On cosmological expansion and local physics

We find an exact convergence in the local dynamics described by two supposedly antagonistic approaches in modern cosmology: one starting from an expanding universe perspective such as FLRW, the other based on a local model ignoring any notion of expansion, such as Schwarzschild dS. Both models are in complete agreement when the local effects of the expansion are circumscribed to the presence of the cosmological constant. We elaborate in the relevant role of static backgrounds like the Schwarzschild-dS metric in standard form as the most proper coordinatizations to describe physics at the local scale. Finally, making use of an old and too often forgotten relativistic kinematical invariant, we clarify some widespread misunderstandings on space expansion, cosmological and gravitational redshifts. As a byproduct we propose a {\sl unique and unambiguous} prescription to match the local and cosmological expression of a specific observable.


Motivation
Different and opposing views coexist at this moment as to whether the expansion of the universe affects the local dynamics at the scale of the solar system, and the amount of this effect. It is true that the consequence, if any, is too small to be detectable, but the question of principle remains as to what impact has the expansion of the universe on local systems. Here we focus on two -in principle-opposite views that compete in this arena: i) On one side there is the very popular "expanding space" picture which claims that it is the very fabric of 3-space that is growing as time passes by, thus giving rise to the observational effects like the recession of galaxies. Imported to the local 1 ground, albeit perhaps with different nuances, there is the view that such an effect may exist [1,2,3,4]. To put it simply, its effect on the local dynamics 2 of a particle would boil down to an additional repulsive acceleration term, a functional of the scale factor a(t) present in the FLRW background metric, and of the particle's position.
ii) On the other side, other analysis ignore every fact about the expanding universe, reducing its local effect to the presence of a non-vanishing cosmological constant. In doing so one applies locally the Schwarszchild de Sitter metric in its static form, leaving no room whatsoever for the very idea of any possible effect of the expansion itself [5].
It clearly seems that both pictures can't hold simultaneously. We will try in this paper to elaborate in favour of what we think is the correct standpoint. It is based on an obvious fact, to wit, that when examining the adequacy of a metric in order to describe a certain physical situation, one must ensure its consistency with the right hand side (rhs) of Einstein's equations, that is, the matter energy-momentum tensor 3 .
Consider for instance the usual layman question: if space is expanding, does this means that my home is expanding?, followed with the intriguing: but, if my measuring stick is expanding too, how can I measure such an effect in the first place?. If we take a look at the Einstein equations at our local scale, we will find an answer to the former question, which in turn makes the latter void of content. What can one infer from the Einstein equations at our local scale? First and foremost: that, except for the possible presence of a cosmological constant, there is no trace whatsoever of the homogeneous Hubble flow which sources the FLRW metric 4 . The reason is more than obvious: the Hubble flow is the averaged picture of the distribution of matter that only works at much, much larger scales, than the local one considered here. And therefore, the FLRW metric is just a broad-brush, coarse-grained, averaged picture of the real metric of spacetime, only apt to describe phenomena at the cosmological scale. Simply we can not continue to use this concept of Hubble flow at the local scale and insist on its homogeneity. Thus there is no expanding space at all at our, local scale. Take for instance the solar system, and adopt as approximately valid the simplifying assumption of spherical symmetry of the matter external to it, then we find ourselves in the framework of the Einstein Straus approach [6] in which clearly the Hubble flow has no effect -except for its cosmological constant component-on the local system. This paper presents a critical assessment of some of these widespread misunderstandings in cosmology, for a comprehensible introduction see [7], while we investigate under which circumstances local and cosmological physics match. We start by reviewing the de Sitter spacetime and its cosmological incarnations, Sec. 2. In particular we show a change of coordinates to describe the spacetime metric surrounding a comoving observer in geodesic motion. In Sec. 3 we discuss under which circumstances the dynamics for geodesics in a cosmological de Sitter metric is physically equivalent to that induced by the dS component in a static Schwarzschild-dS . This brings the opportunity to discuss, Sec. 4, the role of non-static metrics and the distinguished Schwarzschild-dS coordinatization to describe the local scale. On the other side, at the cosmological scale, we show that the only possible source of the energy-momentum tensor to bring a FLRW metric to a static form is that of a cosmological constant. We end, Sec. 5, by elaborating on the interpretation of gravitational/cosmological redshifts as Doppler effect generalized to General Relativity.
In Appendix A we have gathered the construction of an invariant for the Doppler effect and its generalization to General Relativity. Applications of this generalization to the massive case are discussed in Appendix B.

