Kinematics in spatially flat FLRW space-time
aa r X i v : . [ g r- q c ] F e b Kinematics in spatially flat FLRWspace-times
Ion I. Cot˘aescu
West University of Timi¸soara,V. Pˆarvan Ave. 4, RO-1900 Timi¸soara, Romania
February 8, 2021
Abstract
The kinematics on spatially flat FLRW space-times is pre-sented for the first time in co-moving local charts with physical co-ordinates, i. e. the cosmic time and Painlev´e-type Cartesian spacecoordinates. It is shown that there exists a conserved momentumwhich determines the form of the covariant four-momentum ongeodesics in terms of physical coordinates. Moreover, with thehelp of the conserved momentum one identifies the peculiar mo-mentum separating the peculiar and recessional motions withoutambiguities. It is shown that the energy and peculiar momentumsatisfy the mass-shell condition of special relativity while the re-cessional momentum does not produce energy. In this framework,the measurements of the kinetic quantities along geodesic per-formed by different observers are analysed pointing out an energyloss of the massive particles similar to that giving the photon red-shift. The examples of the kinematics on the de Sitter expandinguniverse and a new Milne-type space-time are extensively anal-ysed.Pacs: 04.62.+v
Keywords: FLRW spec-etime; energy-momentum; conserved quanti-ties; de Sitter expanding universe; Milne-type universe.1
Introduction
The geodesic motion in general relativity can be described in various lo-cal charts (called here frames) as each observer may choose his properframe with preferred coordinates. From the point of view of general rel-ativity all these frames are equivalent as their coordinates are relatedthrough diffeomorphisms under which the mathematical objects trans-form covariantly [1, 2]. However, the general diffeomorphisms are notable to produce conserved quantities such that we must focus mainly onthe isometries which may give rise to conserved quantities via Noethertheorem [3–5]. Thus we must restrict ourselves to a class of observer’sframes related through isometries where we have to apply the methods ofspecial relativity in studying the relative motion but using the isometrygroup instead of the Poincar´e one as in our recent de Sitter relativity[6, 7].Under such circumstances, it is crucial to understand which are thesignificant physical quantities and how these may be related to the con-served quantities generated by isometries. Another problem is how themeasurement depends on the choice of observer’s frame taking into ac-count that the isometries transform the conserved quantities among them-selves such that different observers measure different values of these quan-tities. In this paper we would like to analyse these problems in the simplecase of the spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW)space-times where we focus on the kinetic quantities on time-like or nullgeodesics in frames with physical coordinates.Apparently these are trivial problems, that might be solved from longtime, but in fact these are still actual since the physical coordinates, giv-ing directly the physical distances, are those of Painlev´e type which wereused in static problems but never in the case of the co-moving framesof the spatially flat FLRW space-times. Here we introduce these coor-dinates obtaining the physical co-moving frames with a time-dependentmetric but with spatially flat sections whose Cartesian coordinates givethe physical distances as in Minkowski spece-time. We assume thatthe measured quantities are the components p µ of the covariant energy-momentum four-vector in the physical frames, formed by energy , p and covariant momentum, p , bearing in mind that, in general, these are func-tions of cosmic time.In other respects, the spatially flat FLRW space-times have the Eu-clidean isometry group E (3) formed by space rotations and space transla-tions giving a conserved angular momentum and a conserved momentum, P . The angular momentum is related to the symmetry under rotations2hich is global as we use Cartesian coordinates. The conserved momen-tum which does not coincide with the covariant one is important sincethis generates three prime integrals helping us to derive the energy andcovariant momentum we need for integrating the geodesic equation whichwill be