#### Geometry of the isotropic oscillator driven by the conformal mode

Eur. Phys. J. C
Geometry of the isotropic oscillator driven by the conformal mode
Anton Galajinsky 0
0 School of Physics, Tomsk Polytechnic University , Lenin Ave. 30, Tomsk 634050 , Russia
Geometrization of a Lagrangian conservative system typically amounts to reformulating its equations of motion as the geodesic equations in a properly chosen curved spacetime. The conventional methods include the Jacobi metric and the Eisenhart lift. In this work, a modification of the Eisenhart lift is proposed which describes the isotropic oscillator in arbitrary dimension driven by the one-dimensional conformal mode.
1 Introduction
It would not be an exaggeration to say that, since the
discovery of general relativity, geometry and theoretical physics
go parallel. Given a dynamical system with finitely many
degrees of freedom, it is customary to ask: What is its
geometric description? Vice versa, given a geometric idea, it is
natural to wonder: What would be its physical application?
In general, geometrization of a Lagrangian conservative
system, whose kinetic term involves a positive definite
metric, amounts to reformulating its equations of motion as the
geodesic equations in a properly chosen curved spacetime or
embedding them into the geodesics of a larger theory such
that the time evolution of the extra degrees of freedom is
unambiguously fixed, provided the dynamics of the original
model is known. The Jacobi metric [1] and the Eisenhart lift
[2, 3] represent the conventional tools of that kind (for a recent
application to time-dependent systems see [4]). While the
former method yields a Riemannian metric, the latter gives
a Lorentzian metric thus paving the way for applications in
the general relativistic context.
The goal of this work is to discuss a geometric
formulation for a particular dynamical system which describes the
isotropic oscillator in arbitrary dimension driven by the
onedimensional conformal mode (see Eq. (
1
) below). It naturally
arises if one applies the conventional method of non-linear
realizations [5] to the conformal extension of the Newton–
Hooke algebra [6].1 The model describes a particle moving
along the ellipse such that the periods of its accelerated or
decelerated motion are controlled by a single function of time
variable representing the d = 1 conformal mechanics of de
Alfaro et al. [10]. It is demonstrated below that, by
properly including the conformal mode into the Eisenhart metric
associated with the isotropic oscillator, one can attain a
satisfactory geometric description in terms of a Lorentzian metric
which solves the Einstein equations.
The organization of the paper is as follows: In Sect. 2, the
isotropic oscillator in arbitrary dimension driven by the d = 1
conformal mode is discussed. Its symmetries and the
general solution to the equations of motion are given. Section 3
contains a brief account of the conventional Eisenhart lift
applied to conservative mechanical systems. An embedding
of the isotropic oscillator driven by the one-dimensional
conformal mode into the geodesics of the Eisenhart-like metric
is discussed in Sect. 4. The conformal mode enters the
metric as a scale factor in such a way that the oscillator equation
follows from the geodesics, while the conformal mechanics
arises when imposing the Einstein equations. In Sect. 5, the
Killing isometries of the metric are studied. A set of vector
fields is found which all together form the Newton–Hooke
algebra under the commutator. The requirement that they be
the Killing vector fields of the metric in Sect. 4 reproduces
the conformal mechanics equation on the scale factor. It is
demonstrated that the same condition occurs if one demands
the spacetime to be stationary. We summarize our results
1 It is customary to refer to this algebra as the l = 21 conformal Newton–
Hooke algebra. Note that the equations of motion similar to (
1
) were
first obtained in [7] when constructing dynamical realizations for the so
called l = 1 conformal Galilei algebra. As compared to (
1
), the L12 term
was absent and the frequency of oscillation was bigger by a factor of
two. A geometric formulation for the second equation in (
1
) (with no 1
L2
term) was first proposed in [8]. An alternative parametrization of the
coset space for the l-conformal Newton–Hooke algebra was recently
discussed in [9].
and discuss possible further developments in the
concluding Sect. 6. Throughout the paper summation over repeated
indices is understood.
2 Isotropic oscillator driven by d = 1 conformal mode
Consider the configuration space R1 × Rd parametrized by
the coordinates ρ and xi , i = 1, . . . , d, and the dynamical
system governed by the equations of motion
(
1
)
(
2
)
ρ(t )2 d
dt
d2ρ(t )
dt 2
ρ(t )2 d
dt xi (t )
γ 2
= ρ(t )3 −
ρ(t )
L2 ,
+ γ 2xi (t ) = 0,
where γ and L are positive constants. The second equation
describes the d = 1 conformal mechanics [10], while the first
equation corresponds to the isotropic oscillator in d
dimensions driven by the conformal mode.
