arXiv:2012.14110v2 [gr-qc] 4 May 2021 nrilcs 3.Mauigterdhf stemi ehdt e info get cosm to and method energy main dark measurements. the these on from is Information mainly redshift in obtained movement. light the their t of Measuring for and redshift known Galaxies the well [3]. and is effect case It Doppler inertial small. the not that is frame tensor reference metrical in term nondiagonal rbe fteitrrtto fapaac ftrsaiigdet due arising terms arising. is of o cosmology space appearance physical in of in or cosmolog interpretation used Earth in the the frame on of ba reference synchronous telescopes in problem the Our terms to quadrupole 2]. not coo understan [1, and correspond different mentioned to dipole in be of necessary bodies can example is of tion known movement it well of mechanics the equations classical cosmology in for terms case equally different they the of region is space-time it same as the describe systems ordinate yrttn at sntrli n at odsrb l observatio all describe to wants one if noninertial natural example fram For Earth. is reference frame. Earth rotating non inertial rotating the of the for by of use use forces to case of inertial reasonable in case called not force the so is Coriolis the in the introduce than and must simpler one force is case last usually prefe the frames as In such frames in reference inertial motion has of one mechanics classical In Introduction 1 h ieec ewe eeec rmsi motn o ag dis large for important is frames reference between difference The nGnrlRltvt hr r opeerdcodnt systems coordinate preferred no are there Relativity General In nryA Grib A. Andrey Systems Cosmology Coordinate in Different of Effects Some Abstract Russia Kazan, versity, nain osati iseneutosi made. cosmolog c is effective the equations this of Einstein of in action difference constant the invariant the to of on analogous estimate T synchronous distance Numerical the the leads from effects. to different frames observable system proportional different different the of in in Use appearance resulting the forces made. is inertial cosmology in in used frames ne,6 oso,VO,S.Ptrbr,197,Rsi;yu Russia; 199178, Petersburg, St. V.O., Bolshoy, 61 ences, 8Mia t eesug 916 usa andrei Russia; 191186, Petersburg, St. Moika, 48 4 3 2 1 ..LbcesyIsiueo ahmtc n ehnc,K Mechanics, and Mathematics of Institute Lobachevsky N.I. .FidanLbrtr o hoeia hsc,S.Peter St. Physics, Theoretical for Laboratory Friedmann A. nttt fPolm nMcaia niern fRussian of Engineering Mechanical in Problems of Institute hoeia hsc n srnm eatet h eznU Herzen The Department, Astronomy and Physics Theoretical h nlsso h yaiso ailmvmn ndffrn re different in movement radial of dynamics the of analysis The 1 , 2 n uiV Pavlov V. Yuri and 3 , 4 [email protected] [email protected] sal sdi t othe So it. in used usually y eiprateetis effect important he rdoe.Tedescription The ones. rred sooia constant. osmological clcntn n the and constant ical .Hwvrsmtmsit sometimes However e. fteacceleration the of e br,Russia sburg, xml h centrifugal the example eeec rm formed frame reference ti ieetfo the from different is it znFdrlUni- Federal azan a eue.However used. be can h eeec frame reference the o ftodffrn co- different two If . so h immovable the on ns cdm fSci- of Academy otedifference the to h hsclsense physical the d dnt ytm.In systems. rdinate lgclcntn is constant ological ecs frotating of case he h oa system Solar the f acsweethe where tances krudradia- ckground nrilframes. inertial mto nfar on rmation niversity, ference 1 2

And it is well known from the text book [4], Sec. 88, Problem one, that the forces acting on a particle in a gravitational field are obtained from the analysis of the Christoffel terms in equations. That is why we must write these equations and expressions for such terms to understand the origin of noninertial forces appearing in our case. In this paper we write geodesic equations in the homogeneous expanding Universe in two different reference frames often used in cosmology and investigate the physical meaning of different terms arising in them. The expressions for the energy and momentum are also obtained. In our previous papers [5, 6, 7] we showed that besides the difference in inertial forces there is a difference in possible energies of particles, i.e. existence of particles with negative energies in the nonsynchronous system. The analysis of geodesic equations made by us shows that besides the usual existence of some kind of the viscosity force existing in the synchronous frame leading to deceleration there appears a term leading in some cases to the acceleration. This acceleration occurs however for the cases of nonzero cosmological constant or exotic matter equation of state which seems to be the expectable result.

