aa r X i v : . [ phy s i c s . g e n - ph ] D ec Interpretations of Cosmological Spectral Shifts
Dag Østvang
Department of Physics, Norwegian University of Science and Technology (NTNU)N-7491 Trondheim, Norway
Abstract
It is shown that for Robertson-Walker models with flat or closed space sections, all of the cosmological spectral shift can be attributed to the non-flat connection(and thus indirectly to space-time curvature). For Robertson-Walker models withhyperbolic space sections, it is shown that cosmological spectral shifts uniquelysplit up into “kinematic” and “gravitational” parts provided that distances aresmall. For large distances no such unique split-up exists in general. A number ofcommon, but incorrect assertions found in the literature regarding interpretationsof cosmological spectral shifts, is pointed out.
Although there is in general no dispute about actual predictions coming from universemodels based on General Relativity (GR), interpretations of the nature of the cosmicexpansion/contraction and cosmic spectral shifts predicted by these models, have onthe other hand been subject to some lengthy controversy. (See e.g., [1] and referencestherein.) The controversial question is, when pulses of electromagnetic radiation areemitted and received between “fundamental observers” (FOs) following the cosmic fluid(with no peculiar motions); what is the nature of the resulting spectral shifts?What is completely uncontroversial is the fact that the ratio of the observed andemitted wavelengths λ obs and λ em , respectively, is related to the ratio of the cosmicscale factors at observation and emission a obs and a em , respectively, and the cosmologicalspectral shift z via the formula λ obs /λ em = a obs /a em = 1 + z . However, the controversialpart of said question is to what extent, if any, such cosmological spectral shifts can beinterpreted as Doppler shifts in flat space-time.One school of thought claims that, since the Equivalence Principle (EP) says thatspace-time is locally flat, the nature of cosmological spectral shifts must be interpretedas Doppler shifts in flat space-time in the limit where distances between FOs go tozero. A different view is that, since the FOs are at rest with respect to the cosmicfluid (defining an “expanding frame”) and since cosmological spectral shifts are given1rom the formula shown above rather than from the special-relativistic Doppler formula,cosmological spectral shifts should in principle have nothing to do with Doppler shifts,even for arbitrarily small distances. These interpretations are known as the “kinematic”(in the narrow sense of the word) and the “expanding space” interpretations, respectively.Note that for curved space-times, there is no general agreement between proponents of the“expanding space” interpretation on whether or not cosmological spectral shifts should beentirely attributed to space-time curvature. However, some authors claim this and thusthat cosmological spectral shifts should be interpreted as some sort of “gravitational”spectral shifts whenever space-time is not flat.At first glance, at least for small distances, the difference between these interpretationsseems to be the rather trivial matter of describing the same physics using different frames.Thus, to some people it would seem reasonable to proclaim both interpretations validfor small distances and just representing equivalent points of view. However, beforesuch a solution is endorsed, it must be established that the various interpretations aremathematically consistent. But it turns out that they aren’t, since interpretations maybe associated with geometrical restrictions. In particular, in this paper we show that the“kinematic” interpretation is in general mathematically inconsistent with the geometryof the Robertson-Walker (RW) models, so that this interpretation is not valid generally.On the other hand, we show that what is crucial for interpreting spectral shifts in theRW-models is not the mere existence of an “expanding frame”, but rather how this framerelates to space-time curvature. And as we shall see in the next section, this relationshipdiffers between types of RW-models. That is why the “expanding space” interpretationis not useful in general, either.Besides, since proponents of the “expanding space” interpretation do not in generalagree on how it is related to space-time curvature, even for small distances, it will bemore clarifying to talk about spectral shifts due to space-time curvature rather than the“expanding space” interpretation. However, there is a common point of view claimingthat spectral shifts cannot really be an effect of space-time curvature since, unlike theequation of geodesic deviation, the geodesic equation does not contain components ofthe Riemann tensor, but only connection coefficients. It is true that, unlike tidal effects,spectral shifts cannot be a direct effect of space-time curvature, i.e., representing oper-ational measures of it. But it is indeed possible that spectral shifts may represent an indirect effect of space-time curvature.To see this, define (e.g., via coordinate-parametrisation) specific world lines in a (suf-ficiently small) region of some (curved) space-time geometry. Calculate spectral shiftsobtained by photon signalling between observers moving along the chosen world lines.2ow replace the metric with its flat space-time form in the region, holding the chosenworld lines and the coordinate system fixed. Calculate spectral shifts again, but nowwith the