section[1.2em]1em0.5pc subsection[4em]2em1pc

Kerr-Schild Double Copy of the Coulomb Solution in Three Dimensions

Gökhan Alkaç$\,{}^{a}$, Mehmet Kemal Gümüş$\,{}^{b}$ and Mehmet Ali
Olpak$\,{}^{c}$

aPhysics Engineering Department, Faculty of Engineering,
Hacettepe University, 06800, Ankara, Turkey

bDepartment of Physics, Faculty of Arts and Sciences,
Middle East Technical University, 06800, Ankara, Turkey

cDepartment of Electrical and Electronics Engineering, Faculty of Engineering,
University of Turkish Aeronautical Association, 06790, Ankara, Turkey
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

While the Kerr-Schild double copy of the Coulomb solution in dimensions higher
than three is the Schwarzschild black hole, it is known that it should be a
nonvacuum solution in three dimensions. We show that the static black hole
solution of Einstein-Maxwell theory (with one ghost sign in the action) is the
double copy with the correct Newtonian limit, which provides an improvement
over the previous construction with a free scalar field that does not vanish
at infinity. By considering a negative cosmological constant, we also study
the charged Bañados-Teitelboim-Zanelli black hole and find that the single
copy gauge field is the Coulomb solution modified by a term which describes an
electric field linearly increasing with the radial coordinate, which is the
usual behavior of the Schwarzschild-AdS black hole in higher dimensions when
written around a flat background metric.

###### Contents

  1. 1 Introduction
  2. 2 The Coulomb Solution from the Free Scalar
    1. 2.1 Solution with the correct Newtonian potential
    2. 2.2 Gauge theory single copy
  3. 3 Scalar-Vector Duality and Its Consequences
  4. 4 Einstein-Maxwell Theory in Three Dimensions
    1. 4.1 The most general static solution of Kerr-Schild form
    2. 4.2 Solutions with the correct Newtonian potential ($\Lambda=0$)
    3. 4.3 The charged Bañados-Teitelboim-Zanelli (BTZ) black hole ($\Lambda<0$)
  5. 5 Conclusions
  6. A Scalar-Vector Duality as Described in [80]

##  1 Introduction

After its discovery in the context of scattering amplitudes [1, 2], the double
copy idea has found applications in the classical domain as perturbative and
exact relations between gravity and gauge theories. While perturbative
spacetimes can be obtained by squaring the numerator of some diagrams in Yang-
Mills theory [40, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38,
39], it is possible to establish a map between exact solutions that take a
much simpler form, where certain classical solutions of general relativity
(GR) are in correspondence with solutions of Maxwell’s theory [41, 42, 43, 44,
45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63,
64]. Among different versions of the classical double copy that have been
discovered so far, the Kerr-Schild (KS) double copy [40] has the advantage of
admitting a formulation in arbitrary dimensions. For metrics of the KS form
[65], the Ricci tensor with mixed indices is linear in the perturbation for a
flat [66] and curved [67, 68, 69] background metric. Due to this crucial
property, a map to solutions of Maxwell’s theory defined on the background
spacetime, which has a linear nature, can be established. Alternatively, by
writing the tensorial equations of GR and Maxwell’s theory in $d=4$ in terms
of $2$-component spinors, one can construct a Weyl spinor characterizing the
spacetime from two electromagnetic field strength spinors up to a scalar
field, leading to the Weyl double copy [58]. Also, a class of solutions of GR
and self-dual solutions of Maxwell’s theory can be related through Newman-
Penrose formalism [70], giving rise to the Newman-Penrose map [60]. Although
the last two are limited to $d=4$ in their present formulations, the fact that
they were shown to originate from twistor theory [71, 72, 73] is a sign of a
much richer structure than previously thought, which already found a practical
utility by providing a way to fix the scalar field in the Weyl double copy.

Having a new theoretical tool to investigate the relation between two
seemingly different theories, it is important to consider situations where it
might break in order to understand limitations if there are any. A possible
route in this direction is to work in three dimensions (3d), for which the KS
formulation is available, and this was pursued in [63, 64]. Due to the lack of
degrees of freedom and a Newtonian limit in 3d GR, it is not obvious, at first
sight, how the procedure works.111See [74, 75] for the study of 3d amplitudes
where a degree of freedom is introduced in the gravity side by adding a Chern-
Simons term in the action. As a result, one obtains the amplitudes in
topologically massive gravity as the double copy of the amplitudes of
topologically massive electrodynamics. For the Coulomb solution, which is the
simplest nontrivial solution in the gauge theory side, the gauge boson degrees
of freedom is mapped to the graviton and the KS scalar characterizing the
spacetime metric is directly linked to the Newtonian potential as given in
(2.13). However, without matter coupling, this cannot be realized since the
Newtonian potential vanishes identically. When the problem was tackled by
coupling a free scalar field, a hairy black hole with the desired properties
can be obtained if the scalar is a ghost, which is equivalent to coupling a
spacelike fluid [63]. In order to support the black hole solution, the scalar
should be linear in the azimuthal angle, and therefore, does not vanish at
infinity as suggested by the no-hair theorem. In the linearized theory, the
ghost sign can be removed by a certain generalized gauge transformation;
however, it is still somewhat unsatisfactory to not have a reasonable behavior
of the matter field at infinity. When investigated further in [64], it was
proposed to take the Einstein-Hilbert term with a ghost sign, which does not
introduce a dynamical ghost in the theory and removes the need for a
generalized gauge transformation to get rid of the ghost in the linearized
theory.