2 Isotropic homogeneous Universe and Einstein’s equation

The square of space-time interval of the isotropic cosmological Friedmann model can be written in synchronous frame [4] as

dr2 ds2 = c2dt2 − a2(t) + r2dΩ2 , (1) 1 − Kr2   where c is speed of light, parameter r is changing from 0 to ∞ in open (K = −1) and quasi- Euclidean flat models (K = 0), and from 0 to 1 in closed cosmological model (K = 1), dΩ2 = dθ2 + sin2 θdϕ2. Changing the variables

sin χ, K =1, r = f(χ)= χ, K =0, (2)   sinh χ, K = −1 the (1) can be written also in form

ds2 = c2dt2 − a2(t) dχ2 + f 2(χ)dΩ2 . (3)

In closed model χ is changing varying from 0 to π, in cases K =0 , −1 one has χ ∈ [0, +∞). Using the conformal time η: cdt = a(η) dη, (4) the metric (3) takes the form

ds2 = a2(η) dη2 − dχ2 − f 2(χ)dΩ2 . (5)

The Christoffel symbols 

1 ∂g ∂g ∂g Γ i = gim mk + ml − kl (6) kl 2 ∂xl ∂xk ∂xm   3

in the homogeneous isotropic space-time with metric (5) are

′ 0 a a˙ i a˙ i 0 a˙ α α Γ 00 = = , Γ 0 = δ , Γ = γ , Γ (g )=Γ (γ ) (7) a c j c j αβ c αβ βδ ik βδ νµ where prime denotes the derivative with respect to conformal time η, the dot above symbol is the derivative with respect to time t, γαβ is metric of 3-dimensional space of constant curvature K. Ricci tensor components and the scalar curvature are aa¨ aa¨ a˙ 2 R00 =3 , R = −γ +2 + K , (8) c2 αβ αβ c2 c2    6 aa¨ a˙ 2 R = + + K . (9) a2 c2 c2    Einstein’s equations are 1 G R − Rg +Λg = −8π T , (10) ik 2 ik ik c4 ik where Λ is cosmological constant, G is the gravitational constant, Tik is energy-momentum tensor of background matter. Equations (10) for metric (5) are

2 2 2 a˙ + Kc c 8πG 0 = T0 +Λ , (11) a2 3 c4   3 a¨ a˙ 2 + Kc2 c2 8πG + = T α +Λ . (12) a 2a2 2 3c4 α α=1 ! X From (11), (12) one obtains

2 3 a¨ c 4πG α 0 = Λ+ T − T0 . (13) a 3 c4 α α=1 !! X In comoving coordinates the energy-momentum tensor of background matter in isotropic homogeneous Universe is diagonal

k Ti = diag (ε, −p, −p, −p), (14) where ε and p are the energy density and pressure of background matter. So a¨ Λc2 4πG = H˙ + H2 = − (ε +3p) , (15) a 3 3c2 where H =a/a ˙ is the Hubble “constant”. We call it the “constant” following [8] in spite it is a variable depending on time. The radial distance between points χ = 0 and χ in metrics (3), (5) is D = a(t)χ and it’s the same in the metric (1). If t is fixed then the maximal value of D in the closed model is Dmax = πa(t). In open and flat models D is non limited. Nonzero D can be understood as the distance to the far Galaxy from the place where the observer is located D = 0. The distance D corresponds to the “proper distance” of book [9] and is equal to 4

the distance that would be measured by observers located rather closely in the expanding universe between the origin of and the point with comoving coordinate χ at the same moment of cosmic time t. Take the new coordinates t,D,θ,ϕ (see also [10, 11]). Then one obtains

a˙ 1 a˙ dD = D dt + a dχ, dχ = dD − Ddt (16) a a a   and the interval (3) becomes

