Gravitational redshift

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Template:Expert Gravitational redshift of photons is the change in photon's color toward the red end of the spectrum, and sometimes beyond that to invisible light such as radio, microwave, and infrared radiation. As a photon shoots away from massive objects, such as the sun or the earth, it does not slow down. Instead, it loses momentum and decreases in frequency (see Planck's constant). "Thus, a beam of electromagnetic radiation (visible light included) when directed toward an extremely massive object such as a planet, would be percieved from the object's perspective as a more amplified form of electromagnetic radiation, the type of which depends on the mass of the object. An infinitely massive object's gravitational feild would amplify them into gamma rays or perhaps even a more amplified form which has never been detected due to the inability to measure infinities. This effect is the opposite of gravitational redshift and is known as gravitational blueshift." (Andrew T. Troupis, student)

Gravitational redshift is implied by the Equivalence principle. Albert Einstein defined the [Equivalence principle]] years before he completed the full theory of relativity. Gravitational redshift was also predicted by John Michell, however, by using less precise Newtonian arguments, "Transactions of the Royal Society" (1783). Einstein's contribution was to rediscover the effect, examine it in detail, and recognise that it led inevitably to gravitational time dilation.Observing the gravitational redshift in the solar system is one of the classical tests of general relativity.

First experimental verification

Experimental verification of the gravitational redshift requires good clocks since at Earth the effect is small. The first experimental confirmation came as late as in 1959, in the Pound-Rebka experiment (R.V. Pound and G.A. Rebka, Apparent weight of photons, Phys. Rev. Lett. 4 337-341 (1960)) later improved by Pound and Snider. The famous experiment is generally called the Pound-Rebka-Snider experiment:

• They used a very well-defined "clock" in the form of an atomic transition (of isotopes) which results in the emission of gamma rays forming a very narrow line on the electromagnetic spectrum. The narrowness of the line is caused by the so called Mossbauer effect, where in this case the isotope used is Fe57 emitting photons with the energy of about 14.4 keV. The emitter and absorber were placed in a tower of only 22 meter height at the bottom and top respectively. The observed gravitational redshift z, defined as the relative change in wavelength, the ratio
$z = \frac{\Delta\lambda}{\lambda_e}$
with Δλ = λo − λe the difference between the observed λo and emitted λe wavelength. z is proportional to the difference in gravitational potential. With the gravitational acceleration g of the Earth, c the velocity of light and with a height h=22 m, the prediction
$\Delta\lambda/\lambda = \frac{gh}{c^2} = 2.5\times 10^{-15}$
was obtained with a 1% accuracy. Nowadays the accuracy is measured up to 0.02% Note from the formula above that the loss of energy of the photon is just equal to the difference in potential energy gh.

Gravitational redshift in stars

Photons emitted from a stellar surface on a star of mass M and radius R are expected to have a redshift equal to the difference in gravitational potential. With G the gravitational constant, the gravitational potential at the stellar surface is GM / R and zero at infinity, so

$\frac{\Delta\lambda}{\lambda} = \frac{G}{c^2} \cdot \frac{M}{R}$

where c is the speed of light. The coefficient G/c2 = 7.414×10-29cm/g. For the Sun, M = 2.0×1033g and R = 6.955×1010cm, so Δλ/λ = 2.12×10-6. This means each spectral line should be shifted towards the red end of the spectrum by a little over one millionth of its original wavelength. This small difference was measured for the first time on the Sun in 1962.

In addition, observation of much more massive and compact stars such as white dwarfs have shown that Einstein shift does occur and is within the correct order of magnitude. Recently also the gravitational redshift of a neutron star has been measured from spectral lines in the x-ray range. The result gives the quantity M/R, the mass M and radius R of the neutron star. If the mass is obtained by other means (for example from the motion of the neutron star around a campanion star), one can measure the radius of a neutron star in this way.

Black holes have infinite gravitational redshift

The gravitational redshift of a photon is infinity when it recedes from the event horizon of a black hole called the Schwarzschild radius. In fact a black hole can best be defined as a massive compact object surrounded by an area at which the redshift (as observed from a non-zero distance) is infinitely large.

Gravitational blueshift occurs when a photon is going the opposite direction (i.e. towards a gravitating mass).

• In general the gravitational redshift z for a spherical mass M is given
$1+z = \frac{\sqrt{1-\frac{2GM}{c^2 r} }}{\sqrt{1-\frac{2GM}{c^2 R} }}$
G Gravitational Constant. c the velocity of light. R photon's radius from the center of mass at emmision. r the observer's (the photon absorber's) radius from the center of mass. For significantly large r, the numerator is close to 1. For R approaching 2GM / c2 the redshift $z\rightarrow\infty$. The quantity 2GM / c2 is called the Schwarzschild radius.

When a star is imploding to form a black hole, one never observes the star to pass the Schwarzschild radius. Following this and other evidence, one may even argue that the imploding star never passes the Schwarzschild radius during its finite lifetime (see Hawking Radiation and Shapiro effect). As the star approaches this radius it will appear increasingly redder and dimmer in a very short time. In the past such a star was called a frozen star instead of a black hole. However, in a very short time the collapsing star emits its "last photon" and the object thereafter is black indeed. For most people, the terminology black hole is preferred above frozen star.

Gravitational redshift, the "applied side of general relativity"

Corrections for gravitational redshift are nowadays common practice in many situations. With present-day accuracies, clocks in orbit around the Earth must be corrected for this effect. This is in particular the case with satellite-based navigational systems such as the Global Positioning System (GPS). To get accuracies on the order of 10 m, light travel times with an accuracy of order 30 ns (nanoseconds) have to be measured. Special-relativistic time dilatation (caused by the velocity) and gravitational redshift corrections in these satellites are of order 30000 ns per day.

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