Cosmological aspects of the Eisenhart–Duval lift
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Abstract
A cosmological extension of the Eisenhart–Duval metric is constructed by incorporating a cosmic scale factor and the energymomentum tensor into the scheme. The dynamics of the spacetime is governed by the Ermakov–Milne–Pinney equation. Killing isometries include spatial translations and rotations, Newton–Hooke boosts and translation in the null direction. Geodesic motion in Ermakov–Milne–Pinney cosmoi is analyzed. The derivation of the Ermakov–Lewis invariant, the Friedmann equations and the Dmitriev–Zel’dovich equations within the Eisenhart–Duval framework is presented.
1 Introduction
The Eisenhart–Duval (ED) lift [1, 2, 3] provides a convenient framework for treating timedependent dynamical systems and their symmetries [4]. Such systems include timedependent harmonic oscillators (TDHO’s). TDHO’s and their symmetries have a large literature. In particular, they are conveniently studied using the Ermakov–Milne–Pinney (EMP) equation [5, 6, 7].
An important tool for solving the EMP equation is a constant of the motion known as the Ermakov–Lewis invariant – which is in fact an exact invariant for any TDHO. It is then natural to inquire about the relationship between the ED lift and the EMP equation. Can one obtain the Ermakov–Lewis invariant using the ED lift?
There is also a considerable body of work in which methods based on the EMP equation have been applied to cosmology [8, 9, 10, 11, 12, 13, 14, 15], in particular to finding solutions and the symmetries of the Friedmann equations. This leads to the further question of what insights can be gained into cosmology using the ED lift?
A first attempt to incorporate a cosmic scale factor into the ED scheme has been reported recently [16]. Yet, because these considerations relied upon the EMP equation with a constant frequency, the resulting spacetime turned out to be stationary. It is then natural to wonder whether a dynamical spacetime with cosmological features can also be constructed along the same lines.
The purpose of this paper is to address the questions raised above, focusing on the cosmological aspects of the ED lift.
The paper is organized as follows. In Sect. 2 we provide a thorough review of the EMP equation for readers who may not be familiar with the rather extensive literature on this subject.
In Sect. 3.1 a brief account of the ED lift is given. It is shown that any Newtonian mechanical system can be represented by a metric which solves the Einstein equations provided the energy momentum tensor is chosen in a suitable way. In Sect. 3.2 the Ermakov–Lewis constant is obtained by using a conformal transformation of the associated ED metric. An explicit formula which depends upon a solution of the EMP equation is presented.
Then in Sect. 3.3 the ED lift involving a scale factor analogous to that in the Friedmann metrics is given. It is shown that the dynamics of the spacetime is governed by the EMP equation. Section 3.4 contains a detailed derivation of the conformal symmetries of this modified metric. Geodesic motion in EMP cosmoi is analyzed in Sect. 3.5.
Sect. 4.1 contains some introductory material on Friedmann–Lemaître–Robertson–Walker (FLRW) metrics, establishing the terminology and the conventions. The Einstein equations reduce to the Friedmann and Raychaudhuri equations; in the case of radiation and dark energy, they reduce to the EMP equation.
Sect. 4.2 deals with conformal symmetries and what we refer to as temporal diffeomorphisms, which may be used to provide explicit solutions to the Friedmann equations. These cases include the reduction to a timeindependent harmonic oscillator.
Sect. 4.3 contains a derivation of the ED lift of geodesic motion in FLRW spacetimes. This allows describing the system’s dynamical symmetries, and relating them to the null geodesics of the EMP spacetime of Sect. 3.5. The result is generalized to obtain the lift of the Dmitriev–Zel’dovich equations.
Finally in Sect. 4.4 we regard the Friedmann equations as a dynamical system with a constraint and provide a generalised ED lift of it.
In Sect. 5 we summarize our results.
Throughout the paper summation over repeated indices is understood.
2 The Ermakov–Milne–Pinney Equation

In dynamics: using its solutions to map problems involving TDHO’s to problems involving timeindependent harmonic oscillators (TIHO’s).