2 HD ds2 = 1 − c2dt2 +2HD dDdt − dD2 − a2f 2(D/a) dΩ2. (17) c   !

Note that metric (17) is not singular on the surface D = c/H, in spite of g00 = 0. The 4 4 2 det (gik) = −a f (D/a) sin θ is zero only for D = 0 or θ = 0, π, where the coordinate singularities are conditioned by use of spherical coordinates of 3-space. The surface D = c/H is analogous to the static limit for rotating black hole (see [6]). It is also called apparent horizon in cosmology [12]. The energy-momentum tensor for background matter in coordinates t, D can be found from formula Tik =(p + ε)uiuk − pgik (18) (see [4]) and expressions for four-velocities ui of the background matter (with χ = const)

HD ui = 1, , 0, 0 , u = (1, 0, 0, 0) . (19) c i   That is why one has 0 β β T0 = ε, Tα = −pδα (20) and therefore ε and p are the energy density and pressure of background matter in coordi- nates t, D also. The same result can be obtained by coordinate transformation of the tensor (14) from one coordinate system to another. The difference in the form of the energy-momentum tensor in coordinates t, D is manifested in the appearance of non-diagonal terms. Note that for small distances (DH/c ≪ 1, D/a ≪ 1) the metric (17) becomes the metric of comoving spherical coordinate system

ds2 = c2dt2 − dD2 − D2dΩ2. (21)

If the object is located far from the observer (DH/c ≪ 1 is not correct) one cannot go to the diagonal form (21).

3 Free movement in different coordinate systems

Let us write the equation for the geodesic lines

d2xi dxk dxl = −Γ i , (22) dλ2 kl dλ dλ 5

where λ is the affine parameter on the geodesic. The terms in the right hand side are similar to inertial forces in nonrelativistic classical mechanics. First study the radial movement in coordinates η, χ. Equations of radial in these coordinates taking into account (7) are

2 2 d2η a˙ dη dχ + + =0, (23) dλ2 c dλ dλ "    # d2χ a˙ dη dχ +2 =0. (24) dλ2 c dλ dλ In order to give the interpretation of dissipative terms in (24) let us find the energy and momentum of the particle in the corresponding coordinate system. Note that the geodesic equations can be obtained from the Lagrangian [13] g dxi dxk L = ik . (25) 2 dλ dλ Generalized momenta are by definition

k def ∂L dx p = = g , (26) i ∂x˙ i ik dλ i i i where nowx ˙ = dx /dλ. The value of pip is conserved due to Euler-Lagrange equations d ∂L ∂L − =0. (27) dλ ∂x˙ n ∂xn For time-like geodesics the affine parameter can be taken as λ = τ/m so that τ is the of the particle and then

ik 2 2 pipkg = m c . (28)

n If the metric components gik do not depend on some coordinate x then the correspond- ing canonical momentum (the corresponding covariant component) is conserved in motion along the geodesic due to Euler-Lagrange equations: ∂g ∂L dxl ik =0 ⇒ p = = g = const. (29) ∂xn n ∂x˙ n nl dλ Note that contravariant components of the 4-momentum correspond to the 4-velocities multiplied by the particle mass pn = mdxn/dτ and are generally not conserved even in case when metric does not depend in the corresponding coordinates. The covariant radial component of the momentum of the particle in the homogeneous isotropic expanding space with metrics (3) and (5) are dχ dχ p = −a2(t) = −ma2(t) . (30) χ dλ dτ The covariant radial component of the momentum for metric (17) is dD dt p p = − + HD = χ . (31) D dλ dλ a 6

One can see from (30), (31) that pD depends on the velocity dχ/dτ but not on the value of χ. Contravariant radial component for metric (17) is dD dχ dt pD = m = ma + mDH . (32) dτ dτ dτ For D → 0 the metric (17) becomes the metric of comoving coordinate system (21), so let D us callp ˜ = ma dχ/dτ = −pχ/a the “physical” radial component of the momentum of the particle. For radial movement pχ = const, because the metric components g00, g01, g11 don’t depend on χ (see (5) and (29)). So after expanding of space in k-times, the “physical” momentum −pχ/a, becomes smaller in k-times. The energy defined by translations in time η is equal to dη E = p c = mca2 . (33) η η dτ For metric (17) due to (26) one obtains

