# Bertrand’s Theorem and the Double Copy of Relativistic Field Theories

Dijs de Neeling  Van Swinderen Institute for Particle Physics and Gravity,
University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Bernoulli Institute for Mathematics, Computer Science and Artificial
Intelligence,
University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands
Diederik Roest  Van Swinderen Institute for Particle Physics and Gravity,
University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Marcello Seri  Bernoulli Institute for Mathematics, Computer Science and
Artificial Intelligence,
University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands
Holger Waalkens  Bernoulli Institute for Mathematics, Computer Science and
Artificial Intelligence,
University of Groningen, Nijenborgh 9, 9747 AG Groningen, The Netherlands

###### Abstract

Which relativistic field theories give rise to Kepler dynamics in the two-body
problem? We consider a class of Hamiltonians that is the unique relativistic
extension of the Kepler problem preserving its so(4) algebra, and have orbits
related through time reparametrisation to orbits of the original Kepler
problem. For three explicit examples, we give a natural interpretation in
terms of spin-0,-1 and -2 interacting field theories in 5D. These are
organically connected via the classical double copy, which therefore preserves
maximal superintegrability.

##  I Introduction

The Laser Interferometer Space Antenna (LISA) is expected to open the door to
observations of binary black holes systems with long inspirals and extreme
mass ratio inspirals, both of which can be approached fruitfully from an
analytical point of view Babak  _et al._ (2017); Berry  _et al._ (2019);
Cheung  _et al._ (2023). The non-relativistic limit of these binaries in the
center of mass frame is the Kepler problem. The properties of the latter are,
even among classical systems, very special. It is one of only two central
potential problems whose bounded orbits are all closed, the other being the
isotropic harmonic oscillator, a result known as Bertrand’s theorem and going
back to 1873 Santos  _et al._ (2007); Goldstein  _et al._ (2001). This is a
consequence of the large, ‘hidden’ so(4) symmetry of the system, leading to
maximally superintegrable dynamics.

General Relativity breaks this enhanced so(4) symmetry down to so(3) (or even
further in case of spinning objects), as do most other relativistic theories.
However, there exist relativistic dynamical systems with the same symmetry
group as Kepler to all orders. The simplest of these are test-particle
limits111This is to be expected: while it is far from trivial to write down
the Hamiltonian for a multi-worldline system explicitly, the test-particle
case can be viewed as a one-body problem, as the large mass is taken to be
non-moving. and include the ’branonium’ systems identified in string theory
Burgess  _et al._ (2003, 2004). More recently, the same hidden symmetry was
found in $\mathcal{N}=4$ super-Yang-Mills Caron-Huot and Henn (2014); Alvarez-
Jimenez  _et al._ (2018) and $\mathcal{N}=8$ supergravity Caron-Huot and
Zahraee (2019). Additionally, in the latter theory, the two-body system is
known to possess the symmetry to first order in the Post-Newtonian expansion,
while it preserves the related non-precession of orbits to third order in the
Post-Minkowskian expansion Parra-Martinez  _et al._ (2020). Finally, also the
Kaluza-Klein monopole scattering of Gibbons and Manton (1986); Gibbons and
Ruback (1987) should be noted222Another relativistic so(4)-conserving two-body
system is the time-asymmetric scalar-vector theory studied by Duviryak
(1996)..

As we have shown recently by explicit construction, Bertrand’s theorem can be
extended beyond classical central potentials: at least up to and including 5th
PN order, there exists a unique class of relativistic two-body Hamiltonians
with the appropriate classical limit that displays additional symmetry.
Moreover, all elements of this class can be related to functions of the Kepler
Hamiltonian through canonical transformations Neeling  _et al._ (2024). An
all-order argument was subsequently put forward by Davis and Melville in Davis
and Melville (2023) and will be outlined below.

In the present work, we further investigate this class of relativistic
Hamiltonians possessing Keplerian symmetry to all orders, having the correct
special-relativistic and non-relativistic limits. Our focus will be on three
examples of the class, for which we propose a realisation in terms of
fundamental forces. We find that all these can be lifted to 5D systems, and
subsequently linked to classical theories via the Eisenhart lift. This
provides for a geometric interpretation of the link333 We generalise this link
between classical and relativistic dynamics to multi-center systems in the
Appendix A, along with a rigorous mathematical proof of the relation. with
Keplerian dynamics.

The 5D formulation also allows for a connection between the three examples via
the double copy. Known primarily as a relation of scattering amplitudes in
gravity appearing as the ‘square’ of gauge theory amplitudes Bern  _et al._
(2010), we will instead employ the so-called classical double copy, similarly
linking background solutions of these theories Monteiro  _et al._ (2014). More
specifically, we will use it to connect dynamical systems in the sense of
Gonzo and Shi (2021). Interestingly, the three examples of relativistic
dynamics with Keplerian dynamics are exactly linked in this way. This provides
a precise sense in which hidden symmetries and maximal
superintegrability444Another, less symmetric example is provided by the
integrability of the Kerr space-time that also carries over Ball  _et al._
(2023). carry over under the classical double copy.

