Gravitational redshift
This loss of energy corresponds to a decrease in the wave frequency and increase in the wavelength, known more generally as a redshift.
Light escaping from the surface of the Sun was predicted by Einstein in 1911 to be redshifted by roughly 2 ppm or 2 × 10−6.
[9] Navigational signals from GPS satellites orbiting at 20000 km altitude are perceived blueshifted by approximately 0.5 ppb or 5 × 10−10,[10] corresponding to a (negligible) increase of less than 1 Hz in the frequency of a 1.5 GHz GPS radio signal (however, the accompanying gravitational time dilation affecting the atomic clock in the satellite is crucially important for accurate navigation[11]).
, and the corresponding redshift is roughly 10−16 (0.1 parts per quadrillion) per meter of change in elevation and/or altitude.
In astronomy, the magnitude of a gravitational redshift is often expressed as the velocity that would create an equivalent shift through the relativistic Doppler effect.
In such units, the 2 ppm sunlight redshift corresponds to a 633 m/s receding velocity, roughly of the same magnitude as convective motions in the Sun, thus complicating the measurement.
[12] Observing the gravitational redshift in the Solar System is one of the classical tests of general relativity.
[13] Measuring the gravitational redshift to high precision with atomic clocks can serve as a test of Lorentz symmetry and guide searches for dark matter.
Einstein's theory of general relativity incorporates the equivalence principle, which can be stated in various different ways.
On Earth's surface (or in a spaceship accelerating at 1 g), the gravitational redshift is approximately 1.1×10−16, the equivalent of a 3.3×10−8 m/s Doppler shift for every 1 m of altitude.
The result is that frequencies and wavelengths are shifted according to the ratio where This can be related to the redshift parameter conventionally defined as
In the case where neither the emitter nor the observer is at infinity, the transitivity of Doppler shifts allows us to generalize the result to
, thus For example, the gravitational blueshift of distant starlight due to the Sun's gravity, which the Earth is orbiting at about 30 km/s, would be approximately 1 × 10−8 or the equivalent of a 3 m/s radial Doppler shift.
For an object in a (circular) orbit, the gravitational redshift is of comparable magnitude as the transverse Doppler effect,
and the wave vector of a photon leaving the gravitational field in radial direction the energy equation becomes Using
[17][citation needed] James W. Brault, a graduate student of Robert Dicke at Princeton University, measured the gravitational redshift of the sun using optical methods in 1962.
[19][20] Measuring the solar redshift is complicated by the Doppler shift caused by the motion of the Sun's surface, which is of similar magnitude as the gravitational effect.
[21] In 2018, the star S2 made its closest approach to Sgr A*, the 4-million solar mass supermassive black hole at the centre of the Milky Way, reaching 7650 km/s or about 2.5% of the speed of light while passing the black hole at a distance of just 120 AU, or 1400 Schwarzschild radii.
Independent analyses by the GRAVITY collaboration[22][23][24][25] (led by Reinhard Genzel) and the KECK/UCLA Galactic Center Group[26][27] (led by Andrea Ghez) revealed a combined transverse Doppler and gravitational redshift up to 200 km/s/c, in agreement with general relativity predictions.
[28] In 2024, Padilla et al. have estimated the gravitational redshifts of supermassive black holes (SMBH) in eight thousand quasars and one hundred Seyfert type 1 galaxies from the full width at half maximum (FWHM) of their emission lines, finding log z ≈ −4, compatible with SMBHs of ~ 1 billion solar masses and broadline regions of ~ 1 parsec radius.
The Pound–Rebka experiment of 1959 measured the gravitational redshift in spectral lines using a terrestrial 57Fe gamma source over a vertical height of 22.5 metres.
This rate of the discrepancy is sufficient to substantially impair the function of GPS within hours if not accounted for.
An excellent account of the role played by general relativity in the design of GPS can be found in Ashby 2003.
[33] In 2010, an experiment placed two aluminum-ion quantum clocks close to each other, but with the second elevated 33 cm compared to the first, making the gravitational red shift effect visible in everyday lab scales.
[34][35] In 2020, a group at the University of Tokyo measured the gravitational redshift of two strontium-87 optical lattice clocks.
[36] The measurement took place at Tokyo Skytree where the clocks were separated by approximately 450 m and connected by telecom fibers.
[37][38] In October 2021, a group at JILA led by physicist Jun Ye reported a measurement of gravitational redshift in the submillimeter scale.
The measurement is done on the 87Sr clock transition between the top and the bottom of a millimeter-tall ultracold cloud of 100,000 strontium atoms in an optical lattice.
Einstein's prediction was confirmed by many experiments, starting with Arthur Eddington's 1919 solar eclipse expedition.
To calculate the changes in frequency in a nearly static gravitational field, only the time component of the metric tensor is important, and the lowest order approximation is accurate enough for ordinary stars and planets, which are much bigger than their Schwarzschild radius.