In this paper, we aim to present an alternative for the matter coupling with a
better behavior at infinity, which, as we will see, also provides a beautiful
connection to a well-known solution of 3d black hole physics. In Sec. 2, we
will make a review of the construction of [63] by emphasizing the points that
will be relevant to our later discussion. In Sec. 3, considering the on-shell
duality of a scalar and a gauge vector together with the KS ansatz for the
metric, we will find that the same type of solution can be obtained in
Einstein-Maxwell theory. In Sec. 4, we will obtain the most general static
solution of the KS form by introducing a cosmological constant. When the
cosmological constant is zero, we will show that a charged black hole solution
with the correct Newtonian limit, which also gives rise to the Coulomb
solution as its single copy, can be obtained when a ghost sign is used in the
action. The electric field corresponding to the gauge field in the gravity
side vanishes at infinity, and therefore, provides the promised improvement.
For a negative cosmological constant, the charged Bañados-Teitelboim-Zanelli
black hole [76] follows from the most general solution without taking any
ghost sign in the action. We end this section by studying the gauge theory
single copy of the solution. Finally, we present our conclusions in Sec. 5.

##  2 The Coulomb Solution from the Free Scalar

###  2.1 Solution with the correct Newtonian potential

In [63], the Coulomb solution was obtained as the single copy of the static
black hole solution of GR coupled to a free scalar with the following action:

$S=\int\mathop{}\\!\mathrm{d}^{3}x\sqrt{-g}\left[\frac{\zeta_{1}}{\kappa^{2}}R-\frac{\zeta_{2}}{2}\left(\partial\varphi\right)^{2}\right],\qquad\kappa^{2}=8\pi
G$ (2.1)

where $\zeta_{i}=\pm 1$ ($i=1,2$) control the sign of the kinetic terms and
take a negative value for a ghost graviton or a dilaton [64]. The field
equations which follow from the action (2.1) are

$\displaystyle R_{\mu\nu}$
$\displaystyle=\zeta\frac{\kappa^{2}}{2}\partial_{\mu}\varphi\partial_{\nu}\varphi,$
(2.2)
$\displaystyle\partial_{\mu}(\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}\varphi)$
$\displaystyle=0,$ (2.3)

where $\zeta=\zeta_{1}\zeta_{2}$. Let us consider the following static KS
metric around the Minkowski space

$g_{\mu\nu}=\eta_{\mu\nu}+\phi(r)\,k_{\mu}k_{\nu}\,,$ (2.4)

where the vector $k_{\mu}$ is null and geodesic with respect to both the
background metric $\eta_{\mu\nu}$ and the full metric $g_{\mu\nu}$ (see chap.
32 of [77] for a detailed discussion of important properties). Writing the
background line element in polar coordinates

$\eta_{\mu\nu}\mathop{}\\!\mathrm{d}x^{\mu}\mathop{}\\!\mathrm{d}x^{\nu}=-\mathop{}\\!\mathrm{d}t^{2}+\mathop{}\\!\mathrm{d}r^{2}+r^{2}\mathop{}\\!\mathrm{d}\theta^{2},$
(2.5)

the vector $k_{\mu}$ can be written as follows

$-k_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}=\mathop{}\\!\mathrm{d}t+\mathop{}\\!\mathrm{d}r,$
(2.6)

and the line element in the KS coordinates becomes

$\displaystyle\mathop{}\\!\mathrm{d}s^{2}$
$\displaystyle=\eta_{\mu\nu}\mathop{}\\!\mathrm{d}x^{\mu}\mathop{}\\!\mathrm{d}x^{\nu}+\phi(r)(k_{\mu}\mathop{}\\!\mathrm{d}x^{\mu})^{2}$
$\displaystyle=-\left[1-\phi(r)\right]\mathop{}\\!\mathrm{d}t^{2}+\left[1+\phi(r)\right]\mathop{}\\!\mathrm{d}r^{2}+2\phi(r)\mathop{}\\!\mathrm{d}t\mathop{}\\!\mathrm{d}r+r^{2}\mathop{}\\!\mathrm{d}\theta^{2}.$
(2.7)

If one assumes222In [63], the authors directly use the KS scalar yielding the
correct Newtonian potential and conclude that the solution should be sourced
by a free scalar. Here, we give the derivation in a way that will be useful in
our later discussion.

$\partial_{\mu}\varphi=(0,0,c),\qquad\qquad c=\text{constant},$ (2.8)

the equation for the scalar field (2.3) is satisfied independent of the KS
scalar $\phi$ as follows:

$\partial_{\mu}(\sqrt{-g}\,g^{\mu\nu}\partial_{\nu}\varphi)=\partial_{\mu}(\sqrt{-\eta}\,\eta^{\mu\nu}\partial_{\nu}\varphi)=0,$
(2.9)

where we have used $\det g=\det\eta$, $g^{\theta\theta}=\eta^{\theta\theta}$
and $g^{t\theta}=g^{r\theta}=0$.