de Sitter spacetime: static and cosmological incarnations
Since de Sitter (dS) spacetime plays a relevant role throughout this paper, we review in the sequel its main features. It can be defined [8] by a 4-dimensional embedding in a flat, 5dimensional Minkowski spacetime, M (1,4) , with coordinates Z = (x 0 , x 4 , x 1 , x 2 , x 3 ) with Lorentzian metric Then the dS background is described by the hyperboloid submanifold with the cosmological constant Λ > 0. One can define coordinates (T, x 1 , x 2 , x 3 ) for the dS submanifold by so that (2.2) is satisfied and the metric induced on the quadric by the ambient metric (2.1) becomes Two remarks are in order: i) these coordinates do not cover the whole hyperboloid (2.2).
ii) A static observer in this background, (R, θ, ϕ) constant, is constantly accelerating. Starting from (2.4), we introduce, under the guidance of some physical principles, the cosmological, spherically symmetric versions of dS. The key point is to obtain a radial coordinate such that the observers comoving with it are time-like geodesics. Thus our first task is to find the radial geodesics for the background (2.4). In fact we can study a more general case. Consider spherically symmetric static metrics of the form which include Schwarzschild, dS, AdS, Schwarzschild-dS or Schwarzschild-AdS metrics. We look for the equations for the radial time-like geodesics (T (s), R(s), θ 0 , ϕ 0 ) in terms of proper time s. The 4-velocity, V (s) = (T ′ (s), R ′ (s), θ 0 , ϕ 0 ), and the proper time condi- . Implementing this into the geodesic Thus for (2.4) we have being C a constant. Time-like radial geodesics are classified according to the sign of this constant. Assuming conventional initial conditions at s = 0, the different solutions to (2.7) are: On the whole we are proposing three different changes of coordinates (T, R) → (t, r) that can be read from (2.8) by writing t instead of s and R(t, r) and T (t, r) in place of R(s), T (s). The metric (2.4) is written, in the new coordinates, as which are the well known cosmological dS metrics with the flat, hyperbolic and spherical slices, respectively. The physical arguments used to produce these changes of coordinates will be a guiding principle in the discussions in the next section.

Local effects of the expansion
Let us consider the general FLRW cosmology, with metric being σ a constant. Although, as mentioned above, this metric is a valid description of spacetime at the cosmological scale let's us elaborate on the outcomes assuming for a while its correctness at the local scale. It is straightforward to quantify the local effect of being in such cosmological background. One should find a suitable description of the time evolving physical distance between a comoving observer at radial coordinate r and the center r = 0. Without entering into fine details, is is clear that a function like R(t) = a(t) r accomplishes this goal 5 . In fact one realises that for σ = 0, it gives the distance from the location (r, θ, ϕ) to the origin r = 0, obtained with the metric induced from (3.10) on the equal time hypersurface 6 . Clearly this new time dependent radial variable satisfies [4,9,10], and thus it makes the case that the effect of the expansion on the local dynamics, either at the scale of the planetary orbits or at that of the electronic orbits of an atom, is just a repulsive 7 radial acceleration, proportional to the distance R(t) and to the time-time component of the Ricci tensor a ′′ (t) a(t) . In fact, the more complete discussion in [2], with the use of Fermi coordinates, leads to the same results. Thus one concludes that, if such FLRW (3.10) models were valid at the local scale, then (3.11) would capture the effect of the expansion. But, as argued above, except for the possible presence of a cosmological constant, there is no trace of the Hubble flow at the local scale -just check the rhs of Einstein equations. This would make the general derivation of (3.11) just an interesting and valuable academic exercise, except that it turns to be right in a qualified and restricted sense: a contribution to the rhs of (3.11) remains at the local scale when the Hubble flow includes a cosmological constant component. In such case, the cosmological constant will remain as the only local effect of the Hubble flow, because it is present at all scales. In 5 It is worth noticing that, after checking with (2.9), the trajectories R(s) in (2.8) can be written in this form. 6 For σ = 0 that would not be exactly the distance, but it will remain a good approximation as long as σr 2 << 1. 7 As long as a ′′ (t) > 0.
addition to that, in the case that a(t) corresponds to dS or AdS spacetimes, and in view of (2.8) and (2.9), the variable R in (3.11) is nothing but the radial variable in the static coordinatization (2.4).
The cosmological dS spacetimes have been studied in Sec. 2 and the procedure to deal with AdS spacetime is analogous. In all these cases the same relation holds, a ′′ (t) a(t) = Λ 3 , with Λ positive for dS and negative for AdS. Thus, either for dS or AdS, (3.11) becomes Up to here we have analysed the effects of the cosmological constant in the framework of cosmological FLRW models. Now let us turn our attention to the approach [5] where the presence of a cosmological constant at the solar system scale is examined by using a static version of the Schwarzschild-dS metric [11], in which (2.5) is materialized with The radial geodesic equation is obtained from (2.6), which contains a Newtonian leading term in addition to the acceleration coming from the presence of Λ, which we isolate, It is centrifugal for dS and centripetal for AdS. Results in (3.12) and (3.15) match, but this could be just a formal coincidence because of the notation. So we must look closely at the physical meaning attached to (3.12) and (3.15). This has been already pursued in Sec. 2, where it was shown that the time coordinate t in the cosmological dS settings (2.9) is just the proper time s of the radial geodesics of (2.4). We conclude therefore that (3.12) and (3.15) are equations with identical content.
The first lesson we extract form this analysis is that equation (3.11) is indeed correct if one restricts its application to the only acceptable case, that of a Hubble flow including a cosmological constant component.
The second lesson is that, at the local scale, the effect of the Hubble flow is completely captured with a static metric, of which (3.13) is a good example, and there is no need to implement a time dependent metric.
We elaborate slightly more in this second point in the following.

