Gravitational redshift

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https://en.wikipedia.org/wiki/Gravitational_redshift 

 

The gravitational redshift of a light wave as it moves upwards against a gravitational field (produced by the yellow star below). The effect is greatly exaggerated in this diagram.

In Einstein's general theory of relativity, the gravitational redshift is the phenomenon that clocks deeper in a gravitational well tick slower when observed from outside the well. More specifically the term refers to the shift of wavelength of a photon to longer wavelength (the red side in an optical spectrum) when observed from a point at a higher gravitational potential. In the latter case the 'clock' is the frequency of the photon and a lower frequency is the same as a longer ("redder") wavelength.

The gravitational redshift is a simple consequence of Einstein's equivalence principle (that gravity and acceleration are equivalent) and was found by Einstein eight years before the full theory of relativity.

Observing the gravitational redshift in the solar system is one of the classical tests of general relativity. Gravitational redshifts are an important effect in satellite-based navigation systems such as GPS. If the effects of general relativity were not taken into account, such systems would not work at all.

Prediction by the equivalence principle and general relativity

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

${\displaystyle {\frac {\Delta \lambda }{\lambda }}\approx {\frac {g\Delta y}{c^{2}}},}$

where ${\displaystyle \Delta y}$ 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.

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, ${\displaystyle d\tau ^{2}=\left(1-r_{\text{S}}/R\right)dt^{2}+\ldots }$, where ${\displaystyle d\tau }$ is the clock time of an observer at distance R from the center, ${\displaystyle dt}$ is the time measured by an observer at infinity, ${\displaystyle r_{\text{S}}}$ is the Schwarzschild radius ${\displaystyle 2GM/c^{2}}$, "..." represents terms that vanish if the observer is at rest, ${\displaystyle G}$ is Newton's gravitational constant, ${\displaystyle M}$ the mass of the gravitating body, and ${\displaystyle c}$ the speed of light. The result is that frequencies and wavelengths are shifted according to the ratio

${\displaystyle {\frac {\lambda _{\infty }}{\lambda _{\text{e}}}}=\left(1-{\frac {r_{\text{S}}}{R_{\text{e}}}}\right)^{-{\frac {1}{2}}},}$

where

• ${\displaystyle \lambda _{\infty }\,}$ is the wavelength of the light as measured by the observer at infinity,
• ${\displaystyle \lambda _{\text{e}}\,}$ is the wavelength measured at the source of emission, and
• ${\displaystyle R_{\text{e}}}$ radius at which the photon is emitted.

This can be related to the redshift parameter conventionally defined as ${\displaystyle z=\lambda _{\infty }/\lambda _{\text{e}}-1}$. 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 ${\displaystyle \lambda _{1}/\lambda _{2}=\left[\left(1-r_{\text{S}}/R_{1}\right)/\left(1-r_{\text{S}}/R_{2}\right)\right]^{1/2}}$. The redshift formula for the frequency ${\displaystyle \nu =c/\lambda }$ is ${\displaystyle \nu _{o}/\nu _{\text{e}}=\lambda _{\text{e}}/\lambda _{o}}$. When ${\displaystyle R_{1}-R_{2}}$ is small, these results are consistent with the equation given above based on the equivalence principle.

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 ${\displaystyle R_{\text{e}}}$ is larger than ${\displaystyle r_{\text{S}}}$. 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.

In the Newtonian limit, i.e. when ${\displaystyle R_{\text{e}}}$ is sufficiently large compared to the Schwarzschild radius ${\displaystyle r_{\text{S}}}$, the redshift can be approximated as

${\displaystyle z\approx {\frac {1}{2}}{\frac {r_{\text{S}}}{R_{\text{e}}}}={\frac {GM}{c^{2}R_{\text{e}}}}}$

Experimental verification

Initial observations of gravitational redshift of white dwarf stars

A number of experimenters initially claimed to have identified the effect using astronomical measurements, and the effect was considered to have been finally identified in the spectral lines of the star Sirius B by W.S. Adams in 1925. However, measurements by Adams have been criticized as being too low and these observations are now considered to be measurements of spectra that are unusable because of scattered light from the primary, Sirius A. The first accurate measurement of the gravitational redshift of a white dwarf was done by Popper in 1954, measuring a 21 km/s gravitational redshift of 40 Eridani B.