With this at hand, one can now check the gravity equations (2.2). The
independent nonzero components of the Ricci tensor read

$\displaystyle R_{tt}$
$\displaystyle=\frac{\left[\phi(r)-1\right]\left[r\,\phi^{\prime\prime}(r)+\phi^{\prime}(r)\right]}{2r},$
$\displaystyle R_{tr}$
$\displaystyle=\frac{\phi(r)\left[r\,\phi^{\prime\prime}(r)+\phi^{\prime}(r)\right]}{2r},$
$\displaystyle R_{rr}$
$\displaystyle=\frac{\left[\phi(r)+1\right]\left[r\,\phi^{\prime\prime}(r)+\phi^{\prime}(r)\right]}{2r},$
$\displaystyle R_{\theta\theta}$ $\displaystyle=r\,\phi^{\prime}(r),$ (2.10)

and the only nonzero component of the right-hand-side of the equations is

$\text{(RHS)}_{\theta\theta}=4\pi\zeta Gc^{2},$ (2.11)

From the $\theta\theta$ component, one finds the solution for the KS scalar as

$\phi=b+4\pi\zeta Gc^{2}\log(r),\qquad b=\text{constant},$ (2.12)

which also satisfies the remaining components. The constant $c$ and the
parameter $\zeta$ can be fixed by considering the Newtonian limit. The
Newtonian potential is given by

$\Phi=-\frac{1}{2}(1+g_{00})=-\frac{\phi}{2},$ (2.13)

and in order to mimic the Newtonian gravity

$\vec{g}=-\vec{\nabla}\Phi=-\frac{GM}{r}\hat{r},$ (2.14)

the KS scalar should be in the following form:

$\phi=-2GM\log(r)+\text{constant}.$ (2.15)

Therefore, we need to fix the parameters as follows:

$c=\sqrt{\frac{M}{2\pi}},\qquad\qquad\zeta=-1,$ (2.16)

with which, the KS scalar becomes

$\phi=-2GM\log(r)+b.$ (2.17)

Since $\zeta=-1$, one should choose the “wrong sign” for one of the kinetic
terms in the action (2.1). While the scalar was chosen to be a ghost in [63],
introducing the Einstein-Hilbert (EH) term with the negative sign has the
advantage that it does not propagate any physical degree of freedom [64].

In order to fix the integration constant $b$, we write the metric (2.4) in the
Boyer-Lindsquit (BL) coordinates by the following coordinate transformation:

$\mathop{}\\!\mathrm{d}t\rightarrow\mathop{}\\!\mathrm{d}t+\frac{\phi(r)}{1-\phi(r)}\,\mathop{}\\!\mathrm{d}r,$
(2.18)

which leads to the line element

$\mathop{}\\!\mathrm{d}s^{2}=-f(r)\mathop{}\\!\mathrm{d}t^{2}+\frac{\mathop{}\\!\mathrm{d}r^{2}}{f(r)}+r^{2}\mathop{}\\!\mathrm{d}\theta^{2},\qquad
f(r)=1-\phi(r),$ (2.19)

for a generic KS scalar. In our case, the metric function becomes

$f(r)=1-b+2GM\log(r).$ (2.20)

In order to recover the Minkowski spacetime when the black hole mass vanishes
($M=0$), one should take $b=0$. Therefore, the KS scalar and the metric
function are given by

$\displaystyle\phi(r)$ $\displaystyle=-2GM\log(r),$ (2.21) $\displaystyle
f(r)$ $\displaystyle=1+2GM\log(r).$ (2.22)

We refer the reader to [63] for an analysis of the motion of massive particles
where the authors show that stable orbits exist for a certain range of
parameters. In Sec. 4, we will show that it is true for the solution of
Einstein-Maxwell theory.

###  2.2 Gauge theory single copy

The gauge theory single copy for a generic matter coupling can be obtained by
considering the trace-reversed gravity equations

$R^{\mu}_{\ \nu}=\frac{\kappa{{}^{2}}}{2}\left[T^{\mu}_{\ \nu}-\delta^{\mu}_{\
\nu}T\right],\qquad\qquad T=T^{\mu}_{\ \mu},$ (2.23)

where $T_{\mu\nu}$ is the energy-momentum tensor. For a KS metric (2.4), the
Ricci tensor with mixed indices reads

$R^{\mu}_{\
\nu}=\frac{1}{2}\left[\partial^{\alpha}\partial^{\mu}(\phi\,k_{\nu}k_{\alpha})+\partial^{\alpha}\partial_{\nu}(\phi\,k^{\mu}k_{\alpha})-\partial^{\alpha}\partial_{\alpha}(\phi\,k^{\mu}k_{\nu})\right],$
(2.24)

which is linear in the perturbation. If $k^{0}=+1$ and one identifies
$A_{\mu}=\phi\,k_{\mu}$, the $\mu 0$ component can be written as

$R^{\mu}_{\ 0}=\frac{1}{2}\partial_{\nu}F^{\nu\mu},$ (2.25)

where $F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$ is the field strength tensor of
the gauge field $A_{\mu}$. Therefore, the $\mu 0$ component of the gravity
equations can be mapped to Maxwell’s equations as follows:

$\partial_{\nu}F^{\nu\mu}=gJ^{\mu},$ (2.26)

where the source is given by

$J^{\mu}=2\left[T^{\mu}_{\ 0}-\delta^{\mu}_{\ 0}T\right],$ (2.27)

and the gauge coupling is obtained by $\frac{\kappa^{2}}{2}\rightarrow g$
[40].

Application of this procedure to our solution gives the single copy gauge
field as333We write the solution with a different normalization than [63, 64]
to simplify the solution of Einstein-Maxwell theory that we will give in Sec.
4. The Maxwell action is taken as (3.6) and we formulate the scalar-vector
duality in Sec. 3 accordingly.

$A_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}=\phi
k_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}=Q\log
r\,(\mathop{}\\!\mathrm{d}t+\mathop{}\\!\mathrm{d}r),$ (2.28)

where we have made the replacement $2GM\rightarrow Q$. This is just the
Coulomb solution in a gauge where $A_{\mu}A^{\mu}=0$, and the source is

$J^{\mu}\partial_{\mu}=Q\,\delta^{2}(\vec{r})\,\partial_{t},$ (2.29)

which corresponds to a charged particle in the flat spacetime.

