The One-Way Speed of Light and the Milne Universe
PPublications of the Astronomical Society of Australia (PASA)doi: 10.1017/pas.2020.xxx.
The One-Way Speed of Light and the Milne Universe
Geraint F. Lewis and Luke A. Barnes Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney. NSW 2006, Australia Western Sydney University, Locked Bag 1797, Penrith South DC, NSW 2751, Australia
Abstract
In Einstein’s Special Theory of Relativity, all observers measure the speed of light, c , to be the same.However, this refers to the round trip speed, where a clock at the origin times the outward and returntrip of light reflecting off a distant mirror. Measuring the one-way speed of light is fraught with issues ofclock synchronisation, and, as long as the average speed of light remains c , the speeds on the outwardand return legs could be different. One objection to this anisotropic speed of light is that views of thedistant universe would be different in different directions, especially with regards to the ages of observedobjects and the smoothness of the Cosmic Microwave Background. In this paper, we explore this inthe Milne universe, the limiting case of a Friedmann-Robertson-Walker universe containing no matter,radiation or dark energy. Given that this universe is empty, it can be mapped onto flat Minkowskispace-time, and so can be explored in terms of the one-way speed of light. The conclusion is that thepresence of an anisotropic speed of light leads to anisotropic time dilation effects, and hence observersin the Milne universe would be presented with an isotropic view of the distant cosmos. Keywords: cosmology: theory
Central to Einstein’s Special Theory of Relativity isthat all inertial observers will measure an identical valueof the speed of light, c (Einstein, 1905). However, asnoted by Einstein himself, this refers to the averageof a round-trip journey for light that is reflected off adistant mirror, and, as long as the average speed is c ,the outward and inward velocities could be different(see extensive review in Anderson et al., 1998). Whilstthis might seem strange, anisotropy in the speed oflight would result in anisotropy in time dilation effects,ensuring that synchronisation of distant clocks remainsfraught. Hence, no experimental measurement of theone-way speed of light is possible.An objection to differing one-way speeds of light mightbe observations of the distant universe, where we clearlyhave, on average, an isotropic view, seeing young galaxiesat high redshift, and the smoothness and uniformity ofthe Cosmic Microwave Background over the sky; surelythe anisotropy in the speed of light would be imprintedon this view? In this paper, we will tackle this question byconsidering an idealised cosmological model, the Milneuniverse, and will explore an extreme case where thespeed of light is infinite in one direction, and c/ ∗ [email protected] the other. The layout of this paper is as follows: InSection 2 we present the mathematics of differing one-way speeds of light, and will present the Milne universein Section 3. We discuss the Milne universe with differingone-way speeds of light in Section 4, whilst presentingour conclusions in Section 5. In the following, we willset c , the average round trip speed of light, to unity. In considering differing one-way speed of light mod-els, the underlying transformations of coordinates aremodified. In the following, we follow the mathemati-cal formalism of Anderson et al. (1998). We considerdiffering one-way speeds of light related to c by c ± = c ∓ κ ≡ ∓ κ (1)Setting κ = 0 corresponds to an isotropic speed of light,whereas κ = 1 presents the extreme case where the c + = ∞ and c − = 1 /
2. To preserve the observations ofspecial relativity, a coordinate velocity v = dxdt in theisotropic c case ( κ = 0) is mapped to a new velocity˜ v = d ˜ xd ˜ t = γ ˜ γ v (2)1 a r X i v : . [ g r- q c ] D ec Lewis & Barnes
Figure 1.