In quantum mechanics: using its solutions to map problems involving the onedimensional timeindependent Schrödinger equation into Schrödinger equations which have explicit solutions.
2.1 Comparison of two time dependent harmonic oscillators
2.2 The Ermakov–Milne–Pinney equation and the Ermakov–Lewis invariant
One may regard the Lewis invariant as an exact form of the adiabatic invariant for a TDHO, and as such, has obvious applications to the quantum theory as will be seen shortly.
Subsequent to the paper by Lewis, the idea was extended to charged particles moving in a spatially uniform magnetic field and a time dependent harmonic potential [25, 26]. This is in effect covered by the theorems of Larmor and Kohn which allow one to eliminate the magnetic field by passing to a rotating frame [27, 28, 29]. Despite its great experimental, astronomical and technological importance, we shall not pursue this issue further.
We mention for completeness that generalizations of the Ermakov–Lewis invariant (II.13b) were obtained by considering modifications of the pair of EMP Eqs. (II.1a) and (II.10) [30, 31].
 The orbit in the plane is an ellipse.$$\begin{aligned} \omega= & {} \mathrm{constant} =\omega _0 ,\nonumber \\ r^2= & {} (A \cos \omega _0t + B \sin \omega _0 t)^2+ \frac{h^2}{A^2 \omega _0^2} \sin ^2 \omega _0 t .\nonumber \\ \end{aligned}$$(II.18)
 $$\begin{aligned} \omega = \frac{b}{t^2}, \qquad \theta  \theta _0= \frac{b}{t}, \qquad r= \sqrt{\frac{h}{b}}\, t =\frac{ \sqrt{ hb}}{\theta _0 \theta } . \end{aligned}$$(II.19)
 which yields a logarithmic spiral.$$\begin{aligned}&\omega = \frac{b}{t},\quad r^2 = bt ,\quad b=\frac{h}{\sqrt{b^2\frac{1}{4} }},\nonumber \\&b \ne \frac{1}{2}, \quad \theta =\theta _0 + \frac{h}{b}\ln t , \quad r= b \exp {\frac{b}{h} (\theta \theta _0)},\nonumber \\ \end{aligned}$$(II.20)
 where \(J_n , Y_n\) are Bessel functions. If \(k>0\) , then \(r \rightarrow 0 \) as \( t \rightarrow \infty \), and if \(k<0\) then \(r \rightarrow \infty \) as \( t \rightarrow \infty \).$$\begin{aligned} \omega= & {} bt^k , \, r^2 =\pi h A t \Bigl ( J^2_n( 2 A b^2 t^{k+1} ) + Y^2_n( 2 A b^2 t^{k+1}) \Bigr ) ,\nonumber \\&n= \frac{1}{2}(k+1) , \quad \end{aligned}$$(II.21)
2.3 The onedimensional Schrödinger equation and the Liouville–Green–Jeffreys–Wentzel–Kramers–Brillouin approximation
The relation to the Liouville–Green–Jeffreys–Wentzel–Kramers–Brillouin method may be seen by noting that (II.3) and (II.4), (II.8), and (II.7), are identical to (2), (6), (3) in Sect. 17.122 of [35], which contains a systematic procedure for obtaining asymptotic expansions, provides a list of references and justifies the addition of the names of Liouville, Green and Jeffreys to the better known Wentzel, Brillouin, and Kramers.
We remark en passant that the EMP equation may be incorporated into the formalism of supersymmetric quantum mechanics [36, 37].
2.4 The Ermakov–Milne–Pinney equation and Madelung’s hydrodynamic transcription
2.5 Bose–Einstein condensates
Among the many applications of the EMP equation is one to Bose–Einstein condensates (BEC’s) [44, 45].
3 The Eisenhart–Duval Lift and the Ermakov–Milne–Pinney Equation
3.1 Eisenhart–Duval lift of Newtonian mechanics
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).
3.2 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].
3.3 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 energymomentum tensor is chosen in a proper way.
3.4 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 nonrelativistic 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 rederive the abovementioned symmetries in the specific case of our EMP spacetime of Sect.3.3.
From Eq. (III.43) one concludes that the \(\kappa \), \(\dot{\lambda }(t)\), \(\epsilon \), and \(\rho _i(t)\)transformations give rise to conformal Killing vectors,^{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 \(so(2,2+d)\).
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).
As it follows from (III.41), the isometries of (III.44) also involve the translation in the vdirection, 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 GalileiCarroll “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 timedependent 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].
3.5 Geodesic motion in Ermakov–Milne–Pinney cosmoi
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 timelike geodesics.
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.
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.
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 timedependent \(\gamma \) 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 timedependent oscillator, with frequency \(\gamma (t)^2\), finding an analytic solution may be complicated.
4 Friedmann–Lemaître–Robertson–Walker spacetimes
There has been considerable interest in the past few years in applying the ideas circling around the EMP equation to problems in cosmology, both classical and quantum, [8, 9, 10, 11, 12, 13, 14, 15]. In this section we shall establish some connections. Before doing so we shall begin by establishing our notations and conventions.
4.1 Matter models

\(\gamma =1\) corresponds to pressure–free matter.

