Expanding Space the Root of all Evil.pdf

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arXiv:0707

.0380v1

[astro-ph]

3Jul2007

Expanding Space: the Root of all Evil?

Matthew J. Francis1,4, Luke A. Barnes1,2, J. Berian James1,3 & Geraint F.Lewis1

1Institute of Astronomy, School of Physics, University of Sydney, NSW 2006, Australia2Institute of Astronomy, Madingley Rd, Cambridge, UK

3Institute for Astronomy, Blackford Hill, Edinburgh EH9 3HJ, U.K.4Email: [email protected]

Abstract: While it remains the staple of virtually all cosmological teaching, the concept of expandingspace in explaining the increasing separation of galaxies has recently come under fire as a dangerousidea whose application leads to the development of confusion and the establishment of misconceptions.In this paper, we develop a notion of expanding space that is completely valid as a framework for thedescription of the evolution of the universe and whose application allows an intuitive understandingof the influence of universal expansion. We also demonstrate how arguments against the concept ingeneral have failed thus far, as they imbue expanding space with physical properties not consistentwith the expectations of general relativity.

Keywords: cosmology: theory

1 Introduction

When the mathematical picture of cosmology is firstintroduced to students in senior undergraduate or ju-nior postgraduate courses, a key concept to be graspedis the relation between the observation of the redshiftof galaxies and the general relativistic picture of the ex-pansion of the Universe. When presenting these newideas, lecturers and textbooks often resort to analo-gies of stretching rubber sheets or cooking raisin breadto allow students to visualise how galaxies are movedapart, and waves of light are stretched by the expan-sion of space. These kinds of analogies are appar-

ently thought to be useful in giving students a men-tal picture of cosmology, before they have the abilityto directly comprehend the implications of the formalgeneral relativistic description. However, the academicargument surrounding the expansion of space is not asclear as standard explanations suggest; an interestedstudent and reader of New Scientist may have seenMartin Rees & Steven Weinberg (1993) state

...how is it possible for space, which is ut-terly empty, to expand? How can noth-ing expand? The answer is: space doesnot expand. Cosmologists sometimes talkabout expanding space, but they shouldknow better.

while being told by Harrison (2000) thatexpansion redshifts are produced by theexpansion of space between bodies thatare stationary in space

What is a lay-person or proto-cosmologist to make ofthis apparently contradictory situation?

Whether or not attempting to describe the obser-vations of the cosmos in terms of expanding space is auseful goal, regardless of the devices used to do so,

is far from uncontroversial. Recent attacks on thephysical concept of expanding space have centred onthe motion of test particles in the expanding universe;Whiting (2004), Peacock (2006) and others claim thatexpanding space fails to adequately explain the motionof test particles and hence that it should be abandoned.But what, exactly, is at fault? Crucially, these claimsrely on falsifying predictions made from using expand-ing space as a tool to guide intuition, to bypass the fullmathematical calculation. However, the very meaningof the phrase expanding space is not rigorously defined,despite its widespread use in teaching and textbooks.Hence, it is prudent to be wary of predictions based

on such a poorly defined intuitive frameworks.In recent work, Barnes et al. (2006) solved the test

particle motion problem for universes with arbitraryasymptotic equation of state w and found agreementbetween the general relativistic solution and the ex-pected behaviour of particles in expanding space. Wesuggest that the apparent conflict between this workand others, for instance Chodorowski (2006b), lies pre-dominantly in differing meanings of the very concept ofexpanding space. This is unsurprising, given that it isa phrase and concept often stated but seldom definedwith any rigour.

In this paper, we examine the picture of expand-ing space within the framework of fully general rela-

tivistic cosmologies and develop it into a precise def-inition for understanding the dynamical properties ofFriedman-Robertson-Walker (FRW) spacetimes. Thisframework is pedagogically superior to ostensibly sim-pler misleading formulations of expanding space ormore general schemes to picture the expansion of theUniverse such as kinematic models and approxi-mations to special relativity or Newtonian mechanics,since it is both clearer and easier to understand as wellas being a more accurate approximation. In particu-

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2 Publications of the Astronomical Society of Australia

lar, it must be emphasised that the expansion of spacedoes not, in and of itself, represent new physics that isa cause of observable effects, such as redshift. Ratherthe expansion of space is an intuitive framework forunderstanding the effects of General Relativity.