flat space-time geometry. If the results are different for the two cases, this mustcertainly be due to space-time curvature via the non-flat connection.In particular, it may be possible that the latter calculation will yield no spectral shiftat all. In such a case, it is rather obvious that the spectral shift should unambiguouslybe interpreted as purely “gravitational”, i.e., as an effect purely due to space-time cur-vature. We show in the next section that this situation arises for any RW-model withflat or spherical space sections with the FOs playing the role as “preferred” observersoperationally defining cosmological spectral shifts. To understand what is actually meant by “kinematic” and “gravitational” spectral shiftsin context of the RW-models, it is necessary to define these concepts mathematically.Such definitions should be formulated together with a recipe for spectral shift split-upinto “kinematic” and “gravitational” parts. It would perhaps seem natural to insistthat said definitions must be based on a general spectral shift split-up coming fromsome geometrical procedure being valid for all RW-models. However, it is shown in thissection that such an approach cannot be justified if the definition of “kinematic” spectralshift is required to be based on the definition of spectral shifts in Special Relativity (SR).Abandoning this requirement is certainly possible, but then the definitions of “kinematic”and “gravitational” spectral shifts will be only formal and misleading, and thus not veryuseful for interpretations.The mathematical framework considered in this paper is given by the usual 4-dimensionalsemi-Riemann manifold ( M , g ). In addition it is required that (at least a subset of)( M , g ) can be foliated into a continuous sequence of 3-dimensional spatial hypersurfaces S ( x ) parameterized by a time function x . The fundamental observers are “preferred”observers defined from the foliation by the criterion that their world lines are everywherecontinuous and orthogonal to S ( x ). The choice of foliation (and thus of time coordinate)is required to be unambiguously made from purely geometrical selection criteria.Since this paper is about the RW-manifolds, the analysis presented here is restrictedto one specific choice of selection criteria. This specific choice of selection criteria picksout space-time manifolds that can be foliated into a set of hypersurfaces such that thespatial geometry is everywhere isotropic and homogeneous. Moreover, the unit normalvector field to the hypersurfaces should not be a (time-like) Killing vector field. This3ast criterion excludes static manifolds with topology S× R equipped with a foliationdetermined from the product topology (here S is one of the space geometries R , S or H ). These selection criteria uniquely yield the RW-manifolds each equipped with theprescribed “preferred” foliation, determining “preferred” hypersurfaces. (In the next sec-tion, we will also consider other foliations of “open” RW-manifolds than the “preferred”ones as useful for specific calculations.)Given ( M , g ) and a foliation of it into spatial hypersurfaces S ( x ), spectral shiftsobtained by exchanging light signals between nearby FOs are unambiguously determinedfrom the space-time geometry. Moreover, this holds irrespectively of any particular choiceof field equations, so it is not necessary to assume the validity of the GR field equa-tions. Consequently, the results obtained in this paper depend only on the geometryof space-time with no extra assumptions. In particular, no particular relationship be-tween geometrical quantities and matter sources is assumed to hold. Nevertheless, wewill by convention call spectral shifts entirely due to space-time curvature for “gravita-tional”spectral shifts. (See Definition 1 below for such a situation.)To calculate spectral shifts in general, there exists a simple geometric procedure, asfirst pointed out by Synge. That is, imagine a pulse of electromagnetic radiation beingemitted at some given event and subsequently observed at some other given event. Then,by parallel-transporting the 4-velocity u e of the emitter along the null curve connectingthe given events, the parallel-transported 4-velocity of the emitter can be projected intothe local rest frame of the observer. This yields a 3-velocity that can be inserted intothe special-relativistic Doppler formula to give the desired spectral shift. (For the fullmathematical details of this procedure, see [2].) This procedure works for any relativisticspace-time (and even for cases where the space-time geometry is not semi-Riemann [3]),and implies that it is always possible to interpret spectral shifts as due to the Dopplereffect in curved space-time, without any geometrical restrictions whatsoever. Moreover,this procedure illustrates that what is relevant for calculating spectral shifts are theconnection coefficients, since these enter into the mathematical expression for parallel-transport. Any non-zero values of the connection coefficients may arise due to the choiceof coordinates, a non-flat connection, or both.For any RW-manifold, given some coordinate system covering (part of) it, the con-nection coefficients relevant for spectral shifts obtained from photon signalling betweenFOs are uniquely determined from the evolution with time of the spatial geometry h ofthe “preferred” foliation in the direction of the unit vector field n normal to the hyper-surfaces. That is, the relevant quantity is given from the extrinsic