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The FLRW spacetimes are the most plausible models of our universe in various periods of evolution. The actual universe is observed as being spatially flat with a reasonable accuracy. For this reason, we focus here on such manifolds for which we consider many types of frames with Cartesian or spherical coordinates searching for measurable quantities expressed in terms of physical coordinates of Painlevé type.
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The Painlevé - Gullstrand coordinates [9, 10] were proposed for studying the Schwarzschild black holes. Similar coordinates can be introduced in any isotropic manifold
$ (M,g) $ having frames$ \{x\} = \{t,{\vec x}\} $ with flat space sections. In these frames, the coordinates,$ x^{\mu} $ ($ \alpha,\mu,\nu,\cdots = $ $ 0,1,2,3 $ ), may be formed by the cosmic time t and either Cartesian space coordinates$ {\vec x} = (x^1,x^2,x^3) $ or associated spherical ones$ (r,\theta,\phi) $ with Euclidean metric$ {\rm d}s^2_E = {\rm d}{\vec x}\cdot $ $ {\rm d}{\vec x} = {\rm d}r^2+r^2 {\rm d}\Omega^2 $ , where$ {\rm d}\Omega^2 = {\rm d}\theta^2+\sin^2\theta\, {\rm d}\phi^2 $ . For example, the line element$ {\rm d}s^2 = f(r)\, {\rm d}t_s^2-\frac{{\rm d}r^2}{f(r)}-r^2 {\rm d}\Omega^2\,, $
(1) of any static frame,
$ \{t_s,r,\theta,\phi\} $ , with static time$ t_s $ , can be put in Painlevé-Gullstrand form,$ {\rm d}s^2 = f(r){\rm d}t^2+ 2\sqrt{1-f(r)}\,{\rm d}t{\rm d}r-{\rm d}r^2-r^2 {\rm d}\Omega^2\,; $
(2) substituting this in Eq. (1),
$ t_s = t+\int {\rm d}r \frac{\sqrt{1-f(r)}}{f(r)}\,, $
(3) where t is the cosmic time.
Similar space coordinates, known as proper coordinates, called here more intuitively the physical Cartesian space coordinates and denoted by
$ \vec{x} $ , are associated with the cosmic time, t, defining the physical frame$ \{t,\vec{x}\} $ . These coordinates can be introduced in any spatially flat FLRW spacetime through those of the conformal Euclidean frame$ \{t_c,{\vec x}_c\} $ , i.e., the conformal time,$ t_c $ , and the comoving Cartesian coordinates,$ \vec{x}_c $ , giving the line element$ {\rm d}s^2 = a(t_c)^2\left({\rm d}t_c^2-{\rm d}{\vec x}_c\cdot {\rm d}{\vec x}_c\right)\,. $
(4) Substituting the physical coordinates
$ \{t,{\vec x}\} $ as$ t_c = \int \frac{{\rm d}t}{a(t)}\,, \quad {\vec x}_c = \frac{{\vec x}}{a(t)}\,. $
(5) we obtain the line element of the physical frame,
$ {\rm d}s^2 = \left(1-\frac{\dot a(t)^2}{a(t)^2}\, {{\vec x}}^2\right){\rm d}t^2+ 2\frac{\dot a(t)}{a(t)}\, {\vec x}\cdot {\rm d}{\vec x}\,{\rm d}t -{\rm d}{\vec x}\cdot {\rm d}{\vec x}\,, $
(6) where
$ a(t) = a[t_c(t)] $ is the usual FLRW scale factor while$ \frac{\dot a(t)}{a(t)} = \frac{1}{a(t)}\frac{{\rm d} a(t)}{{\rm dt}} = \frac{1}{a(t_c)^2}\frac{{\rm d} a(t_c)}{{\rm d}t_c}\,, $
(7) is the Hubble function for which we do not use a special notation. The inverse transformation
$ \{t,{\vec x}\}\,\to\,\{t_c,{\vec x}_c\} $ is obvious,$ t = \int a(t_c){\rm d}t_c\,, \quad {\vec x} = a(t_c){\vec x}_c\,. $
(8) We suppose that the function
$ a(t) $ is smooth such that the transformations (8) and (5) are diffeomorphisms.The metric (6) is of an observer at rest in origin (
$ {\vec x} = 0 $ ) and having a dynamic apparent horizon,$ \Big|{\vec x}_h(t)\Big| = \frac{a(t)}{\dot a(t)}\,. $
(9) This particularity makes this metric less popular in cosmology; however, it is useful in astrophysics for studying how different observers record the relative geodesic motion measuring directly the physical coordinates. Another advantage of this metric is that this is approaching to the Minkowski one in a neighbourhood of
$ {\vec x} = 0 $ . However, in cosmology one prefers the FLRW coordinates$ \{t,{\vec x}_c\} $ with the well-known line element$ {\rm d}s^2 = {\rm d}t^2-a(t)^2 {\rm d}{\vec x}_c\cdot {\rm d}{\vec x}_c\,, $
(10) because the comoving space coordinates comply with the homogeneity of the universes with flat space sections. In what follows we use mainly physical frames resorting to the conformal ones as an auxiliary tool when these offer technical advantages.