2 HD dt dD E = p c = mc2 1 − + mHD . (34) D t c dτ dτ   ! Let us call the “physical” energy E the energy measured by the observer in the reference frame of the background matter in which at the given moment the particle is at the origin. Going to the limit D → 0 in (34) we obtain dt E E = mc2 = η . (35) dτ a

Note that E is equal to ED only for D → 0. From (33) and (35) equation (24) can be written as d2χ E dχ +2H =0. (36) dτ 2 mc2 dτ So radial movement in expanding Universe for the observer with coordinate χ is similar to movement in viscous medium with viscosity proportional to the Hubble constant. Writing (36) in the form d2χ 2E ma = − Hp˜D, (37) dτ 2 mc2 one obtains that the “inertial” force for the coordinate χ is equal to the doubled specific energy of the body with the minus sign multiplied on the ratio of the “physical” momentum of the particle to the Hubble time tH =1/H. From equations (23), (24) we find, that

2 d2χ dχ adχ + H 4+ =0. (38) dt2 dt cdt   ! For case of non-relativistic movement relative to background matter one has |adχ/cdt|≪ 1 and therefore d2χ dχ ma ≈ −4maH = −4Hp˜D. (39) dt2 dt 7

So for the observer using not the proper time of the moving particle but the coordinate time t, the “inertial” force is equal to the ratio of the “physical” momentum of the particle to the Hubble time tH multiplied by minus four. One can find the dependence of the radial component in time in case of radial movement of point mass m with fixed conserved radial component of momentum −pχ at point χ0 in time t0 from the equations (28), (30)

t pχ dt χ(t)= χ0 + . (40) 2 m 2 pχ tZ0 a 1+ mca r   Now let us consider radial movement in coordinates (t, D). Changing the variables in (23), (24), one obtains the radial geodesic equations in coordinates (t, D)

2 d2t H dD dt + − DH =0, (41) dλ2 c2 dλ dλ   2 2 d2D DH2 dD dt dt + − DH − (H˙ + H2)D =0. (42) dλ2 c2 dλ dλ dλ     These equations can be obtained directly from (22) using the Christoffel symbols in met- ric (17)

2 3 2 0 D H 0 DH 0 H Γ 00 = , Γ 01 = − , Γ 11 = , c3 c2 c 2 4 2 3 2 1 D D H 2 1 D H 1 DH Γ 00 = − H˙ − H , Γ 01 = − , Γ 11 = . (43) c2 c2 c3 c2   i dxk dxl Note that due to [4], Sec. 87 terms mΓ kl dλ dλ play the role of forces acting on the particle with mass m in gravitational field. It is evident that such forces depend on the choice of the reference frame. In absence of gravitation due to the possibility of inertial reference frame one can discriminate inertial forces from other forces. In general case such differentiation is impossible. One of the solution of the system of equation (41), (42) is a dt D = D0, = const. (44) a0 dλ This solution describes the of the particle at rest in synchronous with the back- ground matter reference frame (3): χ = χ0 and proper time is equal to the coordinate time t (see also [14]). The equation of movement (42) by using (31), (35) can be written as

2 d2D p 2 E = −DH2 D +(H˙ + H2)D . (45) dτ 2 mc mc2     In nonrelativistic case the first term proportional to the square of the velocity corresponds to the force of resistance for movement in medium. The force of this resistance is propor- tional to the square of the Hubble constant and the distance from the point of observation. 8

In nonrelativistic case pD ≈ mv, where v is the velocity of the moving body relative to background matter. So the first term is in (v/c)2 times smaller than the second. The second term in nonrelativistic case (E ≈ mc2) does not depend on the velocity and is proportional to the distance D from the coordinate origin, i.e. corresponds to some cosmological constant. Consider this issue more exactly. Let us write the interval for the case of “point” mass M in presence of cosmological constant [15] (Kottler metric [16])

2GM Λr2 dr2 ds2 = 1 − − c2dt2 − − r2(dθ2 + sin2 θdφ2). (46) c2r 3 2GM Λr2   1 − − c2r 3 In nonrelativistic case (see Sec. 99 in [4]) this metric leads to gravitational potential

