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Some Effects of Different Coordinate Systems in Cosmology

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Some Effects of Different Coordinate Systems in Cosmology 1 Some Effects of Different Coordinate Systems in Cosmology Andrey A. Grib1,2 and Yuri V. Pavlov3,4 1 Theoretical Physics and Astronomy Department, The Herzen University, 48 Moika, St. Petersburg, 191186, Russia; andrei [email protected] 2 A. Friedmann Laboratory for Theoretical Physics, St. Petersburg, Russia 3 Institute of Problems in Mechanical Engineering of Russian Academy of Sci- ences, 61 Bolshoy, V.O., St. Petersburg, 199178, Russia; [email protected] 4 N.I. Lobachevsky Institute of Mathematics and Mechanics, Kazan Federal Uni- versity, Kazan, Russia Abstract The analysis of the dynamics of radial movement in different reference frames used in cosmology is made. Use of different frames leads to the difference in inertial forces resulting in different observable effects. The important effect is the appearance in the system different from the synchronous one of the acceleration proportional to the distance analogous to the action of the cosmological constant. Numerical estimate of the difference of this effective cosmological constant and the invariant constant in Einstein equations is made. 1 Introduction In classical mechanics one has inertial reference frames as preferred ones. The description of motion in such frames usually is simpler than in the case of use of noninertial frames. In the last case one must introduce the so called inertial forces for example the centrifugal force and the Coriolis force in case of the rotating reference frame. However sometimes it is not reasonable to use the inertial frame. For example noninertial reference frame formed by rotating Earth is natural if one wants to describe all observations on the immovable arXiv:2012.14110v2 [gr-qc] 4 May 2021 Earth. In General Relativity there are no preferred coordinate systems. If two different co- ordinate systems describe the same space-time region they equally can be used. However as it is the case for classical mechanics it is necessary to understand the physical sense of different terms in equations of movement of bodies in different coordinate systems. In cosmology the well known example of dipole and quadrupole terms in background radia- tion can be mentioned [1, 2]. Our telescopes on the Earth or in space of the Solar system correspond not to the synchronous reference frame in cosmology usually used in it. So the problem of the interpretation of appearance of terms arising due to the reference frame used in physical cosmology is arising. The difference between reference frames is important for large distances where the nondiagonal term in metrical tensor is not small. It is well known for the case of rotating reference frame that the Doppler effect and the redshift of light in it is different from the inertial case [3]. Measuring the redshift is the main method to get information on far Galaxies and their movement. Information on dark energy and cosmological constant is obtained mainly from these measurements. 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 geodesic 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 metric (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 coordinate system 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.
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