##  II Relativistic Bertrand’s Theorem

It is natural to wonder what lies beyond the celebrated Bertrand’s theorem of
classical mechanics, once moving into the realm of relativistic physics. For
reasons that will be outlined in what follows, we will consider the following
(implicit) definition for our class of relativistic Hamiltonians:

$\displaystyle
H^{2}=m^{2}c^{4}+p^{2}c^{2}-\left(\frac{1}{r}\right)F\left(mc^{2}+H\right)\,,$
(1)

where $F\left(x\right)$ is a continuous and smooth function that encodes the
nature of the attractive force. Note that this class of Hamiltonians reduces,
in the absence of the attractive force ($F\rightarrow 0$), to the special
relativistic free particle. When perturbatively including the attractive
force,

$\displaystyle F(x)=f_{0}+f_{1}x+\tfrac{1}{2}f_{2}x^{2}+\ldots\,,$ (2)

we see the coefficients must have dimensions
$\left[f_{i}\right]=M^{(2-i)}L^{(3-i)}T^{(i-2)}$. Specializing to
$f_{2}={8GM}/{c^{2}}$ as the only nonzero component for instance, with $M$
some reference mass, the resulting Hamiltonian reads in a post-Newtonian
expansion

$\displaystyle H=mc^{2}$ $\displaystyle+\frac{p^{2}}{2m}-\frac{8GMm}{r}$
$\displaystyle-\tfrac{1}{8}\frac{p^{4}}{m^{3}c^{2}}+\frac{32G^{2}M^{2}m}{r^{2}c^{2}}+\mathcal{O}({\frac{1}{c^{4}}})\,.$
(3)

In the non-relativistic limit, the system therefore reduces to Kepler (up to a
constant rest-mass energy).

The relativistic corrections of this class of Hamiltonians retain the special
property that all bounded trajectories are closed orbits. At a heuristic
level, this can be seen in the following way. As energy is conserved, the
argument of the force term in (1) will take a specific, constant value for a
given trajectory. For this trajectory, the force is therefore given by some
constant $F$, and the Hamiltonian reads

$\displaystyle H=mc^{2}\sqrt{1+2H_{\rm Kep}/mc^{2}}\,,$ (4)

for a Kepler Hamiltonian

$\displaystyle H_{\rm Kep}=\frac{p^{2}}{2m}-\frac{1}{r}\frac{F}{2mc^{2}}\,.$
(5)

It is therefore natural to expect that this class of systems has closed
orbits. This is confirmed by the presence of a relativistic generalisation of
the Laplace-Runge-Lenz vector, that takes the form

$\displaystyle\vec{A}=\vec{p}\times\vec{L}-\frac{1}{2c^{2}}F\left(mc^{2}+H\right)\hat{r}\,,$
(6)

in terms of angular momentum $\vec{L}=\vec{r}\times\vec{p}$ and the unit
position vector $\hat{r}$. This vector is responsible for the maximal
superintegrability and hence closed orbits in the non-relativistic case.

The dynamics of the above theories can be seen as a non-standard way of
stacking Kepler energy levels: the effective gravitational coefficient is set
by $F$ and hence is orbit-dependent. In a hypothetical solar system governed
by such an attractive force, planets would fail to satisfy the universal
Kepler’s third law relation between period and radius. As discussed in more
detail in the Appendix A, a particular time reparametrisation of (1) connects
it to a new Kepler Hamiltonian, for which Kepler’s third law does hold for all
orbits with the same universal proportionality constant.

One issue with the above heuristic argument is that it only applies to
separate energy levels and not the complete phase space. It would therefore be
desirable to have an alternative perspective, that applies to the union of all
orbits. Indeed this appears to be possible: we have demonstrated (with an
explicit construction) that this class of Hamiltonians is the unique extension
of the Kepler system that combines the special relativistic and the non-
relativistic limits (when $F\rightarrow 0$ and $c\to\infty$, respectively)
that has the so(4) hidden symmetry and closed orbits, up to and including 5PN
order. It is therefore the natural generalisation of Bertrand’s theorem to the
relativistic domain (at least for the Kepler system). Moreover, this system is
canonically equivalent to Kepler in a neighbourhood around an energy level, at
least up to 5PN order Neeling  _et al._ (2024).

A closely related claim has been put forward by Davis and Melville in Davis
and Melville (2023), who employ a reorganisation of PN corrections into powers
of $1/r$ multiplied by functions of the Kepler Hamiltonian. They argue that
the function multiplying the $1/r$ part of this expansion can be made
vanishing (their so-called ‘LRL gauge’) by means of canonical transformations.
Subsequently, they demonstrate that all higher-order powers of $1/r$ similarly
have to vanish for the system to have closed orbits and hidden symmetries.
This then implies that the class of Hamiltonians (1) is canonically conjugate
to Kepler to all PN orders.

Having introduced the class of Hamiltonians as the relativistic generalisation
of the Kepler system that has hidden symmetries, in the next sections we turn
to their physical interpretation; in other words, which forces generate this
kind of dynamics? We will show that there are three cases - where we take $F$
to be constant, linear or quadratic in its argument - where a natural
interpretation in terms of 4D relativistic field theories presents itself.
Moreover, all three cases allow for an uplift to a 5D system. Different
reductions of this 5D perspective allow one to prove the canonical equivalence
between the classical and relativistic systems to all orders in the PN
expansion.