The redshift of Sirius B was finally measured by Greenstein et al. in 1971, obtaining the value for the gravitational redshift of 89±19 km/s, with more accurate measurements by the Hubble Space Telescope, showing 80.4±4.8 km/s.

Terrestrial tests

The effect is now considered to have been definitively verified by the experiments of Pound, Rebka and Snider between 1959 and 1965. 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 paper was the first determination of the gravitational redshift which used measurements of the change in wavelength of gamma-ray photons generated with the Mössbauer effect, which generates radiation with a very narrow line width. The accuracy of the gamma-ray measurements was typically 1%.

An improved experiment was done by Pound and Snider in 1965, with an accuracy better than the 1% level.

A very accurate gravitational redshift experiment was performed in 1976, where a hydrogen maser clock on a rocket was launched to a height of 10,000 km, and its rate compared with an identical clock on the ground. It tested the gravitational redshift to 0.007%.

Later tests can be done with the Global Positioning System (GPS), which must account for the gravitational redshift in its timing system, and physicists have analyzed timing data from the GPS to confirm other tests. When the first satellite was launched, it showed the predicted shift of 38 microseconds per day. 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.

Later astronomical measurements

James W. Brault, a graduate student of Robert Dicke at Princeton University, measured the gravitational redshift of the sun using optical methods in 1962.

In 2011 the group of Radek Wojtak of the Niels Bohr Institute at the University of Copenhagen collected data from 8000 galaxy clusters and found that the light coming from the cluster centers tended to be red-shifted compared to the cluster edges, confirming the energy loss due to gravity.

Early historical development of the theory

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 (see: emission theory) 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 (1801), 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.

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 (the side away from the direction of acceleration) was slower than the clock rate at the "top" (the side toward the direction of acceleration). Nowadays, this can be easily shown in accelerated coordinates. The metric tensor in units where the speed of light is one is:

${\displaystyle ds^{2}=-r^{2}dt^{2}+dr^{2}\,}$

and for an observer at a constant value of r, the rate at which a clock ticks, R(r), is the square root of the time coefficient, R(r)=r. The acceleration at position r is equal to the curvature of the hyperbola at fixed r, and like the curvature of the nested circles in polar coordinates, it is equal to 1/r.

So at a fixed value of g, the fractional rate of change of the clock-rate, the percentage change in the ticking at the top of an accelerating box vs at the bottom, is:

${\displaystyle {R(r+dr)-R(r) \over R}={dr \over r}=gdr\,}$

The rate is faster at larger values of R, away from the apparent direction of acceleration. The rate is zero at r=0, which is the location of the acceleration horizon.

Using the equivalence principle, Einstein concluded that the same thing holds in any gravitational field, that the rate of clocks R at different heights was altered according to the gravitational field g. When g is slowly varying, it gives the fractional rate of change of the ticking rate. If the ticking rate is everywhere almost this same, the fractional rate of change is the same as the absolute rate of change, so that:

${\displaystyle {dR \over dx}=g=-{dV \over dx}\,}$

Since the rate of clocks and the gravitational potential have the same derivative, they are the same up to a constant. The constant is chosen to make the clock rate at infinity equal to 1. Since the gravitational potential is zero at infinity:

${\displaystyle R(x)=1-{V(x) \over c^{2}}\,}$

where the speed of light has been restored to make the gravitational potential dimensionless.

The coefficient of the ${\displaystyle dt^{2}}$ in the metric tensor is the square of the clock rate, which for small values of the potential is given by keeping only the linear term:

${\displaystyle R^{2}=1-2V\,}$

and the full metric tensor is:

${\displaystyle ds^{2}=-\left(1-{2V(r) \over c^{2}}\right)c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}}$

where again the C's have been restored. This expression is correct in the full theory of general relativity, to lowest order in the gravitational field, and ignoring the variation of the space-space and space-time components of the metric tensor, which only affect fast moving objects.

Using this approximation, Einstein reproduced the incorrect Newtonian value for the deflection of light in 1909. But since a light beam is a fast moving object, the space-space components contribute too. After constructing the full theory of general relativity in 1916, Einstein solved for the space-space components in a post-Newtonian approximation and calculated the correct amount of light deflection – double the Newtonian value. Einstein's prediction was confirmed by many experiments, starting with Arthur Eddington's 1919 solar eclipse expedition.

