Interpretations of cosmic expansion: anchoring conceptions and misconceptions
IInterpretations of cosmic expansion: anchoringconceptions and misconceptions
M P¨ossel Haus der Astronomie and Max Planck Institute for Astronomy, K¨onigstuhl 17,69117 Heidelberg, GermanyE-mail: [email protected]
Abstract.
Teaching cosmology at the undergraduate or high school level requiressimplifications and analogies, and inevitably brings the teacher into contact withat least one of the pedagogical interpretations of the expanding universe. The byfar most popular interpretation holds that galaxies in an expanding universe arestationary, while space itself expands and thus causes the growing distances thatcharacterize cosmic expansion. The alternative relativistic explosion interpretationregards cosmic expansion as a pattern of (relativistic) galaxy motion. The aim of thisarticle is to discuss the two competing interpretations from the perspective of potentialstudent preconceptions, taking into account both beneficial anchoring conceptions andpotentially harmful preconceptions that can lead to misconceptions.
Keywords : cosmology, cosmic expansion, cosmological redshift, anchoring conceptions
This is the version of the article before peer review or editing, as submitted by the author to thejournal
Physics Education . IOP Publishing Ltd is not responsible for any errors or omissions in thisversion of the manuscript or any version derived from it. The Version of Record is available online athttps://doi.org/10.1088/1361-6552/aba3b1
1. Introduction
At the graduate level, teaching about the expanding universe can rely on the appropriatemathematical tools of general relativity. There, the central mathematical object isthe metric describing the geometry of Friedmann-Lemaˆıtre-Robertson-Walker (FLRW)spacetimes, from which all the basic properties of expanding universes can be derived.Together with the Friedmann equations, which state how the energy and mattercontent of the universe determines the evolution of cosmic expansion, they providethe foundation for our modern cosmological models.Where these mathematical tools are not available, notably at the undergraduateor high school level, teaching cosmology must rely on simplified models and analogies— something that is true more generally when teaching about general relativity [1, 2]. a r X i v : . [ phy s i c s . e d - ph ] A ug nterpretations of cosmic expansion: anchoring conceptions and misconceptions interpretation of cosmic expansion, that is, on a conceptualframework that provides elementary descriptions such as “galaxies are moving awayfrom each other.”The most common such framework is the expanding space interpretation , whichposits that galaxies participating in cosmic expansion are at rest, but that space betweenthem is expanding. An alternative is the relativistic motion interpretation , whichregards cosmic expansion as a pattern of (relativistic) galaxy motion in space. Thereis a considerable body of literature on the merits and problems of each interpretation[3, 4, 5, 6, 7, 8, and references therein], but the discussion has mostly centered onthe physical and mathematical aspects. The aim of this article is to examine thetwo interpretations from a complementary perspective, focussing on the connectionswith basic preconceptions that pupils are likely to bring to cosmology from everydayexperience as well as from high-school and university-level physics.Such connections are of interest for teaching since preconceived knowledge can leadto misconceptions, but it can also aid understanding through “anchoring conceptions” inthe sense of Clement, Brown & Zietsman: elements of “an intuitive knowledge structurethat is in rough agreement with accepted physical theory,” which allow for anchoringnew material in learners’ existing frameworks of knowledge [9]. While the presentwork does not include an empirical component, the results suggest ways of extendingexisting studies of student’s cosmological conceptions and misconceptions [10, 11, 12]by explicitly taking into account the different roles such preconceptions play for the twocompeting interpretations.