A tale at the local scale
Let us take the two versions of dS spacetime, (a) the static (2.4) and (b) the expanding (2.9). While static observers in the former do not observe time evolution in their spacetime, static observers in the latter do notice an expanding universe. The way out to this apparent paradox is that static observers in (a) and (b) do not share the same physical properties. As a matter of fact the static observers in (a) are constantly accelerating, whereas the static observers in (b) are geodesics. What is the most convenient coordinatization in order to facilitate the quantitative description of physical measurements? To find the answer it may be helpful if we consider Schwarszchild spacetime and examine some coordinate systems available to describe it. Before getting into details, we may establish two consecutive stages as regards the connection between the mathematical coordinates, quite arbitrary because of diffeomorphism invariance, and the measuring devices.
(I) In the first stage, observables in General Relativity 8 (GR) are defined through a gauge fixing i.e. after a common choice, made by all the observers, of a coordinatization [12,13,14].
(II) The second stage is the trickiest one: to connect the coordinates with the physical measuring devices. That is, given the observational features that the particular users are focusing on, to optimize its mathematization through the adoption of coordinate descriptions suited to their measuring devices.
Now consider the solar system scale. As regards considerations of static versus nonstatic descriptions, we must point out that the success of the standard coordinatization for the Schwarszchild metric makes such choice the preferable option. As a consequence of Birkoff's theorem the correccions to the solar system Newton dynamics are perturbative terms, and these obtain maximal simplicity -just a single term-within the standard Schwarszchild coordinatization [16], whereas for instance using isotropic coordinates -also static-the corrections are distributed among several terms. As said, static observers in the standard form for Schwarzschild metric are constantly accelerating. In fact, and probably with some degree of retrospection, it is intuitively clear that coordinates for which a static observer is constantly accelerating have a good physical content to describe, precisely, geodesic motion. The reason is inspired in Newtonian physics, in which ideal static observers, placed around the sun, have an internal experience of constant acceleration (to keep them at rest).
By passing let us mention that the authors of [17] have provided with an intrinsic definition of distance, disconnected from Newtonian physics, showing the physical preference of the standard form of the Schwarzschild metric in order to describe the geodesics associated to planetary orbits in general relativity.
If in addition to the Schwarszchild picture we include the presence of a cosmological constant, we may consider Schwarzschild-dS in the forms (a) static, (3.13), and (b) time dependent 9 . According to the observations derived from the previous analysis in Sec. 3, in which we saw that the additional acceleration induced by the cosmological constant is already accounted for in the static version of dS, it seems quite clear that the static description (a) is the candidate to be the most convenient one in order to describe physical phenomena at the local scale.
We have considered hitherto the solar system scale up to the galactic scale. In addressing the cosmological scale, we find that there is a drastic limitation for the FLRW metrics that admit static versions. This is the subject of the next section.

