Red Shift in Light P.K. SRIVASTAVA Physics Department, DAV(PG) College, Kanpur [email protected] ABSTRACT The phenomenon of redshift of light is taught in undergraduate classes. The redshift of light occurs in three different contexts called Doppler redshift, cosmological redshift and gravitational redshift. In each case, the dynamics is distinct and involves different physical concepts. These must be discussed together so that one gets a comprehensive understanding of the subject. In this article, we describe each of these phenomena in a pedagogical manner without any advanced mathematics. The phenomenon of ‘red shift’ more often frequency of sound due to relative motion of alludes to the Doppler Effect in light. Doppler source towards or away from the observer is Effect is popularly known in context of sound known as Doppler Effect. Similar effect is waves. The pitch (or frequency) of the siren of observed in light also. The observed frequency a train (or a car) is higher when it is of light, emitted by a source, changes due to approaching a listener standing on ground and relative motion between the source and the suddenly changes to a lower value as the observer. However, as we shall discuss in this engine of train (or the car) passes by the article, the phenomenon of red shift in light is listener and moves away. This change in very rich; besides Doppler red shift there also Physics Education • October − December 2009 291 occur the phenomena of cosmological and light, which requires no medium for gravitational red shifts. The descriptions of all propagation, is same in both the cases. these phenomena lead to several other Doppler Effect in light essentially requires interesting concepts. a relativistic analysis as described below; after all when light source is in motion, one should better do special relativity. Though relativistic 1. Doppler Redshift Doppler Effect is discussed in the texts dealing 2,3 In case of Doppler Effect in sound waves, if with special relativity it is instructive to medium (say air) is still, the source is at rest, recapture its basic steps. Suppose frames S and and observer is moving away from the source S' are attached to observer and source with constant speed v along line joining the respectively. Let S' be moving away from S source and the observer, then the observed (rest frame) with constant velocity v, along apparent frequency ν by the observer is given common X-X' axis. The source emits by monochromatic, harmonic waves of frequency ν0 in frame S'. The first concept that goes into ν = [(u − v)/u]ν0 (1) the analysis is that we can regard the light source as a clock which ticks regularly at time where, ν0 is the frequency of sound if the source is at rest with respect to the observer interval τ0 = 1/ ν0, emitting a harmonic pulse and u is speed of sound in still medium. On (sine wave) of light at each tick. If space-time other hand, if the observer is at rest and the coordinates of emission of two consecutive sound source is moving away from the pulses in S' frames are (x1', t1') and (x2', t2'), observer, then observed frequency is: then x2' = x1' and t2' = t1' + τ0. For the observer S these events occur at coordinates (x1, t1) and ν = [u/(u − v)] ν (2) 0 (x2, t2). Lorentz transformations between S and Thus, while in both of the above cases, the S' yield relative velocity v between the source and the ∆x = x2 – x1 = γv τ0 observer is same, the observed frequencies differ depending upon whether the source is ∆t = t2– t1 = γ τ0 moving or the observer. This asymmetry, where γ = (1− v2/c2)1/2 is the Lorentz factor. which is apparently against the principle of The observer in frame S finds that two light relativity, arises because sound waves travel in 1,2 pulses reach him separated by time interval τ and with respect to a material medium. such that Asymmetry lies in the fact that in one case, source is at rest with respect to medium while τ = ∆t + ∆x/c = γ(1 + v/c)τ0 (3) in the other case, it moves relative to medium. The time τ contains two terms: ∆t because two The two cases are therefore analyzed pulses are emitted at different times and (∆x/c) differently: when the observer moves away because second pulse has to travel an extra from the source, it receives lesser number of distance ∆x in order to reach the observer. The wave pulses per second than emitted by the observed frequency by observer S at rest source thereby effectively observing less therefore is frequency, while when the source moves away from the observer, undulations in medium get ν = 1/τ = [1/γ(1 + v/c)]ν0 stretched thereby effectively increasing 1/2 wavelength. In contrast, Doppler Effect