2. Interpretations of cosmic expansion
Modern cosmological models describe a family of idealised galaxies, said to be “in theHubble flow,” in a universe that is taken to be homogeneous on average. At each pointin space, the Hubble flow defines a local standard of rest. Pick out two arbitraryHubble-flow galaxies, and their distances will change over time in proportion to auniversal cosmic scale factor, often called a ( t ). Although there are re-collapsing FLRWspacetimes, we will in the following concentrate on cosmic expansion, where a ( t ) growsover time. An immediate consequence is the Hubble-Lemaˆıtre relation: At the presenttime, the recession velocity v rec , that is, the change over time of the distance d betweentwo Hubble-flow galaxies, is related to d as v rec = H · d, (1)with H the Hubble constant. Edwin Hubble’s empirical version, published in 1929, wasinstrumental for the acceptance of the concept of an expanding universe in cosmology.With the FLRW spacetimes formulated by Friedmann and Lemaˆıtre, and refined laterby Robertson, Walker and others, it became clear that for our own universe, generalrelativity predicts a time t ini in the past where a ( t ini ) = 0: the big bang singularity. nterpretations of cosmic expansion: anchoring conceptions and misconceptions t ini , wherethe universe was filled with a hot and dense plasma, make up a major portion of moderncosmological research.The most common framework for interpreting FLRW spacetimes is the expandingspace interpretation . An excellent review can be found in the seminal papers of Davisand Lineweaver, which are centered around popular misconceptions of cosmic expansion[4, 13]. A thorough treatment can also be found in the book by Harrison [14]. At thecore of this interpretation is the notion that galaxies are at rest in space. Changinginter-galaxy distances, then, are not due to galaxy motion through space. Instead, theyare the consequence of space between the galaxies expanding.In the expanding space interpretation, light from distant galaxies is redshifted“[b]ecause expanding space stretches all light waves as they propagate” [13]. In fact,in FLRW spacetimes, the wavelength λ e with which the light is emitted by a distantgalaxy at time t e and the wavelength λ r with which the light is received at our owngalaxy at a later time t r are related to the cosmic scale factor as1 + z ≡ λ r λ e = a ( t r ) a ( t e ) , (2)where the left-hand equation defines the redshift z . This direct proportionality lendsplausibility to the light-stretching interpretation.Alternatively, in the relativistic explosion interpretation , galaxies are movingthrough space, and the cosmological redshift can be interpreted as a Doppler shift: as aconsequence of observers in relative motion measuring the wavelength of the same lightsignal, and coming to different conclusions. Crucially, the relative radial velocity v rad ofa galaxy moving directly away from us in the course of cosmic expansion is not the sameas the recession velocity v rec defined by the Hubble-Lemaˆıtre law (1). It is a relativisticradial velocity derived using the general-relativistic notion of parallel transport [3, 8].In terms of this relativistic radial velocity, which is always subluminal, v rad < c , thecosmological redshift can be written using the same formula as the special-relativisticredshift for radial velocity, λ r λ e = (cid:115) c + vc − v , (3)which for FLRW spacetimes turns out to be an alternative way of writing the redshiftformula (2).The earliest reference to this interpretation appears to be due to Lanczos in 1923,based on calculations using de Sitter’s cosmological solution [15], and thus predating thedescription of the more general FLRW spacetimes. Synge derives the Doppler redshiftformula in his 1960 general relativity book [16], and the result has been brought tothe attention of more recent readers e.g. by Narlikar [3] and by Bunn and Hogg [7]. Asign of the relative obscurity of this interpretation is that variations thereof have beenre-discovered, independently, by different authors over the past decades [17, 6, 18, 19].It should be stressed that the disagreement between the two interpretationsdoes not extend to the underlying mathematics of FLRW spacetimes, nor to the nterpretations of cosmic expansion: anchoring conceptions and misconceptions
3. Moving vs. getting carried away
A considerable advantage of the expanding space interpretation is its close relationwith what I will refer to collectively as expanding-substrate models : teaching modelsin which the space of an expanding universe is represented by a medium or substrate.These include a stretching rubber band which represents a one-dimensional universe,with painted-on dots for galaxies, Fig. 1, and an inflating rubber balloon with stickersfor galaxies, as a two-dimensional cosmos, Fig. 2. The models allow for (limited)quantitative measurements of the Hubble-Lemaˆıtre relation [20, 21]. We ourselveshave no everyday experience of space expanding, but there are relevant preconceptionsdirectly related to these expanding-substrate models. Why, for instance, do the distancesbetween galaxy-stickers on the rubber balloon change? They are being (a) pulled alongby the expanding surface, which requires (b) that the surface itself is expanding and (c)that the stickers are affixed to the surface.How are these preconceptions-by-proxy related to the underlying physics of cosmicexpansion? Part of the answer, culminating in the question of to what extent it evenmakes sense to ascribe properties to space directly, would lead us into rather deepphilosophical waters, related to Einstein’s version of Mach’s principle [22], far beyondthe scope of this article. Instead, consider a much simpler approach: Real galaxies canhave non-zero peculiar velocities relative to their local Hubble-flow standard of rest. Forexample, when the center of mass of a galaxy cluster follows the Hubble flow, individualgalaxies orbiting within the cluster are in motion relative to that center. At any point inspace, a galaxy with peculiar motion up to the local speed of light is just as compatiblewith cosmic expansion as a galaxy in the Hubble flow. Whatever the “medium” thatexpands in this interpretation, it cannot be one to which galaxies can be said to be“affixed.”What about galaxies getting “pulled along”? Consider a galaxy which has justthe right peculiar velocity to be momentarily at rest relative to our own galaxy. If youexpect that galaxy to begin moving away from us, getting pulled along with the Hubble nterpretations of cosmic expansion: anchoring conceptions and misconceptions Figure 1.