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Our principal objective is to derive the equations of the time-like and null geodesics and the associated kinetic quantities in the physical frame
$ \{t,{\vec x}\}_{O} $ of an observer staying at rest in the origin O. We search for the components$p^{\mu} \!=\! \dfrac{{\rm d}x^{\mu}}{{\rm d}\lambda}$ of the four-momentum$(p^0, {\vec p} )$ formed by the measured energy$ p^0 $ and covariant momentum$ {\vec p} $ . Here$ \lambda $ is an affine parameter related to the mass m of a particle moving freely along a geodesic as$ {\rm d}s = m\,{\rm d} \lambda $ .We start with an intermediate step, focusing first on the components
$ p^{\mu}_c = \dfrac{{\rm d}x^{\mu}_c}{{\rm d}\lambda} $ of the four-momentum in the conformal frame$ \{t_c,{\vec x}_c\}_{O} $ of our observer where we meet the simple prime integral,$ a(t_c)^2\left[p_c^0(t_c)^2-{\vec p}_c(t_c)^2\right] = m^2\,, $
(11) resulted from the line element (4). In other respects, we may exploit the fact that the spatially flat FLRW spacetimes have
$ E(3) $ isometries formed by global rotations,$ x^i \to R^I_j x^j $ ($ i,j,k,\cdots = 1,2,3 $ ), and space translations$ \begin{array}{*{20}{l}} {{t_c}} = {{t_{c'}}{\mkern 1mu} ,}\\ {x_c^i} = {x_c^{\prime {\kern 1pt} i} + {\xi ^i}{\mkern 1mu} ,} \end{array} \to \begin{array}{*{20}{l}} t = {t'{\mkern 1mu} ,}\\ {{x^i}} = {{x^{\prime {\kern 1pt} i}} + {\xi ^i}{\mkern 1mu} a(t){\mkern 1mu} .} \end{array} $
(12) The associated Killing vectors
$ k_{(i)} $ have the components$ k^0_{(i)} = 0 $ and$ k^j_{(i)} = \delta_{ij} $ in the frame$ \{t_c,{\vec x}_c\}_{O} $ giving rise to the conserved quantities$ P^i = - k_{(i)\,j}\frac{{\rm d}x_c^j}{{\rm d}\lambda} = a(t_c)^2\frac{{\rm d}x_c^i}{{\rm d}\lambda}\,, $
(13) representing the components of the conserved momentum
$ \vec{P} $ which is different from the covariant momentum$ {\vec p}(t) $ . Then, by using the prime integrals (11) and (13) we derive the energy and covariant momentum in this frame as$ p_c^0(t_c) = \frac{{\rm d}t_c}{{\rm d}\lambda} = \frac{1}{a(t_c)} \sqrt{m^2+\frac{{P}^{2}}{a(t_c)^2}}\,, $
(14) $ p_c^i(t_c) = \frac{{\rm d}{x_c^i}}{{\rm d}\lambda} = \frac{P^i}{a(t_c)^2}\,, $
(15) where we denote
$ P = |{\vec P}\,| $ . The geodesic results from integrating the equation$ \frac{{\rm d}x_c^i}{{\rm d}t_c} = \frac{p_c^i(t_c)}{p^0_c(t_c)}\,, $
(16) that yields
$ x^i_c(t_c) = x_{c0}^i +\frac{P^i}{m}\int_{t_{c0}}^{t_c}\frac{{\rm d}t_c}{\sqrt{a(t_c)^2+\dfrac{P^2}{m^2}}}\,. $
(17) We may conclude that any time-like geodesic is determined completely by its conserved momentum
$ {\vec P} = {\vec n}_P P $ and the initial condition$ {\vec x}_c(t_{c0}) = {\vec x}_{c0} $ . However, this equation must be integrated in each particular case, but for the massless particles (with$ m = 0 $ ), we have the universal solution$ {\vec x}_c(t_c) = {\vec x}_{c0}+{\vec n}_P(t_c-t_{c0})\,, $
(18) giving the null geodesics on any FLRW spacetime.
The corresponding physical quantities measured by the observer O in his proper frame,
$ \{t,{\vec x}\}_{O} $ , may be obtained by substituting the physical coordinates according to Eq. (5). Thus we obtain the covariant components,$ p^0(t) = \frac{{\rm d}t}{{\rm d}\lambda} = \sqrt{m^2+\frac{{P}^{2}}{a(t)^2}}\,, $
(19) $ p^i(t) = \frac{{\rm d}{x^i}}{{\rm d}\lambda} = \frac{P^i}{a(t)}+x^i(t)\,\frac{\dot a(t)}{a(t)}\sqrt{m^2+\frac{{P}^{2}}{a(t)^2}}\,, $
(20) which represent the measured energy and covariant momentum in the point
$ [t, {\vec x}(t)] $ of the time-like geodesic,$ {\vec x}(t) = {\vec x}_0\,\frac{a(t)}{a(t_0)} + \frac{\vec P}{m}\,a(t)\int_{t_0}^{t}\frac{{\rm d}t}{a(t)\sqrt{a(t)^2+\dfrac{P^2}{m^2}}}\,, $
(21) which is passing through the space point
$ {\vec x}(t_0) = {\vec x}_0 $ at the initial time$ t_0 $ . In the physical frame$ \{t,{\vec x}\}_{O} $ the equation of the null geodesics,$ {\vec x}(t) = {\vec x}_0\,\frac{a(t)}{a(t_0)}+ {\vec n}_P\, a(t)\left[t_c(t)-t_c(t_0)\right]\,, $
(22) results from Eq. (18).