In Sections 2.1 to 2.4 we detail this working pic-ture of expanding space, before Section 2.5 exploresthis definition in terms of the motion of free parti-cles in an expanding universe. The application of thisapproach to dispel arguments against the expansionspace is demonstrated in Section 2.6, before the con-clusions are presented in Section 3.

2 Expanding Space

In understanding the concept of expanding space, itis important to examine the basic premise of generalrelativity, neatly packaged in John Wheelers adage

matter tells Spacetime how to curve, andSpacetime tells matter how to move

which sets out the dynamical relationship between thegeometry of spacetime and the density and pressuresof fluids contained therein.

However, if the prominent cosmologists quoted inthe previous section, will ask how can space, whichis ultimately empty, expand, we must also ask thequestion of how this nothingness of the vacuum canbe curved? By reducing Wheelers adage to

matter tells matter how to move

the concept of spacetime, just like the aether, can bebanished as being non-existent and unnecessary (e.g.Chodorowski 2006b). Such a picture is not as hereti-cal as it seems; Weinberg (1972), in his classic text ongeneral relativity, questions the whole geometric pic-ture of relativity, and the language it encompasses, asan unfortunate hangover which is not necessary.

It is enlightening to realise that this situation oc-curs in many branches of physics. For example, interms analogue to Wheelers adage, electromagnetismcan be reduced to

charges tell charges how to move

but the employed framework contains the concepts ofelectric and magnetic fields which are as intrinsicallyunobservable as spacetime. Furthermore, these fieldsobey strict mathematical relationships, through theequations of Maxwell, and many researchers can pic-ture the evolution of these fields in dynamical circum-stances, even though it is just charges telling chargeswhat to do.

Hence, we arrive at the view point that while gen-

eral relativity is just matter telling matter how tomove, its framework contains deformable and stretchyspacetime. As with electromagnetism, this field is notintrinsically detectable, but does obey strict mathe-matical relations. Similarly, as correct as it is to thinkof electromagnetism in terms of electric and magneticfields, we can think of general relativity in terms of thedynamical entity of spacetime as long as we developour intuition in terms of the underlying mathematics,and not try to match the properties of spacetime to the

properties of dough or rubber; just as it would makeno sense to attempt to construct a physical or thoughtexperiment to attempt to prove or disprove the realexistence of magnetic fields, it is similarly meaninglessto discuss the expansion of space in these terms.

To illustrate how short this pragmatic formalismfalls of being platitude, one need look no further thanAbramowicz et al. (2006), in which a thought experi-ment of laser ranging in an FRW Universe is proposedto prove that space must expand. This is sensiblyrefuted by Chodorowski (2006b), but followed by aspurious counter-claim that such a refutation likewiseproves space does not expand. The exercise is futile:what matters on a technical level are predictions forobservable quantities, which of course are the sameregardless of how the problem is pictured and whatco-ordinate system is chosen. The expansion of spaceis no more extant than magnetic fields are, and existsonly as a tool for understanding the unambiguous pre-dictions of GR, not a force-like term in a dynamicalequation. A recent example of the dangers of thinkingof expanding space as a real physical theory is con-tained in Table 2 of Lieu (2007) in which the expan-sion of space is lumped together with the Big Bang,

Dark Energy, Dark Matter and Inflation as a physicaltheory demanding verification.We can certainly agree that this kind of misuse of

the term expansion of space is fallacious and mostcertainly dangerous. But throwing the baby of an in-tuitive framework out with the bathwater of miscon-ceptions leaves us only with bare mathematics, whichin the case of general relativity is particularly dauntingfor the uninitiated, and useless as a conceptual device.

2.1 The Cosmological Picture

We turn now to outlining the way in which the ex-pansion of space can be retained as a useful pedagog-ical device, while avoiding the pitfalls of misleadingformulations. It is worth starting from first princi-ples and asking what the general relativistic pictureof cosmology actually contains. The adoption of thecosmological principle, in that the Universe is homoge-neous and isotropic, restricts the form of the underly-ing geometry of the Universe, expressed in terms of theFRW metric. With this metric, the continuity equa-tion demonstrates that in other than finely-tuned orcontrived examples, the density and pressures of cos-mological fluids must change over cosmic time, and itis this change that represents the basic property of anexpanding (or contracting) universe.