Although we achieved the Coulomb solution as the single copy, the construction
has the undesired feature that the scalar field in the gravity side is linear
in the azimuthal angle. This is not unexpected due to the existence of the
scalar hair (see Appendix of [64] for a detailed discussion of no-hair theorem
for free scalar fields). One way to obtain a matter configuration which is
well behaved at infinity is to consider the coupling of a gauge vector since
it will yield a global charge, i.e., the electric charge, and therefore,
produces no hair. In $d\geq 4$, this is just the Reissner-Nordström black hole
solution of Einstein-Maxwell theory, whose metric can be also written in the
KS form around Minkowski background.

##  3 Scalar-Vector Duality and Its Consequences

In this section, we will present the duality between a free scalar and a gauge
vector in three dimensions by following Sec. 7.8 of [78], and then, discuss
its consequences for the classical double copy. In $d$-dimensions, the number
of on-shell degrees of freedom of a $p$-form gauge field is
$C(d-2,p)=\frac{(d-2)!}{p!(d-p-2)!}$. Due to the identity
$C(d-2,p)=C(d-2,d-p-2)$, a $p$-form and a $(d-p-2)$-form in $d$-dimensions
have the same number of degrees of freedom. In $d=3$, this implies that a
scalar ($p=0$) and a vector gauge field ($p=1$) have the same number of
degrees of freedom, which is one. Indeed, one can also show that the free
field equations are equivalent and the solutions are in one-to-one
correspondence. In order to see that, let us consider the following flat space
action

$S=\int\mathop{}\\!\mathrm{d}^{3}x\sqrt{-\eta}\left[\frac{1}{8\pi}f_{\mu\nu}f^{\mu\nu}+\frac{1}{2\sqrt{2\pi}}\epsilon^{\mu\nu\rho}f_{\mu\nu}\partial_{\rho}\varphi\right],$
(3.1)

where $\epsilon^{\mu\nu\rho}=\frac{1}{\sqrt{-\eta}}{\varepsilon^{\mu\nu\rho}}$
is the Levi-Civita tensor and we take the Minkowski spacetime in polar
coordinates (2.5) for later convenience. The equation for $\varphi$ gives

$\epsilon^{\mu\nu\rho}\partial_{\mu}f_{\nu\rho}=0,$ (3.2)

which implies $f_{\mu\nu}=2\partial_{[\mu}a_{\nu]}$, i.e., $f_{\mu\nu}$ is the
field strength tensor of a gauge field $a_{\mu}$. Checking the equation for
$f_{\mu\nu}$ gives how it is related to the scalar $\varphi$ as follows:

$f_{\mu\nu}=-\sqrt{2\pi}\,\epsilon_{\mu\nu\rho}\partial^{\rho}\varphi\implies\partial_{\mu}\varphi=\frac{1}{2\sqrt{2\pi}}\,\epsilon_{\mu\nu\rho}f^{\nu\rho}.$
(3.3)

Inserting the expression for $f_{\mu\nu}$ into the action (3.1) yields the
action for a free scalar

$S_{\text{scalar}}=\int\mathop{}\\!\mathrm{d}^{3}x\sqrt{-\eta}\,\left[\frac{1}{2}\left(\partial\varphi\right)^{2}\right],$
(3.4)

with the field equation

$\partial_{\mu}(\sqrt{-\eta}\,\eta^{\mu\nu}\partial_{\nu}\varphi)=0.$ (3.5)

In the same way, one can also eliminate $\varphi$ from the action (3.1) by
using the expression in (3.3), which yields the action for a free gauge field

$S_{\text{vector}}=\int\mathop{}\\!\mathrm{d}^{3}x\sqrt{-\eta}\,\left[-\frac{1}{8\pi}f_{\mu\nu}f^{\mu\nu}\right],$
(3.6)

where the field equation is given by

$\partial_{\nu}(\sqrt{-\eta}\,\eta^{\nu\alpha}\eta^{\mu\beta}f_{\alpha\beta})=0.$
(3.7)

We are now in a position to discuss the implications of the duality. As we
have seen in (2.9), the scalar field configuration given in (2.8) is a
solution when the spacetime is curved and endowed with the metric (2.4), or
equivalently, flat and endowed with the Minkowski metric in polar coordinates
(2.5). Our analysis shows that the actions for the free scalar (3.4) and the
free vector (3.6) are equivalent and the solutions to free field equations
(3.4,3.6) are in one-to-one correspondence where the relation between the
solutions is given in (3.3). For the solution of the scalar field given in
(2.8), this implies that, for the corresponding vector solution, the only
nonzero component of the field strength tensor is

$f_{tr}\propto\frac{1}{r},$ (3.8)

which is the electric part. Introducing the electric charge as the
proportionality constant, the gauge field and the field strength tensor can be
written as

$\displaystyle a_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}$
$\displaystyle=q\log(r)\mathop{}\\!\mathrm{d}t,$
$\displaystyle\frac{1}{2}\,f_{\mu\nu}\,\mathop{}\\!\mathrm{d}x^{\mu}\wedge\mathop{}\\!\mathrm{d}x^{\nu}$
$\displaystyle=\frac{q}{r}\,\mathop{}\\!\mathrm{d}r\wedge\mathop{}\\!\mathrm{d}t.$
(3.9)

Similar to the scalar case, this field configuration is also a solution when
the spacetime is curved and endowed with the metric (2.4) as follows:

$\partial_{\nu}(\sqrt{-\eta}\,\eta^{\nu\alpha}\eta^{\mu\beta}f_{\alpha\beta})=\partial_{\nu}(\sqrt{-g}\,g^{\nu\alpha}g^{\mu\beta}f_{\alpha\beta})=0.$
(3.10)

As emphasized in [78], although the equivalence is true for the simplest
kinetic actions, it is not guaranteed to hold in a more general setup. We have
proven that the scalar solution in curved spacetime with a KS metric implies
that the electric field configuration (3.9) is also a solution in such a
spacetime. However, one should keep in mind that this statement is independent
of the KS scalar $\phi(r)$. Therefore, when coupled to gravity, the matter
equations will definitely be satisfied if the metric can be put in the KS form
(2.4); however, the KS scalars in these two cases might differ. Indeed, one
immediately sees that the static black hole solution obtained by coupling the
vector field to gravity should be a charged black hole solution. In the next
section, we will study the black hole solution with this new matter coupling.