A special and delicate problem is that of the tachyons whose kinetic quantities on space-like geodesics can be obtained by substituting
$ m^2\,\to\,-m^2 $ in the above equations. Then the energy$ p^0_{\rm tach}(t) = \sqrt{-m^2+\frac{{P}^{2}}{a(t)^2}}\,, $
(23) is real valued only when
$ a(t)<\dfrac{P}{m} $ . This means that, in expanding universes, a tachyon with conserved momentum$ {\vec P} $ disappears when$ a(t) $ reaches the value$\dfrac{P}{m}$ . This may survive only in collapsing universes for smaller values of the scale factor. As here we focus only on expanding geometries, we ignore the space-like geodesics remaining to study the time-like and null ones.The covariant momentum defined by Eq. (20) can be split as
$ {\vec p}(t) = \vec{\hat p}(t)+ \vec{\bar p}(t) $ where$ \vec {\hat p} = \frac{\vec P}{a(t)}\,,\quad \vec{\bar p} = {\vec x}(t) \,\frac{\dot a(t)}{a(t)}\, p^0(t)\,, $
(24) are the peculiar and respectively recessional momenta. We must stress that this splitting can be performed only in the physical frames where the prime integral derived from the line element (6) gives the familiar identity
$ p^0(t)^2- \vec{\hat p}(t)^2 = m^2\,, $
(25) which is just the mass-shell condition of special relativity satisfied by the energy and peculiar momentum. Therefore, we may conclude that the peculiar momentum produces the entire energy of the geodesic motion as in special relativity. Thus for
$ {\vec P} = 0 $ and$ p^0(t) = m $ , the particle remains at rest in the point${\vec x}(t) = \dfrac{a(t)}{a(t_0)}{\vec x}_0$ but moving with the recessional momentum$ \vec{\bar p} $ with respect the observer O.We must emphasize that these properties hold only in physical frames because, in the conformal frames with comoving space coordinates, this separation is not possible, and the recessional momentum remains hidden, provided that the whole covariant momentum (15) is proportional to the conserved one. Moreover, in these frames, the momenta satisfy inappropriate dispersion relations, depending explicitly on time, as in Eq. (11) or the identity
$ p^0_c(t)^2-a(t)^2 {\vec p}_c(t)^2 = m^2\,, $
(26) that holds in FLRW comoving frames
$ \{t,{\vec x}_c\} $ .In physical frames, other interesting kinetic quantities can be derived as, for example, the velocity
$ {\vec v}(t) = \frac{{\vec p}(t)}{p^0(t)} = {\vec x}(t)\,\frac{\dot a(t)}{a(t)}+ \frac{\vec{\hat p}(t)}{\sqrt{m^2+\vec{\hat p}(t)^2}}\,, $
(27) where the first term is the recessional velocity due to the space evolution, complying with the velocity law, which is sometimes confused with the Hubble one [11, 12]; the second term is the peculiar velocity, which depends on the peculiar momentum as in special relativity.
The covariance under rotations, which behave here as a global symmetry, gives rise to the conserved angular momentum, which depends only on the peculiar momentum
$ {\vec L} = {\vec x}(t)\land {\vec p}(t) = {\vec x}(t)\land\hat {\vec p}(t)= \frac{{\vec x}(t)\land {\vec P}}{a(t)} = \frac{{\vec x}_0\land {\vec P}}{a(t_0)}\,, $
(28) and it can be related to the initial condition. Obviously, this vanishes when the observer O stays at rest in a space point of the measured geodesic.
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The physical quantities
$ {p^0}(t) $ ,$ \vec{\hat p}(t) $ and$ \vec{\bar p}(t) $ are functions of time; however, the last one depends explicitly on coordinate such that the experimental results will depend on the relative position between the detector and the source of the measured particle. However, this does not impede as the peculiar and recession contributions can be separated at any time without ambiguities. Nevertheless, to avoid confusions, we take care of this dependence, searching for suitable positions of the observer's proper frames to obtain intuitive results.The example we would like to discuss here is of two observers measuring the motion of a massive particle on a time-like geodesic which is passing through the origins O and
$ O' $ of their proper frames,$ \{t,{\vec x}\}_{O} $ and$ \{t,{\vec x}'\}_{O'} $ . We assume that at the initial time$ t_0 $ the origin$ O' $ is translated with respect to O as$ {\vec x}(t_0) = {\vec x}\,'(t_0)+{\vec d}(t_0) $ where$ {\vec d}(t_0) = $ $ {\vec d}\,a(t_0) $ depends on the translation parameters of Eq. (12) denoted now by$ \xi^i = d^i $ . Then it is convenient to introduce the unit vector$ {\vec n} $ of the direction$ OO' $ such that$ {\vec d} = {\vec n}\, d $ .Our experiment starts in this layout at
$ t_0 $ when the observer$ O' $ launches a particle of mass m, momentum$ {\vec p} = -{\vec n}\, p $ and energy$ p^0 = \sqrt{m^2+p^2}\,, $
(29) on the geodesic
$ O'\to O $ . The problem is to obtain which are the energy and momentum of this particle measured in the origin O at the final time$ t_f $ when the particle reaches this point. For solving this problem we first consider for the conserved momentum that can be derived in$ O' $ as$ {\vec P}' = {\vec P} = {\vec n}_P P = {\vec p}\,a(t_0)\,, $
(30) such that
$ P = p\, a(t_0) $ and$ {\vec n}_P = -{\vec n} $ . Then by substituting these results in Eqs. (19) and (20) we obtain the energy and momentum in O,$ p^0(t_f) = \sqrt{m^2+p^2 \frac{a(t_0)^2}{a(t_f)^2}}\,, $
(31) $ {\vec p}(t_f) = \vec{\hat p}(t_f) = -{\vec n}\, p\,\frac{a(t_0)}{a(t_f)}\,, $
(32) where
$ t_f $ solves the equation$ {\vec n}\cdot{\vec x}(t_f) = 0 $ with$ {\vec x}(t) $ given by Eq. (21) that can be written as$ \frac{P}{m}\int_{t_0}^{tf}\frac{{\rm d}t}{a(t)\sqrt{a(t)^2+\dfrac{P^2}{m^2}}} = d\,, $
(33) where d is the time-independent translation parameter defined above. Solving the integral (33) we obtain an algebraic equation that can be solved in term of the final time obtaining the function
$ t_f (P,t_0) $ . We observe that this function must be singular in$ P = 0 $ for preventing the left handed term of this equation on vanishing in this limit. Once we have the value of$ t_f $ , we can calculate the propagation time$ t_f-t_0 $ , the distance$ d(t_f) $ between O and$ O' $ at$ t_f $ , and the final peculiar velocity$ \hat v(t_f) $ of the particle arriving in O. According to Eqs. (12) and (31), we obtain$ { d}(t_f) = { d}\, a(t_f) = { d}(t_0)\,\frac{a(t_f)}{a(t_0)}\,, $
(34) $ \hat v(t_f) = \left( 1+\frac{m^2}{p^2}\frac{a(t_f)^2}{a(t_0)^2} \right)^{-\frac{1}{2}}\,, $
(35) completing thus the collection of the kinetic quantities related to this problem.