It is straightforward to verify that Eq. (
1
) hold invariant
under the infinitesimal transformations
t = t + a − bL2(cos (2t /L) − 1) + cL sin (2t /L),
ρ (t ) = ρ(t ) + (bL sin (2t /L) + c cos (2t /L)) ρ(t ),
1
xi (t ) = xi (t ) + ρ(t ) cos (t /L)αi
1
+ ρ(t ) L sin (t /L)βi + ωi j x j (t ),
which provide a realization of the conformal extension of the
Newton–Hooke group [6]. The parameters (a, b, c, αi , βi , ωi j
= −ω ji ) are associated with the time translation,
special conformal transformation, dilatation, spatial translations,
Newton–Hooke boosts, and spatial rotations, respectively,
while L is identified with the characteristic time [11]. In order
to verify the structure relations of the conformal Newton–
Hooke algebra, it suffices to consider the variations of fields
δρ(t ) = ρ (t ) − ρ(t ),
δxi (t ) = xi (t ) − x (t ),
(
3
)
and compute their commutators.2
Making use of the conserved charges associated with the
symmetry transformations (
2
), one can then build the general
solution to (
1
) by purely algebraic means
ρ(t) =
(DL sin (t/L) + C cos (t/L))2 + (γ L sin (t/L))2
C
,
(
4
)
2 When evaluating the commutators, it is to be remembered that the
variations act upon the fields ρ(t) and xi (t) and do not affect the
temporal coordinate t.
γ L,
The increase/decrease of ρ(t ) causes the
deceleration/acceleration of a particle along the ellipse.
Worth mentioning is that the minimum point of the
potenγ 2 2
tial U (ρ) = ρ2 + ρL2 , which occurs at
1 1
xi (t ) = ρ(t ) L sin (t /L)Pi − ρ(t ) cos (t /L)Ki ,
where D, C, Pi , Ki are constants of integration.
The shape of the trajectory in the Newton–Hooke
reference frame is readily found if one uses the alternative
parametrization of the curve
ρ(t )2 ddt = ddϕ , ddϕt = ρ12 ,
ϕ(t ) = γ1 arctan DC + (D2 + γ 2)L tan (t /L) , (
6
)
γ C
which turns the first equation in (
1
) into that describing an
ordinary isotropic oscillator. The orbit is thus an ellipse
xi (t ) = μi cos (γ ϕ(t )) + νi sin (γ ϕ(t )),
μi and νi being constants of integration. According to (
4
), the
conformal mode is an oscillating function which determines
the periods of accelerated/decelerated motion of the particle
along the ellipse. Most easily this is illustrated by examining
the circular motion (μ2 = 1, ν2 = 1, (μ, ν) = 0) for which
(
5
)
(
7
)
(
8
)
(
9
)
(
10
)
(
11
)
provides a particular solution to the second equation in (
1
)
and turns the first equation into
d2xi (t )
dt 2
1
+ L2 xi (t ) = 0.
An ordinary isotropic oscillator is thus a particular instance
of (
1
). Equation (
10
) also arises if one goes over to the
noninertial Newton–Hooke reference frame3
xi = ρ(t )xi ,
which correlates with Eq. (
5
) above.
3 This reference frame also indicates the Lagrangian formulation
L = ρ˙2 − γρ22 − ρL22 + (ρxi ).(ρxi ). − ρ2Lx2ixi which reproduces
the equations of motion (
1
). We thank S. Krivonos for pointing this out
to us. Note that the Eisenhart metric associated with such a Lagrangian
does not solve the Einstein equations unless ρ takes a fixed value as in
(
9
).
3 The Eisenhart lift
The Eisenhart lift [2,3] suggests a geometric formulation for
a conservative mechanical system with d degrees of freedom
xi , i = 1, . . . , d, and the potential energy U (x ) in terms of
geodesics of the Lorentzian metric
gμν (y)d yμd yν = −2U (x )dt 2 − dt ds + d xi d xi
(
12
)
defined on a (d + 2)-dimensional spacetime parametrized
by yμ = (t, s, xi ), where t is identified with the temporal
variable of the Newtonian mechanics and s is an extra
coordinate.
Computing the Christoffel symbols
i
tt = ∂i U,
s
ti = 2∂i U,
where we have split the index μ = (t, s, i ), i = 1, . . . , d, and
abbreviated ∂μ = ∂y∂μ , and analyzing the geodesic equations
d2 yλ
dτ 2 +
λ d yμ d yν
μν dτ dτ
= 0,
d yμ d yν
gμν dτ dτ
= ,
(14)
where = 0 for the null geodesics and = −1 for the
timelike geodesics, one concludes that t is affinely related to the
proper time τ
where κ is a positive constant, while xi obeys Newton’s
equation (passing from τ to t via (15))
d2t
dτ 2 = 0
⇒
The latter determines the so called Bargmann structure on
the manifold [3]. Equation (19) also implies that the
spacetime (
12
) admits a geodesic null congruence with vanishing
expansion, shear and vorticity and thus belongs to the Kundt
class.