determined by the initial condition and its conserved momentum.Moreover, the conserved momentum helps us to separate the peculiar mo-mentum, proportional with P , from the recessional one finding that theenergy and peculiar momentum in the physical co-moving frame satisfythe mass-shell condition of special relativity. All these results concerningthe kinematics in co-moving physical frames are presented in the firstpart on the next section.It remain to find how different observers measures the covariant en-ergy and covariant momentum as long as in the FLRW geometries underconsiderations the translations are isometries transforming the covariantfour-vectors and the conserved quantities. In the last part of the next sec-tion we investigate how these quantities are measured by two observersstaying in two different points of the same geodesic pointing out that,in contrast with the Minkowski space-time, the observer position deter-mines the form and the meaning of the measured quantities. Thus wededuce that the massive or massless particles lose some energy duringpropagation which in the massless case is related to the redshift.The third section is devoted to a well-known example, namely thede Sitter expanding universe whose geodesics we studied in differentframes including the physical one but without paying attention to theenergy and covariant momentum [8]. The de Sitter manifold has ten in-dependent conserved Killing vectors which generate conserved quantitiesamong them we extract the conserved momentum relating thus the con-served quantities to the measured ones for understanding the role of theconserved quantities in the de Sitter kinematics. The conclusion is thatthe conserved energy coincides with the measured energy in some pointsof geodesics while the other conserved quantities, including the conservedmomentum, work together for closing the mass-shell condition. The men-tioned problem of two observers is also discussed for time-like and nullgeodesics pointing out the energy loss and redshift.A new example whose kinematics was never studied is presented inSec. 4. This is a spatially flat FRLW space-time with a Milne-type scalefactor which, in contrast to the genuine Milne universe, has gravitationalsources determining its expansion. We inspect briefly the kinematics onthis manifold observing that this behave somewhat complementary withrespect to the de Sitter one. Finally we present some concluding remarks.In what follows we use the Planck natural units and denote the con-3erved quantities with capital letters. The FLRW space-times are the most plausible models of our universein various periods evolution. The actual universe is observed as beingspatially flat with a reasonable accuracy. For this reason we focus here onthese manifolds for which we consider many types of co-moving frameswith Cartesian or spherical coordinates looking for measurable quantitiesexpressed in terms of physical coordinates of Painlev´e type.
The Painlev´e - Gullstrand coordinates [9, 10] were proposed for studyingthe Schwarzschild black holes. Similar coordinates can be introduced inany isotropic manifold (
M, g ) having frames { x } = { t, x } with flat spacesections. In these frames the coordinates, x µ ( α, µ, ν, ... = 0 , , ,
3) maybe formed by the cosmic time t and either Cartesian space coordinates x = ( x , x , x ) or associated spherical ones ( r, θ, φ ) with Euclidean met-ric ds E = d x · d x = dr + r d Ω where d Ω = dθ + sin θ dφ . Forexample, the line element ds = f ( r ) dt s − dr f ( r ) − r d Ω , (1)of any static frame, { t s , r, θ, φ } , with static time t s , can be put in Painlev´e-Gullstrand form, ds = f ( r ) dt + 2 p − f ( r ) dtdr − dr − r d Ω , (2)substituting in Eq. (1) t s = t + Z dr p − f ( r ) f ( r ) , (3)where t is the cosmic time.Similar coordinates, we call here simply Painlev´e or physical coordi-nates, can be introduced in any spatially flat FLRW space-time startingwith the conformal Euclidean co-moving frame { t c , x c } with the line el-ement ds = a ( t c ) (cid:0) dt c − d x c · d x c (cid:1) . (4)4ere we may substitute the physical coordinates { t, x } defined as t = Z a ( t c ) dt c , x = a ( t c ) x c . (5)obtaining the new line element ds = (cid:18) − ˙ a ( t ) a ( t ) x (cid:19) dt + 2 ˙ a ( t ) a ( t ) x · d x dt − d x · d x , (6)where a ( t ) = a [ t c ( t )] is the usual FRLW scale factor while˙ a ( t ) a ( t ) = 1 a ( t ) da ( t ) dt = 1 a ( t c ) da ( t c ) dt c , (7)is the Hubble function for which we do not use a special notation. Theinverse transformation { t, x } → { t c , x c } is obvious t c = Z dta ( t ) , x c = x a ( t ) . (8)We suppose that the function a ( t ) is smooth such that the transforma-tions (5) and (8) are diffeomorpkisms.The metric (6) is time-dependent laying out an evolving horizon at | x h | = a ( t )˙ a ( t ) which makes it less popular despite of the fact that thesecoordinates are just the physical ones, namely the cosmic time t and theCartesian physical space coordinates x . Another advantage of this metricis that this is approaching to the Minkowski one in a neighbourhood of x = 0.In many applications one prefers the FLRW coordinates { t, x c } withthe well-known line element ds = dt − a ( t ) d x c · d x c , (9)pointing out occasionally the physical distances by multiplying the coor-dinates with the scale factor a ( t ). For avoiding this artifice we forget herethe FLRW coordinates using directly the physical Painlev´e coordinatesand resorting to the conformal ones as an auxiliary tool when these offertechnical advantages. Our principal objective is to derive the equation of the time-like and nullgeodesics as well as the associated kinetic quantities in the physical co-moving frame { t, x } O which is the proper frame of an observer staying at5est in the origin O . We look for the components p µ = dx µ dλ of the covariantfour-momentum ( p , p ) formed by the measured energy p and covariantmomentum p . Here λ is an afine parameter related as ds = m dλ to themass m of a particle moving freely along a geodesic.We start with an intermediate step, focusing first on the components p µc = dx µc dλ of the covariant momentum in the conformal co-moving frame { t c , x c } O of our observer where we have the simple prime integral, a ( t c ) (cid:2) p c ( t c ) − p c ( t c ) (cid:3) = m , (10)resulted from the line element (4). In other respects, we may exploitthe fact that the spatially flat FLRW space-times have E (3) isometriesformed by global rotations, x i → R Ij x j ( i, j, k, ... = 1 , , t c = t ′ c ,x ic = x ′ ic + ξ i , → t = t ′ ,x i = x ′ i + ξ i a ( t ) , (11)whose associated Killing vectors k ( i ) have the components k i ) = 0 and k j ( i ) = δ ij in the frame { t c , x c } O giving rise to the conserved quantities P i = − k ( i ) j dx jc dλ = a ( t c ) dx jc dλ , (12)representing the components of the conserved momentum which is dif-ferent from the covariant momentum p ( t ). Then by using the primeintegrals (10) and (12) we derive the energy and covariant momentum inthis frame as p c ( t c ) = dt c dλ = 1 a ( t c ) s m + P a ( t c ) , (13) p ic ( t c ) = dx ic dλ = P i a ( t c ) , (14)where we denote P = | P | . The geodesic results simply as dx ic dt c = p ic ( t c ) p c ( t c ) → x ic ( t c ) = x ic + P i m Z t c t c dt c q a ( t c ) + P m , (15)concluding that any time-like geodesic is determined completely by itsconserved momentum P = n P P and the initial condition x c ( t c ) = x c .6his equation must be integrated in each particular case but for themassless particles (with m = 0) we have the universal solution x c ( t c ) = x c + n P ( t c − t c ) , (16)giving the null geodesics on any FLRW space-time.The corresponding physical quantities measured by the observer O inhis physical proper frame, { t, x } O , may be obtained by substituting thephysical coordinates according to Eq. (8). Thus we find the covariantcomponents, p ( t ) = dtdλ = s m + P a ( t ) , (17) p i ( t ) = dx i dλ = P i a ( t ) + x i ( t ) ˙ a ( t ) a ( t ) s m + P a ( t ) , (18)which represent the measured energy and covariant momentum in thepoint [ t, ~x ( t )] of the time-like geodesic x ( t ) = x a ( t ) a ( t ) + P m a ( t ) Z tt dta ( t ) q a ( t ) + P m (19)which is passing through the space point x ( t ) = x at the initial time t . In the physical frame { t, x } O the equation of the null geodesics, x ( t ) = x a ( t ) a ( t ) + n P a ( t ) [ t c ( t ) − t c ( t )] , (20)results from Eq. (16).A special problem is that of tachyons whose kinetic quantities onspace-like geodesics can be obtained by substituting m → − m in theabove equations. Then the