Eq. (31) shows that, in expanding universes, a part of energy is lost during the propagation. This can be measured by the relative energy loss we define here for the first time as
$ e = 1-\frac{p^0(t_f)}{p^0(t_0)} = 1-\frac{1}{p^0} \sqrt{m^2+p^2 \frac{a(t_0)^2}{a(t_f)^2}}\,, $
(36) because
$ p^0(t_0) = p^0 $ as it results from Eqs. (19) and (30). This phenomenon is similar to the redshift of the photons with$ m = 0 $ for which we recover the Lemaître equation [13, 14] of Hubble's law [15] as$ \frac{1}{1-e} = 1+z = \frac{p^0(t_0)}{p^0(t_f)} = \frac{a(t_f)}{a(t_0)}\,, $
(37) where z is the usual redshift defined as the relative dilation of the wave length. As expected, for
$ m = 0 $ , the final velocity$ \hat v(t_f) = 1 $ is the speed of light.All the results presented here can be exploited effectively only in concrete geometries where the geodesic, Eq. (21), can be integrated. In what follows, we discuss two such examples starting with one of the most studied geometries.
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The first example is the expanding portion of the de Sitter spacetime defined as the hyperboloid of radius
$ 1/\omega_{\rm H} $ in the five-dimensional flat spacetime$ (M^5,\eta^5) $ of coordinates$ z^A $ (labelled by the indices$ A,\,B,\cdots = 0,1, $ $ 2,3,4 $ ) having the metric$ \eta^5 = {\rm diag}(1,-1,-1,-1,-1) $ . The coordinates$ \{x\} $ can be introduced giving the set of functions$ z^A(x) $ , which solve the hyperboloid equation,$ \eta^5_{AB}z^A(x) z^B(x) = -\frac{1}{\omega_{\rm H}^2}\,. $
(38) where
$ \omega_{\rm H} $ is the Hubble de Sitter constant in our notations.There are conformal frames,
$ \{t_c,{\vec x}_c\} $ [16], associated to physical ones,$ \{t,{\vec x}\} $ , having the conformal and cosmic times related as$ t_c = -\frac{1}{\omega_{\rm H}}{\rm e}^{-\omega_{\rm H} t} $
(39) and the scale factors
$ a(t) = {\rm e}^{\omega_{\rm H} t}\; \to\; a(t_c) = -\frac{1}{\omega t_c}\,, $
(40) defined for
$ t\in {\Bbb R} $ and$ t_c<0 $ corresponding to the expanding portion. In this case, the Hubble function becomes the constant$\omega_{\rm H}$ . In addition, this manifold allows even astatic frame$ \{t_s,{\vec x}\} $ with the line element (1) where$ f(r) = 1-\omega_{\rm H}^2 r^2 $ and$ t_s = t-\ln f(r) $ . -
The de Sitter manifold has a rich isometry group which is the gauge group
$ SO(1,4) $ of the embedding manifold$ (M^5,\eta^5) $ that leave invariant its metric and implicitly Eq. (38). Therefore, given a system of coordinates defined by the functions$ z = z(x) $ , each transformation$ {\frak g}\in SO(1,4) $ defines the isometry$ x\to x' = \phi_{\frak g}(x) $ derived from the system of equations$ z[\phi_{\frak g}(x)] = {\frak g}z(x) $
(41) that holds for any type of coordinates because these isometries are defined globally. The set of frames related through these isometries play the role of the inertial frames similar to those of special relativity.