The general relativistic picture also allows the def-inition of privileged, co-moving observers (said to re-

side in the Hubble flow) within the expanding universe,those at rest with respect to the cosmological fluids; inour Universe, we know we are not one of these priv-ileged observers as our measured CMB dipole revealsour peculiar motion with respect to the backgroundphotons. Being at rest with regards to the cosmic fluid,the proper time for these privileged observers ticks atthe same rate as cosmic time and hence the watchesof all privileged observers are synchronised. In an ex-panding universe, the change of the metric implies that


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the physical distance between any two privileged ob-servers increases with time, and consequently, if eightof these co-moving observers are used to define the cor-ners of a cube, the volume of the cube must increasewith time.

Remembering that the FRW metric describes ahomogeneous universe filled with a fluid of uniformdensity, and assuming that test observers can measuretheir velocity with respect to that fluid, we can nowdescribe the formal statement of the phenomenon werefer to as expanding space:

The distance between observers at rest withrespect to the cosmic fluid increases withtime.

Since two bodies, both at rest with respect to thefluid defining the FRW metric, find the distance be-tween them has increased after a certain time interval,it seems sensible to suggest that there is more spacebetween them than there was previously. It may bemisleading to suggest that the space that was therestretched itself as the universe expanded. Perhaps abetter description, in simple terms, is to suggest thatmore space appeared, or welled up between the twoobservers, however this is a largely semantic distinc-

tion.We are also in a good position to understand why

the expansion can be thought of locally in kinemati-cal, even Newtonian terms. We can imagine attach-ing a Minkowski frame to each point in the Hubbleflow. The local cosmological fluid is stationary withrespect to this frame. Whilst only p erfectly accuratein an infinitesimally small region, the Minkowski framecan be used as an approximation for regions muchsmaller than the Hubble radius. The Hubble flow isthen viewed as a purely kinematical phenomenon objects recede because they have been given an ini-tial velocity proportional to distance. This does notargue against expanding space: the equivalence prin-

ciple guarantees that any free-falling observer in anyGR spacetime can use SR locally.1

2.2 Local expansions

At the global level, Peacock (2006) suggests that theexpansion of space is uncontroversial since

the total volume of a closed universe is awell-defined quantity that increases withtime, so of course space is expanding.

1The kinematical view can be useful, but remains only alocal approximation. The exception is the Milne model: inan empty universe we can make a coordinate transformationthat exchanges the FRW metric for the Minkowski metric[see Harrison (2000), p. 88], effectively extending our local

Minkowski frame to all spacetime. This is only possible be-cause there is no cosmological fluid to define the rest frameof the universe. Hence the Milne model cannot be used tomake general comments on the nature of the cosmologicalexpansion, cf. Chodorowski (2006a). Recently it has beenclaimed by Chodorowski (2006b) that conformal transfor-mations of general FRW metrics can produce a commonglobal frame describing the entire spacetime, analogous tothe common Minkowski frame in the Milne model. Thiswill be examined in a future contribution (Lewis et al., inprep).

but questions whether

this concept has a meaningful local coun-terpart?. . . Is the space in my bedroom ex-panding, and what would this mean?

Retaining the relativistic picture of expanding space, itis easy to address the question of what happens to Pea-cocks bedroom, namely it will evolve as determinedby the relativistic equations. But as ever, knowledgeof the scenario, and particularly the initial conditions,

is vital; the walls of the bedroom are held togetherby electromagnetic forces and hence are not followinggeodesics, and the distribution of matter has collapsedand is not uniform, and so the underlying geometryof spacetime in this region needs to be calculated; itwould not be represented by the FRW spacetime ofthe homogeneous and isotropic universe. Clearly, if theuniverse were homogeneous on scales smaller than Pea-cocks bedroom, and the walls were not held togetherby electromagnetic or other forces, and the particlesmaking up the wall were at rest with the cosmologi-cal fluid which, importantly, requires that they not beinitially at rest with respect to one another, then in-deed as the universe expands the total volume of the

bedroom would increase. The many conditions listedabove are (at least approximately) true for galaxiesnot bound in common groups and hence they behavein ways that can be understood and predicted via theframework of expanding space.