A tale at the cosmological scale
Let us consider the general FLRW cosmology (3.10). We will answer the question as to whether and when such background admits a static metric 10 . With this aim, since we know that comoving observers in a static metric are constantly accelerating, we should look for the radial trajectories of the constantly accelerated observers. One obtains a first change of variables r(t, R) such that, after a second change t(T, R) we end up with an static metric, being R its new radial coordinate. Examining the metric coefficient for the two-sphere it is clear that a(t) r(t, R) must be a function of R only, the simplest choice being R itself.
Thus one can identify r(t, R) = R a(t) . The requirement of being static translates to that of a constantly accelerated observer for constant R. As a consequence one must consider the trajectory X = {t, R a(t) , θ 0 , ϕ 0 } . Its 4-velocity wrt proper time s reads 16) and the acceleration is computed as Next we should compute the Jerk, defined as the Fermi-Walker covariant derivative of the 4-velocity [20,21] and implement the equation Jerk = 0, which describes the constantly accelerated trajectory. Due to the symmetry of our setting, one can check that the vanishing of the Jerk is equivalent to the constancy of the norm of the acceleration. One obtains (4.17) Requiring |A| = constant will in general give solutions for a(t) containing R dependences. This translates contrariwise to our starting point, (3.10), into r dependences in a(t) 11 . The only way in order for the equation |A| = constant not to display an R dependence for its solution a(t), is that the coefficients of different powers of R in the numerator and denominator of (4.17) must sustain a relation of proportionality. With these considerations we set up the following relation which yields the condition We recognize in the above equation the condition for (3.10) to be a maximally symmetric spacetime [19] 12 . In addition (4.19) is the EOM for the Lagrangian L σ = a ′ (t) 2 − σ a(t) 2 , which associated Hamiltonian, H σ = a ′ (t) 2 + σ a(t) 2 , is a constant of motion that is nothing but being Λ the cosmological constant. Using this fact (4.19) boils down to 13 11 Notice however that a(t) may indeed depend on σ, which is a parameter already present in the metric. 12 We can check that the solutions to (4.19), see below, satisfy the requirement |A| = constant. 13 Notice that this equation is derived form the Lagrangian L Λ = a ′ (t) 2 + Λ 3 a(t) 2 , in which case σ appears as the new Hamiltonian constant of motion. Thus equations (4.19) and (4.20) are equivalent, and so they are their parent Lagrangians L σ and L Λ . 14 contains three different cases, depending on the sign of Λ:

Its solution
(I) The case Λ > 0 is dS spacetime which has already been dealt with in Sec. 2.
(II) The vanishing Λ case is Minkowski spacetime in Milne coordinatization [22] 15 (4.21) (III) The case Λ < 0 is AdS spacetime. It needs σ < 0 and its equal time slices are 3-hyperboloids All these solutions share the same description with a static metric, (4.23) To summarize, the only source of matter-energy compatible with bringing a given FLRW cosmology to a static metric description is the cosmological constant.

Is the 3-space expanding?
Let us give a tentative definition, within the FLRW models, of what is meant by expanding 3-space: it is the idea, or belief, that there is a physical process of some sort -acting perhaps at an ultra-micoscopic scale-that is producing the homogeneous growing of the equal-time hypersurfaces of the background (3.10), as dictated by the scale factor a(t).
Take a family of non-interacting test particles following radial time-like geodesics inside a Minkowski spacetime. Suppose we adopt Milne's form (4.21) for the metric. Hence test particles are comoving observers with scale factor H t describing the time-increasing separation between them. Should one conclude from this picture that the 3-space is expanding? [24]. Obviously such notion is a pure artifact of the chosen coordinatization. 14 We skip the trivial solution a(t) = constant, σ = Λ = 0, which is Minkowski spacetime in standard coordinatization. 15 The change of variables from Minkowski, ds 2 = −dT 2 + dR 2 + R 2 dΩ 2 , to Milne coordinates is given by T (t, r) = √ 1 + H 2 r 2 t , R(t, r) = HrT .
On the other hand, there is a strong observational evidence that the galaxies are moving apart from each other. Provided the intergalactic void is not so different from the strict void described by (4.21) 16 , should we believe that the 3-space in our universe expands whereas that in (4.21) does not?. We find neither compelling reason nor need to believe in the expansion of 3-space as defined above. This picture is not sensible and receives the final blow when realizing that if taken seriously, then one is bound to accept the absurd consequence that this growing process holds at the local scale, for which there is no basis at all when one looks at the rhs of Einstein's equations.