Space-time diagram for the situation where the speed of light is equal in both directions (left) and the limiting case wherethe speed of light is c/ where γ = 1 √ − v and ˜ γ = 1 − κv √ − v (3)The relative time dilation between two observers in thecase where the one-way speed of light is given by d ˜ t d ˜ t = 1˜ γ (4)Figure 1 presents an illustration of the impact of theanisotropic speed of light. The left-hand side of thisfigure presents the familiar case where the speed of lightis equal in both directions with the green-dashed linerepresenting a light cone for an observer at the origin.The grey lines represent the worldlines of massive objectsmoving relative to the observer at the origin at v = 0 . The discussion in the previous section is wrapped in thelanguage of special relativity, whereas the cosmologicaldescription of the universe relies on Einstein’s GeneralTheory of Relativity. Whilst the typical approach tostudying cosmology is to begin with the Friedmann-Robertson-Walker (FRW) metric (e.g. Hobson et al.,2006), this is just a convenient choice of coordinates and other choices can be made. Infeld & Schild (1945)demonstrated that cosmological models can be cast in akinematic form, where by motion through space, coupledwith gravitational potentials, replaces the picture ofexpanding space (e.g. Lewis et al., 2007).The focus of this paper will be the limiting case ofthe FRW metric, namely the Milne universe in whichthe universe is empty, devoid of any matter, radiationor energy (Milne, 1933). A key feature of of the Milneuniverse is that, while it is spatially curved, its space-time is flat and can, therefore, be directly mapped intothe Minkowski metric (see Chodorowski, 2005)We begin by describing the Milne universe inMinkowski space-time. At t = 0, a collection of mas-sive test particles (quaintly referred to as galaxies ) areejected in all directions from the origin ( x = 0) with arange of velocities. In the limiting case of an empty uni-verse, we disregard gravity (the effect of these galaxieson space-time), and so the galaxies maintain a constantvelocity. Because faster galaxies move further in a givenperiod of time, the further away we look, the faster thegalaxies are moving and the more their light is redshifted;this gives the Hubble law for any of the galaxies.Assuming an isotropic speed of light, consider twogalaxies that are emitted with the same speed in oppositedirections. In order to measure their positions, we (still atthe origin) send a light beam after them at time t . Thelight bounces off the galaxy at ( x g , t g ) and returns to usat t . With an isotropic speed of light, the light reachedthe galaxy halfway between t and t . The distance to he Cosmological One-Way Speed of Light Figure 2.
Space-time diagram for the Milne universe in FRW coordinates. The horizontal dashed grey line denotes now in cosmic time,whilst the sold grey lines are comoving objects at x = 1 , , ,
50 and 100. The blue lines represent the past light cone for an observer atthe origin today and the time where they cross the comoving objects is the age we observe them at today; clearly, due to the symmetryof the situation, the view in opposite directions will be the same, with more distant objects appearing younger. The two black dotsdenote emission from x = 5 that is observed at the origin today, whilst the red dashed line represents the age of the universe when thelight from these sources is emitted. the galaxy is half of the total light travel time: t g = 12 ( t + t ) (5) x g = 12 ( t − t ) . (6)This second expression effectively defines the radar dis-tance to an object (c.f. Lewis et al., 2008). If the galaxystarted a clock as it departed the origin, the time onthat clock when our photon arrives is given by, τ p = q x g − t g = √ t t . (7)From this, we infer that the galaxy is moving with speed v g = x g /t g = ( t − t ) / ( t + t ).Now, suppose that the speed of light is c + = ∞ inthe positive x -direction and c − = 1 / x -direction, as shown in Figure 1 (right). As before, wesend a beam of light after the galaxy at t and it returnsto us at t . Right-hand Galaxy (moving in the positive direction):
The light travels instantaneously to the galaxy at t ,and returns to us at speed 1 / t gr = t (8) x gr = 12 ( t − t ) . (9)For this anisotropic universe, the formula to calculatethe proper time is, τ gr = q t gr + 2 x gr t gr = √ t t , (10) as above. The inferred speed of the galaxy is v gr = x gr /t gr = − ( t − t ) /t , which is related to the inferredvelocity for the isotropic case by Equation (2). Left-hand Galaxy (moving in the negative direction):
The light travels at speed c/ t , andreturns instantaneously to us, t gl = t (11) x gl = −
12 ( t − t ) . (12)For this anisotropic universe, the formula to calculatethe proper time is, τ gl = q t gl + 2 x gl t gl = √ t t , (13)as above. The inferred speed of the galaxy is v gl = | x gl | /t gl = ( t − t ) /t , which is related to the inferredvelocity for the isotropic case by Equation (2).Importantly, the redshift of the returning light thatis observed by us is the same in all three cases: 1 + z g = p t /t , so the observed universe is identical whetherthe speed of light is identical in all directions, or isanisotropic. However, for the anisotropic advocate, galax-ies with the same redshift in different directions arelocated at the same distance, but emitted the light weobserve at different times. They have different velocities,and so they conclude that the left side of the universe isexpanding faster than the right hand side.What does this Milne universe look like in an ex-panding space framework? We begin with the the FRWmetric, ds = − dt + a ( t ) (cid:2) dx + R o S k ( x/R o ) d Ω (cid:3) (14) Lewis & Barnes
Figure 3.