This leads to an important point, namely that weshould not expect the global behaviour of a perfectlyhomogeneous and isotropic model to be applicable whenthese conditions are not even approximately met. Theexpansion of space fails to have a meaningful localcounterpart not because there is some sleight of handinvolved in considering the two regimes but becausethe physical conditions that manifest the effects de-scribed as the expansion of space are not met in theaverage suburban bedroom.

2.3 Dark Energy

In a matter-dominated universe, the statement in thepreceding section regarding the metric in the regionof a collapsed object being unlike the FRW metric isstraightforward. However, if the universe is dominated(as we believe ours currently is) by an energy that bydefinition is homogeneous, or only inhomogeneous onvery large scales then we must be more careful. Inthis case the dominant driver of the specifics of theexpansion rate will apply equally on all scales. How-ever, so long as the equation of state w of the darkenergy obeys the condition w 1 the energy den-

sity will not increase with time and bound structureswill remain bound and stable. Effectively the regionof spacetime inside a bound structure will in fact bematter-dominated, even though the global mean den-sity is dark energy-dominated.

2.4 The Value of Analogies

What efficacy then, if any, do the common expand-ing universe analogies have? The balloon-with-dots


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or bread-with-raisins analogies, like any analogies, areuseful so long as we are aware of what they success-fully illustrate and what constitutes pushing the anal-ogy too far. They show how a homogeneous expan-sion inevitably results in velocity being proportionalto distance, and also gives an intuition for how theexpansion of the universe looks the same from everypoint in the universe. They illustrate that the uni-verse does not expand into previously existing emptyspace; it consists of expanding space. But using theseanalogies to visualise a mechanism like a frictional orviscous force is taking the analogy too far. They cor-rectly demonstrate the effects of the expansion of theuniverse, but not the mechanism. That they fail atsome level is hardly surprising: were representing 4-dimensional pseudo-Riemannian manifolds with partysupplies. We cant manipulate frames like gravity can.

2.5 The Challenge of Particle Mo-tion

We now turn to the issue of test particle motion, sincethis is at the heart of many of the attacks on expand-ing space. The classic thought experiment used is the

tethered galaxy problem (Harrison 1995). In this,a test galaxy in an expanding universe is held at restwith respect to the origin at a cosmological distance.By Hubbles law, we would expect this galaxy to bereceding, however we prevent this, artificially holdingthe test galaxy in place. The question is, when thegalaxy is released, what does it do? Since critics ofthe expanding space concept argue that the Newto-nian analogue of expanding space is the presence ofsome kind of viscous force, dragging the galaxies apartlike objects carried along by a river: therefore, in thisthought experiment, the test particle should pick upa velocity away from the origin due to the expandingriver of space. In fact, what the particle does oncebeing released depends on the acceleration of the uni-verse. If the scale factor is accelerating the particlemoves away but if it is decelerating the particle movestowards the origin [see Barnes et al. (2006) for the fulldetails]. That acceleration is important in the questionof the future velocity of a particle is a concept that astudent of the most elementary physics is comfortablewith. Since expanding space has apparently misleadour intuition so severely this appears to demonstratethe dangers of this interpretation. But is this a fairtest of expanding space? If the balloon or baking-bread analogy is used to attempt to picture this situa-tion then indeed the incorrect answer is easily reached.However, as mentioned, this is pushing these analogiestoo far, since they are useful in picturing what an ex-

panding universe looks like, but do not speak of whatdrives that expansion.This is the central issue and point of confusion.

Galaxies move apart because they did in the past,causing the density of the Universe to change and there-fore altering the metric of spacetime. We can describethis alteration as the expansion of space, but the keypoint is that it is a result of the change in the mean en-ergy density, not the other way around. The expansionof space does not cause the distance between galax-

ies to increase, rather this increase in distance causesspace to expand, or more plainly that this increase indistance is described by the framework of expandingspace. There is therefore no need to look for Newto-nian analogues to the expansion of space, since it is aneffect rather than a cause. In any case, why should webe seeking Newtonian analogues when we know gen-eral relativity describes the situation well, can be de-scribed in simple terms and any Newtonian view willbreak down at at a non-trivial limit? Whiting (2004)describes the tethered galaxy problem in Newtonianterms and uses Newtonian equations to make predic-tions about the asymptotic behaviour of the test par-ticle. However, as shown in Barnes et al. (2006) theseequations deviate significantly from the general rela-tivistic results which begs the question of why New-tonian analogues should be sought for fundamentallyrelativistic problems?