##  III General Relativity

We will first discuss the quadratic case, $F(x)=\frac{1}{2}f_{2}x^{2}$, and
its interpretation in terms of general relativity. As the constant $f_{2}$
must have the dimension of length, it is natural to suppose this is given in
terms of a mass scale set by some reference mass M, such that $f_{2}=2r_{M}$,
with $r_{M}=\frac{4GM}{c^{2}}$ (while the factor of $4$ will be clear in a
moment). The Hamiltonian reads

$\displaystyle
H^{2}=m^{2}c^{4}+p^{2}c^{2}-\left(\frac{r_{M}}{r}\right)(mc^{2}+H)^{2}\,.$ (7)

It has been shown555Compare equation (4.22) in Neeling  _et al._ (2024) with
$H\to H+m$. that this system arises when an extremal particle (with equal mass
and charge) is orbiting a specific background of the 4D Einstein-Maxwell-
dilaton system, with bulk Lagrangian density

$\displaystyle\mathcal{L}=\frac{\sqrt{-g}}{16\pi}\left(\frac{c^{3}}{G}R-\frac{2c^{3}}{G}(\partial\phi)^{2}-\frac{e^{-2a\phi}}{c}F^{2}\right)\,,$
(8)

where we take the units such that the gauge potential $A_{\mu}$ has
$\left[A^{2}\right]=MLT^{-2}$. For closed orbits, the dilaton coupling of this
system has to take precisely the Kaluza-Klein value $a=\sqrt{3}$ that allows
for the bulk Lagrangian to be uplifted to 5D. This suggests that also the
particle Hamiltonian (7) has a 5D interpretation. Indeed this is the case;
upon making the identifications

$\displaystyle P_{0}=\frac{H}{c}\,,\quad P_{5}=mc\,,\quad(\text{\it space-
like})\,,$ (9)

the Hamiltonian (7) corresponds to geodesic motion along a null geodesic,

$\displaystyle g^{AB}P_{A}P_{B}=0\,.$ (10)

The metric is the uplift of the Einstein-Maxwell-dilaton background

$\displaystyle
ds^{2}=\eta_{AB}dx^{A}dx^{B}+\left(\frac{r_{M}}{r}\right)(dx^{0}-dx^{5})^{2}\,,$
(11)

and is a so-called Bargmann space. Because of the harmonicity of the potential
$\frac{r_{M}}{r}$, this space is Ricci-flat, i.e. it solves the vacuum
Einstein equations Duval  _et al._ (1991). This spacetime is an example of a
pp-wave (plane-fronted wave with parallel rays) due to the presence of the
covariantly constant null-vector $\partial_{0}+\partial_{5}$.

Interestingly, the same 5D perspective can also be related to the non-
relativistic Kepler case; this is referred to as the Eisenhart-Duval lift
Cariglia (2012); Cariglia  _et al._ (2014). Starting in 5D and making the
alternative identifications

$\displaystyle\frac{P_{0}-P_{5}}{\sqrt{2}}=\frac{\bar{H}_{\rm
Kep}}{c}\,,\quad\frac{P_{0}+P_{5}}{\sqrt{2}}=mc\,,\quad(\text{\it null}\,)\,,$
(12)

the vanishing norm of the 5D momentum leads to a particular non-relativistic
Kepler system

$\displaystyle\bar{H}_{\rm Kep}=\frac{p^{2}}{2m}-\frac{r_{M}mc^{2}}{r}\,.$
(13)

Though in different guises, the above two reduced systems therefore are one
and the same, up to time reparametrisation (or conformal transformations, in
the case of null-geodesics Zhang  _et al._ (2019)). Yet the reductions
correspond to what appear to be distinct physical systems, dependent on the
different assignments (9) or (12) for mass and energy, as discussed in
generality in Appendix A.

The 5D formulation allows for a geometric perspective on the so(4) hidden
symmetry of the Kepler system, as these can be seen to be generated by an
interplay of Killing vectors and tensors Duval  _et al._ (1991); Cariglia  _et
al._ (2014). A vector field $K$ is a Killing vector field if

$\nabla_{(a}K_{b)}=0\,.$ (14)

and generate conserved quantities $K^{a}p_{a}$ along geodesic motion. A
rank-$n$ Killing tensor $K_{a_{1}\dots a_{n}}$ is defined analogously by
Lindström and Sarıoğlu (2022)

$\nabla_{(a_{1}}K_{a_{2}\dots a_{n+1})}=0\,,$ (15)

and leads to a conserved quantity $K^{a_{1}\dots a_{n}}p_{a_{1}}\cdots
p_{a_{n}}$ along geodesics Walker and Penrose (1970). Notice that the arrow of
implication goes both ways: any conserved quantity that can be written as a
tensor contracted with momenta also implies the existence of a Killing tensor.
Conserved quantities of quadratic or higher order in momenta are symmetries of
the phase space that cannot be constructed as the lift of a symmetry of
configuration space, and are referred to as hidden symmetries Cariglia (2014).

The above concepts can be applied to the 5D incarnation of the Kepler problem.
The Bargmann space (11) has three Killing vectors associated with angular
momentum. In addition, it has three Killing tensors that generate the Laplace-
Runge-Lenz vector reading

$\vec{A}=(\vec{p}\crossproduct(\vec{r}\crossproduct\vec{p}))-\frac{1}{2}(P_{0}+P_{5})^{2}\frac{r_{M}c^{2}}{r}\vec{r}\,,$
(16)

which is quadratic in momenta.

Let us conclude this section with two remarks on the 5D symmetry algebra.
Firstly, when we take the limit $M\to 0$ toward flat space, the second term in
the above vector vanishes, implying the quadratic Killing tensors become
reducible, as they are now built completely out of the Killing vectors related
to the conserved angular momentum $\vec{L}=\vec{r}\crossproduct\vec{p}$ and
the momentum $\vec{p}$. The latter in this case is conserved as well, since
the space regains translational symmetry in the three spatial directions of
$\vec{r}$.