energy p ( t ) = s − m + P a ( t ) , (21)is real valued only when a ( t ) < Pm . This means that in expanding uni-verses a tachyon with conserved momentum P disappears when a ( t )reaches the value Pm this surviving only in collapsing universes for smallervalues of the scale factor. As here we focus only on expanding geometrieswe ignore the space-like geodesics remaining to study the time-like andnull ones. 7he momentum defined by Eq. (18) can be split as p ( t ) = ˆ p ( t ) + ¯ p ( t )where ˆ p = P a ( t ) , ¯ p = x ( t ) ˙ a ( t ) a ( t ) p ( t ) , (22)are the peculiar and respectively recessional momenta. The prime integralderived from the line element (6) gives the familiar identity p ( t ) − ˆ p ( t ) = m , (23)which is just the mass-shell condition of special relativity satisfied bythe energy and peculiar momentum along the geodesic. Thus we seethat in the physical co-moving frame the peculiar momentum can beseparated naturally being proportional with the conserved momentum.Moreover, this produces the entire energy of the geodesic as in specialrelativity. Thus for P = 0 and p ( t ) = m , the particle remaining at restin the point x ( t ) = a ( t ) a ( t ) x but moving with the recessional momentum¯ p with respect the observer O . We must stress that these propertieshold only in the co-moving frames with physical coordinates since inother frames this separation is not possible while the momenta satisfydispersion relations depending explicitly on time as in Eq. (10) or identity p ( t ) − a ( t ) p c ( t ) = m that holds in FLRW coordinates { t, x c } .Hereby other interesting kinetic quantities can be derived as, for ex-ample, the velocity v ( t ) = p ( t ) p ( t ) = x ( t ) ˙ a ( t ) a ( t ) + ˆ p ( t ) p m + ˆ p ( t ) ) , (24)whose first term is the recessional velocity due to the space evolution,complying with the velocity law which is confused sometimes with theHubble one [11, 12]. The second term is the peculiar velocity whichdepends on the peculiar momentum as in special relativity.The covariance under rotations, which behave here as a global sym-metry, gives rise to the conserved angular momentum, that depends onlyon the peculiar momentum L = x ( t ) ∧ p ( t ) = x ( t ) ∧ ˆ p ( t ) = x ( t ) ∧ P a ( t ) = x ∧ P a ( t ) , (25)and can be related to the initial condition. This vanishes when the ob-server O stays at rest in a space point of the measured geodesic.8 .3 Two observers problem The physical quantities p ( t ), ˆ p ( t ) and ˆ p ( t ) are functions of time butthe last one depends explicitly on coordinate such that the experimen-tal results will depend on the relative position between the detector andthe measured particle. However, this is not an impediment as the pe-culiar and recession contributions can be separated at any time withoutambiguities. Nevertheless, for avoiding confusions we take care on thisdependence looking for suitable positions of observer’s frames in order toobtain intuitive results.The example we would like to discuss here is of two observers mea-suring the motion of a massive particle on a time-like geodesic which ispassing through the origins O and O ′ of their proper co-moving frames { t, x } O and { t, x ′ } O ′ . We assume that at the initial time t the ori-gin O ′ is translated with respect to O as x ( t ) = x ′ ( t ) + d ( t ) where d ( t ) = d a ( t ) depends on the translation parameters of Eqs. (11) de-noted now by ξ i = d i . Then it is convenient to introduce the unit vector n of the direction OO ′ such that d = n d .Our experiment starts in this lay out at the time t when the observer O ′ lunches a particle of mass m , momentum p = − n p and energy p = p m + p on the geodesic O ′ → O . The problem is to find which are theenergy and momentum of this particle measured in the origin O at thefinal time t f when the particle reach this point. For solving this problemwe look first for the conserved momentum that can be derived in O ′ as P = n P P = p a ( t ) → P = p a ( t ) , n P = − n . (26)Then by using Eqs. (17) and (18) we obtain the momentum and energymeasured in O , p ( t f ) = s m + p a ( t ) a ( t f ) , (27) p ( t f ) = ˆ p ( t f ) = − n p a ( t ) a ( t f ) , (28)where t f is the solution of the equation n · x ( t f ) = 0 with x ( t ) given byEq. (19). This equation can be written simply as Pm