Can the tethered galaxy problem by understoodin the context of expanding space? We contend thatthe answer is yes. The key is to carefully examine theinitial conditions of the particle. Whiting (2004) andPeacock (2006) have in mind that the initial conditionsof the problem describe a particle dropped innocently

into the universe. It has no proper velocity and thusno prejudiceit is free to go wherever expanding spacewishes to take it. This is certainly true from a kine-matic, Newtonian perspective: the particle is at restin our chosen inertial frame and approaches the origindue to the gravitational attraction of the matter be-tween the particle and the origin. This is locally validand even useful, but it is not how to understand thescenario from an expanding space perspective. Themotion of the particle must be analysed with respectto its local rest frame of the test particles, provided bythe Hubble flow. In this frame, we see the original ob-server moving at vrec,0 and the particle shot out of thelocal Hubble frame at vpec,0, so that the scenario re-sembles a race. Since their velocities are initially equal,

the winner of the race is decided by how these veloci-ties change with time. In a decelerating universe, therecession velocity of the original observer decreases,handing victory to the test particle, which catches upwith the observer.

The difference between the kinematic and expand-ing space interpretation is well illustrated by Figure1 of Davis, Lineweaver, & Webb (2001). Figure 1ashows the kinematic perspective the observer andthe tethered particle are at rest with respect to eachother, and gravitational attraction will bring them to-gether. Figure 1b shows the scenario as seen from thelocal rest frame of the tethered particle, i.e. a racebetween the original observer and the test particle.

The original observer should view the initial condi-tions of the test particle, not as neutral, but as a bat-tle between motion through space and the expansionof space. The expansion of space has been momentar-ily nullified by the initial conditions, so we must askhow the expansion of space changes with time.

We contend that this explanation successfully in-corporates test particle motion into the concept of ex-panding space. In particular, it shows why it is wrongto expect, on the basis of the balloon analogy, that


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expanding space would carry the particle away. Thealternative is either to give up on a physical conceptentirely, so that the only rationale for the cosmologi-cal facts is that thats what the maths tells us, orto formulate a new framework into which these factsand more can be accommodated. The first option isunsavoury, the second unlikely, unless one wants to dis-card GR entirely and formulate cosmology using onlyNewtonian ideas (Tipler 1996).

2.6 Using Expanding Space

In this section, we examine in detail the employmentof the concept of expanding space in a number of cos-mological scenarios.

2.6.1 Superluminal Recession Velocities

By failing to place a limit on the range of validity ofHubbles Law, the FRW metric implies that there isno speed limit on recession velocities, seeming to vi-olate a fundamental principle of relativity by imply-ing superluminal motion. This is a frequent cause ofconcern and confusion. In terms of the proper dis-

tance D defined as D = a and the cosmic time tin the FRW metric then the differential dD

dtmost cer-

tainly can exceed unity, and hence by this definitionof velocity, represents superluminal motion. However,as shown in Grn & Elgary (2006) and Page (1993)we can, by a co-ordinate transformation, describe sim-ple Minkowski space-time as the FRW metric of anempty universe, known as the Milne model. In theMinkowski special relativistic co-ordinates we of coursecannot have superluminal motion. However in the newMilne model co-ordinates we do find that dD

dt> 1

beyond a certain distance from the origin. Thus wehave apparently described superluminal motion in aspacetime that we know cannot permit such a phe-nomenon. Via conformal transformations, it is possi-ble (see e.g. Chodorowski (2006b)) to make a simi-lar transformation between general FRW metrics andconformally related Minkowski-like metrics. Again inthe FRW case dD