Given an isometry
$ x\to x' = \phi_{{\frak g}(\xi)}(x) $ depending on the group parameter$ \xi $ there exists an associated Killing vector,$ {k} = \partial_{\xi}\phi_{\xi}|_{\xi = 0} $ (which satisfy the Killing equation$ {k}_{\mu;\nu}+{k}_{\nu;\mu} = 0 $ ). Thus, in a canonical parametrization of the$ SO(1,4) $ group, with real skew-symmetric parameters$ \xi^{AB} = -\xi^{BA} $ , any infinitesimal isometry can be written as$ \phi^{\mu}_{{\frak g}(\xi)}(x) = x^{\mu}+ \xi^{AB}k^{\mu}_{(AB)}(x)+\cdots $ . Starting with the general definition of the Killing vectors in the pseudo-Euclidean spacetime$ (M^5,\eta^5) $ , we may consider the following identity$ K^{(AB)}_C{\rm d}z^C = z^A{\rm d}z^B-z^B{\rm d}z^A = k^{(AB)}_{\mu}{\rm d}x^{\mu}\,, $
(42) giving the covariant components of the Killing vectors in an arbitrary frame
$ \{x\} $ of the de Sitter manifold as$ k_{(AB)\,\mu} = \eta^5_{AC}\eta^5_{BD}k^{(CD)}_{\mu} = z_A\partial_{\mu}z_B-z_B\partial_{\mu}z_A\,, $
(43) where
$ z_A = \eta_{AB}z^B $ .The classical conserved quantities along the time-like geodesics have the general form
$ {\cal{ K}}_{(AB)}(x,{\vec P}) = \omega_{\rm H} k_{(AB)\,\mu} p^{\mu} $ , where$ p^{\mu} $ are the components of the covariant four-vector defined above. The conserved quantities with physical meaning [17] are the energy,$ E = \omega_{\rm H} k_{(04)\,\mu}p^{\mu} $ , the angular momentum components,$ L_i = \frac{1}{2}\,\varepsilon_{ijk}k_{(jk)\,\mu}p^{\mu} $ , and the components$ K_i = k_{(0i)\,\mu} p^{\mu} $ and$ R_i = k_{(i4)\,\mu} p^{\mu} $ of two vectors related to the conserved momentum$ {\vec P} $ and its associated dual momentum$ {\vec Q} $ :$ {\vec P} = -\omega_{\rm H} ({\vec R}+{\vec K})\,, \quad {\vec Q} = \omega_{\rm H} ({\vec K}-{\vec R})\,. $
(44) satisfying the identity
$ E^2-\omega_{\rm H}^2 {{\vec L}}^2-{\vec P}\cdot {\vec Q} = m^2\,, $
(45) corresponding to the first Casimir invariant of the
$ so(1,4) $ algebra [17]. In the flat limit,$ \omega_{\rm H}\to 0 $ and$ -\omega_{\rm H} t_c\to 1 $ , we have$ {\vec Q} \to {\vec P} $ such that this identity becomes just the usual null mass-shell condition$ E^2-{\vec P}^2 = m^2 $ of special relativity.Note that the conserved quantities transform among themselves under de Sitter isometries including the simple translations, which in this case transform the energy and dual momentum as we have shown recently [6].
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The coordinates of the physical frame
$ \{t,{\vec x}\}_{O} $ are introduced by the functions$ \begin{aligned}[b] &z^0(x) = \frac{1}{2\omega_{\rm H}}\left[{\rm e}^{\omega_{\rm H} t}-{\rm e}^{-\omega_H t}\left(1 - \omega_{\rm H}^2{\vec x}^2\right)\right]\,, \\ &z^i(x) = x^i \,, \\ &z^4(x) = \frac{1}{2\omega_{\rm H}}\left[{\rm e}^{\omega_{\rm H} t}+{\rm e}^{-\omega_{\rm H} t}\left(1 - \omega_{\rm H}^2{\vec x}^2\right)\right]\,, \end{aligned} $
(46) giving the line element
$ {\rm d}s^2 = (1-\omega_H^2 {{\vec x}}^2){\rm d}t^2+2\omega_H {\vec x}\cdot {\rm d}{\vec x}\,{\rm d}t -{\rm d}{\vec x}\cdot {\rm d}{\vec x}\,, $
(47) with observer's horizon at
$ |{\vec x}_h| = \omega_{\rm H}^{-1} $ , such that the condition$ \omega_{\rm H}|{\vec x}|<1 $ is mandatory. In this frame, the equation of a time-like geodesic can be obtained by solving the integral of Eq. (21) that yields [8]$\begin{aligned}[b] {\vec x}(t) =& {\vec x}_0{\rm e}^{\omega_{\rm H}( t-t_0)}+{\vec n}_P\frac{{\rm e}^{\omega_{\rm H} t}}{\omega_{\rm H} P}\left(\sqrt{m^2+P^2{\rm e}^{-2\omega_{\rm H}t_0}}\,\right.\\ &-\left.\sqrt{m^2 +P^2{\rm e}^{-2\omega_{\rm H} t}}\right)\,, \end{aligned} $
(48) which is determined by the conserved momentum
$ {\vec P} = {\vec n}_P P $ and the initial condition$ {\vec x}(t_0) = {\vec x}_0 $ fixed at time$ t_0 $ .The conserved quantities in an arbitrary point
$ (t,{\vec x}(t)) $ of this geodesic can be expressed as [6, 8]$ E = \omega_{\rm H}\, {\vec x}(t)\cdot {\vec P}\, {\rm e}^{-\omega_{\rm H} t}+\sqrt{ m^2+{P}^{2}{\rm e}^{-2\omega_{\rm H}t}}\,, $
(49) $ {\vec L} = {\vec x}(t)\land {\vec P}{\rm e}^{-\omega_{\rm H} t}\,, $
(50) $ {\vec Q} = 2\omega_{\rm H}\, {\vec x}(t)E {\rm e}^{-\omega_{\rm H} t}+{\vec P} {\rm e}^{-2\omega_{\rm H} t}\left[1-\omega_{\rm H}^2{\vec x}(t)^2\right]\,. $
(51) satisfying the identity (45). Moreover, Eqs. (19) and (20) give the energy and covariant momentum components,
$ p^0(t) = \frac{{\rm d}t}{{\rm d}\lambda} = \sqrt{m^2 +P^2 {\rm e}^{-2\omega_{\rm H}t}}\,, $
(52) $ p^i(t) = \frac{{\rm d}x^i}{{\rm d}\lambda} \!=\! {\rm e}^{-\omega_{\rm H}t}P^i \!+\!\omega_{\rm H} x^i(t) \sqrt{m^2 \!+\!P^2 {\rm e}^{-2\omega_{\rm H}t}}\,, $