Z tft dta ( t ) q a ( t ) + P m = d (29)where d is the time-independent translation parameter defined above.Solving Eq. (29) we obtain a function t f ( P, t ) which must be singular in9 = 0 for preventing the left handed term of this equation on vanishing inthis limit. Once we have the value of t f we can calculate the propagationtime t f − t , the distance d ( t f ) between O and O ′ at the time t f and thefinal peculiar velocity ˆ v ( t f ) of the particle arriving in O . According toEqs. (11) and (27) we find d ( t f ) = d a ( t f ) = d ( t ) a ( t f ) a ( t ) , (30)ˆ v ( t f ) = (cid:18) m p a ( t f ) a ( t ) (cid:19) − , (31)completing thus the collection of kinetic quantities related to this prob-lem.Eq. (27) shows that in expanding universes a part of energy is lostduring the propagation. This can be measured by the relative energyloss defined as e = 1 − p ( t f ) p ( t ) . (32)This phenomenon is similar to the redshift of the photons with m = 0for which we recover the Lemaˆıtre equation [13, 14] of Hubble’s law [15]as 11 − e = 1 + z = p ( t ) p ( t f ) = a ( t f ) a ( t ) , (33)where z is the usual redshift defined as the relative dilation of the wavelength. As was expected for m = 0 the final velocity ˆ v ( t f ) = 1 is thespeed of light.All the results presented here can be exploited effectively only inconcrete geometries where the geodesic equation can be integrated. Inwhat follows we discuss two such examples starting with one of the moststudied geometries. The first example is the expanding portion of the de Sitter space-timedefined as the hyperboloid of radius 1 /ω H in the five-dimensional flatspacetime ( M , η ) of coordinates z A (labelled by the indices A, B, ... =0 , , , ,
4) having the metric η = diag(1 , − , − , − , − { x } can be introduced giving the set of functions z A ( x ) which solvethe hyperboloid equation, η AB z A ( x ) z B ( x ) = − ω H . (34)10here ω H is the Hubble de Sitter constant in our notations.There are co-moving frames with conformal coordinates { t c , x c } orwith physical ones { t, x } having the scale factors a ( t ) = e ω H t → t c = − ω H e − ω H t → a ( t c ) = − ωt c , (35)defined for t ∈ R and t c < ω H . In addition,this manifold allows even a static frame { t s , x } with the line element (1)where f ( r ) = 1 − ω H r and t s = t − ln f ( r ). The de Sitter manifold has a rich isometry group which is just the gaugegroup SO (1 ,
4) of the embedding manifold ( M , η ) that leave invariantits metric and implicitly Eq. (34). Therefore, given a system of coordi-nates defined by the functions z = z ( x ), each transformation g ∈ SO (1 , x → x ′ = φ g ( x ) derived from the system of equations z [ φ g ( x )] = g z ( x ) (36)that holds for any type of coordinates which means that these isome-tries are defined globally. The sets of local charts related through theseisometries play the role of the inertial frames similar to those of specialrelativity.Given an isometry x → x ′ = φ g ( ξ ) ( x ) depending on the group parame-ter ξ there exists an associated Killing vector, k = ∂ ξ φ ξ | ξ =0 (which satisfythe Killing equation k µ ; ν + k ν ; µ = 0). Thus in a canonical parametriza-tion of the SO (1 ,
4) group, with real skew-symmetric parameters ξ AB = − ξ BA , any infinitesimal isometry can be written as φ µ g ( ξ ) ( x ) = x µ + ξ AB k µ ( AB ) ( x ) + ... . Starting with the general definition of the Killing vec-tors in the pseudo-Euclidean spacetime ( M , η ), we may consider thefollowing identity K ( AB ) C dz C = z A dz B − z B dz A = k ( AB ) µ dx µ , (37)giving the covariant components of the Killing vectors in an arbitraryframe { x } of the de Sitter manifold as k ( AB ) µ = η AC η BD k ( CD ) µ = z A ∂ µ z B − z B ∂ µ z A , (38)where z A = η AB z B . 11he classical conserved quantities along the time-like geodesics havethe general form K ( AB ) ( x, P ) = ω H k ( AB ) µ p µ where p µ are the compo-nents of the covariant four-vector defined above. The conserved quanti-ties with physical meaning [16] are, the energy E = ω H k (04) µ p µ , the an-gular momentum components, L i = ε ijk k ( jk ) µ p µ , and the components K i = k (0 i ) µ p µ and R i = k ( i µ p µ of two vectors related to the conservedmomentum P and its associated dual momentum Q as, P = − ω H ( R + K ) , Q = ω H ( K − R ) . (39)satisfying the identity E − ω H L − P · Q = m , (40)corresponding to the first Casimir invariant of the so (1 ,