dtmake exceed unity, while in the

conformal co-ordinates the speed is limited. Whilesome authors (e.g. Chodorowski (2006b) and Page(1993)) have argued that this demonstrates that su-perluminal recession is impossible, others, for instanceGrn & Elgary (2006) have argued that superluminalrecession is a fundamental consequence of the FRWmetric. As pointed out in Barnes et al. (2006), manyof the debates surrounding expanding space turn out tobe based on different definitions of poorly defined con-cepts, in this case the term superluminal. If we mean

by superluminal that the motion described in the co-ordinates of the Minkowski (or conformal Minkowski-like) frame defined by extending the local inertial frameof an given observer is greater than unity then every-one agrees that this does not occur. On the otherhand, if we take the FRW co-ordinates it is clear thatthere is no limit on the recession velocity: if we chooseto call this superluminal motion, then it indeed oc-curs. The debate seems to boil down to whether thisshould or should not be given the name superluminal

but crucially the physical predictions made by eithercamp will be identical. What matters is not what wecall the phenomenon but whether the understandingan individual has of a given term reflects reality and itis clear that not all the authors mentioned above heldcommon meanings of the term superluminal.

What does matter is that we have a frameworkfor understanding the consequences of the FRW met-ric that is unambiguous and easy to understand. Inthe seminal work of Davis & Lineweaver (2004) sev-eral common mistakes regarding recession velocitiesare examined. The authors take a strong view thatrecession velocities really are superluminal but moreimportantly show the types of mistakes that can bemade by on the one hand using the FRW formalismand on the other hand making ad hoc corrections toprevent apparent superluminal motion in the FRW co-ordinates. The key point to take from this is that onemust be consistent; if we use the very convenient FRWmetric we must be aware that the recession speed is notlimited in any way. If this is uncomfortable alternativeco-ordinates may be adopted, and if used consistentlywill return the same physical predictions as the cor-rectly used FRW co-ordinates.

We will now outline how velocities can be treatedconsistently and clearly within the framework of ex-panding space. Consider a test particle moving radi-ally with coordinate velocity (t). The proper velocityof the object as measured by an observer at the originis:

rp = R(t)(t) + R(t) (t) (1)

The first term is the same as for a particle in the Hub-ble flow at the same co-moving coordinate and dependson the rate of increase of the scale factor. It is zero foran object at the origin or in a static universe. Now,consider the second term: the time measured on aclock () attached to the particle is given by the FRWline element as

c2d2 = c2dt2 R2(t)d2 (2)

d

dt

2= 1

R(t) (t)

c

2(3)

Since is observable it must be real (zero for a pho-ton): (d)2 0 implies that |R(t) (t)| c. Thus,the velocity of the particle due its motion relative tothe Hubble flow (or equivalently the homogeneous fluiddefining the FRW metric) must be less than the speedof light; its velocity due to the increase of the scalefactor is not restricted in this way.

We interpret R(t)(t) as the increase in distanceto the object due to the expansion of the space be-

tween the observer and test particle (recession veloc-ity), and R(t) (t) as the velocity of object due to itsmotion through the local rest frame (peculiar velocity).As previously mentioned, we can consider attaching aMinkowski frame to each point in the Hubble flow.Then the speed of light limits the speed of an objectthrough space. But since there is no global Minkowskiinertial frame (except for in an empty universe), therelative motion of different regions of the Hubble flowsees no speed limit. Note that the kinematical view


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sees no difference between recession and peculiar ve-locities, and thus cannot explain this result. As an il-lustration, for light moving radially away from the ori-gin: vpec = c, so that rp = c+H(t)rp > c. An observerwho insists on extending their Minkowski frame intoexpanding space will encounter light travelling fasterthan light!

Note that while the above has ascribed a velocityto be due to the expansion of space, we again stressthat this is a useful description, rather than a physi-cal cause or law. The physics in operation is generalrelativity and the ultimate cause of the evolution ofthe quantities (rp, rp, , ,R, R) is the characteristics(summarised by the equation of state) and initial con-ditions of the energy in the Universe.

While the picture of expanding space possesses dis-tant observers who are moving superluminally, it isimportant not to let classical commonsense guide yourintuition. This would suggest that if you fired a photonat this distant observer, it could never catch up, butintegration of the geodesic equations can reveal other-wise (this is very clear in the conformal representationof FRW universes (Chodorowski 2006b); this will beexamined more deeply in a future contribution (Lewis

et al in prep). Hence, again, what is important is notthe statement of superlumininal motion, but implica-tions for observations, and these must be independentof the framework in which you choose to work.