Cosmological and gravitational redshift vs. Doppler effect
Take two test observers in Minkowski spacetime, A and B, simultaneously departing from the origin of coordinates at T = 0 and traveling radially in different directions. They correspond, see (4.21), to comoving observers located at fixed {r a , θ a , ϕ a } and {r b , θ b , ϕ b } in the Milne coordinatization. One can compute the redshift of a photon emitted by A and detected by B using the Special Relativity (SR) formulas for the Doppler effect, when sender and receiver are in different inertial reference systems. Although this redshift can also be computed as a byproduct of the "expansion of space" picture, it is clearly nothing else than a Doppler effect. The Doppler effect as introduced above in the framework of SR can be generalized to GR as relating the frequencies of the emitted and received photons by arbitrary sources and observers. This was done long time ago, first in [26,27] and later re-elaborated [28,29]. Once this generalization of the Doppler effect to curved spacetime is at our disposal, all these considerations on a redshift that is no longer Doppler but only cosmological, or gravitational, become meaningless. To avoid misunderstandings: It is not wrong to talk on cosmological redshift, or gravitational redshift, what is wrong is to claim that they are something conceptually different from the Doppler redshift, now understood as an extension to GR of the SR effect. This extension is reviewed in the Appendix A.
In the next two subsections we examine the cosmological and gravitational redshifts within the framework of this generalization. It is worth noticing in these derivations the relevance of identifying an affine parameter for the photon trajectory. In Appendix B we discuss the massive particle case, and show that we recover in the massless limit the results shown below, thus becoming an alternative derivation of the frequency shifts for the photon, only relying on the notion of proper time.

5.2.1
The cosmological redshift as a General Relativity Doppler effect As is elaborated in depth in App. A the spectral shift, z, relates the frequency ν s of the photon as seen by the Source at the time of emission, to the frequency ν o of the photon as seen by the Observer at the time of reception 17 . This relation can be cast in an invariant way as (A.39) In the sequel we shall apply the previous expression, (5.24), to verify that it describes the cosmological redshift for the FLRW, (3.10). We consider both the Source and the Observer as comoving in the Hubble flow. The Source is located at r = 0, in this way the spherical symmetry imposes that all geodesics passing through r = 0 must be radial. In addition, the maximal symmetry of the equal time slices allows that any non-radial geodesic become automatically included in the analysis through a change of coordinates in the equal-time slices. With the aforementioned conditions, the photon trajectory is given by X(t) = t, r(t), θ 0 , φ 0 , with r(t)=0 and velocity wrt coordinate time d X d t = (1, r ′ (t), 0, 0). The velocity wrt the affine parameter s is, where h is as yet an unknown function. The equation for h(t) is found by inserting U(t) into the geodesic equation, from which we obtain a differential equation for .
(5.27) 17 In both cases the frequency is proportional to the kinetic energy. 18 Lacking of the concept of proper time for the massless particle, this is the only way to ensure that the quotient in (5.24) is indeed an invariant. 19 The arbitrary constant factor of the solution will be discussed later on.
Imposing the light-like condition d X d t 2 = 0, which becomes a(t) 2 r ′ (t) 2 1 − σ r(t) 2 = 1 , one determines the trajectory 20 which holds for both positive or negative σ and also in the limit of vanishing σ. Thus we end up with The comoving Source at the time of emission t = 0 has proper velocity V s (0) = (1, 0, 0, 0), . On the other hand, the comoving Observer at the time t of . All in all, from (5.24) we get which is the standard formula for the cosmological redshift. This result shows that the cosmological redshift is just the manifestation of the Doppler effect, once extended from SR to GR. Nothing more, nothing less. Of course one can use the rubber balloon picture [25] as a metaphor, but to claim that such a picture is necessary in the sense of our tentative definition given above is a mistake.