##  4 Einstein-Maxwell Theory in Three Dimensions

###  4.1 The most general static solution of Kerr-Schild form

Motivated by the results of the previous section, we consider Einstein-Maxwell
theory with a cosmological constant described by the following action:

$S=\int\mathop{}\\!\mathrm{d}^{3}x\sqrt{-g}\left[\frac{\zeta_{1}}{\kappa^{2}}\left(R-2\Lambda\right)-\frac{\zeta_{2}}{8\pi}f_{\mu\nu}f^{\mu\nu}\right],\qquad\kappa^{2}=8\pi
G,$ (4.1)

where, similar to the scalar case, $\zeta_{i}=\pm 1$ ($i=1,2$) control the
sign of the kinetic terms and take a negative value for a ghost graviton or a
vector. The field equations arising from the action (4.1) are given by

$\displaystyle R_{\mu\nu}-2\Lambda g_{\mu\nu}$
$\displaystyle=\zeta\frac{\kappa^{2}}{8\pi}\left(2f_{\mu}^{\
\alpha}f_{\nu\alpha}-g_{\mu\nu}f_{\alpha\beta}f^{\alpha\beta}\right),$ (4.2)
$\displaystyle\partial_{\nu}(\sqrt{-g}\,f^{\nu\mu})$ $\displaystyle=0,$ (4.3)

where $\zeta=\zeta_{1}\zeta_{2}$. As discussed in the previous section, for a
static metric in the KS form (2.4), the solution for the vector field is given
in (3.9). With this at hand, one can solve the gravitational field equations
(4.2). Introducing the cosmological constant modifies the left-hand side as
follows:

$\displaystyle\text{(LHS)}_{tt}$
$\displaystyle=\frac{\left[\phi(r)-1\right]\left[r\,\phi^{\prime\prime}(r)+\phi^{\prime}(r)-4\Lambda
r\right]}{2r},$ $\displaystyle\text{(LHS)}_{tr}$
$\displaystyle=\frac{\phi(r)\left[r\,\phi^{\prime\prime}(r)+\phi^{\prime}(r)-4\Lambda
r\right]}{2r},$ $\displaystyle\text{(LHS)}_{rr}$
$\displaystyle=\frac{\left[\phi(r)+1\right]\left[r\,\phi^{\prime\prime}(r)+\phi^{\prime}(r)-4\Lambda
r\right]}{2r},$ $\displaystyle\text{(LHS)}_{\theta\theta}$
$\displaystyle=r\,\phi^{\prime}(r)-2\Lambda r^{2},$ (4.4)

and the only nonzero component of the right-hand-side of the equations is

$\text{(RHS)}_{\theta\theta}=2\zeta Gq^{2},$ (4.5)

From the $\theta\theta$ component, the KS scalar can be solved as

$\phi(r)=C+\Lambda r^{2}+2\zeta Gq^{2}\log(r),\qquad C=\text{constant}$ (4.6)

which also solves the other components of the field equations. In order to
give a physical meaning to the integration constant $C$, we again write the
metric in the BL coordinates (2.19) via the transformation (2.18), which leads
to the following metric function:

$f(r)=1-C-\Lambda r^{2}-2\zeta Gq^{2}\log(r).$ (4.7)

Having found the most general solution, we are now ready to investigate
physically interesting possibilities.

###  4.2 Solutions with the correct Newtonian potential ($\Lambda=0$)

In order to obtain solutions with the correct Newtonian potential, we take
$\Lambda=0$ since only the logarithmic term is needed. In the Newtonian limit,
the gravitational field in terms of the KS scalar can be obtained from Eqs.
(2.13)-()2.14), which yields

$\vec{g}=\frac{1}{2}\vec{\nabla}\phi.$ (4.8)

For the KS scalar given in (4.6) with $\Lambda=0$, we obtain

$\vec{g}=\frac{\zeta Gq^{2}}{r}\hat{r},$ (4.9)

which shows that in order to preserve the attractive nature of the
gravitational force, one should have $\zeta=-1$, i.e., either the EH term or
the vector kinetic term in the action (4.1) should carry a ghost sign (see
Appendix A for more details).

Note that our solution should be a charged black hole and the gravitational
attraction is provided by the electric charge. Therefore, the integration
constant should be a function of the mass of the black hole $C=C(M)$. In order
to fix the constant, we again need to check the metric function in the BL
coordinates given in (4.7) and demand that the metric reduces to the Minkowski
metric [$f(r)=1$] when the mass and the charge are set to zero ($M\rightarrow
0$, $q\rightarrow 0$). This constraint can be satisfied by taking the mass
term with different signs as follows:

$f^{\pm}(r)=1\pm 8GM+2Gq^{2}\log(r),$ (4.10)

where for both choices $f(r)$ has a single zero, and therefore, admits one
event horizon.