Secondly, note that none of the conserved quantities related through Noether’s
theorem to the so(4) symmetry of the Kepler problem is generated by a
genuinely conformal Killing tensor or vector. All conserved functions commute
with the 5D geodesic Hamiltonian regardless of its value. As such, it is clear
that the relation between Kaluza-Klein dynamics and the Kepler system still
holds for _massive_ geodesics, though the effect on the dynamics is of
negligible consequence, as it results in the addition of a fixed constant to
the Hamiltonians after reduction.

##  IV Electromagnetism

We now turn to the second element of the class of Hamiltonians (1) for which
we propose an interpretation as coming from a relativistic field theory, being
the case with a linear function $F(x)=f_{1}x$. In this case the Hamiltonian
reads

$\displaystyle
H^{2}=m^{2}c^{4}+p^{2}c^{2}-\left(\frac{1}{r}\right)f_{1}(mc^{2}+H)\,.$ (17)

Upon making the space-like replacements (9), this becomes

$\displaystyle\eta^{AB}P_{A}P_{B}-\left(\frac{1}{r}\right)\frac{f_{1}}{c}(P_{5}+P_{0})=0\,.$
(18)

In contrast to the quadratic case, the above cannot be written in terms of a
Lorentzian metric solving the Einstein equations666It can be written as a
momentum norm on a Finsler space, a generalisation of pseudo-Riemannian space
Beil (1987).. Instead, the natural interpretation of the lifted form is in
terms of an electromagnetic force, linear in momenta.

This can be understood in terms of the classical double copy, which maps the
specific class of so-called Kerr-Schild backgrounds of General Relativity to
solutions of Maxwell’s equations in electromagnetism. Among more obvious
solutions such as the Kerr and Schwarzschild black holes, also time-dependent
pp-waves have been shown to have a classical double copy structure Monteiro
_et al._ (2014). The spacetime we are interested in is a subset of this, being
a time-independent pp-wave.

The class of Kerr-Schild solutions takes the form

$g_{AB}=\bar{g}_{AB}+r_{s}\phi\,l_{A}l_{B}\,,$ (19)

and consist of a background metric $\bar{g}$ (which should separately satisfy
Einstein’s equations) plus a deformation in terms of a harmonic scalar
function, $\Box\phi(r)=0$, and a vector $l_{A}$ that is null with respect to
$\bar{g}$ and therefore also total metric $g$. The Schwarzschild radius reads
$r_{s}=2GM/c^{2}$. The Kerr-Schild double copy then says that the vector
$A_{A}=\phi\,l_{A}$ is a solution of the Maxwell equations and $\phi(r)$ is a
solution of the scalar field equation.

The Bargmann space introduced above is exactly of Kerr-Schild form, with a
flat background metric and SO(3) rotationally symmetric function, $\phi=2/r$.
This gravitational solution therefore generates, via the double copy
procedure, the following solution of Maxwell’s equations, now including all
constants:

$A_{A}=\frac{g}{4\pi}\tilde{q}\phi(r)l_{A}\,,$ (20)

where we introduced $g$ as electromagnetic coupling and $\tilde{q}$ as a
charge. This single copy is itself an electromagnetic pp-wave and $l_{A}$ is
its wave vector.

Given the interesting dynamics of geodesics in the Bargmann space-time, can we
expect similar behaviour in the Maxwell configuration? In other words, does
the hidden symmetry and maximal superintegrability survive the double copy?
Interestingly, this turns out indeed to be the case: the Hamiltonian with
linear force function (17) has an interpretation as a charged particle in the
single copy background (20). Specifically, after setting
$f_{1}=\frac{g^{2}}{4\pi}q\tilde{q}$ and uplifting to 5D via (9), the
Hamiltonian of the linear case can be written as

$\displaystyle\eta^{AB}\tilde{P}_{A}\tilde{P}_{B}=0\,,\ \text{with
}\tilde{P}_{A}=P_{A}-\frac{gq}{c}A_{A}\,,$ (21)

in terms of the modified momentum $\tilde{P}$ of charged particles in the
vacuum EM background (20). This places the linear case on a par with the
quadratic one, with interpretations in terms of electromagnetic and
gravitational force fields, respectively.

With the 5D interpretation in hand, one can consider its dimensional
reductions. The null reduction straightforwardly leads to the usual non-
relativistic Hamiltonian, which in this case is interpreted as the Coulomb
system. In contrast, the space-like reduction leads to a relativistic field
theory with coupled degrees of freedom. In order to see this, separate the
5-dimensional vector field in parts $A_{A}=(A_{\mu},A_{5}=\chi)$. The field
Lagrangian then reads

$\mathcal{L}=-\frac{1}{4}F_{AB}F^{AB}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}(\partial_{\mu}\chi)(\partial^{\mu}\chi)\,,$
(22)

implying a theory with a scalar $\chi$ and vector $A_{\mu}$. The particle
Lagrangian can be found by taking

$\mathcal{L}_{p}=-\frac{1}{2h}\eta_{AB}\dot{x}^{A}\dot{x}^{B}+\frac{gq}{c}A_{A}\dot{x}^{A}\,,$
(23)

which can be Legendre transformed to write it in terms of $P_{5}$ instead of
$\dot{x}_{5}$. One then has

$\mathcal{L}_{p}=\frac{1}{2h}\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}-\frac{h}{2}\left(\frac{gq}{c}\chi-
P_{5}\right)^{2}-\frac{gq}{c}A_{\mu}\dot{x}^{\mu}\,.$ (24)

Solving the equation for the auxiliary variable $h$, picking the branch giving
the correct kinetic term, we find the Lagrangian