The distinguished vector field ξ μ can be used to construct
the trace-free energy–momentum tensor4
d
Tμν = 2π
(y)2ξμξν , T μμ = 0,
where (y)2 is an arbitrary function (the energy density).
1
Because the only non-vanishing component of ξμ is ξt = − 2 ,
the energy–momentum tensor has only one nonzero
component
Taking into account that ξμ is covariantly constant, one
can compute the covariant derivative
∂ρ ξμξν ,
and verify that the energy–momentum tensor is conserved
provided does not depend on s, i.e. = (t, x ).
Finally, making use of the Christoffel symbols (
13
), one
finds that the Ricci tensor has only one nonzero component
Rtt while the scalar curvature vanishes
Given the general solution to (16), the dynamics of s is fixed
from the condition that the geodesic is null or time-like (again
passing from τ to t )
ds d xi d xi
dt = dt dt − 2U − κ2 ,
The conventional Newtonian mechanics is thus recovered by
implementing the null reduction along s [2,3].
A remarkable property of the Eisenhart metric is that it
holds invariant under the transformation
s = s + λ,
where λ is a constant, which gives rise to the covariantly
constant null Killing vector field
ξ μ∂μ = ∂s ,
∇μξν = 0, ξ 2 = 0.
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
Rtt = ∂i ∂i U,
R = 0.
Given the Eisenhart metric (
12
) and the energy–momentum
tensor (20), the conventional Einstein equations
1
Rμν − 2 gμν (R + 2 ) = 8π Tμν
imply that the contribution of the cosmological term
necessarily vanishes = 0 thus reducing (25) to
Rμν = 8π Tμν ,
2
=
4π G
d
r (x ),
only (t t )-component of which is non-trivial.
A particularly interesting example of the Eisenhart
geometry occurs it one takes 2 in (20) to be s, and t -independent
4 The factor 2dπ , d being the dimension of the x-subspace, is chosen
for further convenience.
and interprets G as the Newtonian constant and r (x ) as the
mass density. Then the Einstein equations (26) reproduce the
Newtonian equation for the gravitational potential
U (x ) = 4π Gr (x ).
4 Conformal mode as a scale factor in the Eisenhart
metric
Having reviewed the Eisenhart geometric description of
conservative mechanical systems, let us discuss its possible
generalization to encounter systems like (
1
). Given the form of
the x -orbit and the impact of ρ(t ) upon it, a natural question
arises whether a geometric description of (
1
) exists in which
the conformal mode plays the role of a cosmic scale factor.
Consider the following generalization of the Eisenhart
metric (
12
):
gμν (y)d yμd yν = − γρ2(xti)x2i dt 2 − dt ds
+ ρ(t )2d xi d xi + 2q xi dt d xi ,
where γ , q are constants. A few comments are in order.
Firstly, for a fixed value of t the line element in
the d-dimensional slice parametrized by xi is given by
ρ(t )2d xi d xi . Hence ρ(t )2 may be viewed as a cosmic scale
factor.
Secondly, the metric admits a covariantly constant null
Killing vector field realized as in (19) and hence belongs to
the Kundt class. Choosing (y) in Eq. (20) in the form
(28)
(29)
(30)
(31)
(32)
1
(y) = L ,
d2ρ γ 2 ρ
dt 2 = ρ3 − L2 ,
where L is a positive constant, and imposing the Einstein
equations (26) one obtains the restriction on the scale factor
which precisely coincides with the second equation in (
1
).
To the best of our knowledge, this is the first example in
the literature that the conformal mechanics of de Alfaro et
al. [10] shows up in the general relativistic context.
Thirdly, computing the Christoffel symbols
(33)
(34)
where κ is a positive constant, the evolution of xi (t ) is
determined by the equation which precisely coincides with the
first equation in (
1
), while s is fixed from the condition that
the geodesic is null or time-like
s˙ = −
γ 2xi xi
ρ2
+ ρ2x˙i x˙i + 2q xi x˙i − κ2 ,
where = 0 for the null geodesics and = −1 for the
time-like geodesics. Given the general solution in (
4
), the
dynamics of the extra variable s is unambiguously fixed by
the first order ordinary differential equation (34). Thus the
original dynamical system (
1
) is recovered if one implements
the null reduction along s.
Note that, as compared to the conventional Eisenhart
prescription, in which coordinates parametrizing the spacetime
are associated with degrees of freedom of the original
dynamical system, the conformal mode ρ(t ) enters the metric (29)
as a specific scale factor whose time evolution is governed
by the Einstein equations.