curvature tensor K defined by (in component notation using a general coordinate system { x µ } and using4instein’s summation convention, see, e.g., ref. [4], p. 256) K µν ≡ £ n h µν = 12 (cid:16) h µν , α n α + h αν n α , µ + h µα n α , ν (cid:17) , (1)where £ n denotes the Lie derivative in the n -direction and where a comma denotesa partial derivative. Using a spherically symmetric hypersurface-orthogonal coordinatesystem { x , χ, θ, φ } where n µ = (1 , , , ds = − ( dx ) + a ( x ) (cid:16) dχ + Σ ( χ ) d Ω (cid:17) , d Ω ≡ dθ + sin θdφ , (2)where a ( x ) is the scale factor of the hypersurfaces and whereΣ( χ ) = sin χ for hypersurfaces with spherical geometry ,χ for hypersurfaces with flat geometry , sinh χ for hypersurfaces with hyperbolical geometry . (3)Using the form (2) of the metric, equation (1) for the extrinsic curvature of the “preferred”hypersurfaces (embedded into the RW-manifolds) takes the form (with ˙ a ≡ dadx ) K µν = 12 ∂∂x h µν = 1 c H ( x ) h µν , H ( x ) ≡ ˙ aa c, (4)where H ( x ) is the Hubble parameter. In a hypersurface-orthogonal coordinate system(such as used in equation (2)), the nonzero components of the spatial metric h can befound directly from the spatial part of the space-time metric g , yielding the nonzerocomponents of K from equation (4). Note that K is a tensor field on space (since thescalar product K · n = ), and that H ( x ) = c K µµ is a scalar field (constructed from the“preferred” foliation). This means that H ( x ) is not a coordinate-dependent quantity,despite the fact that the relationship between H ( x ) and the connection coefficientscertainly is.Now it turns out that there exists an expression for the intrinsic Riemann curvaturetensor P of the hypersurfaces in terms of the space-time Riemann curvature tensor R and the extrinsic curvature tensor K . This is the well-known Gauss equation (see, e.g.,ref. [4], p. 258), and in component notation it reads P αβγδ = R λρµν h αλ h ρβ h µγ h νδ + K αδ K βγ − K αγ K βδ . (5)Moreover, contracting equation (5) twice and using equation (4), we get P = R + 2 R αβ n α n β − H /c , ⇒ H = c h G αβ n α n β − P i , (6)5here R and P are the scalar curvatures of space-time and space, respectively, and where R αβ and G αβ are the components of the Ricci tensor and the Einstein tensor on space-time, respectively. Equation (6) represents a well-known constraint equation as part ofthe initial-value problem applied to the RW-manifolds equipped with the “preferred”foliation.Now we see from equation (4) that there can be no spectral shift (detected by photonsignalling between FOs) if K = . Therefore, to make sense of any “kinematic” part ofthe spectral shift having a similarity with spectral shifts in SR, it must be possible tohave a limit where the relevant part of R may be neglected but such that K = . If sucha limit does not exist, the spectral shift must be entirely due to space-time curvature(i.e., “gravitational”). Whether or not such a limit exists can be found from equation(6). (The Weyl tensor vanishes identically for the RW-manifolds, so the Ricci tensor(or the Einstein tensor) captures all aspects of space-time curvature.) We thus have thedefinition Definition 1
Assume as given a semi-Riemann manifold ( M , g ) of RW-type and a fo-liation of it into “preferred”isotropic and homogeneous spatial hypersurfaces S ( x ) (withunit normal vector field n ) defined from equation (2). Also denote any hypersurface met-ric by h with extrinsic curvature tensor K , intrinsic Riemann curvature tensor P andintrinsic curvature scalar P . The space-time Einstein curvature tensor is denoted by G .Then, if it is not possible to have G αβ n α n β arbitrary small independent of P with K = ,any spectral shift obtained by photon signalling between FOs is entirely due to space-timecurvature. If a situation like that described in Definition 1 occurs, any definitions and spectralsplit-ups that allow for a non-zero “kinematic” spectral shift do not make sense, sincethe correspondence with spectral shifts in SR will be lost. This is why the approach ofstarting with general definitions of “kinematic” and “gravitational” spectral shifts validfor any RW-models cannot be justified, since, as we shall see, the situation described inDefinition 1 occurs for all RW-models where the “preferred” foliation consists of flat orspherical hypersurfaces.To prove that the situation described in Definition 1 occurs for the case of flat hyper-surfaces, it is obvious from equation (6) that it is not possible to have a flat RW-manifoldwith flat spatial sections (i.e., P ≡ ) and still have H ( x ) =0. That is, the requirements P = 0, G = mean that equation (6) is satisfied only for H ( x ) = 0. Note that this isnot in any way a coordinate-dependent result. Thus we arrive at the conclusion that tohave a RW-manifold with flat space sections and at the same time H ( x ) =0, space-time6ust be curved. This means that according to Definition 1, spectral shifts observed byexchanging photons between FOs in a RW-manifold with flat spatial sections are entirelydue to space-time curvature. Since this result holds irrespective of distances betweenFOs, we are forced to interpret the relevant spectral shifts as purely “gravitational” forall RW-manifolds with flat spatial sections.A similar result holds for the closed RW-manifolds (with spherical spatial sections).In this case P = a >
0, and equation (6) yields that it is not possible to have G αβ n α n β arbitrary small independent of P such that H >