(53) that can be measured by the observer O in its proper frame
$ \{t,{\vec x}\}_{O} $ . The conserved quantities are related to the measured ones as$ E = \omega_{\rm H}\, {\vec x}(t)\cdot \vec{\hat p}(t)+p^0(t)\,, $
(54) $ {\vec L} = {\vec x}(t)\land \vec{\hat p}(t)\,, $
(55) $ {\vec P} = \vec{\hat p}(t)\,{\rm e}^{\omega_{\rm H} t} $
(56) $ {\vec Q} = {\rm e}^{-\omega_{\rm H} t}\left\{2\omega_{\rm H}\, {\vec x}(t)E +\vec{\hat p}(t) [1-\omega_{\rm H}^2{\vec x}(t)^2]\right\}\,. $
(57) Hereby we conclude that the conserved quantities depend only on position and peculiar momentum. Among them, only E and
$ {\vec L} $ can be measured while$ {\vec P} $ and$ {\vec Q} $ are not accessible directly, with their role pertaining to only closing the invariant (45) as$ E^2-\omega_{\rm H}^2 {\vec L}^2-{\vec P}\cdot {\vec Q} = p^0(t)^2-\vec{\hat p}(t)^2 = m^2\,. $
(58) For example, a measurement in observer's origin O gives
$ E = p^0 $ ,$ {\vec L} = 0 $ ,$ {\vec P} = \vec{\hat p}\,{\rm e}^{\omega_{\rm H} t} $ , and$ {\vec Q} = \vec{\hat p}\,{\rm e}^{-\omega_{\rm H} t} $ such that$ {\vec P}\cdot{\vec Q} = {\vec{\hat p}}\,^2 $ . Note that there is a natural choice of the initial moment,$ t_0 = 0 $ , for which we have$ \vec{\hat p}(0) = {\vec P} = {\vec Q} $ , and the calculations become simpler.Now we can revisit the problem presented in Sec. II.C, searching for the value of
$ t_f $ that solves Eq. (33). Taking into account that now$ {\vec P} = {\vec p}\,{\rm e}^{\omega_{\rm H} t_0} = -{\vec n}\, p\,{\rm e}^{\omega_{\rm H} t_0} $ and$ {\vec x}_0 = {\vec d}(t_0) = {\vec n} d(t_0) $ , we obtain the identity$ \frac{a(t_0)^2}{a(t_f)^2} = {\rm e}^{-2\omega_{\rm H}(t_f-t_0)}= \frac{1}{p^2}\left({p^0}-\omega_{\rm H} d(t_0) p\right)^2-\frac{m^2}{p^2}\,, $
(59) which may be substituted in Eq. (31) leading to the final result
$ p^0(t_f) = p^0-\omega_{\rm H} {d}(t_0){p}= p^0+\omega_{\rm H} {\vec d}(t_0)\cdot {\vec p}\,, $
(60) $ {\vec p}(t_f) = \vec{\hat p}(t_f) = -{\vec n}\sqrt{p^0(t_f)^2-m^2}\,, $
(61) expressed exclusively in terms of physical quantities with
$ p^0 $ given by Eq. (29). Hereby we deduce the relative energy loss$ e = \omega_{\rm H} d(t_0)\frac{p}{p^0} = \omega_{\rm H} d(t_0) v\,, $
(62) proportional to the initial velocity v of the particle lunched by
$ O' $ . In the case of the massless photons,$ v = 1 $ recovering the energy loss producing the redshift. It remains to derive the final distance and velocity, which take the form$ d(t_f) = d(t_0)\,\frac{p}{|{\vec p}(t_f)|}\,, $
(63) $ \hat v(t_f) = \left( 1+\frac{m^2}{|{\vec p}(t_f)|^2} \right)^{-\frac{1}{2}}\,, $
(64) as it results from Eqs. (34), (35), and (59).
For understanding the role of the conserved quantities in this experiment we must specify that the observers O and
$ O' $ record different conserved quantities because the translation is an isometry, which changes the components of the conserved quantities apart from the conserved momentum that is not affected by these isometries [6]. Moreover, as the origins of these frames are on the geodesics, both the angular momenta measured in O and$ O' $ vanish. We denote by E,$ {\vec Q} $ the remaining conserved quantities measured in O and by$ E' $ ,$ {\vec Q}' $ those recorded in$ O' $ bearing in mind that$ {\vec P}' = {\vec P} = {\vec p}\,{\rm e}^{\omega_{\rm H} t_0}\,. $
(65) The values observed in
$ O' $ can be deduced from Eqs. (49) and (51) for$ {\vec x}' = 0 $ obtaining the previous mentioned result,$ E' = p^0 $ and$ {\vec Q}' = {\vec p}\,{\rm e}^{-\omega_{\rm H} t_0} $ . The observer O prefers to search for the conserved quantities at$ t_0 $ because he knows that these do not change along the geodesic. Thus, he records$ E = p^0(t_f)\,, $
(66) $ {\vec Q} = {\rm e}^{-\omega_{\rm H} t_0}\left[2\omega_{\rm H} {\vec d}(t_0) E+{\vec p}\left( 1-\omega_{\rm H}^2 {\vec d}(t_0)^2\right)\right]\,, $
(67) as it results from Eqs. (49) and (51) for
$ {\vec x}_0 = {\vec d}(t_0) $ , verifying that$ {\vec P}\cdot{\vec Q} = {\vec p}(t_f)^2 $ for closing again the identity (45). Note that the relation among the conserved quantities$ E, {\vec P},\cdots $ and$ E', {\vec P}',\cdots $ can be derived directly according to the transformation rule under isometries we have discussed recently [18, 19]. -
The de Sitter null geodesics of the photons with
$ m = 0 $ that read$ {\vec x}(t) = {\vec x}_0\, {\rm e}^{\omega_{\rm H}(t-t_0)}+{\vec n}_{P}\frac{{\rm e}^{\omega_{\rm H}(t-t_0)}-1 }{\omega_{\rm H}}\,$
(68) are interesting being involved in the theory of the redshift. The energy and covariant momentum denoted now by