4) algebra [16].In the flat limit, ω H → − ω H t c →
1, we have Q → P such that thisidentity becomes just the usual null mass-shell condition E − P = m of special relativity.Note that the conserved quantities transform among themselves underde Sitter isometries including the simple translations which in this casetransform the energy and dual momentum as we have shown recently [6]. The coordinates of the physical co-moving frame { t, x } O are introducedby the functions z ( x ) = 12 ω H (cid:2) e ω H t − e − ω H t (1 − ω H x ) (cid:3) ,z i ( x ) = x i , (41) z ( x ) = 12 ω H (cid:2) e ω H t + e − ω H t (1 − ω H x ) (cid:3) . giving the line element ds = (1 − ω H x ) dt + 2 ω H x · d x dt − d x · d x , (42)having the horizon at | x h | = ω − H such that the condition ω H | x | < x ( t ) = x e ω H ( t − t ) + n P e ω H t ω H P (cid:16) √ m + P e − ω H t − √ m + P e − ω H t (cid:17) , (43)12etermined by the conserved momentum P = n P P and the initial con-dition x ( t ) = x fixed at the time t .The conserved quantities in an arbitrary point ( t, x ( t )) of this geodesiccan be expressed as [6, 8], E = ω H x ( t ) · P e − ω H t + √ m + P e − ω H t , (44) L = x ( t ) ∧ P e − ω H t , (45) Q = 2 ω H x ( t ) Ee − ω H t + P e − ω H t [1 − ω H x ( t ) ] . (46)satisfying the identity (40). Moreover, the Eqs. (17) and (18) give theenergy and covariant momentum components, p ( t ) = dtdλ = √ m + P e − ω H t , (47) p i ( t ) = dx i dλ = e − ω H t P i + ω H x i ( t ) √ m + P e − ω H t , (48)that can be measured by the observer O in his proper frame { t, x } O . Theconserved quantities are related to the measured ones as E = ω H x ( t ) · ˆ p ( t ) + p ( t ) , (49) L = x ( t ) ∧ ˆ p ( t ) , (50) P = ˆ p ( t ) e ω H t (51) Q = e − ω H t (cid:8) ω H x ( t ) E + ˆ p ( t )[1 − ω H x ( t ) ] (cid:9) . (52)Hereby we conclude that the conserved quantities depend only on posi-tion and peculiar momentum. Among them only E and L can be mea-sured while P and Q are not accessible directly, their role consisting onlyin closing the invariant (40) as E − ω H L − P · Q = p ( t ) − ˆ p ( t ) = m . (53)For example, a measurement in observer’s origin O gives E = p , L = 0, P = ˆ p e ω H t and Q = ˆ p e − ω H t such that P · Q = ˆ p . Note that thereis a natural choice of the initial moment, t = 0, for which we haveˆ p ( t ) = P = Q and the calculations become simpler.Now we can revisit the problem of Sec. 2.3 looking for the value of t f which solves the equation (29). Taking into account that now P = p e ω H t = − n p e ω H t and x = d ( t ) = n d ( t ) we obtain the identity a ( t ) a ( t f ) = e − ω H ( t f − t ) = 1 p (cid:0) p − ω H d ( t ) p (cid:1) − m p , (54)13hich may be substituted in Eq. (27) leading to the final result p ( t f ) = p − ω H d ( t ) p = p + ω H d ( t ) · p , (55) p ( t f ) = ˆ p ( t f ) = − n q p ( t f ) − m , (56)expressed exclusively in terms of physical quantities. Hereby we deducethe relative energy loss e = ω H d ( t ) pp = ω H d ( t ) v , (57)proportional with the initial velocity v of the particle lunched by O ′ .In the case of the massless photons v = 1 recovering the energy lossproducing the redshift. It remains to derive the final distance and velocitywhich take the form d ( t f ) = d ( t ) p | p ( t f ) | , (58)ˆ v ( t f ) = (cid:18) m | p ( t f ) | (cid:19) − , (59)as it results from Eqs. (30), (31) and (54).For understanding the role of the conserved quantities in this ex-periment we must specify that the observers O and O ′ record differentconserved quantities since the translation is an isometry which changesthe components of the conserved quantities apart from the conservedmomentum which is not affected by these isometries [6]. Moreover, asthe origins of these frames are on the geodesics, both the angular mo-menta measured in O and O ′ vanishes. We denote by E, Q the remainingconserved quantities measured in O and by E ′ , Q ′ those recorded in O ′ bearing in mind that P ′ = P = p e ω H t . (60)The values observed in O ′ can be deduced from Eqs. (44) and (46)for x ′ = 0 obtaining the previous mentioned result, E ′ = p and Q ′ = p e − ω H t . The observer O prefers to look for the conserved quantities atthe time t since his knows