2.6.2 Is everything expanding?

An extension of the argument against global expansiongiven in section 2.2 is that is should be undetectable,since everything will simply expand with it. However,this is not the case: consider a normal object, bywhich we mean one consisting of many particles, heldtogether by internal forces. Suppose that the centreof the object travels along a radial geodesic c(t) inFRW spacetime. Suppose further that the front of the

object travels along a trajectory f(t) that keeps it ata constant proper distance (L) from the centre, i.e.

R(t)f(t) R(t)c(t) = L (a constant) (4) f(t) = c(t) + L

R(t)(5)

The back of the object will move along an analogouspath. Then the coordinate trajectory f(t) is not ageodesic of FRW spacetime. The foremost particle willexperience a four-force, which can be calculated bysubstituting Equation (5) into the equation of motionof a particle experiencing a four-force fa:

d2xa

d2 + abc

dxb

d

dxc

d =fa

m (6)

The observed force in the radial direction is given byprojecting f1 onto an orthonormal basis; the final re-sult is equation (1) of Harrison (1995) with U(t) =H(t)L for all time. In the case of L small (comparedto c/H, the Hubble radius), we have that the radialforce F is:

F = mL RR

(7)

This result tells us how not to understand expand-ing space. Expanding space does not stretch rigidrulers how could it? It is just a trick with iner-tial frames. The internal, interatomic forces in rigidobjects are able to maintain the objects dimensions;Dicke & Peebles (1964) [see also Carrera & Giulini (2006)]argue that EM forces do just this. Objects are held to-gether by forces that pull their extremities through asuccession of rest frames.

It is worth considering what would happen to anobject if there were no electromagnetic forces holdingit together. Consider an object of many particles withno internal forces. It is shot away from the origin ( =0) with speed v0, the first particle leaving at time t0and the last at t0 + t0. The length of the object isl0 = v0t0. The object travels to an observer in theHubble flow at , who measures its speed relative tohim (vf) and the time of arrival of the first ( tf) andlast particle (tf + tf) in order to measure its length(lf = vftf). Following Barnes et al. (2006):

=

Ztf

t0

dt

R

1 + C0R2=

Ztf+tf

t0+t0

dt

Rp

1 + CfR2

(8)

where C0 and Cf are calculated from the initial condi-tions for each particle. If we assume that t0 and tfare small, it follows that we can assume C0 = Cf Cand then rearrange the limits of the integral to give2

Zt0+t0t0

dt

R

1 + CR2=

Ztf+tftf

dt

R

1 + CR2

(9)

t0R(t0)

p1 + CR2(t0)

=tf

R(tf)p

1 + CR2(tf)(10)

Then, following the method of Barnes et al. (2006) tocalculate vf = (tf)R(tf) and substituting for C wehave that

lfl0

=vftfv0t0

=R(tf)

R(t0)(11)

Hence, the length of the object has increased in pro-portion with the scale factor.

This result answers the question: what if an ob-ject had no internal forces, leaving it at the mercy ofexpanding space? This is a rather strange object it would very quickly be disrupted by the forces of ev-eryday life. Nevertheless, it is a useful thought exper-iment. The above result shows that the object, beingsubject only to expanding space, has been stretched inproportion with the scale factor. These are essentiallycosmological tidal forces.

We therefore have clear, unambiguous conditionsthat determine whether an object will be stretched bythe expansion of space. Objects will not expand withthe universe when there are sufficient internal forces tomaintain the dimensions of the object.

2This part of the derivation is similar to the derivationof the cosmological redshift directly from the FRW metric;see, among many others, Hobson, Efstathiou, & Lasenby(2005), p. 368.


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2.6.3 Why arent galaxies or clusters pulled

apart by the expansion of space?