Although we have already ensured the correct Newtonian limit, it is
interesting to have a closer look at the properties of the metric as done in
[63] for the scalar case. For this purpose, we check the geodesic motion of a
timelike particle described by the equation

$\frac{1}{2}E^{2}=\frac{1}{2}\left(\frac{\mathop{}\\!\mathrm{d}r}{\mathop{}\\!\mathrm{d}t}\right)^{2}+V^{\pm}_{\text{eff}},\qquad
V^{\pm}_{\text{eff}}=\frac{1}{2}\left(\frac{L^{2}}{r^{2}}+1\right)f^{\pm}(r),$
(4.11)

where $E$ and $L$ are the energy and the angular momentum of the particle
which are defined through the timelike and the angular Killing vectors
$\xi^{\mu}_{(t,\theta)}$ as follows:

$E=-g_{\mu\nu}\,\xi^{\mu}_{(t)}u^{\nu},\qquad
L=g_{\mu\nu}\,\xi^{\mu}_{(\theta)}u^{\nu},$ (4.12)

where $u^{\mu}$ is the velocity the particle. The Newtonian limit of the
effective potential $V^{\pm}_{\text{eff}}$ can be obtained by neglecting the
$L^{2}G$ terms as follows444For a general analysis, one should write the
logarithmic term in both $V_{\text{eff}}$ and $V_{\text{N}}$ by introducing a
length scale as $\log(\frac{r}{r_{0}})$. We set $r_{0}=1$ for simplicity.:

$V^{\pm}_{\text{N}}=\frac{1}{2}\pm 4GM+\frac{L^{2}}{2r^{2}}+Gq^{2}\log(r).$
(4.13)

The metric function obtained by coupling a free scalar (2.22) and the ones we
obtained by coupling a gauge vector (4.10) have the same functional form
[$f(r)=A+B\log(r)$, $A,B$: constant]. Therefore, all the physically important
properties of the solution that is discussed in [63] are also valid for our
solutions. They can be summarized as follows:

  1. 1.

The Newtonian potential $V_{\text{N}}$ has an infinite barrier at short
distances and matches with the effective potential $V_{\text{eff}}$ at long
distances.

  2. 2.

A timelike particle cannot escape to infinity due to the logarithmic
divergence of the potential as $r\rightarrow\infty$.

  3. 3.

The effective potential $V_{\text{eff}}$ develops a local maximum
($V^{\text{max}}_{\text{eff}}$) and a local minimum
($V^{\text{min}}_{\text{eff}}$) when the angular momentum of the particle $L$
is larger than a certain value ($L_{min}$). A timelike particle moves along a
stable orbit provided that $V^{\text{min}}_{\text{eff}}\leq
E<V^{\text{max}}_{\text{eff}}$. On the other hand, the Newtonian potential
$V_{\text{N}}$ always admit stable orbits.

  4. 4.

When $E=V^{\text{min}}_{\text{eff}}$ and $L>L_{\text{min}}$, timelike
geodesics form circular orbits, i.e., one has
$\frac{\mathop{}\\!\mathrm{d}r}{\mathop{}\\!\mathrm{d}t}=0$ in (4.11).

  5. 5.

Since the central potential is not that of an inverse-square central force
[$V(r)=-\frac{k}{r}$, $k$: constant] or a radial harmonic oscillator
[$V(r)=\frac{1}{2}kr^{2}$, $k$: constant], Bertrand’s theorem assures that
there will be precession for orbits with $E>V^{\text{min}}_{\text{eff}}$.

In Fig. 1, we show that properties 1-4 hold for the metric functions
$f^{\pm}(r)$ given in (4.10) by tuning the parameters such that
$L_{\text{min}}=1$. Having shown that the qualitative properties of the metric
is the same, we refer the reader to [63] where the authors present timelike
geodesics, and also, show that more precession is observed in the relativistic
orbits when compared to the Newtonian orbits.

Figure 1: First row shows the effective potential $V_{\text{eff}}^{-}$ and the
Newtonian potential $V_{\text{N}}^{-}$ for the metric function $f^{-}(r)$ with
$GM=\frac{1}{16}$ and $Gq^{2}=\frac{1}{4}$. The second row shows the effective
potential $V_{\text{eff}}^{+}$ and the Newtonian potential $V_{\text{N}}^{+}$
for the metric function $f^{+}(r)$ with $GM=\frac{1}{16}$ and
$Gq^{2}=\frac{3}{4}$. For both cases, timelike geodesics are stable orbits
when $L>L_{\text{min}}=1$.

The KS scalars corresponding to the metric functions (4.10) are given by

$\phi^{\pm}(r)=\mp 8GM-2Gq^{2}\log(r),$ (4.14)

and lead to the following single copy gauge field:

$A_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}=\phi
k_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}=(\pm 8GM+2Gq^{2}\log
r)\,(\mathop{}\\!\mathrm{d}t+\mathop{}\\!\mathrm{d}r).$ (4.15)

The constant factor does not play a role and this is just the Coulomb solution
(2.28) with the identification $2Gq^{2}\rightarrow Q$, i.e., the electric
charge in the gravity side $q$ yields a positively charged point particle in
the gauge theory. This is a remarkable difference compared to the higher
dimensional cases.555Various aspects of the charged black holes solutions in
the context of the KS double copy are discussed in [43] and the source terms
for $d=4$ are given in [44] . In dimensions higher than three ($d\geq 4$), the
static solution of the Einstein-Maxwell theory, the Reissner-Nordström black
hole, has the following KS scalar:

$\phi(r)=\frac{2GM}{r^{d-3}}-\frac{Gq^{2}}{r^{2(d-3)}},$ (4.16)

where $M$ and $q$ are the mass and the electric charge of the black hole
respectively. The gauge field in the gravity side and the corresponding field
strength tensor are given by