The Milne universe presented in Figure 2, but now mapped into the flat space-time coordinates. The comoving objects havebeen mapped into sloped lines, whereas synchronised lines of constant cosmological time have been mapped into hyperbola. There areclear similarities between this and the left-hand space-time diagram presented in Figure 1. where a ( t ) is the normalised scale factor, such that a ( t o ) = 1 and t o is the present age of the universe. Thefunction, S k ( x ) is sin( x ), x , and sinh( x ) for a spatiallyclosed, flat and open universe respectively. The angularterms, which are related to the surface of a 3-sphere, aregiven by d Ω = dθ + sin θ dφ . The present day scalefactor, R o , in an open universe is given by R o = 1 H o √ − Ω o (15)where H o is the present day Hubble Constant and Ω o is the present day total energy density. For the Milneuniverse, Ω o = 0, and so R o = H − o , and the normalisedscale factor a ( t ) = t/t o .It is instructive to construct a space-time diagramfor the Milne universe (Figure 2) which shows the in-stantaneous proper distance to a comoving observer ata spatial coordinate, x , given by D ( t ) = a ( t ) x versesthe cosmological time, t ; as an illustration, comovingobservers are presented at x = 1 , , ,
50 and 100. Also,presented in blue, is the past lightcone for an observerat the spatial origin at the present time. The path of alight ray in these coordinates is governed by dxdt = ± t o t (16)Remembering that in these coordinates, the proper timesof comoving observers are synchronised with the cosmictime, t , the observer at the origin will see distant objectswith an age given by their crossing of the past light cone,and, given the symmetry of the situation, the originobservers view will be symmetrical, with more distantobjects appearing younger. Examining Figure 2 suggests that the Milne universe isvery different to the flat space-time of special relativity.However, given that it has no material content, the un-derlying space-time structure of the two are the same,and so we should be able to undertake a coordinatetransformation between the two. Note that this is differ-ent to the conformal representation of FRW universes(e.g. Harrison, 1991), which straightens light rays to 45 o ,as there can be a complex relationship between universalconformal time and the experienced proper time.We follow Chodorowski (2007) by firstly defining χ = x/R o and dt = R o adη such that Equation 14 can bewritten as ds = R o a ( η ) (cid:2) − dη + dχ + sinh ( χ ) d Ω (cid:3) (17)At this stage, we define a coordinate transformationfrom ( η, χ ) to new coordinate ( T, R ) through T = Ae η cosh( χ ) (18) R = Ae η sinh( χ ) (19)where A is a constant. With these Ae η = p T − R and tanh( χ ) = RT (20)with these transformations, and a little algebra, Equa-tion 17 can be written as ds = − dT + dR + R d Ω (21)which is just the flat space-time of special relativitywith polar coordinates over the spatial part. Clearly, in he Cosmological One-Way Speed of Light Figure 4.
