Gravitational redshift


In physics and general relativity, gravitational redshift is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy corresponds to a decrease in the wave frequency and increase in the wavelength, known more generally as a redshift. The opposite effect, in which photons gain energy when travelling into a gravitational well, is known as a gravitational blueshift. The effect was first described by Einstein in 1907, eight years before his publication of the full theory of relativity. Observing the gravitational redshift in the Solar System is one of the classical tests of general relativity.
Gravitational redshift can be interpreted as a consequence of the equivalence principle or as a consequence of the mass–energy equivalence and conservation of energy, though there are numerous subtleties that complicate a rigorous derivation. A gravitational redshift can also equivalently be interpreted as gravitational time dilation at the source of the radiation: if two oscillators are operating at different gravitational potentials, the oscillator at the higher gravitational potential will tick faster; that is, when observed from the same location, it will have a higher measured frequency than the oscillator at the lower gravitational potential.

Magnitudes

To first approximation, gravitational redshift is proportional to the difference in gravitational potential divided by the speed of light squared,, thus resulting in a very small effect. 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. Navigational signals from GPS satellites orbiting at altitude are perceived blueshifted by approximately 0.5 ppb or 5 × 10−10, corresponding to a increase of less than 1 Hz in the frequency of a 1.5 GHz GPS radio signal. On the surface of the Earth the gravitational potential is proportional to height,, and the corresponding redshift is roughly 10−16 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. The GPS satellite gravitational blueshift velocity equivalent is less than 0.2 m/s, which is negligible compared to the actual Doppler shift resulting from its orbital velocity. In astronomical objects with strong gravitational fields the redshift can be much greater; for example, light from the surface of a white dwarf is gravitationally redshifted on average by around /c.

Prediction by the equivalence principle and general relativity

Uniform gravitational field or acceleration

Einstein's theory of general relativity incorporates the equivalence principle, which can be stated in various different ways. One such statement is that gravitational effects are locally undetectable for a free-falling observer. Therefore, in a laboratory experiment at the surface of the Earth, all gravitational effects should be equivalent to the effects that would have been observed if the laboratory had been accelerating through outer space at g. One consequence is a gravitational Doppler effect. If a light pulse is emitted at the floor of the laboratory, then a free-falling observer says that by the time it reaches the ceiling, the ceiling has accelerated away from it, and therefore when observed by a detector fixed to the ceiling, it will be observed to have been Doppler shifted toward the red end of the spectrum. This shift, which the free-falling observer considers to be a kinematical Doppler shift, is thought of by the laboratory observer as a gravitational redshift. Such an effect was verified in the 1959 Pound–Rebka experiment. In a case such as this, where the gravitational field is uniform, the change in wavelength is given by
where is the change in height. Since this prediction arises directly from the equivalence principle, it does not require any of the mathematical apparatus of general relativity, and its verification does not specifically support general relativity over any other theory that incorporates the equivalence principle.
On Earth's surface, the gravitational redshift is approximately, the equivalent of a Doppler shift for every 1 m of altitude.

Spherically symmetric gravitational field

When the field is not uniform, the simplest and most useful case to consider is that of a spherically symmetric field. By Birkhoff's theorem, such a field is described in general relativity by the Schwarzschild metric,, where is the clock time of an observer at distance R from the center, is the time measured by an observer at infinity, is the Schwarzschild radius, "..." represents terms that vanish if the observer is at rest, is the Newtonian constant of gravitation, the mass of the gravitating body, and the speed of light. The result is that frequencies and wavelengths are shifted according to the ratio
where
  • is the wavelength of the light as measured by the observer at infinity,
  • is the wavelength measured at the source of emission, and
  • is the radius at which the photon is emitted.
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. The redshift formula for the frequency is. When is small, these results are consistent with the equation given above based on the equivalence principle.
The redshift ratio may also be expressed in terms of a escape velocity at, resulting in the corresponding Lorentz factor:
For an object compact enough to have an event horizon, the redshift is not defined for photons emitted inside the Schwarzschild radius, both because signals cannot escape from inside the horizon and because an object such as the emitter cannot be stationary inside the horizon, as was assumed above. Therefore, this formula only applies when is larger than. When the photon is emitted at a distance equal to the Schwarzschild radius, the redshift will be infinitely large, and it will not escape to any finite distance from the Schwarzschild sphere. When the photon is emitted at an infinitely large distance, there is no redshift.