$\mathcal{L}_{p}|_{h}=-\sqrt{-\left(\frac{gq}{c}\chi-
P_{5}\right)^{2}\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}-\frac{gq}{c}A_{\mu}\dot{x}^{\mu}\,.$
(25)

The solutions of the fields are, as in (20), given by

$\chi=\frac{g}{4\pi}\tilde{q}\frac{2}{r}\,,\qquad
A_{0}=-\frac{g}{4\pi}\tilde{q}\frac{2}{r}\,.$ (26)

This is exactly the theory found by Alvarez-Jimenez  _et al._ (2018),
describing a non-minimally coupled scalar-vector theory, having a LRL
symmetry. It has its origin in $\mathcal{N}=4$ supersymmetric Yang-Mills,
which similarly enjoys an additional symmetry in the the limit that one of the
particles is much larger than the other Caron-Huot and Henn (2014); Sakata
_et al._ (2017).

Due to the 5D interpretation of this theory, we now have a natural link
between the non-relativistic Coulomb system and this specific relativistic
Maxwell-axion field theory. The relation between both symplectic reductions
implies that the energy-level equivalence between the two theories necessarily
extends to all orders in the PN expansion, with the time reparametrisation as
discussed in Appendix A.

##  V Nordström gravity

The last Hamiltonian in the class (1) that we propose a field theoretic
interpretation for, is the one with function $F(x)=f_{0}$ constant:

$\displaystyle H^{2}=m^{2}c^{4}+p^{2}c^{2}-\left(\frac{1}{r}\right)f_{0}\,.$
(27)

The Hamiltonian of this system is a simple function of the Kepler Hamiltonian,
and as such naturally has a hidden symmetry and closed orbits. Given the
sequence of double and single copy in the previous two sections, however, it
is natural to investigate its interpretation as a zeroth copy relativistic
field theory. This will feature an attractive scalar field, as first proposed
by Nordström in 1912 Nordström (1912).

The scalar ‘zeroth copy’ is an Abelian version of the bi-adjoint scalar. The
constant $f_{0}$ in the potential term now has the dimension of energy squared
times length, so we set $f_{0}=r_{M}m_{s}^{2}c^{4}$, with $r_{M}$ as before
and $m_{s}$ some mass. Lifting the Hamiltonian to 5D yields

$\displaystyle\eta_{AB}P^{A}P^{B}=m_{s}^{2}c^{2}\frac{r_{M}}{r}\,,$ (28)

and hence a position-dependent mass. Legendre transforming this to the
Lagrangian of a geodesic yields

$\mathcal{L}_{p}=\frac{\eta_{AB}\dot{x}^{A}\dot{x}^{B}}{2h}+m_{s}^{2}c^{2}\frac{h}{2}\frac{r_{M}}{r}\,,$
(29)

where we have included the auxiliary Einbein $h$. It can be solved for to
generate

$\mathcal{L}_{p}|_{h}=m_{s}c\sqrt{\phi(r)\eta_{AB}\dot{x}^{A}\dot{x}^{B}}\,,$
(30)

with the scalar field $\phi(r)=\frac{r_{M}}{r}$ minimally coupled and now
dimensionless, and solving the field equation in a vacuum.

This completes the trilogy of double copy related relativistic field theories,
at least from the 5D perspective: the Hamiltonians with constant, linear and
quadratic functions correspond to the natural sequence of spin-$0,1,2$
exchange field theories.

Again, it will be interesting to investigate the dimensional reductions. As
before, the null reduction directly generates a non-relativistic system akin
to the Kepler and Coulomb ones. The space-like reduction, instead, generates a
coupling to a new scalar field profile. After dimensional reduction of (29)
and subsequent elimination of the Einbein $h$, we find

$\mathcal{L}_{p}|_{h}=-m_{s}c\sqrt{\bar{\phi}(r)(-\eta_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu})}\,,$
(31)

with the scalar now given by

$\displaystyle\bar{\phi}(r)=1-\frac{r_{M}}{r}\,.$ (32)

This particular profile for the scalar field then produces exactly closing
time-like curves, and it is the Lagrangian giving rise to $H$ in equation (27)
with proper time parametrisation.

In general, scalar fields generate a perihelion precession that is given to
first post-Newtonian order by

$\Delta\theta=-\frac{1+a_{2}}{6}\Delta\theta_{\rm GR}\,,$ (33)

with $\Delta\theta_{\rm GR}$ the perihelion shift in GR and $a_{2}$ is a
parameter that characterises the coupling of the scalar field (at quadratic
level) to the probe particle Deruelle (2011). How does this and similar
results relate to our system displaying no precession? Whereas $a_{2}=0$ in
Nordström’s final theory (see Sundrum (2003)), $a_{2}=1$ in the only-scalar
limit of Davis and Melville (2023) and Nordström’s first theory (see Nordström
(1912); Ravndal (2004)) and indeed, working out the square-root expansion we
have $a_{2}=-1$ for the scalar (32).

At this point it is also interesting to compare to the classification of
Bertrand spacetimes, as pioneered by Perlick Perlick (1992). These spacetimes
are defined by having the special property of having closed orbits; however,
Bertrand spacetimes are not required to solve the source-free Einstein’s
equations and generically need to be supplemented with non-trivial energy-
momentum tensors. In contrast, in all three cases that we have discussed, the
backgrounds satisfy the source-free bulk equations of motion of the three
relativistic field theories. The latter case of the zeroth copy includes the
coupling to the harmonic scalar field in Nordström (1912). From the particle’s
perspective, this is equivalent to coupling to a conformally flat metric with
overall factor (32); this therefore has to be a Bertrand space-time. We have
checked that the general class of solutions of Perlick (1992) indeed includes
this as the unique conformally flat possibility777Specifically, the mapping is
from the lower signs in equation (13) of Perlick (1992) with
$D\to-2/r_{M}^{2}$, $K\to 4/r_{M}^{4}$ and $G\to 0$ and rescaling the time
$t\to t\;r_{M}/\sqrt{2}$..