The gravitational redshift as a General Relativity Doppler effect
Once the Doppler effect has been properly extended to GR, we can conclude that cosmological and gravitational redshifts have a common origin. This result was already derived in [26,27,28] and in the following we make a detailed treatment for the general Schwarszchild-dS spacetime.
Consider the general metric (2.5) with f (R) given in (3.13). We are interested in the radial emission of a photon from a location (R 0 , θ 0 , φ 0 ) and its latter detection along the same radial line at (R 1 , θ 0 , φ 0 ) with R 1 > R 0 . The Source (Observer) position and velocity 20 Note that Source : , 0, 0, 0) , Observer : , 0, 0, 0) . At any T > 0, with R(T ) = R 1 , the scalar products are found to be Adapting (5.30) to the case at hand which is the usual formula for the gravitational redshift for radial photons in the Swcharzschild metric. Here it is also valid for Schwarzschild-dS or Schwarzschild-AdS backgrounds.
Notice that contrariwise to the previous discussion on the cosmological redshift the Source and Observer are no longer geodesics but constantly accelerating. One can nevertheless consider the Source as belonging to a radial geodesics which happens to be at R 0 when t = 0 and such that R ′ (0) = 0, and the same can be done at time t for the Observer. The GR Doppler formula still holds because the quotient of the scalar products in (5.30) is always an invariant regardless of the fact that Source and Observer be geodesics or not. In addition, once the Doppler effect has been extended to GR, the curvature of spacetime contributes to this effect, making it detectable even in cases where Source and Observer are at rest 23 .

Concluding remarks
In the previous pages we have focused in relating local and cosmological physics: i) On one side we have matched the radial EOM of a FLRW-dS model with the contribution from a cosmological constant of an static Schwarzschild-dS, providing evidence that both models describe the same dynamics. In this way, some approaches in the literature that until now seemed to be incompatible can be reconciled when applied to plausible physical scenarios. ii) On the other side we reviewed an old, but not yet as popular as it deserves, unified presentation of the GR Doppler effect, with a single formula encompassing all circumstances. It's common theme being that of energy gain or loss for a particle, either massive or massless, in geodesic motion from the Source to the Observer. Under this common theme, all energy shifts, including the cosmological and gravitational ones, appear as particular cases of this GR Doppler effect. Our presentation has the novelty to include in the same framework the massless as well as the massive case, showing how to retrieve the former from the latter by taking the appropriate limit. In this sense, the role of the affine parameter in the massless case can be circumvented. We claim that the matching of an invariant at the local and cosmological scale provides an unambiguous and unique way to relate observables at both scales. derivation of this effect and examine what is basically equivalent: the energy gain/loss of a particle in free motion from the Source to the Observer. This is more general that just Doppler, because it includes the massive case.
We consider the inertial reference system of the Observer, placed at the origin of spatial coordinates, whereas the Source and the massive particle are moving with respect to it at speeds v , u respectively. Their respective 4-velocities wrt proper time are with v = | v|, u = | u|. It is assumed that the particle intersects the Source and Observer trajectories at different points in Minkowski spacetime. If we set the mass of the particle to m = 1 its energy can be expressed in terms of an invariant form from both rest systems, Observer and Source and therefore, the invariant ratio of energies, E s /E o , for the massive particle is In the massless limit, u → 1, we get the standard formula for the energy shift of the photon 25 , which, for α = π, gives the usual longitudinal Doppler redshift when the motions of Source and Observer are aligned and in opposite directions. Summing up, in the SR framework, the Doppler effect or in general, the ratio of the particle's energy seen from the Source rest frame to the particle's energy seen from the Observer rest frame 26 is always described by the invariant with U s = U o = U in this case. 25 Obviously the same result is obtained by using directly for the massless particle the velocity U = (1, ω) with | ω| = 1 . 26 Be the particle either massive or massless.

A.2 ...to General Relativity
There are many definitions within the SR framework that can be extended to GR. Take for instance the geodesic motion, which is extended to GR by basically replacing the ordinary derivative for the covariant one. Or the concept of the constantly accelerated observer, that can be brought to GR by keeping the requirement of constancy [21,31,32] for some curvature scalars that generalize the Frenet-Serret formalism [33]. These cases bear in common that only the point and its neighborhood in a world line trajectory are necessary ingredients. Other concepts, like the Doppler effect, require more refined considerations because points of different trajectories are involved. Luckily enough (A.38) is easily exported to GR. In such case the particle travels through a geodesic with a 4velocity computed either wrt proper time for massive particles or wrt an affine parameter for massless ones. Its velocity U is typically different when evaluated at the Source location, U s , than when evaluated at the point of reception by the Observer, U o . Unlike the massive case, the affine parameter for photons is only determined up to an arbitrary constant factor, the consequence being that whereas for the massive case both scalar products, V s · U s and 27 , it is only their quotient which is invariant for massless particles The above expression captures the Doppler effect and its extension to the massive case as a ratio between some data from the emission event, V s · U s , to some data from the reception event, V o · U o 28 . We submit that (A.39) must be taken as the definition of the Doppler effect -interpreted as an energy shift and also extended to the massive case-in GR.
Let us notice that in adopting (A.39), hence including a computational prescription, for the evaluation of the GR Doppler effect, the notion of the relative velocity between Source and Observer, which is crucial in the SR derivation, has disappeared. 27 We mean invariants under general changes of coordinates. In the passive interpretation of diffeomorphisms a scalar computed at a given point becomes an invariant, in the sense that its value is independent of the coordinates used to describe such point [34]. 28 One can go one step further and parallel transport the data form the Source to the Observer's location.
Since U s is transported to U o , one can see that the whole effect originates from the fact that V s is not transported to V o .

