The kinematic origin of the cosmological redshift

The kinematic origin of the cosmological redshiftEmory F. Bunn and David W. HoggCitation: Am. J. Phys. 77, 688 (2009); doi: 10.1119/1.3129103View online: http://dx.doi.org/10.1119/1.3129103View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v77/i8Published by the American Association of Physics TeachersRelated ArticlesThe Standard Model appearing in Minute Physics video vignettes on YouTubeyoutube.com/watch?v=HVO0HgMi6LcPhys. Teach. 50, 253 (2012)Embeddings and time evolution of the Schwarzschild wormholeAm. J. Phys. 80, 203 (2012)Schwarzschild and de Sitter solutions from the argument by Lenz and SommerfeldAm. J. Phys. 79, 662 (2011)Visualizing circular motion around a Schwarzschild black holeAm. J. Phys. 79, 63 (2011)When action is not least for orbits in general relativityAm. J. Phys. 79, 43 (2011)Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/Journal Information: http://ajp.aapt.org/about/about_the_journalTop downloads: http://ajp.aapt.org/most_downloadedInformation for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.htmlDownloaded 10 Aug 2012 to 138.100.250.132. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission

The kinematic origin of the cosmological redshiftEmory F. Bunn aDepartment of Physics, University of Richmond, Richmond, Virginia 23173David W. HoggCenter for Cosmology and Particle Physics and Department of Physics, New York University,4 Washington Place, New York, New York 10003Received 7 August 2008; accepted 13 April 2009A common belief about big-bang cosmology is that the cosmological redshift cannot be properlyviewed as a Doppler shift that is, as evidence for a recession velocity but must be viewed in termsof the stretching of space. We argue that, contrary to this view, the most natural interpretation of theredshift is as a Doppler shift, or rather as the accumulation of many infinitesimal Doppler shifts. Thestretching-of-space interpretation obscures a central idea of relativity, namely that it is always validto choose a coordinate system that is locally Minkowskian. We show that an observed frequencyshift in any spacetime can be interpreted either as a kinematic Doppler shift or a gravitational shiftby imagining a suitable family of observers along the photon’s path. In the context of the expandinguniverse, the kinematic interpretation corresponds to a family of comoving observers and hence ismore natural. © 2009 American Association of Physics Teachers.DOI: 10.1119/1.3129103I. INTRODUCTIONMany descriptions of big-bang cosmology declare that theobserved redshift of distant galaxies is not a Doppler shiftbut is due to the “stretching of space.” The purpose of thispaper is to examine the meaning of such statements and toassess their validity. We wish to make clear at the outset thatwe are not suggesting any doubt about either the observationsor the general-relativistic equations that successfullyexplain them. Rather, our focus is on the interpretation: giventhat a photon does not arrive at the observer convenientlylabeled “Doppler shift,” “gravitational shift,” or “stretchingof space,” when can or should we apply these labels?Arguably, an enlightened cosmologist never asks thisquestion. In the curved spacetime of general relativity, thereis no unique way to compare vectors at widely separatedspacetime points, and hence the notion of the relative velocityof a distant galaxy is almost meaningless. Indeed, theinability to compare vectors at different points is the definitionof a curved spacetime. 1–4 In practice, however, the enlightenedview is far from universal. The view presented bymany cosmologists and astrophysicists, particularly whentalking to nonspecialists, is that distant galaxies are “really”at rest and that the observed redshift is a consequence ofsome sort of “stretching of space,” which is distinct from theusual kinematic Doppler shift. In these descriptions, statementsthat are artifacts of a particular coordinate system arepresented as if they were statements about the universe, resultingin misunderstandings about the nature of spacetime inrelativity.In this paper, we will show that the redshifts of distantobjects in the expanding universe may be viewed as kinematicshifts due to relative velocities, and we will argue thatif we are forced to interpret the redshift, this interpretation ismore natural than any other.We begin with examples of the description of the cosmologicalredshift in the first three introductory astronomy textbookschosen at random from the bookshelf of one of theauthors.