The Milne universe presented in Figure 2, but mapped into the case with an anisotropic speed of light. Again, the greydashed line corresponds to the present time in the cosmological time of the Milne universe, whereas the the red dashed line is thecosmological time for a pair of emitters on either side of the sky. As can be seen, the observer at the origin is presented with an isotropicview of the sky, even through the speed of slight is anisotropic. these new coordinates, light rays travel at 45 o , comovingobservers are represented as straight lines with a slopegiven by tanh( χ ). Additionally, the relationship betweenthe proper time experienced by an observer at the origin, dT , and the comoving observer, dτ , is given by dτdT = s − (cid:18) dRdT (cid:19) = q − tanh ( χ ) (22)To illustrate this, we map the situation presented inFigure 2 into the ( T, R ) coordinates. This is illustratedin Figure 3. As noted above, the comoving observers tosloped lines, and as can be seen, these asymptote to 45 o as all motion is bounded by the speed of light. In these co-ordinates, the comoving observers in the Milne universe,which are moving apart due to expanding space, aretransformed into objects moving with velocities relativeto each other through flat space-time. The key pointsthat synchronous lines in the Milne universe, which arelines of constant universal time, have been mapped intohyperbola in the ( T, R ) coordinates.The remaining question is: how do lines of constantcosmological time map onto the situation where the one-way speed of light is anisotropic? Hence we undertakethe transformation of the situation in Figure 3 throughthe mathematics present in Section 2, noting that thevelocity is given by v = dRdT , and present the result inFigure 4. Again, the extreme case is considered, so thespeed of light in one direction is 1 / / κ . To considerthis, we explore the form of the synchronous lines inthe FRW universe (lines of constant cosmological time)in the Milne universe when we consider the anisotropicspeed of light. In Figure 5 the emitters from the previousdiscussion are presented as filled circles, but now valuesof κ of 0 to 1 in steps of 0 . κ increases, the shape of this synchronous linesbecomes more asymmetrical. Also shown, as filled circles,is the location of the emitters for each of the consideredvalues of κ ; importantly, it should be noted that theirlocation in ˜ R is fixed.We can explore the impact of differing values of κ interms of geometry in the space-time diagram. Consider Lewis & Barnes
Figure 5.
The line of simultaneity in the emitter in the FRW coordinates (Figure 2), mapped into the anisotropic velocity of lightcoordinates (Section 4) from κ = 0, the isotropic case represented as a hyperbola, to the extreme case, with κ = 1, both presented inbolder red, with intermediate cases, in steps of κ = 0 .
2, presented in lighter red. The filled circles represent the location of the emitter inthese coordinates for each of the cases. Note that the spatial location of the emitter in the ˜ R coordinate is independent of κ . an observer located at ˜ T = ˜ t o at the origin, and emitterswho have experienced a proper-time, ˜ τ e since leavingthe origin. For an arbitrary value of κ , and noting thatthe location of the emitter when a photon is emitted,(˜ t r , ˜ r e ), is given by ˜ r e = ˜ v ˜ t e (23)and noting that, from Equation 4, that˜ t e = 1 − kv √ − v ˜ τ e (24)then, using the definition of the velocity in these coordi-nates (Equations 2),˜ r e = v √ − v ˜ τ e (25)which is independent of κ , as expected (for more expla-nation, see the detailed discussion of the transformationspresented in Anderson et al., 1998). Hence, the speed ofa light ray connecting the emitter and the observer isgiven by d ˜ rd ˜ t = ∆˜ r ∆˜ t = ˜ r e ˜ t o − ˜ t e (26)Noting that this speed equal ± κ = 0,this expression becomes d ˜ rd ˜ t = − sgn ( v )1 + sgn ( v ) κ (27)where sgn ( v ) is the sign of the velocity. When κ = 1, thisrecovers the velocity of light as being 1 / κ , the observer will see an isotropicuniverse. In his formulation of the special theory of relativity,Einstein chose the convention that the speed of light isisotropic and so equal in all directions. He also acknowl-edged that the physical predictions of his theory will beunchanged if the speed of light was anisotropic, as longas the average round-trip speed is equal to c . In thispaper, we have considered the question of the impact ofthe one-way speed of light on cosmological observations,addressing the suggestion we should observe differentsides of the sky possessing different ages if light speedwas unequal. By examining the simplest FRW universe,namely the empty Milne universe, it is seen that theanisotropic speed of light results in anisotropic timedilation effects that compensate for the differing lighttravel times. In this universe, any observer would seean isotropic universe around them, even if the speed oflight was not.The anisotropic speed of light advocate must concludethat galaxies that are a given distance away have a fasterrecession speed in one direction than in the other, andthe universe is expanding faster to the right or to theleft. However, the dependence of redshift on the speedof light means that this does not change the appearanceof the night sky. One side of the sky is not significantlymore redshifted than the other. The initial velocities he Cosmological One-Way Speed of Light We thank the anonymous referee for their positive commentson this paper. GFL thanks Derek Muller (@veritasium) andPetr Lebedev for discussions on the one-way speed of lightthat sparked this current study. These were part of a consul-tation that led to the preparation of an informative YouTubepresentation on the topic (https://youtu.be/pTn6Ewhb27k).