Newtonian limit

In the Newtonian limit, i.e. when is sufficiently large compared to the Schwarzschild radius, the redshift can be approximated as
where is the gravitational acceleration at. For Earth's surface with respect to infinity, z is approximately ; for the Moon it is approximately . The value for the surface of the Sun is about, corresponding to 0.64 km/s.
The z-value can be expressed succinctly in terms of the escape velocity at, since the gravitational potential is equal to half the square of the escape velocity, thus:
where is the escape velocity at.
It can also be related to the circular orbit velocity at, which equals, 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 orbit, the gravitational redshift is of comparable magnitude as the transverse Doppler effect, where, while both are much smaller than the radial Doppler effect, for which.

Prediction of the Newtonian limit using the properties of photons

The formula for the gravitational red shift in the Newtonian limit can also be derived using the properties of a photon:
In a gravitational field a particle of mass and velocity changes it's energy according to:
For a massless photon described by its energy and momentum this equation becomes after dividing by the Planck constant :
Inserting the gravitational field of a spherical body of mass within the distance
and the wave vector of a photon leaving the gravitational field in radial direction
the energy equation becomes
Using an ordinary differential equation which is only dependent on the radial distance is obtained:
For a photon starting at the surface of a spherical body with a Radius with a frequency the analytical solution is:
In a large distance from the body an observer measures the frequency :
Therefore, the red shift is:
In the linear approximation
the Newtonian limit for the gravitational red shift of General Relativity is obtained.

History

The gravitational weakening of light from high-gravity stars was predicted by John Michell in 1783 and Pierre-Simon Laplace in 1796, using Isaac Newton's concept of light corpuscles and who predicted that some stars would have a gravity so strong that light would not be able to escape. The effect of gravity on light was then explored by Johann Georg von Soldner, who calculated the amount of deflection of a light ray by the Sun, arriving at the Newtonian answer which is half the value predicted by general relativity. All of this early work assumed that light could slow down and fall, which is inconsistent with the modern understanding of light waves.
Einstein's 1917 paper on general relativity proposed three tests: the timing of the perihelion of Mercury, the bending of light around the Sun, and the shift in frequency of light emerging from a different gravitational potential, now called the gravitational redshift. Of these, the redshift proved difficult for physicist to understand and to measure convincingly. A confusing mix of complex and subtle issues plague even famous textbook descriptions of the phenomenon.
Once it became accepted that light was an electromagnetic wave, it was clear that the frequency of light should not change from place to place, since waves from a source with a fixed frequency keep the same frequency everywhere. One way around this conclusion would be if time itself were altered if clocks at different points had different rates. This was precisely Einstein's conclusion in 1911. He considered an accelerating box, and noted that according to the special theory of relativity, the clock rate at the "bottom" of the box was slower than the clock rate at the "top". Indeed, in a frame moving with velocity relative to the rest frame, the clocks at a nearby position are ahead by ; so an acceleration makes clocks at the position to be ahead by, that is, tick at a rate
The equivalence principle implies that this change in clock rate is the same whether the acceleration is that of an accelerated frame without gravitational effects, or caused by a gravitational field in a stationary frame. Since acceleration due to gravitational potential is, we get
so – in weak fields – the change in the clock rate is equal to.
The changing rates of clocks allowed Einstein to conclude that light waves change frequency as they move, and the frequency/energy relationship for photons allowed him to see that this was best interpreted as the effect of the gravitational field on the mass–energy of the photon. 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.