• The cosmological redshift “is not the same as a Dopplershift. Doppler shifts are caused by an object’s motionthrough space, whereas a cosmological redshift is causedby the expansion of space.” 5 Emphasis in original.• “A more accurate view than the Doppler effect of theredshifts of galaxies is that the waves are stretched by thestretching of space they travel through … If space isstretching during all the time the light is traveling, the lightwaves will be stretched as well.” 6• “Astronomers often express redshifts as if they were radialvelocities, but the redshifts of the galaxies are not Dopplershifts … Einstein’s relativistic Doppler formula applies tomotion through space, so it does not apply to the recessionof the galaxies.” 7More advanced textbooks often avoid this language. Forinstance, the books by Peacock 8 and Linder 9 give particularlycareful and clear descriptions of the nature of the cosmologicalredshift. However, statements similar to those wehave cited can be found even in some advanced textbooks.For example, a leading advanced undergraduate level textstates that Doppler shifts “are produced by peculiar and notby recession velocities.” 10 In this paper, we argue, as othershave before us, 11–14 that statements such as these are misleadingand foster misunderstandings about the nature ofspace and time.In general relativity, the “stretching of space” explanationof the redshift is quite problematic. Light is governed byMaxwell’s equations or their general relativistic generalization,which contain no “stretching of space term” and noinformation on the current size of the universe. On the contrary,one of the most important ideas of general relativity isthat spacetime is always locally indistinguishable from thenonstretching spacetime of special relativity, which meansthat a photon doesn’t know about the changing scale factorof the universe. 15The emphasis in many textbooks on the stretching-ofspacetimeinterpretation of the cosmological redshift causesreaders to take too seriously the stretching-rubber-sheet analogyfor the expanding universe. For example, it is sometimesstated as if it were obvious that “it follows that all wave-688 Am. J. Phys. 77 8, August 2009 http://aapt.org/ajp © 2009 American Association of Physics Teachers 688Downloaded 10 Aug 2012 to 138.100.250.132. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission

lengths of the light ray are doubled” if the scale factordoubles. 10 Although this statement is correct, it is not obvious.After all, solutions to the Schrödinger equation, such asthe electron orbitals in the hydrogen atom, don’t stretch asthe universe expands, so why do solutions to Maxwell’sequations?A student presented with the stretching-of-space descriptionof the redshift cannot be faulted for concluding, incorrectly,that hydrogen atoms, the Solar System, and the MilkyWay Galaxy must all constantly “resist the temptation” toexpand along with the universe. One way to see that thisbelief is in error is to consider the problem sometimes knownas the “tethered galaxy problem,” 16,17 in which a galaxy istethered to the Milky Way, forcing the distance between thetwo to remain constant. When the tether is cut, does thegalaxy join up with the Hubble flow and start to recede dueto the expansion of the universe? The intuition that says thatobjects suffer from a temptation to be swept up in the expansionof the universe will lead to an affirmative answer, butthe truth is the reverse: unless there is a large cosmologicalconstant and the galaxy’s distance is comparable to theHubble length, the galaxy falls toward us. 13,14 Similarly, it iscommonly believed that the Solar System has a very slighttendency to expand due to the Hubble expansion althoughthis tendency is generally thought to be negligible in practice.Again, explicit calculation shows this belief not to becorrect. 18,19 The tendency to expand due to the stretching ofspace is nonexistent, not merely negligible.The expanding rubber sheet is quite similar to the ether ofprerelativity physics in that although it is intuitively appealing,it makes no correct testable predictions and some incorrectones, such as the examples we have given. It thereforehas no rightful place in the theory. Some authors 20,21 haveargued that considerations such as these do not refute thenotion that space is really expanding. We agree with the calculationsin these papers but differ regarding the most usefullanguage to use to describe the relevant phenomena.In one set of circumstances, the proper interpretation ofthe redshift seems clear. When the curvature of spacetime issmall over the distance and time scales traveled by a photon,it is natural to interpret the observed frequency shift as aDoppler shift. This interpretation is the reason that a policeofficer can give you a speeding ticket based on the readingon a radar gun. As far as we know, no one has successfullyargued in traffic court that there is an ambiguity in interpretingthe observed frequency shift as a Doppler shift. 