##  VI Discussion and conclusion

In this paper, we have studied the possible interactions giving rise to
Keplerian symmetry in field theories. We have presented the unique extension
of the classical Kepler system to a relativistic system with the same so(4)
algebra and the correct special relativistic limit, at least up to 5PN. This
relativistic class of Hamiltonians contains terms naturally interpretable
through a lift to 5 dimensions as spin-0,-1 and -2 interaction. We discussed
how these project to relativistic theories in 4 dimensions, and their relation
to the classical Kepler problem. Moreover, the interactions can be viewed as
zeroth, single and double copy in a classical double copy structure.

The 4D systems with so(4) symmetry corresponding to spin-1 and -2 in 5D are
known in literature to stem from $\mathcal{N}=4$ super Yang-Mills and
$\mathcal{N}=8$ supergravity Alvarez-Jimenez  _et al._ (2018); Caron-Huot and
Zahraee (2019), which can be truncated to the non-minimally coupled scalar-
vector and Einstein-Maxwell-dilaton with $a=\sqrt{3}$ respectively. However,
the spin-0 system in terms of a dilaton field theory, whose resulting
effective metric coincides with a type of Bertrand spacetime Perlick (1992),
appears to have escaped attention so far.

The trajectories in all systems from a 5D perspective can be seen as
reparametrisations of trajectories of a classical Kepler problem. In a similar
vein, all one-body dynamics in the particular relativistic systems discussed
can be linked to classical dynamics. Though we have focused on the particular
case of relativistic 1-center problems with classical symmetry, all the above
constructions hold with any harmonic background potential $\phi$: the
Einstein, Maxwell and scalar field equations are still solved after the 5D
lift. The reduced systems are then through time reparametrisation related to
their classical counterparts.

For example, a multi-center background in Einstein-Maxwell-dilaton theory with
$a=\sqrt{3}$ and extremal objects can be constructed, which is orbited by an
anti-extremal test particle. This is the space-like reduction of a Bargmann
spacetime of which the null-reduction is the classical Newtonian multi-center
system. The relevant harmonic function for $n$ centers (all extremally
charged) is given by
$-\phi(\vec{r})=\frac{r_{M_{1}}}{\absolutevalue{\vec{r}-\vec{r}_{1}}}+\dots+\frac{r_{M_{n}}}{\absolutevalue{\vec{r}-\vec{r}_{n}}}$
Gibbons and Wells (1993), and the Hamiltonian for an anti-extremal test
particle of mass $m$ satisfies

$H^{2}=m^{2}c^{4}+p^{2}c^{2}+\phi(\vec{r})\left(mc^{2}+H\right)^{2}\,.$ (34)

This connection immediately implies these solutions possess the same
integrability properties as their equivalents in non-relativistic mechanics,
that is, the two-center case is integrable, while the systems with a higher
number of centers all display chaotic scattering Bolotin (1984); Knauf and
Taimanov (2004); Knauf (2002), as discussed for more values of the dilaton
coupling in Kluitenberg  _et al._ (2022).

Some further questions present themselves. Firstly, all mentioned so(4)
preserving systems are test-particle limits. It would be interesting to
consider the two-body problem beyond the first order in mass ratio or
relativistic expansions in these theories, as done for example by Parra-
Martinez  _et al._ (2020), who still found no precession of the orbits at
third Post-Minkowskian order for extremal black holes in $\mathcal{N}=8$
supergravity (but did not confirm or exclude persistence of so(4) symmetry).
Modern scattering amplitude methods888For instance, it would be interesting to
investigate whether the effective one-body approach in Kerr-Schild formulation
Ceresole  _et al._ (2023) allows to capture comparable mass ratio effects in
the 5D Kerr-Schild potential. seem to be well-suited to this task, especially
in the light of the double copy, which might allow one to directly export
results from $\mathcal{N}=4$ super Yang-Mills to $\mathcal{N}=8$ supergravity.

Secondly, is it possible to extend the idea of relating relativistic systems
to geodesics in higher dimensional spaces to multi-body systems? This would
need to account for the multi-worldline nature of such systems, possibly by
including more time-like dimensions in the higher dimensional system. It would
be intriguing to see if this alternative perspective could lead to new or
simplifying insights into relativistic dynamics, and whether interesting
connections can be made to classical systems.

## Acknowledgements

We are grateful to Cliff Burgess, Joaquim Gomis, Johannes Lahnsteiner, Scott
Melville, Fernando Quevedo and Ziqi Yan for stimulating discussions. D.N. is
supported by the Fundamentals of the Universe research program within the
University of Groningen.

##  Appendix A Relativistic dynamics as reparametrised classical trajectories

In this more mathematically oriented appendix we provide further details on
the time reparametrisation that relates the (non-)relativistic Hamiltonians,
and make this connection rigorous in terms of the Marsden-Weinstein reduction
of the higher-dimensional phase space.