$ k^0(t) $ and$ {\vec k}(t) $ , respectively, are given by$ k^0(t) = P\,{\rm e}^{-\omega_{\rm H} t} = |\vec{\hat k}(t)|\,, $
(69) $ {\vec k}(t) = {\rm e}^{-\omega_{\rm H} t}{ P}\left({\vec n}_P+\omega_{\rm H}{\vec x}(t)\right) = \vec{\hat k}(t)+\vec{\bar k}(t)\,, $
(70) such that we can separate the peculiar momentum,
$ \vec{\hat k}(t) = $ $ {\rm e}^{-\omega_{\rm H} t}{\vec P} $ , and the recessional momentum,$ \vec{\bar k}(t) = $ $ \omega_{\rm H} {\vec x}(t) P {\rm e}^{-\omega_{\rm H} t} = \omega_{\rm H} {\vec x}(t) k^0(t) $ .Considering again the problem presented in Sec. II.C, we assume that now the observer
$ O' $ emits a photon of energy k and momentum$ {\vec k} = -{\vec n}k $ . Under such circumstances, Eq. (59) gives$ \frac{a(t_0)}{a(t_f)} = {\rm e}^{-\omega_{\rm H}(t_f-t_0)} = 1-\omega_{\rm H} d(t_0)\,, $
(71) allowing us to derive the quantities observed by O in his proper frame, namely the energy and covariant momentum,
$ k^0(t_f) = k\,\left[1-\omega_{\rm H} {d}(t_0)\right]\,, $
(72) $ {\vec k}(t_f) = \vec{\hat k}(t_f) = -{\vec n}\,k^0(t_f)\,, $
(73) the value of the final time
$ t_f = t_0-\frac{1}{\omega_{\rm H}}\ln\left[1-\omega_{\rm H} d(t_0)\right]\,, $
(74) the final distance between O and
$ O' $ at$ t_f $ ,$ d(t_f) = \frac{d(t_0)}{1-\omega_{\rm H} d(t_0)}\,, $
(75) and the redshift z related to the relative energy loss e observed by O,
$ 1-e = \frac{1}{1+z} = 1-\omega_{\rm H} d(t_0)\,, $
(76) resulted from Eq. (37). We recall that the condition
$ \omega_{\rm H} d(t_0)<1 $ is mandatory.On the null geodesics, the conserved quantities have simpler forms as
$ E = k^0(t_f) = k\,\left[1-\omega_{\rm H} {d}(t_0)\right]\,, $
(77) $ {\vec P} = {\vec k}\,{\rm e}^{\omega t_0}\,, $
(78) $ {\vec Q} = {\vec k}\,{\rm e}^{-\omega_{\rm H} t_0}\left[1-\omega_{\rm H} {d}(t_0)\right]^2\,, $
(79) such that
$ {\vec P}\cdot {\vec Q} = E^2 $ , satisfying the identity (45) with$ m = 0 $ . We observe again that for the special choice$ t_0 = 0 $ we have$ {\vec P} = {\vec k} $ and$ {\vec d}(t_0) = {\vec d} $ which simplifies the calculations and their interpretation. -
Let us finish with an example of a manifold whose kinematics was never studied. This is the spatially flat FLRW manifold M with the Milne type scale factor
$ a(t) = \omega_M t $ defined on the domain$ t\in (0,\infty) $ , whose constant (frequency)$ \omega_M $ is introduced from dimensional reasons [20, 21]. Then, after substituting the Hubble function$ \dfrac{\dot a(t)}{a(t)} = \dfrac{1}{t} $ in Eq. (6), which is independent of$ \omega_M $ , we may write the line element in the physical frame$ \{t,{\vec x}\} $ as$ {\rm d}s^2 = \left(1-\frac{1}{t^2}{\vec x}^{\,2}\right){\rm d}t^2 + 2 {\vec x}\cdot {\rm d}{\vec x}\,\frac{{\rm d}t}{t}-{\rm d}{\vec x}\cdot {\rm d}{\vec x}\,. $
(80) The conformal time
$ t_c\in (-\infty,\infty) $ is defined as$ t_c = \int \frac{{\rm d}t}{a(t)} = \frac{1}{\omega_M} \ln(\omega_M t) \,. $
(81) We obtain
$ a(t_c) = {\rm e}^{\omega_M t_c}\,,$
(82) a function of the line element (4) of the conformal frame
$ \{t_c,{\vec x}_c\} $ .Here, the constant
$ \omega_M $ is a useful free parameter representing the expansion speed of M. We remind the reader that in the case of the genuine Milne universe (of negative space curvature but globally flat), one must set$ \omega_M = 1 $ for eliminating the gravitational sources [2]. In contrast, our spacetime M is produced by isotropic gravitational sources, i.e., the density$ \rho $ and pressure$ \underline{p} $ , evolving in time as [20]$ \rho = \frac{3}{8\pi G}\frac{1}{t^2}\,, \quad \underline{p} = -\frac{1}{8\pi G}\frac{1}{t^2}\,, $
(83) and vanishing for
$ t\to\infty $ . These sources govern the expansion of M that can be better observed in the frame$ \{t, {\vec x}\} $ where the line element (80) lays out an expanding horizon at$ |{\vec x}_h| = t $ . For$ t\to \infty $ , M tends to the Minkowski spacetime and the gravitational sources vanish.We deduce first the equation of the time-like geodesics, solving the integral of Eq. (21), which leads to the final form
$ {\vec x}(t) = \frac{t}{t_0}\,{\vec x}_0\\ + {\vec n}_P\, {t}\ln\left( \frac{t}{t_0}\,\frac{P+\sqrt{P^2+\omega_M^2 m^2 t_0^2}}{P+\sqrt{P^2+\omega_M^2 m^2 t^2}} \, \right)\,, $
(84) for
$ m = 0 $ , we obtain the equation of the null geodesics$ {\vec x}(t) = \frac{t}{t_0}\,{\vec x}_0+ {\vec n}_P\, {t}\ln\left(\frac{t}{t_0}\right)\,. $
(85) The energy, momentum, and velocity need to be derived according to Eqs. (19), (15), and (27). These are complicated formulas that can be manipulated in applications by using algebraic codes on computer.
Furthermore, coming back to the problem of two observers formulated in Sec. II.C, we solve Eq. (33) for deriving the final time
$ t_f $ and the ratio$ \frac{a(t_0)}{a(t_f)} = \frac{t_0}{t_f} = \frac{1}{2p}\left(\nu-\frac{m^2}{\nu}\right)\,, $
(86) where
$ \nu = (p^0+m){\rm e}^{-\omega_M d}\,. $
(87) We recall that p is the scalar initial momentum of the particle of mass m lunched by
$ O' $ at$ t_0 $ while$ p^0 $ is the corresponding energy (29). Then, according to Eqs. (31) and (32), we obtain the final energy and covariant momentum$ p^0(t_f) = \frac{1}{2}\left(\nu+\frac{m^2}{\nu}\right), $
(88) $ {\vec p}(t_f) = -{\vec n}\,\frac{1}{2}\left(\nu-\frac{m^2}{\nu}\right)\,, $
(89) and the energy loss,
$ 1-e = \frac{1}{2p}\left(\nu+\frac{m^2}{\nu}\right)\,. $
(90) The final distance between O and
$ O' $ when the particle arrives in O,$ d(t_f) = d(t_0) \frac{1}{1-e}\,, $
(91) where
$ d(t_0) = d\, a(t_0) = \omega_M t_0 d $ . As in this geometry, the observer's horizon is at t, we must impose the restriction$ d(t_0)<t_0 \to \omega_M d<1 $ in order to keep the final distance inside the observer's horizon,$ d(t_f)< t_f $ .When O and
$ O' $ observe a photon, then they record$ t_f = t_0\, {\rm e}^{\omega_M d} $ ,$ k^0(t_f) = |{\vec k}(t_f)| = k\, {\rm e}^{-\omega_M d} $ and the redshift$ 1+z = {\rm e}^{\omega_M d} $ , which for small values of$ \omega_M d $ can be confused with the de Sitter redshift since the expansion$ \frac{1}{1+z} = {\rm e}^{-\omega_M d} = 1-\omega_M d +{\cal{O}}(\omega_M^2 d^2)\,, $
(92) is somewhat similar to Eq. (76). However, for larger distances, the discrepancy between the linear behavior of the de Sitter redshift and the exponential one in the spacetime M becomes obvious.
Finally, we observe that the Milne-type and ds Sitter universes behave somewhat complementary such that the cosmic time of one of these manifolds behaves as the conformal time of the other one. The self explanatory Table 1 completes this image [21].
M de Sitter t $ {0 < t < \infty }$ $ { - \infty < t < \infty }$ $ {{t_c}}$ ${ - \infty < {t_c} < \infty } $ ${ - \infty < {t_c} < 0} $ ${a(t)} $ $ {{\omega _M}t}$ $ {{{\rm e}^{{\omega _{\rm H}}t}}}$ ${a({t_c})} $ ${{{\rm e}^{{\omega _M}{t_c}}}} $ ${ - \dfrac{1}{{{\omega _{\rm H}}{t_c}}}} $ $ {{\rm{transl}}{\rm{.}}}$ ${{\omega _M}d < 1} $ ${{\omega _{\rm H}}d < 1} $ ${1 + z} $ $ {{{\rm e}^{{\omega _M}d}}}$ ${{{[1 - {\omega _{\rm H}}d({t_0})]}^{ - 1}}} $ Table 1. The two complementary behaviors were observed.
The only similarity is the condition satisfied by the translation parameter d for remaining inside the observer's horizon.
Kinemaitcs in spatially flat FLRW spacetimes
- Received Date: 2021-05-19
- Available Online: 2021-10-15
Abstract: The kinematics on spatially flat FLRW spacetimes is presented for the first time in local charts with physical coordinates, i.e., the cosmic time and proper Cartesian space coordinates of Painlevé-type. It is shown that there exists a conserved momentum that determines the form of the covariant four-momentum on geodesics in terms of physical coordinates. Moreover, with the help of this conserved momentum, the peculiar momentum can be defined, thus separating the peculiar and recessional motions without ambiguity. It is shown that the energy and peculiar momentum satisfy the mass-shell condition of special relativity while the recessional momentum does not produce energy. In this framework, the measurements of the kinetic quantities along geodesics performed by different observers are analyzed, pointing out an energy loss of the massive particles similar to that producing the photon redshift. The examples of the kinematics on the de Sitter expanding universe and a new Milne-type spacetime are extensively analyzed.