22 In theexpanding universe, spacetime curvature is small over regionsencompassing nearby objects, specifically those withz=/1. There should be no hesitation about calling theobserved redshifts Doppler shifts in this case, just as there isnone in traffic court. Surprisingly, however, many peopleseem to believe that the “stretching of space” interpretationof the redshift is the only valid one, even in this limit. Wewill examine the interpretation of redshifts of nearby objectsmore carefully in Sec. II.Aside from low-redshift sources, there is another case inwhich spacetime curvature can be neglected in consideringcosmological redshifts, namely, low-density cosmologicalmodels. An expanding universe with density =0 oftenknown as the Milne model 23 is merely the flat Minkowskispacetime of special relativity expressed in nonstandardcoordinates. 24 In an =0 universe, there are no gravitationaleffects at all, so any observed redshift, even of a very distantgalaxy, must be a Doppler shift. Furthermore, for low butnonzero density 1, the length scale associated withspacetime curvature is much longer than the horizon distance.Hence, spacetime curvature effects that is, gravitationaleffects are weak throughout the observable volume,and the special-relativistic Doppler shift interpretation remainsvalid even for galaxies with arbitrarily high redshifts.The more interesting cases are when z and are notsmall, that is, when the source is sufficiently distant thatgravitational effects are important over the photon’s trajectory.The consensus is that the Doppler shift language mustbe eschewed in this setting. In Sec. III, we review a standardargument 8,13 that even in this case the redshift can be interpretedas the accumulation of infinitesimal Doppler shiftsalong the line of sight, and we further argue that there is anatural way to interpret the redshift as a single noninfinitesimalDoppler shift.A common objection to this claim is that the coordinatevelocity is not related to the redshift in accordance with thespecial-relativistic Doppler formula. 25,26 However, the velocityreferred to in this claim is a mere artifact of a particularchoice of time coordinate. Specifically, it is the rate ofchange of the distance to the object with respect to the cosmictime coordinate, as measured at the present cosmic time.This coordinate velocity is an unnatural quantity to discuss,because it depends on data outside of the observer’s lightcone. The more natural velocity is the velocity of the objectat the time it crossed our past light cone, relative to us today.This velocity is also a coordinate-dependent concept, but aswe will show in Sec. III, the most natural way to specify itoperationally 27,28 leads to a result that is consistent with thespecial-relativistic Doppler formula.In Sec. IV, we widen our focus to consider frequencyshifts in arbitrary curved spacetimes. In any curved spacetimethe observed frequency shift in a photon can be interpretedas either a kinematic effect a Doppler shift or as agravitational shift. The two interpretations arise from differentchoices of coordinates or, equivalently, from imaginingdifferent families of observers along the photon’s path. Wewill describe this construction explicitly and show that thecomoving observers who are usually used to describe phenomenain the expanding universe are the ones that correspondto the Doppler shift interpretation.II. REDSHIFTS OF NEARBY GALAXIES AREDOPPLER SHIFTSWe begin by returning to the parable of the speedingticket, mentioned in Sec. I.A driver is pulled over for speeding. The policeofficer says to the driver, “According to the Dopplershift of the radar signal I bounced off your car,you were traveling faster than the speed limit.”The driver replies, “In certain coordinate systems,the distance between us remained constant duringthe time the radar signal was propagating. In sucha coordinate system, our relative velocity is zero,and the observed wavelength shift was not a Dopplershift. So you can’t give me a ticket.”If you believe that the driver has a legitimate argument,689 Am. J. Phys., Vol. 77, No. 8, August 2009Emory F. Bunn and David W. Hogg 689Downloaded 10 Aug 2012 to 138.100.250.132. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission

1.001.00.80.950.6a0.4t0.900.850.800.00 0.05 0.10 0.15 0.20(a)r0.000.20.00.0 0.2 0.4 0.6 0.8 1.0tFig. 2. The scale factor of a hypothetical “loitering” universe as a functionof time measured in units of the present time. At the time indicated by thedot, a galaxy emits radiation, which is observed at the present time. At thetimes of both emission and observation, the expansion is very slow, yet thegalaxy’s observed redshift is large.T0.050.100.150.200.00 0.05 0.10 0.15 0.20(b)RFig. 1. Spacetime in different coordinate systems. a Plot of an expandingRobertson–Walker universe in the usual comoving coordinates r,t. Coordinatesare expressed in units of the Hubble length and time. The observer islocated at r=0. The dotted curves show the world lines of a particulargalaxy and of a light signal from that galaxy reaching the observer at thepresent time t=1. The dashed and solid curves are contours of constantRiemann normal coordinates R,T; that is, the coordinates that approximateflat spacetime as well as possible near the observer. b The roles of the twocoordinate systems are reversed: the coordinate axes are the Riemann normalcoordinates R,T, and the dashed and solid curves are contours ofconstant comoving coordinates r,t. As time passes, the galaxy moves tolarger values of R; that is, it is not at rest in this coordinate system.Let us begin by reviewing a standard derivation 8,13 of thecosmological redshift. Consider a photon that travels from agalaxy to a distant observer, both of whom are at rest incomoving coordinates. Imagine a family of comoving observersalong the photon’s path, each of whom measures thephoton’s frequency as it goes by. We assume that each observeris close enough to his neighbor so that we can accommodatethem both in one inertial reference frame and usespecial relativity to calculate the change in frequency fromone observer to the next. If adjacent observers are separatedby the small distance r, then their relative speed in thisframe is v=Hr, where H is the Hubble parameter. Thisspeed is much less than c, so the frequency shift is given bythe nonrelativistic Doppler formula/ =−v/c =−Hr/c =−Ht.4We know that H=ȧ/a, where a is the scale factor. We concludethat the frequency change is given by /=−a/a;that is, the frequency decreases in inverse proportion to thescale factor. The overall redshift is therefore given by1+z t et o = at oat e ,5where t e and t o refer to the times of emission and observation,respectively.In this derivation, we interpret the redshift as the accumulatedeffect of many small Doppler shifts along the photon’spath. We now address the question of whether it makes senseto interpret the redshift as one big Doppler shift rather thanthe sum of many small ones.Figure 2 shows a common argument against such an interpretation.Imagine a universe whose expansion rate varieswith time as shown in Fig. 2. A galaxy emits radiation at thetime t e indicated by the dot, and the radiation reaches anobserver at the present time t 0 . The observed redshift is z=at 0 /at e −1=1.5, which by the special-relativistic Dopplerformula would correspond to a speed of 0.72c. At thetimes of emission and absorption of the radiation, the expansionrate is very slow, and the speed ȧr of the galaxy istherefore much less than this value. We can construct modelsin which both ȧt e and ȧt 0 are arbitrarily small withoutchanging the ratio at 0 /at e and hence without changingthe redshift.Upon closer examination, this argument is unconvincing,because the calculated velocities are not the correct velocities.We should not calculate velocities at a fixed instant ofcosmic time either t=t e or t=t 0 . Instead, we should calculatethe velocity of the galaxy at the time of light emissionrelative to the observer at the present time. After all, if adistant galaxy’s redshift is measured today, we wouldn’t expectthe result to depend on what the galaxy is doing todaynor on what the observer was doing long before the age ofthe dinosaurs.In fact, when astrophysicists talk about what a distant objectis doing “now,” they often do not mean at the presentvalue of the cosmic time but rather at the time the objectcrossed our past light cone. For instance, when astronomersmeasure the orbital speeds of planets orbiting other stars, themeasured velocities are always of this sort. There is an excellentreason for this convention: we never have informationabout what a distant object is doing or if the object evenexists at the present cosmic time. Any statement in which“now” is used to refer to the present cosmic time at thelocation of a distant object is not about anything observable,because it refers to events far outside our light cone.691 Am. J. Phys., Vol. 77, No. 8, August 2009Emory F. Bunn and David W. Hogg 691Downloaded 10 Aug 2012 to 138.100.250.132. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission

t1.00.80.60.40.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4rFig. 3. Parallel transport of the velocity four-vector. The solid and dashedlines show the world line of a galaxy and the path of radiation from thegalaxy to the observer in comoving coordinates. The short solid lines showthe galaxy’s four-velocity being parallel transported to the observer alongthe path of the light.In summary, if we wish to discuss the redshift of a distantgalaxy as a Doppler shift, we need to be willing to talk aboutv rel , the velocity of the galaxy then relative to us now. Talkingabout v rel is precisely the sort of thing that the enlightenedcosmologist described in Sec. I refuses to do, because,in a curved spacetime, there is not a unique way to define therelative velocity of objects at widely separated spacetimeevents. Determining the velocity of one object relative toanother involves comparing the two objects’ velocity fourvectors.To be specific, the dot product of these vectors is rel 1−v 2 rel /c 2 −1/2 . To do that, we have to transport one ofthe vectors to the location of the other. In a curved spacetime,the result of such a “parallel transport” depends on thepath along which the vector is transported.Suppose that we wish to defy the purist and talk about v rel .We have to parallel transport the galaxy’s velocity fourvectorto the observer’s location. As others have noted, 27,28the only natural path to choose for this parallel transport isthe path followed by the light that is, the null geodesic joiningthe emission and observation events. If we follow thisprocedure to determine the relative velocity, we find that thevelocity obeys the rulec + v rel= at oc − v rel at e .As noted in Eq. 5, the ratio of scale factors is equal to 1+z, so Eq. 6 is the special-relativistic Doppler formula. Inother words, the relative speed v rel as defined by paralleltransport is related to the observed redshift as it should be ifthe redshift is a Doppler shift.Figure 3 illustrates this process. The solid line is the worldline of a galaxy, and the dashed line is the path of lighttraveling from the galaxy to the observer. The scale factor isas shown in Fig. 2. The short solid lines indicate the galaxy’svelocity four-vector as it is parallel transported along thelight path to the observer. During the times when the universeis expanding slowly the direction of this vector doesn’tchange significantly, but it does when the universe expandsmore rapidly.Equation 6 can be derived by a straightforward calculationusing the rules for parallel transport, but the derivation iseasier if we recast the statement in more physical terms. Aswe did at the beginning of this section, imagine many comovingobservers stationed along the line from the observedgalaxy to the observer. Each observer has a local referenceframe in which special relativity can be taken to apply, and6the observers are close enough together that each one lieswithin the local frame of his neighbor. Observer number 1,who is located near the original galaxy, measures its speed v 1relative to him and gives this information to observer 2. Observer2 measures the speed u of observer 1 relative to her,adds this speed to the speed of the galaxy relative to observer1 using the usual special-relativistic formula,v 2 =v 1 + u1+v 1 u/c 2 , 7and interprets the result as v rel , the speed of the galaxy relativeto her. She passes this information on to the next observer,who follows the same procedure, as does each subsequentobserver. At each stage, the imputed velocity of theoriginal galaxy relative to the observer will match the redshiftof the galaxy in accordance with Eq. 6. The procedurewe have described operationally is equivalent to performingparallel transport of the galaxy’s four-velocity and using thatvector at each stage to calculate v rel .We can express this derivation in yet another way. Considerthe “world tube” obtained by drawing a sphere with asmall radius around each point on the path of the light rayfrom the galaxy to the observer. This region of spacetime canbe considered as flat Minkowski spacetime up to errors oforder 2 . This statement is true for any small