Concluding this section, we note that the L → ∞ limit
of (
1
) yields a dynamical system enjoying the Schrödinger
symmetry [6]. Curiously enough, its geometric description
based upon (29) fails as the corresponding Riemann tensor
turns out to be vanishing.
5 Isometries of the metric
Having fixed the form of the metric, let us establish its
isometry group. The conventional way is to consider the
infinitesimal transformations
and demand the form of (29) to be intact. This yields a
coupled set of first order partial differential equations for
α(t, s, x ), β(t, s, x ), and γ (t, s, x ) whose general solution
determines the Killing vector fields
2 ρ˙ . + 2q ρ˙
ρ ρ
ρ
˙
∂s − ρ xi ∂i , S = ∂s ,
s
tt = −
2γ 2xi xi (ρρ˙ − q)
ρ4
,
s
i j = 2(ρρ˙ − q)δi j ,
i
tt =
γ 2xi
ρ4 ,
i ρ˙
t j = ρ δi j ,
s
ti =
2xi (qρρ˙ + γ 2)
ρ2
,
where the dot designates the derivative with respect to t , and
analyzing the geodesic equations, one concludes that t is
affinely related to the proper time τ
1
Pi = ρ cos (t /L)∂i
1
Ki = ρ L sin (t /L)∂i
+ 2xi
1 ρ
ρ cos (t /L)(q − ρρ˙) − L sin (t /L) ∂s ,
+ 2L xi
1 ρ
ρ sin (t /L)(q −ρρ˙)+ L cos (t /L) ∂s .
(36)
The metric is also invariant under S O(d) rotations acting
upon xi . It is straightforward to verify that H is time-like,
Pi , and Ki are space-like, while S is null and covariantly
constant. Computing the commutators of the vector fields,
one finds
1
[H, Ki ] = Pi , [H, Pi ] = − L2 Ki , [ Pi , K j ] = 2Sδi j .
(37)
This is a variant of the Newton–Hooke algebra [11]5 in which
the covariantly constant null vector field S plays the role
of the central element. As ρ(t ) in (29) is treated as a fixed
function, it does not come as a surprise that the S O(
2,
1
)invariance of the original dynamical system (1) is not
inherited by the metric (29).
Interestingly enough, the Newton–Hooke symmetry (36)
provides another way of obtaining the restriction (31) upon
the scale factor ρ(t ). Indeed, the vector fields (36) prove
to obey the structure relations (37) without imposing any
constraint on ρ(t ). Considering the metric (29) with arbitrary
ρ(t ) and requiring it to admit the Killing vector fields (36),
one immediately gets (31) from the Killing equation.
The fact that (29) admits the time-like Killing vector field
H , provided ρ(t ) obeys (31), implies the spacetime is
stationary. To put it in other words, there exists a coordinate system
in which the metric does not depend on time. This suggests
yet another possibility to arrive at Eq. (31) within the
geometric framework. Consider the metric (29) with arbitrary scale
factor ρ(t ) and implement the coordinate transformation
t = t, xi = ρ(t )xi , s = s + xi xi (ρρ˙ − q),
(38)
which brings it to the form
gμν (y )d y μd y ν = −
γ 2 ρ
ρ4 − ρ¨ xi xi dt 2 − dt ds + d xi d xi .
Requiring the resulting metric to be Lorentzian and
stationary, one gets
γρ42 − ρρ¨ = L12 ,
where L is constant, which reproduces (31). Note that, if
(40) is satisfied, (39) gives the Eisenhart metric associated
with the isotropic oscillator. The coordinate system (38) is
thus the analog of the non-inertial Newton–Hooke reference
frame (
11
).
5 For a more detailed discussion of the Newton–Hooke symmetry and
its conformal extension see [12,13].
(39)
(40)
6 Conclusion
To summarize, in this work we proposed a geometric
formulation for a particular dynamical system which describes the
isotropic oscillator in arbitrary dimension driven by the
onedimensional conformal mode. In contrast to the conventional
Eisenhart prescription, in which coordinates parametrizing
the spacetime are associated with degrees of freedom of the
original dynamical system, the conformal mode enters the
metric as a specific scale factor. The equation which governs
its evolution was obtained in three different ways either by
imposing the Einstein equations, or demanding the Newton–
Hooke isometry group, or requiring the spacetime to be
stationary. To the best of our knowledge, the consideration above
provides the first example in the literature that the conformal
mechanics of de Alfaro, Fubini, and Furlan shows up in the
general relativistic context.
Turning to possible further development, it would be
interesting to generalize this work to the specific chain of isotropic
oscillators driven by d = 1 conformal mode which enjoys
the so called l-conformal Newton–Hooke symmetry [6].
Acknowledgements This work was supported by the Tomsk
Polytechnic University competitiveness enhancement program. We thank S.
Krivonos for communicating to us that the system (
1
) is Lagrangian.
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