GM Λc2r2 ϕ = − − (47) r 6 and to the acceleration of test body d2r GM Λc2r = − + . (48) dt2 r2 3 So in nonrelativistic case the cosmological constant describes the force proportional to the distance to the observable body [17]. From (41), (42) one can obtain the equation of radial movement of the test body in homogeneous isotropic cosmology in coordinates t, D, written as

3 d2D H dD = − DH +(H˙ + H2)D. (49) dt2 c2 dt   Due to dD/dt − DH = adχ/dt the first term describes in analogy with the mechanics of continuous media the drugging of the body by the moving viscous medium if the body is moving in the same direction as medium and it describes deceleration if the body moves in the opposite direction. The arising inertial force is proportional to the cube of the relative velocity. The second term as in case (45) is similar to action of the cosmological constant. Comparing this term with Λ in equation (48), one obtains the “effective” cosmological constant Λeff 3 2 Λeff = (H˙ + H ). (50) c2 Due to (15) one obtains the relation of this effective cosmological constant with the constant present in Einstein’s equation and the energy density and pressure in homogeneous isotropic cosmological model 4πG Λeff =Λ − (ε +3p) . (51) c4 One can use (15) for evaluation (51) in the comoving system because H is the same in our different frames. So nonaccurate use in cosmology of coordinates t, D can lead to the measurement of Λeff instead of true cosmological constant Λ. 9

Let us evaluate Λ − Λeff for real Universe taking p = 0 and the energy density of visible 2 2 and dark matter ≈ 31% of the critical density εc = 3H c /(8πG) [18] so that Λ term (or dark energy) is approximately ≈ 69%:

4πG 3H2 Λ − Λeff = (ε +3p) ≈ 0.31 ≈ 0.22Λ. (52) c4 2c2 So use of t, D coordinates and usual nonrelativistic expression (48) for cosmological con- stant can lead to the error of the order of 20%. If Λ = 0 one can see from (51) that in order to have the true sign “+” for the Λeff as in the case of observable cosmological constant [19]–[21] it is necessary to suppose the dominance of exotic background matter (quintessence etc.) with ε +3p< 0.

4 Conclusion

Our analysis shows that for any measurement of cosmological distances it is necessary to use the dynamical equations in corresponding coordinates to exclude the effects similar to inertial forces. In this paper we show that these forces can lead not only to deceleration of galaxies but also to acceleration. However this acceleration cannot mean that inertial force plays the role of “dark energy” because it occurs only in case of positive cosmological constant which usually is considered as “dark energy”. Nevertheless our analysis shows that due to the effect of the used coordinate system the measured value of the cosmological constant Λeff can be different from Λ — the cosmological constant in Einstein’s equations. It is the matter of fact that our above estimate of 20% varies during the evolution of the Universe. Really, if the Universe would be described by the de Sitter metric, then, taking ε = p = 0, one easily obtains Λeff = Λ. The same is valid for the Kottler metric. In these cases there is no need in “dark energy” with ε +3p < 0. For the Universe described by the metric of the Friedmann stage going to the de Sitter stage the effective slowly varying cosmological constant converges to the value of the invariant cosmological constant. The effective cosmological constant is non invariant being calculated through the Christoffel symbol which is not a tensor. It is proportional to the square of the Hubble constant depending on time and becomes really constant at the De Sitter stage. But then one sees that the invariant constant, which has the same value, also depends on the square of the Hubble constant at some fixed moment of time characterizing the transfer to the De Sitter stage. It cannot have arbitrary value. One of the much disputed matters concerning the observable value of the cosmological 2 constant is its proportionally to H0 value, where H0 is the modern value of Hubble constant (anthropic principle etc.). Strange as it is but term of the same order as we have shown in our paper appears due to use of t, D coordinate system. The message of our paper is not the doubt in existence of nonzero cosmological constant but in careful analysis of what we really observe using our instrumentation and what coordinate system corresponds to it. Acknowledgements The work of Yu.V.P. was supported by the Russian Government Program of Competitive Growth of Kazan Federal University. 10

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