\(\gamma =\frac{4}{3}\) to radiation.

\(\gamma =0\) to a cosmological constant which is equivalent to a constant energy density \(\rho = \frac{\Lambda }{8\pi G}\).

\(\gamma =2 \) is the largest value of \(\gamma \) consistent with the dominant energy condition. If the energy density is positive it is sometimes known as “stiff matter”. If \(\gamma >2\) there exist solutions for which the scale factor blows up in finite time.

\(\gamma = \frac{2}{3}\) corresponds to a gas of cosmic strings.

\(\gamma = \frac{1}{3}\) corresponds to a gas of membranes.

If \(\gamma < \frac{2}{3}\), there exist solutions exhibiting a “big rip”, that is for which the scale factor blows up in finite time [58] .
4.2 Cosmic clocks and temporal diffeomorphisms
A related construction works if \(K=0\) but we include a cosmological constant term or dark energy term.
For more examples of solutions of the Friedmann and Raychaudhuri equations the reader is referred to [65, 66, 67].
4.3 Lifting the FriedmannLemaîtreRobertsonWalker geodesics
Having presented the main features of FLRW cosmologies, now we show how the ED lift can be applied to geodesic motion in FLRW spacetimes and how to derive its dynamical symmetries. We then extend the result to the Dmitriev–Zel’dovich equations.
Another consequence is that the null geodesics of the ED lift just constructed are the same as the null geodesics of the EMP spacetime, described in Sect.3.3.
4.4 The Eisenhart–Duval lift of the Friedmann equations
5 Conclusion

Any Newtonian mechanical system can be described in terms of the EisenhartDuval metric which solves the Einstein equations (III.10). The key ingredient involved in the construction is the energy–momentum tensor (III.6) built out of the covariantly constant null Killing vector field (III.5) and a proper energy density function.

The celebrated Ermakov–Lewis invariant of a timedependent harmonic oscillator can be obtained in purely geometric way by applying the EisenhartDuval lift.

A cosmological extension of the EisenhartDuval metric is constructed by properly incorporating into the scheme a cosmic scale factor and the energymomentum tensor. The evolution of spacetime is governed by the Ermakov–Milne–Pinney equation.

Killing isometries include spatial translations and rotations, Newton–Hooke boosts and translation in the null direction.

The algebra of vector fields which involve both Killing and conformal Killing vectors is identified with \(so(2,2+d)\), the conformal Newton–Hooke algebra being its subalgebra.

Geodesic motion in Ermakov–Milne–Pinney cosmoi is described.

The EisenhartDuval lift of geodesics in the Friedmann–Lemaître–Robertson–Walker spacetimes is found and then generalized to the Dmitriev–Zel’dovich equations.

The derivation of the Friedmann equations within the framework of the Eisenhart–Duval lift is presented.
Footnotes
 1.
The case \(\lambda <0\) is obtained by replacing the Euclidean plane \(\mathbb {E}^2 \) by the Minkowski plane \(\mathbb {E}^{1,1}\) [33].
 2.
 3.
Note that \(\frac{\hbar ^2}{2m}\) in (3) of [36] should be replaced by its inverse.
 4.
The factor \(\frac{d}{2\pi }\), d being the dimension of the x–subspace, is chosen for further convenience.
 5.
Note that a combination of the \(\kappa \) and \(\lambda \) transformations may result in a Killing vector field, provided \(\dot{\lambda }=\kappa \). In view of (III.42), this only happens if \(\Omega (t)={g}/{t}\), where g is a constant.
 6.
From now on we switch back to cosmic time t.
 7.
From now on we drop the tildes.
Notes
Acknowledgements
We would like to thank an anonymous referee for informing us that Ermakov’s paper was preceded by one of Adolph Steen [72, 73] and therefore it would be more appropriate to speak of the SteenErmakovMilnePinney equation. MC was funded by the CNPq under project 303923/20156, and by a Pesquisador Mineiro project no. PPM0063017. AG was supported by the Tomsk Polytechnic University competitiveness enhancement program and the RFBR grant 185205002.
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