0. This result is a consequence ofthe fact that it is not possible to foliate Minkowski space-time into hypersurfaces with S -geometry. So, from Definition 1 we have that spectral shifts observed by exchangingphotons between FOs in a RW-manifold with closed (spherical) spatial sections are entirelydue to space-time curvature. We are then forced to interpret all relevant spectral shiftsas purely “gravitational” for all RW-manifolds with spherical spatial sections as well.We are thus left with open RW-manifolds foliated into hyperbolical hypersurfaces asthe only nontrivial case when it comes to interpretations. In this case P = − a <
0, soit is indeed possible to choose G αβ n α n β arbitrary small independent of P in equation (6)together with H > K = ), so that the situation described in Definition 1 does notoccur. This means that for the case P <
0, it may make sense to define a spectral shiftsplit-up into “kinematic” and “gravitational” parts. In particular it is possible to choose G = , P < H ( x ) >
0, since (part of) Minkowskispace-time can be foliated into hypersurfaces with H -geometry. This special case is the“empty” RW space-time (Milne model), which is just a subset of Minkowski space-timeand thus flat. The line element is given by equations (2) and (3) by setting a ( x ) = x ,i.e., ds = − ( dx ) + ( x ) ( dχ + sinh χd Ω ) . (7)In this case it is obvious that the “kinematic” interpretation is correct since the cosmicexpansion is entirely due to the “preferred” choice of space-time foliation into space andtime. That is, by switching to standard coordinates r ≡ x sinh χ , x ′ ≡ x cosh χ , anotherfoliation is chosen and the line element takes the familiar Minkowski form expressed inspherical coordinates. This means that, by performing a suitable coordinate transfor-mation, it is possible to eliminate the connection coefficients altogether. Moreover, thecosmic redshift can be found locally from the speed w F of a FO relative to a local ob-server moving normal to the x ′ =constant hypersurfaces. We will exploit this fact whentreating general open models in section 3.We may now define a “purely kinematic” spectral shift as one occuring in a RW-7anifold foliated into hyperbolical hypersurfaces for situations where the difference be-tween a non-flat and a flat connection does not matter for photon propagation betweennearby FOs. That is, it may be possible that the contribution to equation (5) from ex-trinsic curvature at some epoch x is identical to the contribution to equation (5) fromextrinsic curvature of a hyperbolic hypersurface with identical geometry but embeddedin Minkowski space-time. (In such a situation, H and P will be identical for the twohypersurfaces, meaning that G αβ n α n β must vanish even at the hypersurface embeddedin curved space-time in order not to violate equation (6).) To find what the latter con-tribution is, it is convenient to use a hypersurface-orthogonal coordinate system as usedin equation (2). One then finds that the contribution to equation (5) from extrinsic cur-vature depends on ˙ a ( x ) but not on a ( x ). Since ˙ a ( x ) = 1 for the Milne model givenin equation (7), we have the definition: Definition 2
Assume as given a semi-Riemann manifold ( M , g ) of RW-type and a fo-liation of it into isotropic and homogeneous spatial hypersurfaces S ( x ) with hyperbolicintrinsic geometry (see equation (2)). Also assume the existence of some hypersurface S ( x ) with spatial metric h ( x ) and extrinsic curvature tensor K ( x ) given from equa-tion (4) (in a hypersurface-orthogonal coordinate system) with | ˙ a ( x ) | = 1 . Then spectralshifts resulting from photon signalling between nearby FOs close to S ( x ) are defined tobe “purely kinematic” in the limit where distances between FOs go to zero and x → x . Definition 2 is based on the fact that for the open RW-models, it may be possible tohave a situation where the contribution to equation (5) from extrinsic curvature at theepoch x is identical to that of a hyperbolic hypersurface with identical geometry in theMilne model at epoch x + b where b is some constant. In such a situation, the relevantconnection coefficients for the open model at epoch x will be identical to those for theMilne model at epoch ˜ x , where the scale factor is given by a (˜ x ) = ˜ x = x + b and suchthat a (˜ x ) = x + b = a ( x ). In the next section, we will give some specific examples ofopen RW-models where this situation occurs. The main result of the previous section was that all
RW-models foliated into flat orspherical hypersurfaces satisfy the situation described in Definition 1. Therefore, all thecosmic spectral shift in these models must be due to space-time curvature, so that aspectral shift split-up into “kinematic” and “gravitational” parts does not make sense forthese RW-models. On the other hand, we show in this section that for RW-models foliated8nto hyperbolical hypersurfaces, such a spectral shift split-up can be defined consistentlyfor small distances and in agreement with Definition 2.To define a split-up of spectral shifts into “kinematic” and “gravitational” parts (validfor RW-models with hyperbolical spatial sections), it is convenient to change the space-time foliation. Note that the change of foliation is made because it makes calculationseasier and the correspondence with the Milne model clearer. Note in particular that theFOs are still being defined as those observers moving normal to the “preferred” foliationgiven from equation (2), and that the defined