To this end, we will start from a slight generalisation of the above, with
Hamiltonians $H_{2}(q,p)$ (introduced in Neeling  _et al._ (2024)) solving an
equation of the form

$-f(H_{2}(q,p))=\frac{p^{2}}{2C_{u}}+\frac{g(H_{2}(q,p))}{C_{u}}\Phi(q)\,,$
(35)

with $f$ and $g$ suitable smooth functions $\mathbb{R}\to\mathbb{R}$ and
$C_{u}$ a nonzero constant. We will show that these are similarly related to
classical Hamiltonians using Marsden-Weinstein reduction, employing a symmetry
of a Hamiltonian system to reduce its dimension by an even number. This
procedure guarantees that the reduced space remains symplectic, since the
symplectic form is given by the restriction of the initial symplectic form to
the new space Marsden and Weinstein (1974).

######  Assumptions 1.

  1. 1.

Let $\Phi:\mathbb{R}^{d}\backslash\Delta\to\mathbb{R}$ be a smooth function,
$\Delta$ a discrete set of isolated points and
$\frac{\partial{\Phi(q)}}{\partial{q}}\neq 0$ for all
$q\in\mathbb{R}^{d}\backslash\Delta$.

  2. 2.

Let $H_{1}:M\to\mathbb{R}$ be a Hamiltonian on
$M=T^{*}(\mathbb{R}^{d}\backslash\Delta)$ with coordinates
$q\in\mathbb{R}^{d}$ and $p\in\mathbb{R}^{d}$, defined by

$H_{1}(q,p)=\frac{p^{2}}{2f(C_{v})}+\frac{g(C_{v})}{f(C_{v})}\Phi(q)\,,$ (36)

for two smooth functions $f,g:\mathbb{R}\to\mathbb{R}$.

  3. 3.

Let $C_{u},C_{v}\in\mathbb{R}$ be nonzero constants such that $f(C_{v})\neq
0$, $g(C_{v})\neq 0$ and

$g^{\prime}(C_{v})\Phi(q)+f^{\prime}(C_{v})C_{u}\neq 0\,,\ \text{for all
}q\in\mathbb{R}^{d}\backslash\Delta.$ (37)

######  Proposition 1.

Given Assumptions 1(1–3), if the Hamiltonian $H_{2}(q,p)$ implicitly defined
by (35) exists for $H_{2}(q,p)=C_{v}$, the trajectories on an energy level
$H_{1}=-C_{u}$ of the first Hamiltonian and $H_{2}=C_{v}$ of the second are in
one-to-one correspondence, up to the time reparametrisation

$\frac{\differential{t_{1}}}{\differential{t_{2}}}=\frac{f(C_{v})}{g^{\prime}(C_{v})\Phi(q)+f^{\prime}(C_{v})C_{u}}\,.$
(38)

###### Proof.

Consider a manifold
$\bar{M}=T^{*}(\mathbb{R}^{2}\crossproduct(\mathbb{R}^{d}\backslash\Delta))$,
with canonical coordinates $(u,v,q;P_{u},P_{v},p)$ and Hamiltonian
$\mathcal{H}:\bar{M}\to\mathbb{R}$ given by

$\mathcal{H}(u,v,q;P_{u},P_{v},p)=g(P_{v})\Phi(q)+f(P_{v})P_{u}+\frac{p^{2}}{2}\,.$
(39)

The proposition is an application of Marsden-Weinstein reduction restricted to
the level set $\mathcal{H}^{-1}(0)$. Since $u$ and $v$ are cyclic coordinates
by construction and the corresponding actions (by translation) are both free
and proper, we can apply the Marsden-Weinstein theorem Marsden and Weinstein
(1974) to symplectically reduce over either of them. We can construct the two
possible reductions using the momentum maps
$\mu_{\star}:=(\mathcal{H},P_{\star}):\bar{M}\to\mathbb{R}^{2}$, where $\star$
denotes either $u$ or $v$, giving two, in principle different base spaces
$B_{\star}=\mu_{\star}^{-1}(0,C_{\star})/\mathbb{R}_{\star}$ and corresponding
projections $\pi_{\star}:\mu_{\star}^{-1}(0,C_{\star})\to B_{\star}$. Here
$(0,C_{\star})$ are assumed to be regular values of $\mu_{\star}$, for now.
Note that the base spaces are odd-dimensional, as the value of $\mathcal{H}$
is fixed, so that the established relation is between energy levels of $H_{1}$
and $H_{2}$.

Consider the regularity requirements $\differential\mathcal{H}\neq 0$ and
$\differential P_{u},\differential P_{v}\neq 0$. The latter are trivially
satisfied, while the former gives

$\displaystyle\differential\mathcal{H}(u,v,q;P_{u},P_{v},p)=\frac{\partial{\Phi}}{\partial{q}}g(P_{v})\differential
q+f(P_{v})\differential P_{u}$
$\displaystyle\qquad+(g^{\prime}(P_{v})\Phi(q)+f^{\prime}(P_{v})P_{u})\differential
P_{v}+p\,\differential p\,.$ (40)

By the Assumptions 1(1–3) this is never vanishing on the domain of
consideration, nor are $\differential\mathcal{H}$ and $\differential
P_{u},\differential P_{v}$ linearly dependent. This shows that the values
$(0,C_{\star})$ are indeed regular.