that these do not change along the geodesic.Thus he records E = p ( t f ) , (61) Q = e − ω H t (cid:2) ω H d ( t ) E + p (cid:0) − ω H d ( t ) (cid:1)(cid:3) , (62)as it results from Eqs. (44) and (46) for x = d ( t ), verifying that P · Q = p ( t f ) for closing again the identity (40). Note that the relation14mong the conserved quantities E, P , ... and E ′ , P ′ , ... can be deriveddirectly according to the transformation rule under isometries we havediscussed recently [19, 20]. The de Sitter null geodesics of the photons with m = 0 that read x ( t ) = x e ω H ( t − t ) + n P e ω H ( t − t ) − ω H , (63)are interesting being involved in the theory of the redshift. The energyand covariant momentum denoted now by k ( t ) and respectively k ( t ) are k ( t ) = P e − ω H t = | ˆ k ( t ) | , (64) k ( t ) = e − ω H t P ( n P + ω H x ( t )) = ˆ k ( t ) + ¯ k ( t ) (65)such that we can separate the peculiar momentum, ˆ k ( t ) = e − ω H t P , andthe recessional one, ¯ k ( t ) = ω H x ( t ) P e − ω H t = ω H x ( t ) k ( t ).Considering again the problem of Sec. 2.3 we assume that now theobserver O ′ emits a photon of energy k and momentum k = − n k . Undersuch circumstances Eq. (54) gives a ( t ) a ( t f ) = e − ω H ( t f − t ) = k [1 − ω H d ( t )] , (66)allowing us to derive the quantities observed by O in his proper frame,namely the energy and covariant momentum, k ( t f ) = k [1 − ω H d ( t )] , (67) k ( t f ) = ˆ k ( t f ) = − n k ( t f ) , (68)the value of the final time t f = t − ω H ln [1 − ω H d ( t )] , (69)the final distance between O and O ′ at the time t f , d ( t f ) = d ( t )1 − ω H d ( t ) , (70)and the redshift z related to the relative energy loss e observed by O ,1 − e = 11 + z = 1 − ω H d ( t ) , (71)15esulted from Eq. (33). We recall that the condition ω H d ( t ) < E = k ( t f ) = k [1 − ω H d ( t )] , (72) P = k e ωt , (73) Q = k e − ω H t [1 − ω H d ( t )] , (74)such that P · Q = E satisfying the identity (40) with m = 0. We observeagain that for the special choice t = 0 we have P = k and d ( t ) = d which simplifies the calculations and their interpretation. Let us finish with an example of manifold whose kinematics was neverstudied. This is the spatially flat FLRW manifold M with the Milne typescale factor a ( t ) = ω M t defined on the domain t ∈ (0 , ∞ ), whose constant(frequency) ω M is introduced from dimensional reasons [17, 18]. Thenwe may write the line element in the physical co-moving frame { t, x } as ds = (cid:18) − t x (cid:19) dt + 2 x · d x dtt − d x · d x , (75)after substituting in Eq. (6) the Hubble function ˙ a ( t ) a ( t ) = t which isindependent on ω M . The conformal time t c ∈ ( −∞ , ∞ ) is defined as t c = Z dta ( t ) = 1 ω M ln( ω M t ) → a ( t c ) = e ω M t c , (76)obtaining the function a ( t c ) of the line element (4) of the conformal co-moving frame { t c , x c } .Here the constant ω M is an useful free parameter representing theexpansion speed of M . We remind the reader that in the case of thegenuine Milne universe (of negative space curvature but globally flat) onemust set ω M = 1 for eliminating the gravitational sources [2]. In contrast,our space-time M is produced by isotropic gravitational sources, i. e. thedensity ρ and pressure p , evolving in time as [17] ρ = 38 πG t , p = − πG t , (77)and vanishing for t → ∞ . These sources govern the expansion of M thatcan be better observed in the frame { t, x } where the line element (75)16ays out an expanding horizon at | x h | = t and tends to the Minkowskispace-time when t → ∞ and the gravitational sources vanish.We deduce first the equation of the time-like geodesics, solving theintegral of Eq. (19), which leads to the final form x ( t ) = tt x + n P t ln tt P + p P + ω M m t P + p P + ω M m t ! , (78)that for m = 0 gives the equation x ( t ) = tt x + n P t ln (cid:18) tt (cid:19) , (79)of the null geodesics. The energy, momentum and velocity have to bederived according to Eqs. (17), (14) and (24). These are complicatedformulas but that can be used in applications by using algebraic codeson computer.Furthermore, coming back to the problem of two observers formulatedin Sec. 2.3., we solve Eq. (29) for deriving the final time t f and the ratio a ( t ) a ( t f ) = t t f = 12 e − ω M d ( p + p ) − e ω M d m p ( p + p ) (80)We recall that p is