spectral shift split-up applies only to theFOs. Observers moving normal to the new foliation only play an auxiliary role.The relevant change of foliation is made by switching to new coordinates r ≡ a ( x )sinh χ , x ′ ≡ a ( x )cosh χ , so that the line element (2) transforms to ds = − (cid:16) ˙ a − − r ( x ′ ) − r ( x ′ ) (cid:17) ( dx ′ ) − rx ′ (cid:16) − ˙ a − − r ( x ′ ) (cid:17) dx ′ dr + (cid:16) − r ( x ′ ) ˙ a − − r ( x ′ ) (cid:17) dr + r d Ω , rx ′ < inf {| ˙ a | − , | ˙ a |} , (8)where ˙ a ≡ dadx = ˙ a ( x ′ , r ) is now a function of both the time coordinate and the radialcoordinate. We can now use equation (8) to find the spectral shift of light emitted by aFO located at the radial coordinate χ (i.e., with coordinate motion drdx ′ = rx ′ ) as observedby a FO located at the origin χ = 0. Moreover, for small values of χ , we will show thatto lowest order in rx ′ , this spectral shift can be written as a sum of “kinematic” and“gravitational” contributions. Note that, since a ( x ) = p ( x ′ ) − r = x ′ + O (2), thechoice of foliation (i.e., the choice of time coordinate) leading to equation (8) is uniqueto first order in the small quantity rx ′ , but not higher. This means that any split-up ofspectral shifts into “kinematic” and “gravitational” parts is limited to small distances.To arrive at the desired spectral shift split-up, we first split up the 4-velocity u F of theFOs into parts normal and tangential to the hypersurfaces x ′ =constant. This split-upreads u F = γ ( c ˜n + w F ) , γ ≡ (1 − w F c ) − / , (9)where ˜n is the unit normal vector field of the hypersurfaces x ′ =constant and w F is the3-velocity difference (with squared norm w F ) between a FO and a local observer movingnormal to these hypersurfaces. Only the r -component of this equation is of interest, and9t reads w r F = drdx ′ dx ′ dτ N + N r N c, cdτ N = N dx ′ = vuut − r ( x ′ ) ˙ a − r ( x ′ ) dx ′ , N r = rx ′ − ˙ a [ ˙ a − r ( x ′ ) ] , (10)where N is the lapse function and N r is the shift vector r -component of observers movingnormal to the hypersurfaces x ′ =constant. The relation of these quantities to the lineelement given by equation (8) can be found from the formula ds = [ N r N r − N ]( dx ′ ) + 2 N r dx ′ dr + ˜ h ij dx i dx j , (11)where ˜ h ij are the components of the hypersurface metric, found explicitly from equation(8). It is straightforward to calculate the speed w F , and we find that w F ≡ q w i F w j F ˜ h ij = tanh χ | ˙ a | c = r | ˙ a | x ′ c. (12)The speed w F can now be put into the special-relativistic Doppler formula to findthe spectral shift as measured by a local observer moving normal to the hypersurfaces x ′ =constant. Applied to the Milne model, this approach yields a local determinationof the cosmological spectral shift in flat space-time. It is thus natural to define a moregeneral “kinematic” spectral shift z k valid for open models, found locally and given by1 + z k ≡ s ± w F /c ∓ w F /c , ⇒ z k = ± r | ˙ a | x ′ + O (2) = ˙ a − H ( x ) rc + O (2) . (13)We see that if | ˙ a |→
1, we have a situation where the spectral shift is defined as “purelykinematic” according to Definition 2, and is identical to the special-relativistic result.Moreover, one may easily see that this definition yields the expected result z k = 0 ifextrapolated to RW-models with flat spatial sections. That is, the coordinate trans-formation r ≡ a ( x ) χ , x ′ ≡ a ( x ) yields the counterpart expression to equation (8) of theline element valid for RW-models with flat space sections. Hence w F ≡ x ′ =constant hypersurfaces as well.Next, we note that an observer moving with constant r -coordinate and a local observermoving normal to the hypersurfaces x ′ =constant will not have coinciding world lines,but will have a 3-velocity difference w . We will now show that the corresponding speed w can be used to define a local determination of “gravitational” spectral shift. To dothat, similar to equations (10) and (12), we find the quantities w r = N r N c = rcx ′ − ˙ a q ( ˙ a − r ( x ′ ) )(1 − r ( x ′ ) ) , ⇒ w = | − ˙ a | r | ˙ a | x ′ (1 − r ( x ′ ) ) c, (14)10nd these expressions vanish in the limit | ˙ a |→
1, as they should for “gravitational” quan-tities. It is thus natural to associate the corresponding spectral shift with space-timecurvature, i.e., it should be due to “gravitational” causes. Since the sign of w r dependson whether 1 − ˙ a is positive or negative, the contribution to the spectral shift with re-spect to the emitting FO will be either negative or positive, respectively. That is, whatenters into the special-relativistic Doppler formula is not the speed w , but rather thequantity w ± defined by w ± ≡ ( ˙ a − r | ˙ a | x ′ (1 − r ( x ′ ) ) c, (15)which may be used to define a “gravitational” spectral shift z g (valid for open models)given by 1 + z g ≡ s ± w ± /c ∓ w ± /c , ⇒ z g = ± r ( ˙ a − | ˙ a | x ′ + O (2) . (16)Again, one may easily check that a similar definition extrapolated to RW-models withflat spatial sections yields the expected lowest-order result z g = H ( x ′ ) rc .If the observer moving with constant r -coordinate emits light that is detected by theFO residing in the spatial origin, the resulting spectral shift will be of higher order inthe small quantity rx ′ , so this contribution can be neglected. (Here, a possible effect ofnonzero ¨ a ≡ d adx may also be neglected if said small quantity is small enough.) This meansthat to lowest order, the total spectral shift measured by the FO residing at the origincan be written as a sum of “kinematic” and “gravitational” contributions, and that thisspectral shift is given by z = z k + z g + O (2) = ± h ( ˙ a −