Let us now reduce the system over the cyclic coordinates $(u.v)$ to obtain new
Hamiltonian functions that are exclusively dependent on the classical
variables $(q,p)$. When we reduce by $v$ we have

$g(C_{v})\Phi+f(C_{v})P_{u}+\frac{p^{2}}{2}=0\,,$ (41)

so that reparametrisation $\lambda\to f(C_{v})\lambda$ results in

$-P_{u}=\frac{p^{2}}{2f(C_{v})}+\frac{g(C_{v})}{f(C_{v})}\Phi(q)\,.$ (42)

This means that we effectively have a Hamiltonian $-P_{u}=H_{1}(q,p)$, which
moreover has a time parameter given by $t_{1}=u=f(C_{v})\lambda$ associated to
it, as is apparent from Hamilton’s equation
$\frac{\differential{u}}{\differential{\lambda}}=\frac{\partial{\mathcal{H}}}{\partial{P_{u}}}$.

If, instead, we reduce phase space $\bar{M}$ over $u$, we end up with

$-f(P_{v})=\frac{p^{2}}{2C_{u}}+\frac{g(P_{v})}{C_{u}}\Phi(q)\,.$ (43)

By assumption, this is solved by $H_{2}(q,p)=C_{v}$. Hamilton’s equation

$\frac{\differential{v}}{\differential{\lambda}}=g^{\prime}(P_{v})\Phi+f^{\prime}(P_{v})C_{u}\,,$
(44)

then implies the associated time $t_{2}$ is given by $\differential
t_{2}=\differential
v=(g^{\prime}(P_{v})\Phi+f^{\prime}(P_{v})C_{u})\differential\lambda$. For
energy levels $H_{1}=-C_{u}$ of the first Hamiltonian and $H_{2}=C_{v}$ of the
second, this means the reparametrisation from the first Hamiltonian to the
second is given by

$\frac{\differential{t_{1}}}{\differential{t_{2}}}=\frac{f(C_{v})}{g^{\prime}(C_{v})\Phi(q)+f^{\prime}(C_{v})C_{u}}\,,$
(45)

which concludes the proof of the relation between the two. ∎

For concreteness we now apply this procedure to the relativistic class of
Hamiltonians

$\displaystyle H^{2}=m^{2}+p^{2}-\left(\frac{1}{r}\right)F\left(m+H\right)\,$
(46)

dropping factors of $c^{2}$ for legibility. As discussed in the main text,
under the replacements (9) these lift up to the level set
$\mathcal{H}^{-1}(0)$ of

$\mathcal{H}=P_{5}^{2}-P_{0}^{2}+p^{2}-\left(\frac{1}{r}\right)F\left(P_{5}+P_{0}\right)\,.$
(47)

Hamilton’s equation tells us that the relation between the time-parameter
$\lambda$ of this new Hamiltonian is related to the old time parameter $t$ by

$\frac{\differential{t}}{\differential{\lambda}}=-2P_{0}-\left(\frac{1}{r}\right)F^{\prime}\left(P_{5}+P_{0}\right)\,,$
(48)

where $t$ is conjugate to $P_{0}$ (and hence $H$).

The canonical transformation to null coordinates

$\displaystyle(x_{+},x_{-})$
$\displaystyle=\tfrac{1}{\sqrt{2}}\left(x_{5}+t,x_{5}-t\right)\,,$ (49)
$\displaystyle(P_{+},P_{-})$
$\displaystyle=\tfrac{1}{\sqrt{2}}(P_{5}+P_{0},P_{5}-P_{0})\,,$ (50)

results in

$-2P_{-}P_{+}=p^{2}-\left(\frac{1}{r}\right)F\left(\sqrt{2}P_{+}\right)\,$
(51)

on level set $\mathcal{H}^{-1}(0)$. Assuming $P_{+}\neq 0$, a time
reparametrisation to $\tilde{\lambda}=2P_{+}\lambda$ reduces it further to

$-P_{-}=\frac{p^{2}}{2P_{+}}-\left(\frac{1}{r}\right)\frac{1}{2P_{+}}F\left(\sqrt{2}P_{+}\right)\,,$
(52)

so the alternative symplectic reduction over $x_{+}$ setting $P_{+}=m_{+}$
results in $-P_{-}=:H_{-}$ defining a Kepler Hamiltonian reading

$H_{-}=\frac{p^{2}}{2m_{+}}-\left(\frac{1}{r}\right)\frac{1}{2m_{+}}F\left(\sqrt{2}m_{+}\right)\,.$
(53)

The relation between both time parametrisation therefore reads

$\frac{\differential{x_{-}}}{\differential{\lambda}}=\frac{\partial{\mathcal{H}}}{\partial{P_{-}}}=2P_{+}\,,$
(54)

where $x_{-}$ is conjugate to $H_{-}$.

Crucially, the Marsden-Weinstein result guarantees that both reductions
preserve the symplectic structure. From the above, it directly follows that
the time reparametrisation that links trajectories on energy levels
$H_{-}=\frac{1}{\sqrt{2}}(H_{0}-m)$ of the Kepler Hamiltonian to those on
energy levels $H=H_{0}$ of the original is given by

$\frac{\differential{x_{-}}}{\differential{t}}=\frac{\sqrt{2}(H_{0}+m)}{-2H_{0}-\left(\frac{1}{r}\right)F^{\prime}\left(m+H_{0}\right)}\,.$
(55)

This transformation maps relativistic systems that violate Kepler’s third law
to Kepler ones that satisfy it.

Note that only when $F(x)$ is homogeneously quadratic, does each energy level
map to the same Kepler background, such that the potential coefficient in the
Hamiltonian becomes linear in the mass of the probe. For all other functions,
the effective Newton’s constant becomes probe-mass dependent and hence would
violate the equivalence between gravitational and inertial mass.

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