One-dimensional rubber band universe getting stretched over time (top tobottom). Galaxies are represented by dots
Figure 2.
A balloon, whose surface represents a two-dimensional universe, beinginflated (left to right). Galaxies are represented by stickers flow, you would be wrong. Depending on the matter content of the model universe, andon cosmic time, even in an expanding universe, that galaxy might instead get pulled towards us [23, and references therein]. What happens depends only on the secondderivative of the scale factor, not on the first derivative, in other words: What happensis independent of how fast our cosmos is expanding at this particular moment.In the relativistic-explosion interpretation, on the other hand, several preconcep-tions concerning motion can serve as helpful anchoring conceptions for these same situ-ations. On the simplest level, relative motion is what we call it when distances between nterpretations of cosmic expansion: anchoring conceptions and misconceptions
4. Light propagation and the cosmological redshift
The most prominent property of light in an expanding universe is the cosmologicalredshift (2) for light reaching us from distant Hubble-flow galaxies. In the expanding nterpretations of cosmic expansion: anchoring conceptions and misconceptions
Figure 3.
When a balloon, whose surface represents a two-dimensional universe, isinflated (left to right), light-waves drawn on the balloon surface are stretched based preconceptions, this correct stretching behaviour is a great plus. The downsideis that the stretching amounts to parts of the wave getting “pulled along,” which inturn depends on the wave structure being affixed to the substrate, which in turn isinconsistent with the light moving freely, and at great speed, relative to the substrate.In the relativistic explosion interpretation, the cosmological redshift is a Dopplershift [3, 7], its magnitude given by the special-relativistic formula (3). This establishesstudents’ pre-existing knowledge of both the special-relativistic and the classical Dopplereffect as anchoring conceptions. It also resolves another potential conflict: If photonsfrom distant galaxies arrive at our own galaxy with less energy than when they wereemitted, where does that energy go? At first glance, that would appear to be in directcontradiction to the conservation of energy. But if a Doppler effect is the explanation,then there is no such concern, since the two energy values refer to different referenceframes.In the expanding space interpretation, that apparent energy loss is less readilyexplained. Those texts that address the problem explicitly link it to the more generalimpossibility of defining global energy conservation in general relativity [14, 27]. Adrawback of this solution is that the problem is also present in the zero-density limitingcase of an empty FLRW spacetime, the so-called Milne universe [28, sec. 16.3], wherephysics is governed by special relativity, and where there most certainly is global energyconservation.In both special and general relativity, light propagation defines an absolute cosmicspeed limit in the sense that no material object or signal can overtake a light signal.This is where the distinction between the recession speed, defined as in (1), and therelativistic radial velocity that is central to the relativistic explosion interpretation iscrucial. Recession speeds become superluminal for distant galaxies. This appears to nterpretations of cosmic expansion: anchoring conceptions and misconceptions not true for recession speeds [29].Last but not least, preconceptions from special relativity can help studentsunderstand why a relativistic explosion need not contradict the homogeneity of ouruniverse. This is easiest to see for the Milne universe, where each galaxy corresponds toan inertial observer, all on an equal footing because of the same relativistic effects thatcombine in special relativity to yield the constancy of the speed of light for all inertialobservers [28, sec. 16.3].
5. Conclusion
Each of the two main interpretations of cosmic expansion is connected with potentialstudent preconceptions — from everyday notions about motion to more advancedconcepts that one encounters in physics courses.For the expanding space interpretation, most of the relevant preconceptions areassociated with the expanding-substrate models. This is a considerable plus both forteaching about the basic pattern of expansion with a universal scale factor and for light nterpretations of cosmic expansion: anchoring conceptions and misconceptions
Acknowledgements
I am grateful to Markus Nielbock and Anna P¨ossel for their help in creating theillustrations, and to Thomas M¨uller for his critical comments on an earlier version ofthis text.
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