neighborhoodaround a geodesic. Within this tube, the observed redshiftcan be explained only as a Doppler shift, because the spacetimeis flat to arbitrary precision. This argument supplies thesimplest proof we can think of that v rel is related to the redshiftby Eq. 6.We do not expect to have convinced purists who refuseeven to talk about v rel to change their minds. They woulddismiss v rel as a mere “coordinate velocity,” not an “actualvelocity,” and take no interest in it. If purists have the courageof their convictions, they would say that it makes nosense to try to attach labels such as “Doppler” or “gravitational”to the observed redshift. This position is unassailable,and we have no wish to argue against it. We do claim, however,that if you wish to try to talk about v rel , then the definitionproposed in this section is the most natural way to doso. Because this definition of v rel results in the Doppler formulaentirely explaining the redshift, we claim that youshould either be a purist and refuse to try to label the cosmologicalredshift, or you should label it a Doppler shift.It is interesting to contrast the position we advocate withanother approach 13,29 that describes the cosmological redshiftas a particular combination of Doppler and gravitationalterms. In that approach, comoving coordinates are used tospecify the velocity of the distant galaxy relative to the observer,and the observed redshift is decomposed into a Dopplerterm based on this velocity and a gravitational blueshift,which can be calculated by considering the gravitational potentialdue to the matter in a sphere centered on the observer.The difference between the two approaches is in the way thevelocity is defined. Because there is no unique specificationof relative velocity for distant objects in curved spacetime,both approaches are correct. We believe that the approachbased on defining v rel through parallel transport is more naturalthan the one based on a particular choice of coordinates.692 Am. J. Phys., Vol. 77, No. 8, August 2009Emory F. Bunn and David W. Hogg 692Downloaded 10 Aug 2012 to 138.100.250.132. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission

IV. KINEMATIC AND GRAVITATIONAL REDSHIFTSIN GENERAL SPACETIMESLet us step back from the specific case of the expandinguniverse and consider the propagation of a photon in an arbitraryspacetime. A photon is emitted at some spacetimepoint P e with a frequency e and is absorbed at P a withfrequency a . The two frequencies are measured by observersin local inertial reference frames at the two spacetimepoints. If there is a frequency shift e a , under whatcircumstances should we describe it as a Doppler shift or asa gravitational shift?A natural way to answer this question is to populate thespace between the photon’s emission and observation with adense family of observers. Let the first observer be at restrelative to the emitter of the photon at the emission event,and let the last be at rest relative to the absorber at the absorptionevent. Finally, let the velocities of observers varysmoothly along the photon’s path. As in Sec. III, we canexplain the observed frequency shift by adding up the shiftsas the photon passes from one observer to the next. We canconstruct two such families of observers: one in which eachof the shifts is a Doppler shift, and one in which each is agravitational shift. By reference to these families, we caninterpret the observed shift as the accumulation of eithermany small Doppler shifts or many small gravitational shifts.The Doppler and gravitational families are not the onlypossibilities. We can construct other families with respect towhom both Doppler and gravitational shifts would be seen tocontribute. As noted in Sec. III, for example, one way ofregarding the cosmological redshift 13,29 splits it into a specificcombination of Doppler and gravitational terms. TheDoppler and gravitational families provide two of many differentways of interpreting the observed shift. In any givensituation, we can argue about which if either of the twofamilies is natural to consider.To construct the Doppler family of observers, we requireall observers to be in free fall. In this case, within each localinertial frame, there are no gravitational effects, and hencethe infinitesimal frequency shift from each observer to thenext is a Doppler shift.To construct the gravitational family of observers, we requirethat each member be at rest relative to her neighbor atthe moment the photon passes by, so that there are noDoppler shifts. Initially, it might seem impossible in generalto satisfy this condition simultaneously with the conditionthat the first and last observers be at rest relative to the emitterand absorber, but it is always possible to do so. One wayto see that it is possible is to draw a small world tube aroundthe photon path as in Sec. III. Within this tube, spacetime isarbitrarily close to flat. We can construct “Rindler elevatorcoordinates,” the special-relativistic generalization of aframe moving with uniform acceleration, within this tube,such that the velocities at the two ends match up correctly. 