielded the net fractional frequency shift due to gravitational time dilation, −(2.1±0.5)×10⁻¹⁵.^{[p 4]}

Over the full ten days of data collection, they calculated a net fractional frequency shift due to gravitational time dilation of −(2.56±0.25)×10⁻¹⁵, which corresponds to the predicted value with an error margin of 10%.^{[p 4]}

In the next several years, the Pound lab published successive refinements of the gravitational redshift measurement, finally reaching the 1% level in 1964.^{[p 9][3]}