in = [(c – v)/(c + v)] ν0 (4) 292 Physics Education • October − December 2009 As beautifully shown by R.P. Feynman,3 us to infer that our universe is expanding in the effect is same if we take source at rest and which all the galaxies are moving away from observer moving; the γ factor plays the trick. each other. However in the non-relativistic limit (v<<c, It is reasonable to believe that any typical neglecting terms of (v/c)2 and higher powers) galaxy shines because of the radiation emitted of the relativistic Doppler Effect in light, the by billions of stars in it and that all typical stars observed apparent frequency is given by same anywhere have similar composition as stars in relation as Eq.1(or 2) as expected, with speed our own galaxy. Therefore spectrum of of sound u replaced by that of light c: radiation from any other galaxy is expected to exhibit same pattern of emission lines as seen ∆ ν = ν0 – ν = (v/c) ν0 (5) in the spectra of our nearby stars or galaxies. Corresponding change in wavelength is The most strong lines usually found in these systems are those belonging to ionized H, He, ∆λ = λ – λ0 = (v/c)λ0 (6) Ca, K, etc. We do find the patterns Thus the observed wavelength increases if corresponding to these lines in spectra of any the light source is moving away from the distant galaxy (situated in any direction) but observer. Since the wavelength of red light is exact values of wavelengths of these lines turn larger in the visible band, this phenomenon is out to be larger by a fixed ratio (for a given called ‘red-shift’, i.e. shift towards larger galaxy) compared to the wavelengths of wavelength. respective lines observed in the laboratory. One of the most important examples of That is, for every observed line, from a Doppler redshift is the Doppler broadening of particular distant galaxy, one finds that λ/λ0 = 1 emission lines from excited atomic (or + z, where z is called the redshift parameter. molecular) gas. At temperature T, different According to Doppler analysis, the value of z is molecules of the gas move randomly in all related to the value of the speed with which directions with different speeds. Any emission that galaxy is moving away from us as given line therefore is Doppler shifted because the by Eq.5. This led to the conclusion that our source is moving. The shift depends upon the universe is expanding. speed of a molecule. The net spectral profile However the interpretation of redshift of galaxies as Doppler shift implies that we are turns out to be Gaussian peaked at frequency ν0 and its full width at half maximum is given as using special relativity, that is, a single Lorentz space-time coordinate system (with ourselves 2 1/2 Δν = ν0(2kT ln2/Mc ) at the origin) all the way up to source galaxy. This interpretation is approximately valid for where ν0 is the frequency if molecules (of mass M) were at rest. nearby galaxies. For distant galaxies, one must take into account space-time curvature and use general relativity to explain galactic red shifts. 2. Cosmological Redshift The general relativity model argues that galaxies are not moving away from us ‘in The second exciting example of redshift in space’. Rather galaxies appear to be receding light is seen in the spectra of the radiation from us because the space itself is expanding emitted by distant galaxies. In the beginning of as a consequence of Big Bang origin of space- the last century it was observed by the time. This aspect is popularly visualized in astronomers that the light emitted by the distant terms of inflation of the (two dimensional) galaxies is red-shifted. This discovery has led surface of a balloon, pock marked with dots Physics Education • October − December 2009 293 representing various galaxies. The balloon Another important example of cosmic example illustrates two points: first that redshift is the cosmic background microwave expansion causes any two dots on its surface to radiation (CMBR) observed today with peak recede from each other as the balloon inflates intensity at wavelength of about 2×10-3m. and secondly that the expanding surface need CMBR corresponds to a redshift of z~1000. It not have any center and edge. Similarly, our simply means that universe has expanded 1000 three-dimensional space is expanding and has times since primeval radiation (with peak at no center and edge. It is a consequence of the wavelength~2×10-6m) from hot big bang Cosmological Principle. decoupled itself from matter and this radiation The Cosmological Principle asserts that has been moving along with expanding space, universe on the whole is both homogeneous getting stretched in the process.