Having dealt with objects that are held together byinternal forces, we now turn to objects held togetherby gravitational force. One response to the questionof galaxies and expansion is that their self gravity issufficient to overcome the global expansion. How-ever, this suggests that on the one hand we have theglobal expansion of space acting as the cause, driving

matter apart, and on the other hand we have gravityfighting this expansion. This hybrid explanation treatsgravity globally in general relativistic terms and lo-cally as Newtonian, or at best a four force tacked ontothe FRW metric. Unsurprisingly then, the resultingpicture the student comes away with is is somewhatmurky and incoherent, with the expansion of the Uni-verse having mystical properties. A clearer explana-tion is simply that on the scales of galaxies the cos-mological principle does not hold, even approximately,and the FRW metric is not valid. The metric of space-time in the region of a galaxy (if it could be calculated)would look much more Schwarzchildian than FRW like,though the true metric would be some kind of chimeraof both. There is no expansion for the galaxy to over-come, since the metric of the local universe has alreadybeen altered by the presence of the mass of the galaxy.Treating gravity as a four-force and something thatwarps spacetime in the one conceptual model is boundto cause student more trouble than the explanation isworth. The expansion of space is global but not uni-versal, since we know the FRW metric is only a largescale approximation.

2.6.4 The Expansion of Space and Red-

shift

The explanation of redshift is a crucial link that needsto be made between cosmological observations and the-ory. A derivation of the balloon analogy is often em-ployed in the teaching of this concept; a wave is sketchedon a balloon and as it is blown up the wavelength isseen to increase as the sketch is stretched along withthe expansion of the underlying space. This is largelyuncontroversial, but care must be taken in ensuringthat the analogy does not mislead. Since we haveshown how b odies held together by electromagneticforces do not expand with the expansion of space, whyshould electromagnetic waves be affected? The key isto make it clear that cosmological redshift is not, as isoften implied, a gradual process caused by the stretch-ing of the space a photon is travelling through. Rathercosmological redshift is caused by the photon being ob-

served in a different frame to that which it is emitted.In this way it is not as dissimilar to a Doppler shift as isoften implied. The difference between frames relates toa changing background metric rather than a differingvelocity. Page 367 of Hobson, Efstathiou, & Lasenby(2005) as well as innumerable other texts shows howredshift can be derived very simply by considering thechange in the orthonormal basis of observers with dif-ferent scale factors in their background metrics. Thisprocess is discreet, occurring at the point of reception

of the photon, rather than being continuous, whichwould require an integral. If we consider a series of co-moving observers, then they effectively see the wave asbeing stretched with the scale factor.

3 Conclusion

Despite (and perhaps in part because of) its ubiquity,

the concept of expanding space has often been artic-ulated poorly and formulated in contradictory ways.That addressing this issue is important must be placedbeyond doubt, as the phrase expansion of space isin such a wide usefrom technical papers, through totextbooks and material intended for school students orthe general publicthat it is no exaggeration to labelit the most prominent feature of Big Bang cosmologies.In this paper, we have shown how a consistent descrip-tion of cosmological dynamics emerges from the ideathat the expansion of space is neither more nor lessthan the increase over time of the distance betweenobservers at rest with respect to the cosmic fluid.

This description of the cosmic expansion should beconsidered a teaching and conceptual aid, rather than

a physical theory with an attendant clutch of physicalpredictions. We have demonstrated the power of thispragmatic conceptualisation in guiding understandingof the universe, particularly in avoiding the traps intowhich we can be lead without rigorous recourse to gen-eral relativity.

The utility of approximation in handling the lesstractable properties of cosmologies is undiminished,but the understanding of physical systems therein willbe hampered wherever full covariance is absent. Allobservational propertieswhether derived in the dy-namically evolving FRW metric or the Minkowski-likeconformal representationmust be the same, indepen-dent of the choice of co-ordinates. As general relativity

approaches its one-hundredth birthday, this is a lessonthat all cosmologists should learn

Acknowledgements

LAB and JBJ are supported by an Australian Post-graduate Award. LAB is also supported by a Univer-sity of Sydney School of Physics Denison Merit Award.JBJ is also generously supported by the University ofEdinburgh. MJF is supported by a University of Syd-ney Faculty of Science UPA scholarship. GFL grate-fully acknowledges support through the University ofCambridge Institute of Astronomy visitor program.Brendon Brewer is thanked for helping suggest the ti-tle. This work has been supported by the AustralianResearch Council under grant DP 0665574.

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