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.

Gravitational time dilation

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Gravitational_time_dilation

Gravitational time dilation is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential (the closer the clock is to the source of gravitation), the slower time passes, speeding up as the gravitational potential increases (the clock getting away from the source of gravitation). Albert Einstein originally predicted this effect in his theory of relativity and it has since been confirmed by tests of general relativity.

This has been demonstrated by noting that atomic clocks at differing altitudes (and thus different gravitational potential) will eventually show different times. The effects detected in such Earth-bound experiments are extremely small, with differences being measured in nanoseconds. Relative to Earth's age in billions of years, Earth's core is effectively 2.5 years younger than its surface. Demonstrating larger effects would require greater distances from the Earth or a larger gravitational source.

Gravitational time dilation was first described by Albert Einstein in 1907 as a consequence of special relativity in accelerated frames of reference. In general relativity, it is considered to be a difference in the passage of proper time at different positions as described by a metric tensor of space-time. The existence of gravitational time dilation was first confirmed directly by the Pound–Rebka experiment in 1959, and later refined by Gravity Probe A and other experiments.

Definition

Clocks that are far from massive bodies (or at higher gravitational potentials) run more quickly, and clocks close to massive bodies (or at lower gravitational potentials) run more slowly. For example, considered over the total time-span of Earth (4.6 billion years), a clock set in a geostationary position at an altitude of 9,000 meters above sea level, such as perhaps at the top of Mount Everest (prominence 8,848 m), would be about 39 hours ahead of a clock set at sea level. This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects.

According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength.

Consider a family of observers along a straight "vertical" line, each of whom experiences a distinct constant g-force directed along this line (e.g., a long accelerating spacecraft, a skyscraper, a shaft on a planet). Let ${\displaystyle g(h)}$ be the dependence of g-force on "height", a coordinate along the aforementioned line. The equation with respect to a base observer at ${\displaystyle h=0}$ is 

${\displaystyle T_{d}(h)=\exp \left[{\frac {1}{c^{2}}}\int _{0}^{h}g(h')dh'\right]}$

where ${\displaystyle T_{d}(h)}$ is the total time dilation at a distant position ${\displaystyle h}$, ${\displaystyle g(h)}$ is the dependence of g-force on "height" ${\displaystyle h}$, ${\displaystyle c}$ is the speed of light, and ${\displaystyle \exp }$ denotes exponentiation by e.

For simplicity, in a Rindler's family of observers in a flat space-time, the dependence would be

${\displaystyle g(h)=c^{2}/(H+h)}$

with constant ${\displaystyle H}$, which yields

${\displaystyle T_{d}(h)=e^{\ln(H+h)-\ln H}={\tfrac {H+h}{H}}}$.

On the other hand, when ${\displaystyle g}$ is nearly constant and ${\displaystyle gh}$ is much smaller than ${\displaystyle c^{2}}$, the linear "weak field" approximation ${\displaystyle T_{d}=1+gh/c^{2}}$ can also be used.

Outside a non-rotating sphere

A common equation used to determine gravitational time dilation is derived from the Schwarzschild metric, which describes space-time in the vicinity of a non-rotating massive spherically symmetric object. The equation is

${\displaystyle t_{0}=t_{f}{\sqrt {1-{\frac {2GM}{rc^{2}}}}}=t_{f}{\sqrt {1-{\frac {r_{s}}{r}}}}=t_{f}{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}=t_{f}{\sqrt {1-\beta _{e}^{2}}}}$

where

• ${\displaystyle t_{0}}$ is the proper time between two events for an observer close to the massive sphere, i.e. deep within the gravitational field
• ${\displaystyle t_{f}}$ is the coordinate time between the events for an observer at an arbitrarily large distance from the massive object (this assumes the far-away observer is using Schwarzschild coordinates, a coordinate system where a clock at infinite distance from the massive sphere would tick at one second per second of coordinate time, while closer clocks would tick at less than that rate),
• ${\displaystyle G}$ is the gravitational constant,
• ${\displaystyle M}$ is the mass of the object creating the gravitational field,
• ${\displaystyle r}$ is the radial coordinate of the observer within the gravitational field (this coordinate is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate; the equation in this form has real solutions for ${\displaystyle r>r_{s}}$),
• ${\displaystyle c}$ is the speed of light,
• ${\displaystyle r_{s}=2GM/c^{2}}$ is the Schwarzschild radius of ${\displaystyle M}$,
• ${\displaystyle v_{e}={\sqrt {\frac {2GM}{r}}}}$ is the escape velocity, and
• ${\displaystyle \beta _{e}=v_{e}/c}$ is the escape velocity, expressed as a fraction of the speed of light c.