1) + 1 i r | ˙ a | x ′ + O (2)= ±| ˙ a | rx ′ + O (2) = H ( x ) rc + O (2) , (17)which is the familiar lowest-order expression for cosmological spectral shifts. Moreover,for small distances the split-up defined in equation (17) is unique. On the other hand, forlarge distances, cosmological spectral shifts in an open RW-model cannot uniquely be splitup into “kinematic” and “gravitational” parts. This is so since other foliations (coincidingwith the foliation defined by the x ′ -coordinate for small distances but differing from itfor large distances) may be equally well be used when defining spectral shift split-up bythe method described above. 11o illustrate the meaning of the split-up defined in equation (17), we finish this sectionwith some simple examples. First, we choose a form of the scale factor consistent with aradiation-dominated universe as predicted by GR, i.e., a ( x ) = √ a ∗ x , ˙ a = 12 r a ∗ x = a ∗ a , z k = 2 ra ∗ + O (2) , z g = [ 12 x − a ∗ ] r + O (2) , (18)where a ∗ represents an arbitrary constant reference scale. We note that z k does notdepend on epoch. Furthermore, we see that z g is positive for early epochs, vanishesfor x = a ∗ /
4, and becomes negative for later epochs. The particular epoch where z g vanishes is (of course) determined by the condition ˙ a = 1. At this epoch, the expansionof the universe momentarily mimics that of the Milne model with a “shifted” scale factorgiven by a ( x ) = x + a ∗ /
4. Hence, since we can neglect the effect of ¨ a for small enoughdistances, in this limit the “kinematic” interpretation of the cosmic redshift will hold,despite the fact that space-time is not flat. However, at earlier epochs ˙ a > a < a ( x ) = ( a ∗ ) ( x ) , ⇒ ˙ a = 23 (cid:16) a ∗ x (cid:17) = 23 r a ∗ a ,z k = 3 r a ∗ ) ( x ) + O (2) , z g = h x − a ∗ ) ( x ) i r + O (2) , (19)where again a ∗ is an arbitrary constant reference scale. We note that, unlike the previousexample, in this case z k depends on epoch. Moreover, z g vanishes for the epoch x = a ∗ ,and at this epoch the universe expands momentarily as an “empty” RW-model with a“shifted” scale factor given by a ( x ) = x + a ∗ . So at this particular epoch the cosmicredshift should be interpreted as a pure “kinematic” effect in flat space-time for smalldistances (even though space-time is not flat). However, this interpretation breaks downfor other epochs.A final example is given where the scale factor is determined by a (positive) cosmo-12ogical constant Λ, i.e., a ( x ) = r
3Λ sinh hr Λ3 x i , ⇒ ˙ a = cosh hr Λ3 x i = r a ,z k = ra q Λ3 a + O (2) , z g = Λ3 ar q Λ3 a + O (2) . (20)We note that in this case, for very early epochs x →
0, the cosmic expansion mimics thatof the Milne model so that z k →∞ and z g → z k decreases exponentially, so it can soon be neglected. Thus, at late epochs, the cosmicredshift should be interpreted as due to space-time curvature (i.e., “gravitational”) withnegligible “kinematic” contribution. The results obtained in section 2 for the flat and closed RW-models were also arrivedat by Roukema [1], using topological methods. That is, by changing the topology ofthe spatial sections of the relevant metrics (given by equations (2) and (3)) from simplyconnected to multiply connected, but without changing the geometry, it was shown thata contradiction arises if spectral shifts are interpreted as due to the Doppler effect in flatspace-time. On the other hand, considering a less general case than for flat and closedRW-models, this contradiction did not occur for open RW-models, except for certainlarge distances. This means that the results obtained in section 3 do not match the cor-responding results in [1], so there seems to be a contradiction. (This would indicate thatusing topological methods is not sufficient for analysing the RW-models with hyperbolicspatial sections.) On the other hand, searching the relevant literature, one finds thatreference [1] is about the only one emphasizing the crucial role of the spatial geometrywhen it comes to interpretations. Otherwise, what has been discussed is the “kinematic”versus the “expanding space” views with no due weight on spatial geometry. It has evenbeen claimed [5] that spatial geometry is irrelevant for interpretations of certain cos-mological gedanken -experiments involving radar distances and spectral shifts, since thecalculated results of such hypothetical experiments do not depend on the spatial partsof the metrics (2). But this argument is flawed since the actual debate is about interpre-tations of models rather than of experimental results. Moreover, there is absolutely noscientific requirement that different interpretations of models should be experimentallydistinguishable. 13nother, common but incorrect assertion is that the effects on spectral shifts of curvedspace-time, as compared to “kinematic” effects, can always be neglected in the RW-models for sufficiently small distances. The argument is that, since one may alwayschoose local coordinates such that the tangent space-time at some event P (given, e.g.,by x = x , χ = 0) takes the standard Minkowski form, and in a (small) neighbourhoodof P approximates the space-time metric to first order in small quantities, the effects ofspace-time curvature can be made negligible in a sufficiently small neighbourhood of P .