the scalar initial momentum of the particle of mass m lunched by O ′ at the time t . Then, according to Eqs, (27) and (28)we find the final energy and covariant momentum p ( t f ) = 12 e − ω M d ( p + p ) + e ω M d m p + p (81) p ( t f ) = − n e − ω M d ( p + p ) − e ω M d m p + p , (82)and the final distance between O and O ′ when the particle arrives in O , d ( t f ) = d ( t ) 2 p ( p + p ) e − ω M d ( p + p ) − e ω M d m , (83)where d ( t ) = d a ( t ) = ω M t d . As in this geometry the horizon is at t we must impose the restriction d ( t ) < t → ω M d < d ( t f ) < t f .When O and O ′ observe a photon then they record t f = t e ω M d , k ( t f ) = | k ( t f ) | = k e − ω M d and the redshift 1 + z = e ω M d which for smallvalues of ω M d can be confused with the de Sitter one since the expansion11 + z = e − ω M d = 1 − ω M d + O ( ω M d ) , (84)17s somewhat similar with Eq. (71). However, for larger distances thediscrepancy between the linear behaviour of the de Sitter redshift andthe exponential one in the space-time M becomes obvious.Finally we observe that the Milne-type and ds Sitter universes be-have somewhat complementary such that the cosmic time of one of thesemanifolds behaves as the conformal time of the other one. The self ex-planatory next table completes this image [18]. M de Sitter t < t = ω M e ω M t c < ∞ −∞ < t < ∞ t c −∞ < t c < ∞ −∞ < t c = − ω H e − ω H t < a ( t ) ω M t e ω H t a ( t c ) e ω M t c − ω H t c transl. ω M d < ω H d <
11 + z e ω M d [1 − ω H d ( t )] − The only similarity is the condition satisfied by the translation parameter d for remaining inside the horizon. We presented here the complete kinematics in co-moving frames withphysical coordinates on spatially flat FLRW space-times, based on theconserved quantities among them the conserved momentum is the cen-tral piece of our approach. In these frames, the geodesics are determinedcompletely by the initial condition and conserved momentum. Moreover,this allows us to separate the peculiar motion from the recessional onesuch that the energy and peculiar momentum satisfy the mass-shell con-dition of special relativity. In this framework we discussed the problemof two observers pointing out the relative energy loss during propagationwhich in the massless case gives the well-known redshift.The first example is the kinematics of the de Sitter expanding universerelated to our previous results concerning the geodesics of this manifold[8]. Here we presented for the first time the measurable quantities ongeodesics in physical co-moving frames showing how these are related tothe rich set of the conserved quantities of this geometry. We observedthat only the conserved energy is related directly to the measured onewhile the conserved momentum and its dual help each other in closing themass-shell relation. Moreover, we pointed out that the meaning of theconserved momentum depends on the choice of the initial time showing18hat we can set this time as the moment in which the conserved momen-tum coincides with the covariant initial momentum. This observation isimportant since the momentum operator of de Sitter quantum mechanicsis related to the conserved momentum [16].The second example we presented here for the first time is the kine-matics of a new manifold we considered recently in quantum theory[17, 18]. This is a spatially flat FLRW space-time with a Milne typescale factor produced by gravitational sources proportional with t − .The geodesic motion on this manifold was studied in physical co-movingframes deriving the kinetic quantities on geodesics and outlined the re-sults of the experiment of two observers including the redshift. Moreover,we argued that this manifold is interesting since it behaves complemen-tary to the de Sitter one having thus two different examples of FRLWkinematics.As a final conclusion we may say that the physical coordinates andthe conserved momentum offer the suitable framework in which we candistinguish without any ambiguity between the recessional motion dueto the background expansion and the peculiar one which behaves just asin special relativity. Thus we may get a new perspective in interpretingthe astrophysical measurements in our actual expanding universe. References [1] S. Weinberg,
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