B.1 The cosmological energy shift for massive particles
We continue in the cosmological FLRW setting (3.10), but considering the emission of a massive particle from a comoving Source located at r = 0. Its geodesic trajectory and velocity wrt proper time, s, are To compute r(t), we formulate the geodesic equation (5.26) with the normalized velocity (B.40), obtaining with C > 0 an integration constant related to the initial condition r ′ (0). Similarly to (5.28), (B.42) holds also for null or negative σ. In addition, the massless case can be recovered in the limit C → 0.
With this at hand the expression for the proper 4-velocity 29 becomes (B.43) where, for convenience, we restored the mass m of the particle and defined p through Let's interpret the invariants: i) At the particular time of emission t = 0 the energy of the massive particle is given by (B.44) 29 Notice that although expressed in terms of the cosmological time it is a proper velocity, U (t) 2 = −1.
Thus p is interpreted as the initial momentum of the particle as measured by the Source 30 .
ii) At the time t of reception we obtain the energy of the particle from the invariant and p(t) is interpreted as the momentum of the particle a time t, as as measured by the Observer.

B.2 The necessary connection between two scales
The result (B.43) has been obtained using the cosmological FLRW metric (3.10), which works at the cosmological scale. Evidently if we just consider a small region in the close neighborhood of the Source there is no trace of the homogeneous Hubble flow that sources the background and therefore (3.10) is not applicable at this scale. Instead, what is applicable at this local scale are the kinematic relations of SR, as it is stated by the equivalence principle. But then the question arises as how can we proceed in order to connect both settings, cosmological and local one. It is not a coordinate transformation because we are talking about different metrics: on one side, the broad-brush FLRW metric, obtained by averaging the density of matter-radiation on very large scales and assuming homogeneity; on the other side, the approximate SR Minkowski metric that holds in every small neighborhood of spacetime. Both pictures are correct, the only caveat being, as said, that they are applicable at completely different scales. To our understanding, there is a unique way to physically connect the two scales: one must retain the values of the invariants found above when moving from the cosmological scale description to the local SR one, or vice versa. Now made explicit, this is the assumption that was already implicit in the previous subsection.
In the local SR frame at the Source we have V s = (1, 0) and U(0) = 1 m m 2 + p 2 , p with p = m v √ 1−v 2 so that V s · U(0) = 1 m m 2 + p 2 . Analogously we will have at the point of reception, at time t, V o · U(t) = 1 m m 2 + p(t) 2 , Thus (B.46) 30 We elaborate on this interpretation in the next subsection.
Notice that in the massless limit, m → 0, we recover (5.30), and the same happens for large momentum, p → ∞, as well.
Once established the connection between the two scales, we infer that p is indeed the momentum of the particle as seen by the comoving Source at the time of emission, and p(t) is indeed the momentum of the particle as seen by the comoving Observer at the time of reception. Thus, independently of the particle being massive or massless, the following relation always holds p(t) a(t) = p(0) a(0) . (B.47)

B.3 The gravitational energy shift for massive particles
Similarly to the massless case, sending and receiving a massive particle will also exhibit a shift in its kinetic energy. With the same setup of subsection 5.2.2, and working directly with proper time s, the trajectory and velocity will be denoted as which implies where E(R 0 ) is the kinetic energy of the emitted particle as seen by the Source and E(R 1 ) is the kinetic energy of the received particle as seen by the Observer. Unlike the cosmological case above, this result is independent of the mass of the particle and it directly admits the massless limit yielding (5.33) in which case these energy ratios can also be read as quotients of frequencies of the photon.