(The EP ensures that such a coordinate system can be found for any metric.) So far, theargument is of course correct. But it is then incorrectly claimed that in such a coordinatesystem, the FOs will define a (radial) velocity field v ( r ) = H r + O (2) (where H is thelocal Hubble parameter) with respect to the FO momentarily residing in P (where r = 0).Since to desired accuracy, v ( r ) by construction represents space-time geodesics defininga velocity field in flat space-time, it is concluded that this proves that cosmic spectralshifts must be interpreted as a purely “kinematic” for small enough distances.The flaw in this reasoning is that the inertial observers defining v ( r ) can in generalnot be identified with the FOs. That is, it is certainly always possible to construct a set ofgeodesics in flat space-time defining a velocity field v with respect to some chosen specificobserver, such that photon signalling between this chosen observer and the observersdefining v mimics Hubble spectral shifts. It is also always possible to identify the chosenobserver with some FO in a curved RW-manifold. But there is absolutely no guaranteethat the FOs in the curved RW-manifold can be identified with the observers defining v .Such observers will in general be some other observers, moving along different geodesicsthan the FOs. In other words, in the curved RW-manifold one started out with, theseother observers will have non-zero peculiar velocities with respect to the FOs. Thiswill obviously not change in the flat space-time approximation, since geodesic deviationcan be neglected for small enough regions. This means that, to be able to interpret thevelocity field v , it is necessary to know the relationship between the FOs and the observersdefining v . That relationship can only be found from the nature of the connection, i.e.,by knowing how well it may be approximated by a flat connection. For example, for thesituation described in Definition 2, a flat connection is a sufficient approximation so thatthe FOs really can be identified with the observers defining v . On the other hand, a flatconnection contributes nothing at all to v for RW-models with flat or spherical spacesections.To see that the effects of a non-flat connection cannot be neglected in general, even forsmall distances, it is illustrating to write the scale factor a ( x ) as a Taylor series around14he event P , i.e., a ( x ) = a ( x ) + ˙ a ( x )[ x − x ] + 12 ¨ a ( x )[ x − x ] + · · · , v ( r ) = ˙ a ( x ) a ( x ) rc + O (2) . (21)Since the relevant connection coefficient for radial motion as obtained from equation(2) is given by Γ χ χ = ˙ aa , we see that the construction of the velocity field v ( r ) in flatspace-time depends only on the fact that this connection coefficient is non-zero. Sincethis is true regardless of the RW-model, one has actually by construction transformed all relevant effects, “kinematic” and curvature effects alike, into v ( r ). In other words,since nothing at all is said regarding the nature of Γ χ χ , and the observers defining v ( r )remain unidentified, the construction of v ( r ) is in fact irrelevant for interpretations of theexpansion.A paper based on the faulty line of reasoning outlined above is [6], claiming thatinterpretations of spectral shifts between FOs for small distances depend on the choice ofcoordinate system and method of calculation. Moreover, it is argued that cosmologicalspectral shifts are most “naturally” interpreted as Doppler shifts in flat space-time forsmall distances. But as we have seen, these claims are simply incorrect. A relatedidea advocated in [6], is that spectral shifts between FOs can equally “naturally” beinterpreted as Doppler shifts in flat space-time even for large distances. To justify thisassertion, the total spectral shift is being thought of as an accumulated effect of manysmall Doppler effects in flat space-time. But this logic will, of course, break down sincespectral shifts between FOs cannot, in general, consistently be interpreted as Dopplershifts in flat space-time even for small distances. On the other hand, the antithesis of [6]is a paper [7] where it is (also incorrectly) argued that cosmic spectral shifts involvingFOs only must “definitely” be interpreted as “gravitational” (with an exception for theMilne model). This claim is based on the specific choice of (discontinuous) scale factor a ( x ) = 1 + θ ( x ), where θ ( x ) is the Heaviside step function. It is then argued that theresulting cosmological spectral shift cannot be interpreted as a Doppler shift in flat space-time since both source and receiver are at rest when the signal is emitted or received.Moreover, it is argued that the sudden “non-local motion” occurring in this exampleshould shed light on the interpretation of cosmological spectral shifts obtained in any“non-empty” RW-model. However, as we have seen in sections 2 and 3, a mere choiceof scale factor without considering spatial geometry is not sufficient for interpretationsof cosmological spectral shifts obtained in the RW-models. Besides, counterexamples tothe claim that cosmological spectral shifts obtained from any non-empty RW-model must“definitely” be interpreted as “gravitational” are presented in section 3 of this paper, forsituations more generally described in Definition 2 (see section 2).15here have also been earlier attempts to split up cosmic spectral shifts into “kine-matic” and “gravitational” parts (for small distances). Such have been based on a Taylorexpansion similar to that shown in equation (21) in combination with a