To illustrate then, without accounting for the effects of rotation, proximity to Earth's gravitational well will cause a clock on the planet's surface to accumulate around 0.0219 fewer seconds over a period of one year than would a distant observer's clock. In comparison, a clock on the surface of the sun will accumulate around 66.4 fewer seconds in one year.

Circular orbits

In the Schwarzschild metric, free-falling objects can be in circular orbits if the orbital radius is larger than ${\displaystyle {\tfrac {3}{2}}r_{s}}$ (the radius of the photon sphere). The formula for a clock at rest is given above; the formula below gives the general relativistic time dilation for a clock in a circular orbit:

${\displaystyle t_{0}=t_{f}{\sqrt {1-{\frac {3}{2}}\!\cdot \!{\frac {r_{s}}{r}}}}\,.}$

Both dilations are shown in the figure below.

Important features of gravitational time dilation

• According to the general theory of relativity, gravitational time dilation is copresent with the existence of an accelerated reference frame. Additionally, all physical phenomena in similar circumstances undergo time dilation equally according to the equivalence principle used in the general theory of relativity.
• The speed of light in a locale is always equal to c according to the observer who is there. That is, every infinitesimal region of space time may be assigned its own proper time and the speed of light according to the proper time at that region is always c. This is the case whether or not a given region is occupied by an observer. A time delay can be measured for photons which are emitted from Earth, bend near the Sun, travel to Venus, and then return to Earth along a similar path. There is no violation of the constancy of the speed of light here, as any observer observing the speed of photons in their region will find the speed of those photons to be c, while the speed at which we observe light travel finite distances in the vicinity of the Sun will differ from c.
• If an observer is able to track the light in a remote, distant locale which intercepts a remote, time dilated observer nearer to a more massive body, that first observer tracks that both the remote light and that remote time dilated observer have a slower time clock than other light which is coming to the first observer at c, like all other light the first observer really can observe (at their own location). If the other, remote light eventually intercepts the first observer, it too will be measured at c by the first observer.
• Gravitational time dilation ${\displaystyle T}$ in a gravitational well is equal to the velocity time dilation for a speed that is needed to escape that gravitational well (given that the metric is of the form ${\displaystyle g=(dt/T(x))^{2}-g_{space}}$, i. e. it is time invariant and there are no "movement" terms ${\displaystyle dxdt}$). To show that, one can apply Noether's theorem to a body that freely falls into the well from infinity. Then the time invariance of the metric implies conservation of the quantity ${\displaystyle g(v,dt)=v^{0}/T^{2}}$, where ${\displaystyle v^{0}}$ is the time component of the 4-velocity ${\displaystyle v}$ of the body. At the infinity ${\displaystyle g(v,dt)=1}$, so ${\displaystyle v^{0}=T^{2}}$, or, in coordinates adjusted to the local time dilation, ${\displaystyle v_{loc}^{0}=T}$; that is, time dilation due to acquired velocity (as measured at the falling body's position) equals to the gravitational time dilation in the well the body fell into. Applying this argument more generally one gets that (under the same assumptions on the metric) the relative gravitational time dilation between two points equals to the time dilation due to velocity needed to climb from the lower point to the higher.

Experimental confirmation

Satellite clocks are slowed by their orbital speed, but accelerated by their distance out of Earth's gravitational well.

Gravitational time dilation has been experimentally measured using atomic clocks on airplanes. The clocks aboard the airplanes were slightly faster than clocks on the ground. The effect is significant enough that the Global Positioning System's artificial satellites need to have their clocks corrected.

Additionally, time dilations due to height differences of less than one metre have been experimentally verified in the laboratory.

Gravitational time dilation has also been confirmed by the Pound–Rebka experiment, observations of the spectra of the white dwarf Sirius B, and experiments with time signals sent to and from Viking 1 Mars lander.