$\displaystyle a_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}$
$\displaystyle=-\frac{q}{r^{(d-3)}}\mathop{}\\!\mathrm{d}t,$ (4.17)
$\displaystyle\frac{1}{2}\,f_{\mu\nu}\,\mathop{}\\!\mathrm{d}x^{\mu}\wedge\mathop{}\\!\mathrm{d}x^{\nu}$
$\displaystyle=(d-3)\frac{\,q}{r^{d-2}}\,\mathop{}\\!\mathrm{d}r\wedge\mathop{}\\!\mathrm{d}t.$
(4.18)

The gauge theory source for the solution is as follows:

$J^{\mu}=\rho\,\delta^{\mu}_{\
0},\qquad\qquad\rho=2GM\delta^{d-1}(\vec{r})-\frac{2(d-3)^{2}Gq^{2}}{r^{2(d-2)}},$
(4.19)

where we see that the mass $M$ of the black hole shows itself as the charge of
a point particle and the electric charge $q$ results in a nonlocalized charge
distribution which vanishes as $r\rightarrow\infty$. Obtaining the Coulomb
solution in $d=3$ is a very peculiar property, which is possible thanks to the
fact that the existence of the electric charge in the gravity side changes the
KS scalar (4.14) such that the modification has the same functional form
[$\log(r)$] with the Coulomb solution in $d=3$. In higher dimensions, as can
be seen in (4.16), the mass term carries the functional form of the Coulomb
solution given in (4.17), and therefore, yields a point charge in the gauge
theory. The modification due to the electric charge has a different functional
form and produces a nonlocalized charge density as described in (4.19).

###  4.3 The charged Bañados-Teitelboim-Zanelli (BTZ) black hole ($\Lambda<0$)

We have shown that the Coulomb solution can be obtained as a gauge theory
single copy by considering Einstein-Maxwell theory where either the EH term or
the vector kinetic term carries a ghost sign. One can introduce the
cosmological constant $\Lambda$ such that solutions with the correct Newtonian
potential are recovered when $\Lambda=0$. Instead, we will study the charged
BTZ black hole whose metric function reads [76]

$f(r)=-8GM+\frac{r^{2}}{\ell^{2}}-2Gq^{2}\log(r).$ (4.20)

Comparing this with the most general solution (4.7) gives that the parameters
should be chosen as follows:

$C=1+8GM,\qquad\Lambda=-\frac{1}{\ell^{2}},\qquad\zeta=1,$ (4.21)

where the last one equation shows that no ghost field is needed to obtain the
solution.

Gravitational field equations with the cosmological constant (4.2) can be
mapped to Maxwell’s equations by again checking the $\mu 0$ component of the
trace-reversed equations and using (2.25), which yield

$\partial_{\nu}F^{\nu\mu}=g\left[J^{\mu}_{(\Lambda=0)}+\bar{J}^{\mu}\right].$
(4.22)

Here, $J^{\mu}_{(\Lambda=0)}$ is the source in the absence of the cosmological
constant, whose general form is given in (2.27). $\bar{J}^{\mu}$ represents
the effect of the cosmological constant on the source and takes the following
form:

$\bar{J}^{\mu}=\rho_{c}\bar{v}^{\mu},\qquad\rho_{c}=\frac{4\Lambda}{g},\qquad\bar{v}^{\mu}=\left(1,0,0\right),$
(4.23)

which is a constant charge density filling all space. The KS scalar
corresponding to the metric function (4.20) is

$\phi(r)=1+8GM+\Lambda r^{2}+2Gq^{2}\log(r),$ (4.24)

with the single copy gauge field

$A_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}=\phi
k_{\mu}\mathop{}\\!\mathrm{d}x^{\mu}=-\left[1+8GM+\Lambda
r^{2}+2Gq^{2}\log(r)\right]\,(\mathop{}\\!\mathrm{d}t+\mathop{}\\!\mathrm{d}r).$
(4.25)

The field strength tensor reads

$\frac{1}{2}\,F_{\mu\nu}\,\mathop{}\\!\mathrm{d}x^{\mu}\wedge\mathop{}\\!\mathrm{d}x^{\nu}=\left[\frac{Q}{r}-\Lambda
r\right]\,\mathop{}\\!\mathrm{d}r\wedge\mathop{}\\!\mathrm{d}t,$ (4.26)

where we have made the replacement $-2Gq^{2}\rightarrow Q$. We see that the
gauge theory single copy of the charged BTZ black hole is the Coulomb solution
($Q<0$) modified by a term which describes an electric field linearly
increasing with the radial coordinate $r$ (since $\Lambda<0$) and the source
is a point charge located in a medium of constant charge density as follows:

$J^{\mu}\partial_{\mu}=\left[Q\,\delta^{2}(\vec{r})+\frac{4\Lambda}{g}\right]\,\partial_{t}.$
(4.27)

It is important to note that this is the usual behavior of the Schwarzschild-
AdS black hole in higher dimensions when written around a flat background
metric [44].

As we see, the Coulomb solution ($Q<0$) modified by the cosmological constant
can be obtained from the well-known charged BTZ black hole without any need
for introducing a ghost. From (4.8), one can calculate the gravitational field
in the Newtonian limit as

$\vec{g}=\left[\frac{\Lambda r}{2}+\frac{Gq^{2}}{r}\right]\hat{r},$ (4.28)

which shows that a negative cosmological constant ($\Lambda<0$) is needed for
an attractive force, which is possible when $r>\sqrt{\frac{2Gq}{-\Lambda}}$.
The geodesics of the charged BTZ black hole exhibit a very rich structure and
the details can be found in [79].