Newtonian ap-proximation to calculate the “gravitational” contribution (a second order blueshift, see,e.g., [8, 9]). However, as we have seen, using equation (21) for this purpose is misguided.Besides, since interpretations of spectral shifts in the RW-models should be based ontheir geometric properties only, without referring to specific dynamical laws, any use ofNewtonian approximations only confuses the issue.A recent attempt of defining said split-up in general (even for large distances) has beenmade in [10]. In that paper, the “recession velocity” is defined as the 3-velocity obtainedby parallel-transporting the 4-velocity of the emitting FO to the observing FO along aspace-like geodesic lying in a hypersurface of constant cosmic time, and then projectingthe resulting 4-velocity into the local rest frame of the observing FO. This “recessionvelocity” then defines the “kinematic” part of the cosmic spectral shift. (But as shown insection 2, this approach does not make sense for RW-models with flat or spherical spacesections since with this definition, there is no correspondence with spectral shifts in SR.)The difference between the total cosmic spectral shift and the “kinematic” spectral shiftis interpreted as a “gravitational” spectral shift. It was shown that this definition of“gravitational” spectral shift agrees with that found in [8, 9] for small distances. Butwhile the effort made in [10] is certainly ingenious, this does not change the fact that theresulting interpretations are in general inconsistent with the geometry of the RW-models,as explained in this paper and in [1]. For several years, a debate has been going on in the scientific literature regarding thenature and interpretation of cosmological spectral shifts. This debate is primarily abouttheoretical models based on GR and whether or not different interpretations of cosmicspectral shifts are consistent with these models.As a general rule, any interpretation is consistent with any theoretical model as longas no logical or mathematical inconsistencies arise. Therefore, different interpretations ofthe same features of any model may in principle be possible. One often hears that this isthe case for the interpretation of cosmic spectral shifts. However, in this paper, we haveshown that the school claiming general validity (at least for sufficiently small distances)of a “kinematic” interpretation of cosmological spectral shifts, is in error. This is sosince geometric properties of the RW-models are inconsistent with such interpretations,16xcept for the Milne model and special epochs in open RW-models. In particular, we haveshown that for flat and closed RW-models, there can be no cosmic expansion withoutthe relevant space-time curvature since otherwise, the Gauss equation would be violated.Therefore, in these models, cosmic spectral shifts must be interpreted as an effect solelydue to space-time curvature. For open models, interpretations are more subtle, sincehere, at least part of the spectral shifts will be “kinematic”.So, is the nature of the cosmic expansion now fully understood and all controversysettled once and for all? This is not likely, since convincing opponents of their erroneousarguments and points of view is very difficult. Besides, an alternative space-time frame-work exists, the so-called quasi-metric framework (QMF), where the cosmic expansion isdescribed as new physics not covered by GR or Newtonian concepts, and its nature dif-fers radically from its counterpart in the RW-models [3]. The QMF describes the natureof the cosmic expansion as “non-kinematic” in the sense that it is not a part of space-time’s causal structure. (Thus one may argue that the nature of the cosmic expansion inthe RW-models is indeed “kinematic” in the broader sense of being part of space-time’scausal structure, without specifying any particular dynamical model.) Moreover, unlikeGR, quasi-metric space-time is by construction equipped by a “preferred” global foliationinto 3-dimensional, simply connected and closed spatial hypersurfaces defining “space”.There is some resemblance to a closed RW-model since the “non-kinematic” expansionalso defines extra space-time curvature via a non-flat connection. That is, just as for theclosed RW-models, in the QMF the cosmic redshift is an effect of space-time curvature.However, a rather unique prediction of the QMF is that gravitationally bound systemsshould expand in general, and this prediction has observational support in the solar sys-tem [11, 12]. (However, the significance of the observations referred to in [12] has beenchallenged in recent years.) This means that it should be possible in principle to test thenature of the cosmic expansion by doing controlled experiments in the solar system. Butbased on the GR prediction that the cosmic expansion in the solar system should be fartoo small to be detectable, both the evidence in favour of local cosmic expansion and thepossibility of doing controlled experiments to test it have been ignored so far.As a final remark, I regret to say that if the scientific discussion regarding popularcosmology were sound, it would not have been necessary for me to write the presentpaper. However, in this field much low-quality and confusing material has been publishedby people who should know better. As a result, several incorrect arguments based onpersonal intuition seem to have been accepted as “mainstream”, misleading people and inparticular students. 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