30The members of the gravitational family are at rest in thesecoordinates. Because they are not in free fall, the members ofthe gravitational family all feel like they are in local gravitationalfields. Because each has zero velocity relative to herneighbor when the photon goes by, each observer interpretsthe shift in the photon’s frequency relative to her neighbor asa gravitational shift.Because the two families exist for any photon path, we canalways describe any frequency shift as either Doppler orgravitational. In some situations, one seems clearly morenatural than the other. When we discuss the Pound–Rebkaexperiment, 31 which measured the redshift of photons movingupward in Earth’s gravitational field, we generallychoose to regard observers fixed relative to the Earth thegravitational family as more natural than free-fall observers,and hence we interpret the measurement as a gravitationalredshift. In contrast, if you were falling past Pound and Rebkain a freely falling elevator as they performed the experiment,you might choose a Minkowskian inertial frame encompassingyou and the entire experiment. Within thisframe, you would, by the equivalence principle, interprettheir results as a Doppler shift. In so doing, you would bechoosing to regard the Doppler family as the natural one,because this family is the one whose behavior is simplest todescribe in your chosen frame. Finally, if you tried to beat aspeeding ticket by claiming that the radar results were due toa gravitational redshift, you would in effect be considering a“gravitational family” of prodigiously accelerating observers,with one at rest relative to the radar gun and one at restrelative to the driver. Needless to say, the police officer whogives you a ticket regards this family as extremely unnatural.In the cosmological context, we almost always work withfreely falling comoving observers, a choice which correspondsprecisely to the Doppler family. This choice is naturalbecause it respects the symmetries of the spacetime andmakes the mathematical description as simple as possible.On the other hand, the gravitational family is so unnatural incosmology that, as far as we know, it has never been used foranything. Because the Doppler family is by far the mostnatural family to work with, it is natural to interpret thecosmological redshift as a Doppler shift, and it is curious thatthis interpretation is frequently frowned upon.V. WHY THE INTERPRETATION MATTERSThere is no “fact of the matter” about the interpretation ofthe cosmological redshift: what one concludes depends onone’s coordinate system or method of calculation. Nonetheless,it is instructive to analyze the differing interpretations ofthe redshift, partly to improve the understanding of cosmology,but more importantly to improve the understanding ofgeneral relativity. That analysis leads us to conclude that themost natural interpretation of the redshift is kinematic.One of the key ideas of general relativity is the importanceof distinguishing between coordinate-independent andcoordinate-dependent statements. Another is the idea thatspacetime is always locally indistinguishable fromMinkowski spacetime. Cosmology instructors, books especiallyat the introductory level, and students often fall intothe fallacy of reifying the rubber sheet; that is, treating theexpanding-rubber-sheet model of space as if it were a realsubstance. This error leads people away from both of thesekey ideas and causes mistaken intuitions such as that theMilky Way Galaxy must constantly “resist the temptation” toexpand with the expanding universe or that the “tetheredgalaxy” described in Sec. I moves away after the tether iscut.The common belief that the cosmological redshift canonly be explained in terms of the stretching of space is basedon conflating the properties of a specific coordinate systemwith properties of space itself. This confusion is precisely theopposite of the correct frame of mind in which to understandrelativity.693 Am. J. Phys., Vol. 77, No. 8, August 2009Emory F. Bunn and David W. Hogg 693Downloaded 10 Aug 2012 to 138.100.250.132. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission

ACKNOWLEDGMENTSThe authors thank Sean Carroll, John Peacock, JimPeebles, Dave Spiegel, and Ned Wright for helpful comments.This work was supported in part by NSF Grant No.AST-0507395 EFB and a research fellowship from the Alexandervon Humboldt Foundation of Germany DWH.a Electronic mail: ebunn@richmond.edu1 J. C. Baez and E. F. Bunn, “The meaning of Einstein’s equation,” Am. J.Phys. 73, 644–652 2005.2 S. Carroll, Spacetime and Geometry: An Introduction to General RelativityBenjamin Cummings, San Francisco, 2003.3 B. F. Schutz, A First Course in General Relativity Cambridge U.P., Cambridge,1985.4 C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation W. H.Freeman, New York, 1973.5 W. J. Kaufmann and R. Freedman, Universe, 5th ed. W. H. Freeman,New York, 1999.6 A. Fraknoi, D. Morrison, and S. Wolff, Voyages Through the Universe,3rd ed. Thomson Brooks/Cole, Belmont, CA, 2004.7 M. A. Seeds, Foundations of Astronomy, 10th ed. Thomson Brooks/Cole, Belmont, CA, 2007.8 J. Peacock, Cosmological Physics Cambridge U.P., Cambridge, 1998.9 E. V. Linder, First Principles of Cosmology Addison-Wesley, Reading,MA, 1997.10 E. Harrison, Cosmology, 2nd ed. Cambridge U.P., Cambridge, 2000.11 M. Chodorowski, “Is space really expanding? A counterexample,” ConceptsPhys. 4, 15–33 2007 arXiv:astro-ph/0601171.12 M. J. Chodorowski, “A direct consequence of the expansion of space?,”Mon. Not. R. Astron. Soc. 378, 239–244 2007.13 J. A. Peacock, “A diatribe on expanding space,” arXiv:0809.4573.14 A. B. Whiting, “The expansion of space: free-particle motion and thecosmological redshift,” Observatory 124, 174–189 2004.15 One might argue against this claim by noting that the general relativisticMaxwell equations contain covariant derivatives, which “know about”the expansion. However, covariant derivatives are locally indistinguishablefrom ordinary derivatives because a local Minkowski frame canalways be chosen.16 E. R. Harrison, “Mining energy in an expanding universe,” Astrophys. J.446, 63–66 1995.17 T. M. Davis, C. H. Lineweaver, and J. K. Webb, “Solutions to the tetheredgalaxy problem in an expanding universe and the observation ofreceding blueshifted objects,” arXiv:astro-ph/0104349.18 M. Sereno and P. Jetzer, “Evolution of gravitational orbits in the expandinguniverse,” Phys. Rev. D 75 6, 064031-1–8 2007.19 F. I. Cooperstock, V. Faraoni, and D. N. Vollick, “The influence of thecosmological expansion on local systems,” Astrophys. J. 503, 61–661998.20 L. A. Barnes, M. J. Francis, J. B. James, and G. F. Lewis, “Joining theHubble flow: Implications for expanding space,” Mon. Not. R. Astron.Soc. 373, 382–390 2006.21 M. J. Francis, L. A. Barnes, J. B. James, and G. F. Lewis, “Expandingspace: The root of all evil?,” Publ. - Astron. Soc. Aust. 24, 95–1022007.22 If you try this argument, let us know how it works out for you.23 E. A. Milne, “A Newtonian expanding universe,” Q. J. Math. 5, 64–721934.24 This fact is an occasional source of confusion, because a low-densityuniverse is an open universe, in which space, as opposed to spacetime,has negative curvature. What matters in the present context is the spacetimecurvature, that is, the deviation of the overall spacetime metric fromthat of special relativity, not the spatial curvature. The latter describes thegeometry of a “slice” through the spacetime at constant time in a particularcoordinate system.25 T. M. Davis and C. H. Lineweaver, “Superluminal recession velocities,”in Cosmology and Particle Physics, edited by R. Durrer, J. Garcia-Bellido, and M. Shaposhnikov American Institute of Physics, Washington,DC, 2001, Vol. 555, pp. 348–351.26 T. M. Davis and C. H. Lineweaver, “Expanding confusion: Commonmisconceptions of cosmological horizons and the superluminal expansionof the universe,” Publ. - Astron. Soc. Aust. 21, 97–109 2004.27 J. L. Synge, Relativity: The General Theory North-Holland, Amsterdam,1960.28 J. V. Narlikar, “Spectral shifts in general relativity,” Am. J. Phys. 62,903–907 1994.29 H. Bondi, “Spherically symmetrical models in general relativity,” Mon.Not. R. Astron. Soc. 107, 410–425 1947.30 W. Rindler, Essential Relativity, 2nd ed. Springer-Verlag, New York,1977.31 R. V. Pound and G. A. Rebka, “Apparent weight of photons,” Phys. Rev.Lett. 4, 337–341 1960.694 Am. J. Phys., Vol. 77, No. 8, August 2009Emory F. Bunn and David W. Hogg 694Downloaded 10 Aug 2012 to 138.100.250.132. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission