Eisenhart–Duval lift of Newtonian mechanics
As originally formulated in [1,2,3], the ED lift provides a geometric description of a mechanical system with d degrees of freedom \(x_i\), \(i=1,\dots ,d\), and the potential energy U(t, x) in terms of geodesics of the Lorentzian metric on a \((d+2)\)–dimensional spacetime
$$\begin{aligned} g_{\mu \nu }(y) dy^\mu dy^\nu =-2 U({t},x) d t^2-dt dv+dx_i dx_i, \end{aligned}$$
(III.1)
where \(y^\mu =(t,v,x_i)\). Computing the Christoffel symbols and analyzing the geodesic equations one finds that t is affinely related to the proper time \(\tau \)
$$\begin{aligned} \frac{d^2t}{d \tau ^2}=0 \quad \Rightarrow \quad \frac{d t}{d \tau }=\kappa , \end{aligned}$$
(III.2)
where \(\kappa \) is a constant, \(x_i\) obeys Newton’s Eq. (passing from \(\tau \) to t)
$$\begin{aligned} \frac{d^2 x_i}{d t^2}+\partial _i U({t},x)=0, \end{aligned}$$
(III.3)
while the dynamics of v is fixed from the condition that the geodesic is null or time–like
$$\begin{aligned} \frac{d v}{d t}=\frac{d x_i}{dt} \frac{d x_i}{dt}-2 U-\frac{\epsilon }{\kappa ^2}, \end{aligned}$$
(III.4)
where \(\epsilon =0\) is for null geodesics and \(\epsilon =-1\) for time–like geodesics. Newtonian mechanics is thus recovered by implementing the null reduction along v [1]. Note that the original construction [1,2,3] dealt with null geodesics only.
The spacetime (III.1) belongs to the Kundt class as it admits the covariantly constant null Killing vector field
$$\begin{aligned} \xi ^\mu \partial _\mu =\partial _v, \qquad \nabla _\mu \xi _\nu =0, \qquad \xi ^2=0. \end{aligned}$$
(III.5)
The latter can be used to construct the trace–free energy–momentum tensor in a geometrically rather appealing wayFootnote 4
$$\begin{aligned}&T_{\mu \nu }=\frac{d}{2\pi } \Omega (y)^2 \xi _\mu \xi _\nu , \qquad {T^\mu }_\mu =0, \nonumber \\&\nabla _\rho T_{\mu \nu }=\frac{d}{\pi } \Omega \, \partial _\rho \Omega \,\xi _\mu \xi _\nu , \end{aligned}$$
(III.6)
where \(\Omega (y)^2\) is an arbitrary function (the energy density). The only non–vanishing component of \(\xi _\mu \) is \(\xi _t=-\frac{1}{2}\), which gives \(T_{tt}=\frac{d }{8\pi } \Omega ^2\) while the rest vanishes. Because \(\xi ^\mu \partial _\mu \Omega =0\) holds true if \(\Omega \) does not depend on v, the energy–momentum tensor is conserved,
$$\begin{aligned} \nabla ^\mu T_{\mu \nu }=0, \end{aligned}$$
(III.7)
provided \(\Omega =\Omega (t,x)\).
The Ricci tensor which derives from (III.1) has only one nonzero component while the scalar curvature vanishes,
$$\begin{aligned} R_{tt}=\partial _i \partial _i U, \qquad R=0. \end{aligned}$$
(III.8)
Given the ED metric (III.1) and the energy–momentum tensor (III.6), the Einstein equations
$$\begin{aligned} R_{\mu \nu }-\frac{1}{2} g_{\mu \nu }(R+2\Lambda )=8\pi T_{\mu \nu } \end{aligned}$$
(III.9)
imply that the contribution of the cosmological term necessarily vanishes, \(\Lambda =0\), thus reducing (III.9) to
$$\begin{aligned} R_{\mu \nu }=8\pi T_{\mu \nu }. \end{aligned}$$
(III.10)
Only the (tt)–component is non–trivial; it links \(\Omega \) to U, as
$$\begin{aligned} \Omega ^2=\frac{1}{d} \partial _i \partial _i U. \end{aligned}$$
(III.11)
A particularly interesting example of the ED geometry occurs if U and \(\Omega \) are t-independent. Setting
$$\begin{aligned} \Omega ^2(x)=\frac{4\pi G}{d} \rho (x) \end{aligned}$$
(III.12)
and interpreting G as Newton’s constant and \(\rho (x)\) as the mass density, one recovers the Newton equation for the gravitational potential.
In conclusion, Newtonian mechanics can be represented in terms of a metric which solves the Einstein Eqs. (III.10) provided the energy–momentum tensor is chosen in the form (III.6).
Lifting the Ermakov–Lewis invariant
The aim of this subsection is to show how applying the ED lift to a TDHO allows one to obtain the Ermakov–Lewis invariant by performing a conformal transformation of the metric [1,2,3].
The equations of motion of a TDHO follow from the Hamiltonian
$$\begin{aligned} H= \frac{1}{2}\Bigl ( m^{-1} (t) p_q^2 + m(t) \omega ^2(t) q^2 \Bigr ) . \end{aligned}$$
(III.13)
Let \(H=-p_t \) and let
$$\begin{aligned} \mathcal{H}(x^\mu ,p_\mu ) = 2 p_t p_v + m^{-1} p_q^2 + m \omega ^2 q^2 p_v^2 = g^{\mu \nu }p_\mu p_\nu \end{aligned}$$
(III.14)
with \(x^\mu = (t,v,q)\), \(p_\mu =(p_t,p_v,p_q)\) . The ED metric on the extended (“Bargmann” [2, 3]) spacetime \(M=\{\mathbb {R}^3, g \}\) is given by,
$$\begin{aligned} g_{\mu \nu }dx^\mu dx^\nu = 2 dt dv + m dq^2 - m \omega ^2 q^2 d t^2 . \end{aligned}$$
(III.15)
The null geodesics, considered as unparametrised curves, of a Lorentzian metric \(g_{\mu \nu }\) are the same as for any conformally related Lorentzian metric. They are given by the Hamiltonian flow on \(T^\star M\) with Hamiltonian
$$\begin{aligned} \mathcal{H} = g^{\mu \nu } p_\mu p_\nu \end{aligned}$$
(III.16)
subject to the constraint
$$\begin{aligned} \mathcal{H}=0. \end{aligned}$$
(III.17)
Hamilton’s equations following from the Hamiltonian (III.16) with constraint (III.17) for two conformally related metrics \(g_{\mu \nu }\) and \(\Omega ^2(x) g_{\mu \nu }\) are identical.
The vector field \(V= \frac{{\partial }}{{\partial }v}\) is a null Killing vector field and thus
$$\begin{aligned} p_v= \mathrm{constant}. \end{aligned}$$
(III.18)
To obtain the motion in 2 spacetime dimensions we perform a Marsden–Weinstein reduction (referred to as “ignoration of cyclic coordinates” in old fashioned books) bearing in mind the constraint (III.17) and setting \(p_v=1\) to recover the Hamiltonian system (III.13). We now set
$$\begin{aligned} Q=\frac{q}{f} \end{aligned}$$
(III.19)
for some function f(t) to be determined. In (Q, t, v) coordinates the metric may be cast in the form
$$\begin{aligned} ds^2 = 2 dt(dv + A) + mf^2 dQ^2 - Q^2 g(t) dt^2, \end{aligned}$$
(III.20)
where A is the one-form
$$\begin{aligned} A = mf\dot{f} Q dQ + \frac{1}{2}mQ^2 (\dot{f}^2 -f^2 \omega ^2) dt + \frac{1}{2}Q^2 g(t) dt \end{aligned}$$
(III.21)
and g(t) is an arbitrary function of time. If A is closed, \(dA=0\), then we may define a new coordinate \(\tilde{v}\) by
$$\begin{aligned} d \tilde{v} = dv + A, \end{aligned}$$
(III.22)
in terms of which the metric becomes
$$\begin{aligned} ds ^2 = 2 dt d \tilde{v} + mf^2 dQ^2 - Q^2 g^2 dt ^2. \end{aligned}$$
(III.23)
Now introducing a new time coordinate \(\tau \) by
$$\begin{aligned} dt = mf^2 d\tau , \end{aligned}$$
(III.24)
we have
$$\begin{aligned} ds ^2 = mf^2 \Bigl \{ 2 d \tau d \tilde{v} + dQ^2 - mf^2 Q^2 g d \tau ^2 \Bigr \} . \end{aligned}$$
(III.25)
The condition for A to be closed is
$$\begin{aligned} \frac{d}{dt}(m \dot{f}) + m\omega ^2 f = \frac{g}{f}. \end{aligned}$$
(III.26)
If
$$\begin{aligned} g = \frac{\Omega ^2 }{mf^2}, \end{aligned}$$
(III.27)
where \(\Omega \) an arbitrary constant, then the metric becomes
$$\begin{aligned} ds^2 = mf^2 \Bigl \{ 2 d \tau d \tilde{v} + dQ^2 - Q^2 d \tau ^2 \Bigr \} . \end{aligned}$$
(III.28)
As noted above, two conformally related metrics have the same null geodesics, although their affine parameters differ. From this we deduce that the null geodesics of the static metric inside the braces in (III.28) are the same as those for our time dependent metric (III.15).
If we now perform a null reduction on the Killing vector \(\frac{{\partial }}{{\partial }\tilde{v}}\), we obtain the Hamiltonian of a TIHO
$$\begin{aligned} \tilde{H} = \frac{1}{2}(p_Q^2 + \Omega ^2 Q^2). \end{aligned}$$
(III.29)
Ermakov–Milne–Pinney cosmology
Now we show that the EMP equation naturally arises if one incorporates a cosmic scale factor into a suitably chosen Bargmann metric, when the energy-momentum tensor is chosen in a proper way.
Consider a \((d+2)\)–dimensional spacetime parametrized by the coordinates \(y^\mu =(t,v,x_i)\), \(i=1,\dots ,d\), and endowed with the Lorentzian metric [16]
$$\begin{aligned} ds^2=-\frac{\gamma ^2 x_i x_i }{a(t)^2} dt^2-dt dv+a(t)^2 dx_i dx_i, \end{aligned}$$
(III.30)
where a(t) is an arbitrary function and \(\gamma \) is a constant. For a fixed value of t the line element in the d–dimensional slice parametrized by \(x_i\) is given by \({a(t)}^2 dx_i dx_i\). Therefore \({a(t)}^2\) may be interpreted as a cosmic scale factor. The metric (III.30) possesses, as does its conventional counterpart (III.1), a covariantly constant null Killing vector field, namely (III.5). The choice of (III.30) will be justified a posteriori in Sect. 4.
Constructing the energy-momentum tensor following the prescription (III.6) and specifying to the class of energy densities which depend on t only, \(\Omega =\Omega (t)\), from Eq. (III.10) one finds
$$\begin{aligned} \ddot{a}+\Omega ^2(t) a=\frac{\gamma ^2}{a^3}. \end{aligned}$$
(III.31)
Thus the dynamics of (III.30) is governed by the EMP equation. The instance of \(\Omega =\text{ const }\) has been discussed recently in [16], in which case the EMP equation reduces to conformal mechanics in one dimension [21].
One can learn more about the geometry of (III.30) by analyzing the geodesic equations. Computing the Christoffel symbols, one concludes that t is affinely related to the proper time,
$$\begin{aligned} \frac{d^2 t}{d \tau ^2}=0\, . \end{aligned}$$
(III.32)
The coordinate \(x_i\) obeys in turn the oscillator–like equation
$$\begin{aligned} a^2 \frac{d}{dt} \left( a^2 \frac{d}{dt} x_i \right) +\gamma ^2 x_i=0, \end{aligned}$$
(III.33)
in which we passed from \(\tau \) to t, while the evolution of v is fixed from the condition that the geodesic be null or time–like. Equation (III.33) prompts one to introduce the conformal time
$$\begin{aligned} a^2(t) \frac{d}{dt}=\frac{d}{d\eta } , \qquad \eta (t)=\int _{t_0}^t \frac{d\tilde{t}}{a^2(\tilde{t})} , \end{aligned}$$
(III.34)
which brings the metric (III.30) to the form
$$\begin{aligned} ds^2=a^2(\eta )\left( -\gamma ^2 x_i x_i d\eta ^2-d\eta dv+dx_i dx_i\right) , \end{aligned}$$
(III.35)
where \(a(\eta )=a(t(\eta ))\), and \(t(\eta )\) is the inverse of \(\eta (t)\) in (III.34). Equation (III.35) is an analog of the flat (k=0) FLRW cosmological model in which the Minkowski metric has been changed into the simplest PP–wave; the Friedmann equation which determines the evolution of the cosmic scale factor is replaced, in this case, by the EMP equation.
To conclude this section we note that the coordinate transformation
$$\begin{aligned} t'=t, \qquad x'_i=a(t) x_i, \qquad v'=v+a(t) \dot{a}(t) x_i x_i \end{aligned}$$
(III.36)
brings (III.30) to the form
$$\begin{aligned} ds^2= \left( \frac{\ddot{a}}{a}-\frac{\gamma ^2}{a^4} \right) x_i x_i dt^2-dt dv+dx_i dx_i, \end{aligned}$$
(III.37)
where we omitted the primes, or, in view of (III.31),
$$\begin{aligned} ds^2= -\Omega (t)^2 x_i x_i dt^2-dt dv+dx_i dx_i \end{aligned}$$
(III.38)
which is the \(d+2\) dimensional Bargmann metric associated with an isotropic oscillator in (d, 1) dimensions with time–dependent frequency \(\Omega (t)\), the motions of which are the projections of the null geodesics of (III.38).
Symmetries as conformal Killing isometries
Finding the symmetries of time dependent harmonic oscillators generated an extensive literature in the early 1980’s, including [46,47,48,49]. A discussion in terms of canonical transformations is in [51]. An alternative approach is presented below in terms of the ED lift. For simplicity, we stick to Eq. (III.38).
Following [2, 3], the symmetries of a non-relativistic system in (d, 1) dimensions can be obtained as a subgroup of the conformal symmetries of the \(d+2\) dimensional Bargmann manifold: one selects those conformal transformations that leave invariant the covariantly constant null vector \({\partial }_v\).
In our case, the symmetries given in Ref. [49] are seen to be consistent with the Schrödinger group in d dimensions – which is a subgroup of the conformal group of the extended (Bargmann) spacetime. Now we re-derive the above-mentioned symmetries in the specific case of our EMP spacetime of Sect.3.3.
Consider indeed a generic infinitesimal transformation
$$\begin{aligned} t'=t+\delta t, \qquad v'=v+\delta v, \qquad x'_i=x_i+\delta x_i , \end{aligned}$$
(III.39)
where \(\delta t\), \(\delta v\), \(\delta x_i\) are arbitrary functions of t, v, x. Demanding (III.38) to be invariant under (III.39) up to a conformal factor,
$$\begin{aligned} ds'^2=(1+\Lambda ) ds^2, \end{aligned}$$
(III.40)
one gets a coupled set of partial differential equations to fix \(\delta t\), \(\delta v\), \(\delta x_i\) and \(\Lambda \). Omitting details, we present the (conformal) isometries of (III.38),
$$\begin{aligned}&t'=t+\lambda (t)+\frac{1}{2} \epsilon x^2-x_i \rho _i(t),\end{aligned}$$
(III.41a)
$$\begin{aligned}&x'_i=x_i+\mu _i (t)+\omega _{ij} x_j+\frac{1}{2} \kappa x_i +\frac{1}{2} \dot{\lambda }(t) x_i+\frac{1}{2} v \epsilon x_i\end{aligned}$$
(III.41b)
$$\begin{aligned}&\qquad +\,\frac{1}{2} x^2 \dot{\rho }_i (t)-x_i x_j \dot{\rho }_j (t)-\frac{1}{2} v \rho _i(t),\end{aligned}$$
(III.41c)
$$\begin{aligned}&v'=v+\nu +2{\dot{\mu }}_i(t) x_i+\frac{1}{2} \ddot{\lambda }(t) x^2-\frac{1}{2} \epsilon \, \Omega ^2x^2 x^2+\kappa v+\frac{1}{2} \epsilon v^2 \end{aligned}$$
(III.41d)
$$\begin{aligned}&\qquad +\,\Omega ^2 x^2 x_i \rho _i(t)-v x_i \dot{\rho }_i (t), \end{aligned}$$
(III.41e)
where \(\epsilon \), \(\nu \), \(\kappa \), \(\omega _{ij}=-\omega _{ji}\) are constant infinitesimal parameters, \(x^2=x_i x_i\), the functions \(\mu _i(t)\), \(\rho _i(t)\), and \(\lambda (t)\) obey the ordinary differential equations
$$\begin{aligned}&\ddot{\mu }_i+\Omega ^2 \mu _i=0, \qquad \ddot{\rho }_i+\Omega ^2 \rho _i=0,\nonumber \\&\lambda ^{(3)}+4 \Omega ^2\, \dot{\lambda }+4 \Omega \dot{\Omega }\, \lambda =0, \end{aligned}$$
(III.42)
while the conformal factor is
$$\begin{aligned} \Lambda =\kappa +\dot{\lambda }(t)+v \epsilon -2 x_i \dot{\rho }_i (t). \end{aligned}$$
(III.43)
Taking into account the order of the differential Eq. (III.42), one finds that the transformations (III.41) involve \(6+4d+\frac{d(d-1)}{2}=\frac{(4+d)(3+d)}{2}\) independent infinitesimal parameters – the same as that of the flat-space conformal group in \(d+2\) dimensions.
From Eq. (III.43) one concludes that the \(\kappa \), \(\dot{\lambda }(t)\), \(\epsilon \), and \(\rho _i(t)\)-transformations give rise to conformal Killing vectors,Footnote 5 while the \(\nu \), \(\mu _i(t)\), \(\omega _{ij}\) and constant \(\lambda \)-transformations generate Killing vectors. In view of (III.42), the isometry with constant parameter \(\lambda \) is only possible for constant frequency \(\Omega \) which corresponds to a stationary spacetime. Time translations are broken in general.
We notice that, while distinct conformal isometries act on the coordinates as in (III.41), the functions \(\lambda (t)\), \(\rho _i(t)\) and \(\mu _i(t)\) are not independent. Obviously, \(\rho _i(t)\) differs from \(\mu _i(t)\) by an inessential constant. If \(\mu \) satisfies the first equation in (III.42), then \(\lambda (t) = \mu ^2(t)\) will satisfy the second. Moreover, if \(\mu _1\) and \(\mu _2\) are independent solutions of the first equation, then the three independent solutions of the second are given by \(\mu _1^2\), \(\mu _2^2\), \(\mu _1 \mu _2\). The deeper reason why this happens is that the conformal Killing vectors form a Lie algebra, which we identify below with \(so(2,2+d)\).
The interpretation of the (conformal) Killing vectors above and their algebra becomes more transparent if one switches to conformal time, (III.35). It is evident that all (conformal) isometries of the PP-wave metric
$$\begin{aligned} ds^2=-\gamma ^2 x_i x_i d\eta ^2-d\eta dv+dx_i dx_i, \end{aligned}$$
(III.44)
will be automatically transmitted to become (conformal) isometries of (III.35), because the two expressions only differ by the cosmic scale factor \(a(\eta )\). In particular, all the symmetry transformations of (III.44), which do not involve \(\eta \) explicitly, will be transformed into the Killing vectors of (III.35), while those affecting \(\eta \) will be transmitted into conformal Killing vectors.
Our clue is that the metric above is conformally flat. This follows from the vanishing of the Weyl tensor. Skipping details we merely mention that it can also be seen, explicitly, by applying the Arnold transformation, see [4, 38, 39].
The group of conformal transformations of any \((1,1+d)\)-dimensional conformally flat spacetime is isomorphic to that of Minkowski space, explaining the “coincidence” we noted earlier.
The explicit form of the (conformal) isometries of (III.44) follows from Eqs. (III.41) and (III.42) at \(\Omega =1\) after the substitution \(t \rightarrow \eta \). In particular, conformal time translations, \(\eta \rightarrow \eta +\theta \), are now symmetries, because \(\gamma =\mathrm{const.}\) in (III.44).
From the first equation in (III.42) one finds,
$$\begin{aligned} \mu _i(\eta )=\alpha _i \cos {(\eta )}+\beta _i \sin {(\eta )}, \end{aligned}$$
(III.45)
where the infinitesimal parameters \(\alpha _i\) and \(\beta _i\) are associated with spatial translations and Newton–Hooke boosts [52]. The second equations generate
$$\begin{aligned} \rho _i(\eta )=\tilde{\alpha }_i \cos {(\eta )}+\tilde{\beta }_i \sin {(\eta )}, \end{aligned}$$
(III.46)
which involve the infinitesimal parameters \(\tilde{\alpha }_i\), \(\tilde{\beta }_i\) and provide contributions to Eq. (III.41) which are nonlinear in \(x_i\). The third equation yields
$$\begin{aligned} \lambda (\eta )=\theta +\sigma \cos {(2\eta )}+\rho \sin {(2\eta )}, \end{aligned}$$
(III.47)
where the infinitesimal parameters \(\theta \), \(\sigma \), \(\rho \), linked to time translations, special conformal transformations, and dilatations form an so(2, 1) subalgebra. Along with spatial rotations described by \(\omega _{ij}\), the \(\mu _i(\eta )\) and \(\lambda (\eta )\)–transformations form the conformal Newton–Hooke algebra. For a detailed discussion of the Schrödinger and conformal Newton–Hooke algebras and their realizations in spacetime see e.g. [53,54,55].
As it follows from (III.41), the isometries of (III.44) also involve the translation in the v-direction, while the set of conformal isometries contains the \(\kappa \), and \(\epsilon \)–transformations.
Note that the \(\kappa \)-transformation in (III.41) and (III.43) is realized in a way analogous to conventional dilatation in the Schrödinger algebra. It derives from the latter by replacing the temporal variable t by the “null” coordinate v; it has appeared before in the context of gravitational waves [56]. The \(\epsilon \)-transformation is in turn an analog of special conformal transformation, again t replaced by v. It is straightforward to verify that, along with the translations in the v–direction, \(v'=v+\nu \), they form an so(2, 1) subalgebra. Interestingly enough and extending the Galilei-Carroll “duality” [57], the latter acts upon the null coordinate v in very much the same way as so(2, 1) entering the conformal Newton–Hooke algebra affects the temporal coordinate t.
To summarize, the algebra of vector fields which involve both Killing and conformal Killing vectors can be identified with \(so(2,2+d)\), the conformal Newton–Hooke algebra being its subalgebra.
Having identified the conformal isometries of the metric (III.38), the symmetries of the underlying classical system in one fewer dimension (i.e., the time-dependent oscillator) could now be derived. Skipping details, we just mention that implementing the null reduction along v, the SO(2, 1) conformal subgroup with parameters \(\nu \), \(\kappa \) and \(\epsilon \) in (III.41) will be broken, allowing us to recover the Schrödinger symmetry found in [49]. The generators are conveniently identified using the formulae in Sect. 3 of [50].
Geodesic motion in Ermakov–Milne–Pinney cosmoi
Having established a link between the EMP equation and the ED lift, let us study the geodesic motion in EMP cosmoi. The analysis is facilitated by switching to the conformal time (III.35) which allows one to solve the geodesic equations by quadrature,
$$\begin{aligned}&\tau =\kappa \int _{\eta _0}^\eta a^2(\tilde{\eta }) d\tilde{\eta }+\tau _0,\end{aligned}$$
(III.48a)
$$\begin{aligned}&x_i(\eta )=\alpha _i \cos {(\gamma \eta )}+\beta _i \sin {(\gamma \eta )}, \end{aligned}$$
(III.48b)
$$\begin{aligned}&v(\eta )=v_0-\frac{1}{2}(\alpha ^2-\beta ^2) \gamma \sin {(2\gamma \eta )}+\alpha \beta \gamma \cos {(2\gamma \eta )}\nonumber \\&\qquad \quad \ \,-\,\epsilon \kappa ^2 \int _{\eta _0}^\eta a^2(\tilde{\eta }) d\tilde{\eta }, \end{aligned}$$
(III.48c)
where \(\tau _0\), \(\kappa \), \(\alpha _i\), \(\beta _i\), \(v_0\) are constants of integration and \(\alpha ^2=\alpha _i \alpha _i\), \(\alpha \beta =\alpha _i \beta _i\). It is assumed that \(\epsilon =0\) for null geodesics and \(\epsilon =-1\) for time-like geodesics.
Before proceeding, it is worth mentioning that, because for null geodesics Eq. (III.48a) admit a particular solution
$$\begin{aligned} x_i(\eta )=0, \qquad v(\eta )=v_0, \end{aligned}$$
(III.49)
the metric (III.35) is formulated in a reference frame comoving with a light signal which travels along the v-axis. This correlates with the fact that (III.35) differs from a PP-wave by a scale factor only.
The first line in (III.48a) defines \(\eta \) as an implicit function of the proper time \(\tau \). Although in most cases of interest the integral \(\int _{\eta _0}^\eta a^2(\tilde{\eta }) d\tilde{\eta }\) cannot be evaluated exactly, Eq. (III.48a) prove to be sufficient to comprehend a qualitative behaviour of geodesics. Indeed, Eq. (III.48b) defines an ellipse. By making use of rotational invariance, one can set the ellipse to lie, say, in the \(x_1 x_2\)-plane. As v evolves with time, geodesics in the EMP cosmology wrap around the elliptic cylinder, v being its axis.
For null geodesics (\(\epsilon =0\)), the trajectory is a closed loop and the motion is periodic (see Fig. 1). For time-like geodesics (\(\epsilon =-1\)), the orbit wraps around the cylinder remaining in a compact region of space for some time, then the \(\epsilon \)-term in Eq. (III.48c) starts dominating and the particle escapes (see Fig. 2). Given the coordinate system in which the metric (III.35) is formulated, this happens because massive particles travel slower than a reference frame comoving with a light signal which propagates along the v-axis.
Little is known about analytic solutions to the EMP equation. Assuming that the energy density \(\Omega ^2(t)\) decreases with time (which happens in an expanding universe), one has (see, e.g., [23] and Sect. 2.4.2 in [34])Footnote 6
$$\begin{aligned}&\Omega ^2(t)=\frac{h^{n-2}}{4 \nu ^2 t^n}, \qquad \nu =\frac{1}{2-n}, \qquad n\ne 2,\end{aligned}$$
(III.50a)
$$\begin{aligned}&a(t)=\sqrt{\pi \gamma \nu t(J_\nu ^2 (\lambda )+Y_\nu ^2 (\lambda ))}, \qquad \lambda ={\left( \frac{t}{h}\right) }^{\frac{1}{2\nu }}, \end{aligned}$$
(III.50b)
where n is a rational number such that \(\nu \) is positive, h is a positive constant, and \(J_\nu \), \(Y_\nu \) are Bessel functions. Note that in these cases a(t) is a monotonically increasing convex function which tends to a fixed nonzero value as \(t\rightarrow 0\).
Other interesting examples are provided by negative integer \(\nu \); then \(\nu \) should be replaced by \(|\nu |\) in the expression for a(t) in Eq. (III.50b). These models are represented by monotonically increasing convex functions starting at \(a(0)=0\). The instance \(t=0\) can be interpreted as the Big Bang. A typical example is shown in Fig. 3.
A notable simplification takes place for \(\ddot{a}=0\), when the solution is expressed in terms of elementary functions
$$\begin{aligned} \Omega (t)=\frac{h}{t^2}, \qquad a(t)=\sqrt{\frac{\gamma }{h}}\, t \qquad \Rightarrow \qquad \Omega (t)=\frac{\gamma }{{a(t)}^2}. \end{aligned}$$
(III.51)
This case corresponds to a linearly expanding universe in which the energy density of matter decreases consistently with the inverse square law. The general solution to the geodesic equations is,
$$\begin{aligned}&t=t_0+\tau /\kappa , \end{aligned}$$
(III.52a)
$$\begin{aligned}&x_i(\tau )=\alpha _i \cos {\left( h/t\right) }+\beta _i \sin {\left( h/t\right) }, \end{aligned}$$
(III.52b)
$$\begin{aligned}&v(\tau )=v_0-\epsilon \kappa ^2 t- \alpha \beta \gamma \cos {\left( 2h/t\right) }\nonumber \\&\qquad \qquad +\,\frac{1}{2}(\alpha ^2-\beta ^2) \gamma \sin {\left( 2h/t\right) }, \end{aligned}$$
(III.52c)
where \(t_0\), \(\kappa \), \(\alpha _i\), \(\beta _i\), \(v_0\) are constants of integration.
Although the graphs representing the cosmic scale factors in the EMP cosmology look quite reasonable, the geodesic motion is apparently unrealistic. This happens because Eq. (III.48b) defines an ellipse. An obvious cure is to generalize the construction to the case of time-dependent \(\gamma \) which will alter the qualitative behaviour of geodesics. It proves sufficient to replace \(\gamma \) in (III.30) by an arbitrary function \(\gamma (t)\) which will then show up on the right hand side of Eq. (III.31), viewed as an algebraic equation to fix \(\gamma (t)\) in terms of a(t) and \(\Omega (t)\). In this way one can model a reasonable geodesic behavior in the generalized EMP cosmoi by properly choosing the cosmic scale factor and the energy density. Note, however, that, as the associated geodesic equations involve a time-dependent oscillator, with frequency \(\gamma (t)^2\), finding an analytic solution may be complicated.