Red Shift in

P.K. SRIVASTAVA Department, DAV(PG) College, Kanpur [email protected]

ABSTRACT The phenomenon of of light is taught in undergraduate classes. The redshift of light occurs in three different contexts called Doppler redshift, cosmological redshift and . In each case, the 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 ‘ shift’ more often of due to relative of alludes to the 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 . 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

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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 . 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 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 ) 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 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 . 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 ∆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 . In contrast, Doppler Effect in = [(c – v)/(c + v)] ν0 (4)

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As beautifully shown by R.P. Feynman,3 us to infer that our is expanding in the effect is same if we take source at rest and which all the are moving away from observer moving; the γ factor plays the trick. each other. However in the non-relativistic limit (v<

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 . decoupled itself from 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. and isotropic as seen from any point (within it) The Doppler shift given by Eq.5 is the and hence all the galaxies are receding not just result from special in which away from us (as we do not constitute any case the speed of receding galaxy cannot special center of the universe) but they are all exceed the (vt1); S(t) is the that represents measure of any receding galaxy depends on how of length in space which changes with time as expansion occurs, that is, on the model of universe we use for study of expansion. space expands. For example, between times t1 and t2, if the universe becomes ten times in size, the observed wavelength too increases ten 3. Gravitational Red Shift times, i.e. z=9. This can’t be interpreted simply as a source moving with speed 9 times that of Another, perhaps more exciting example, of redshift light (at t (or t )). Cosmological in light is the shift caused by motion of light in a 2 1 . When a beam of light moves, say redshift provides information only about upwards from the surface of earth against the change in scale-factor. Cosmic redshift z=9 direction of its (earth’s) gravitational field, its means that universe expanded by 900% wavelength increases. This phenomenon is called between emission and reception of light signal; gravitational redshift. the signal traveled during all this time in an The understanding of gravitational redshift expanding space. It is indeed a phenomenon has several interesting ideas associated with it emerging from general relativity, which of at different levels of sophistication. The course is the right thing to do if we are talking simplest situation, as mentioned above, is to about the motion of cosmos. consider a (weak) uniform gravitational field

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(say of earth) in which two observers P and Q more time. That is, the clock Q runs faster as are sitting at rest with respect to each other, P compared to clock P. Different clocks on surface of earth and Q at a vertical height h (otherwise identical), relatively at rest, run at above P. A light source at P emits light waves different rates depending upon where they are of frequency νP. Observer Q finds that located in . The above analysis indicates frequency of these waves is νQ which is less that clock P runs slower since it lies in the than νP, that is, light gets red shifted as seen by region of higher gravity, being closer to source observer Q. A simple text-book explanation of of gravity. The big result is that gravity slows this phenomenon is given by invoking down the flow of time; it is in Equivalence principle4,5 which says that gravitational field. uniform gravity is equivalent to a non-inertial Eq.8 was obtained in context to uniform frame falling freely with acceleration ‘g’ in a gravity. We can extend it to gravity of a gravity free space. Thus we consider another uniform spherical source of mass M. Consider observer R who is initially sitting at rest along an observer Q situated very far from it, in with Q. The moment P emits a wave, R falls gravity free space, defining a space-time freely. For R, it is gravity free space, and hence coordinate system x-t. The coordinate time t is he observes that the wave is moving with the recorded by his clock. Consider constant frequency νP all the way from P to Q. another observer P situated at distance r from However, as R falls down, he notes that Q is the source so that g = (GM/r2); the time moving up, away from the light source and recorded by the clock at P is its proper time τ. therefore R concludes that the light observed Eq.8. suggests that the clock of P runs slower by Q must be Doppler shifted and is given by (being in gravity) such that Eq. 3: Δτ = Δt(1 – gr/c2) ν = ν (1 – v/c) Q P = Δt(1 − GM/rc2) (9) = ν (1 – gh/c2) (8) P Note that Δτ = Δt when r→∞. We define Δt as where v = gt = gh/c is the speed of Q when the universal coordinate time recorded by a wave reaches him (Q). The above analysis, clock situated very far (r→∞) from all the done within the framework of non-relativistic sources, in gravity free space. The (proper) Newtonian gravity, is a way to interpret time recorded by a clock sitting at distance r gravitational redshift in terms of motion of the from a source of gravity (say a ) is Δτ observer (i.e. as Doppler shift). But the subject which is less than Δt. The clock in gravity runs has deeper significance. slower and the time dilation factor is (1 − We have discussed above that a light source GM/rc2). is essentially a clock which ticks with time- In the language which emerged after period τ = 1/ν. Suppose, therefore, that we have Einstein articulated that gravity is space-time two identical clocks out of which one is given curvature, the gravitational redshift is the to P and the second is taken to Q at height h. manifestation of curvature of time. Time flows Now, clock P ticks N times in time tP = τPN. differently at different positions means that Observer Q receives these N waves in time tQ = time is curved. Since space-time interval τQ N. If νQ < νP, we get τQ > τP and hence tQ > squared is negative of proper time squared, we tP. This implies that during the process of find that for any arbitrary observer at rest emission and receiving of N light waves by P (spatial coordinates being fixed) in (weak) and Q respectively, the clock of Q has recorded gravity Physics Education • October − December 2009 295

Δs2 = -Δτ2 = −(1 − GM/rc2)2 Δt2 potential of the . The total 2 2 (dynamic) energy of the photon is defined as ≈ − (1 − 2GM/rc )Δt (10) the sum of hν and the Note that above analysis, done within energy GMm/r of the photon (whatever it Newtonian framework, shows how Newtonian means), where m= hν/c2 denotes effective (i.e. weak) gravity affects the flow of time. gravitational mass of the photon. Hence energy However it turns out that general relativity conservation implies that applied to strong gravity sources also yield hν = hν + GMm/r = hν(1 + GM/rc2) same result (Eq.10). 0 As mentioned above, curvature of time which leads to Eq.9 up to first order. This, as S. (revealed as gravitational red shift), is Weinberg writes,6 is a way to put quantum produced even by such weak sources as earth. theory, energy conservation, and Newtonian It was verified in laboratory by Robert Pound gravity together. In relativistic theory of and Glen Rebka in 1960 in the famous gravitation (general relativity), radiation energy experiment using Mossbauer resonance density is an intrinsic source of gravity. . They observed the frequency shift of gamma rays emitted by an isotope of (embedded in a cool crystal) at a vertical References separation of 22.5m. Variation of clock rate 1. D. Halliday, R. Resnick, F. Walker, with position in earth’s gravity is also taken Fundamentals of Physics, John Wiley and into account in defining global positioning Sons, New York (1993). system used by satellites. Gravity also curves 2. A.P. French, Special Relativity, MIT Press, space but that is outside the subject of present Mass (1975). article. 3. R.P. Feynman, Lectures in Physics, Vol 1, Addison Wesley, Readings, Mass(1969). Before we close this discussion, we must nd 4. P.K. Srivastava, Mechanics (2 Ed), New Age mention that Eq 8 is often described it terms of 4 International, New Delhi (2007). energy conservation applied to photon. As a 5. B. Shutz, Gravity from the Ground up, photon moves up against the gravity, its energy Cambridge University Press, London (2003). hν0 decreases to hν. What happens to its energy 6. S. Weinberg, Gravitation and , John loss? This is explained in a simple way by Wiley and Sons, New York(1972). saying that it is converted into gravitational

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