##  5 Conclusions

In this paper, we have studied the KS double copy of the Coulomb solution in
3d, which is an important consistency check for the classical double copy due
to the lack of degrees of freedom and a Newtonian limit in GR. The double copy
solution should have the correct Newtonian limit, and in 3d, this can only be
achieved by matter coupling. In [63], the solution was constructed by coupling
to a scalar but has some undesired features. It is a hairy black hole which
requires that either the EH term or the scalar kinetic term carry a ghost
sign, and the scalar field does not vanish at infinity. Making use of the on-
shell duality of a free scalar and a gauge vector, we have shown that a
solution with the correct Newtonian limit can also be obtained as a solution
of Einstein-Maxwell theory such that the single copy is again the Coulomb
solution. While at least one ghost sign is still needed, the electric field in
the gravity side vanishes at infinity, which is an improvement compared to the
scalar case.

When a negative cosmological constant is introduced, the charged BTZ black
hole is a solution without any need for a ghost field, and we have shown that
the single copy gauge field is the Coulomb solution ($Q<0$) modified by a term
describing an electric field whose magnitude linearly increases with the
distance to the point charge. The source is a point particle sitting in a
medium of constant charge density, which is the usual effect of the
cosmological constant. At the expense of this modification, this remarkably
establishes a connection to the well-known black hole solutions in 3d gravity,
which, we believe, shows the potential of 3d KS double copy to have many other
interesting features. We hope to report more on this in the future.

## Acknowledgments

M. K. G. is supported by TÜBİTAK Grant No 118F091. G.A. thanks Mehmet Özkan
for helpful discussions regarding the on-shell duality of $p$-form gauge
fields. We thank Merve Demirtaş Alkaç for her help in creating high-resolution
figures.

##  Appendix A Scalar-Vector Duality as Described in [80]

Note that in our formulation of the scalar-vector duality in Sec. 3, we have
shown that free field equations in flat spacetime are equivalent and a scalar
field which is linear in the azimutal angle ($\varphi=c\,\theta$,
$c=\text{constant}$) implies the Coulomb solution for the gauge vector. This
can be generalized to curved spacetime as long as the metric is in the KS form
(2.4). However; our analysis is limited to the matter equations and the
gravity equations have to be checked independently. The action (3.1) that we
use to establish the duality leads to a free scalar action with a ghost sign
(3.4) and a free Maxwell action with a nonghost sign (3.6), which is enough to
show the equivalence of the matter equations. When coupling to the gravitation
is considered, in order to obtain a black hole solution with the same physical
properties, one needs to have a ghost scalar and a ghost vector (we assume
$\zeta_{1}=1$, i.e., the EH term has the “right sign,” for simplicity.)

Indeed, as shown in [80] recently, the duality can also be formulated such
that the gravitational action with the correct sign for the matter coupling is
obtained directly. The authors consider theories described by Lagrangians of
the form $\mathcal{L}(g^{\mu\nu},R_{\mu\nu},\partial_{\mu}\varphi)$ and study
solutions in the following form

$\mathop{}\\!\mathrm{d}s^{2}=-f(r)\mathop{}\\!\mathrm{d}t^{2}+\frac{\mathop{}\\!\mathrm{d}r^{2}}{f(r)}+r^{2}\mathop{}\\!\mathrm{d}\theta^{2},\qquad\varphi=c\,\theta,\qquad
c=\text{constant},$ (A.1)

which includes the solution that we studied in Sec. 2 with the metric written
in the BL coordinates. It is possible to find theories such that the scalar
equations is automatically satisfied, the equation for the metric function
gets a nontrivial modification and can be solved analytically. They also show
that the same solution can be obtained from a dual Lagrangian of the form
$\mathcal{L}_{\text{dual}}(g^{\mu\nu},R_{\mu\nu},f_{\mu\nu})$. When the matter
fields are related as666We use a different normalization than [80] to obtain
the vector kinetic term with coefficient $\frac{1}{8\pi}$, as we have used
throughout the text.

$f_{\mu\nu}=-\sqrt{2\pi}\,\epsilon_{\mu\nu\rho}\frac{\partial\mathcal{L}}{\partial(\partial_{\rho}\varphi)}$
(A.2)

the dual Lagrangian is given by

$\mathcal{L}_{\text{dual}}=\mathcal{L}-\frac{1}{2\sqrt{2\pi}}\epsilon^{\mu\nu\rho}f_{\mu\nu}\partial_{\rho}\varphi,$
(A.3)

where the original Lagrangian $\mathcal{L}$ should also be written in terms of
$f_{\mu\nu}$ by using the relation (A.2). In [80], nonminimal matter couplings
are used and regular, electrically charged black hole solutions in three
dimensions are obtained. In this work, we only have the kinetic terms, which
are the simplest possible matter couplings. Starting from the Lagrangian

$\mathcal{L}=\frac{1}{\kappa^{2}}R+\frac{1}{2}\left(\partial\varphi\right)^{2},$
(A.4)

leads to the following dual Lagrangian

$\mathcal{L}_{\text{dual}}=\frac{1}{\kappa^{2}}R+\frac{1}{8\pi}f_{\mu\nu}f^{\mu\nu}.$
(A.5)

Both Lagrangians $\mathcal{L}$ and $\mathcal{L}_{\text{dual}}$ support the
solution with the correct Newtonian potential as long as the electric charge
$q$ and the constant $c$ are related as

$c=\frac{q}{\sqrt{2\pi}},$ (A.6)

which is a consequence of (A.2). Using the relation between the mass parameter
$M$ in the scalar case with the constant $c$ (2.16), this implies

$M=q^{2},$ (A.7)

which shows the relation between the coefficient of the logarithmic terms in
the metric functions (2.22) and (4.10).

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