# Rotating black hole with a probe string in Horndeski Gravity

F. F. Santos1<EMAIL_ADDRESS>1Departamento de Física, Universidade
Federal da Paraíba, Caixa Postal 5008, 58051-970, João Pessoa, Paraíba,
Brazil.

###### Abstract

In this work, we present the effect of a probe string on the complexity of a
black hole according to the CA (Complexity equals action) conjecture on
Horndeski’s gravity. In our system, we consider a particle moving on the
boundary of black hole spacetime in ($2+1$)-dimensions. To obtain a dual
description, we need to insertion a fundamental string on the bulk spacetime.
The effect of this string is given by the Nambu-Goto term. For the Nambu-Goto
term, we can analyze the time development of this system, which is affected by
the parameters of Horndeski’s gravity. In our case, we show some interesting
complexity properties for this gravity.

##  I Introduction

Over the past few years, General Relativity has been supported by substantial
evidence and observations, which have been witnessed in many astrophysical
scenarios, these in turn range from Eddington’s measurement of light
deflection in 1919 to the recent direct observation of gravitational waves by
collaboration LIGO Dyson:1920cwa ; Abbott:2016blz . These observational data
served as motivation to consider the modified gravity as a theory that has
given rich support to phenomenology Casalino:2018mna . However, some problems
considered fundamental to be understood in the context of General Relativity,
such as dark matter, dark energy, and the inflationary phase of the Universe.
In recent years, modifications of General Relativity have been proposed;
however, to carry out such modifications of General Relativity, it is
necessary to maintain some of its essential properties, which are second-order
equations of motion resulting from an invariable action by diffeomorphism and
Lorentz’s invariance Heisenberg:2018vsk . By maintaining these properties,
additional degrees of freedom of propagation can be added in the gravity
sector that is consistent to include additional fields such as scalars,
vectors, or tensors. However, to deal with these problems in Einstein’s
gravity, one way was to couple the theory with scalar fields. In this sense,
these efforts led to the development of the now known Galileo theories, which
are theories of scalar tensors Nicolis:2008in . Besides, these studies led to
the rediscovery of Horndeski’s gravity Horndeski:1974wa . This theory was
presented in 1974 by Horndeski for further discussions see Horndeski:1974wa ;
Bruneton:2012zk ; Brito:2018pwe ; Santos:2019ljs , it is characterized by
being a theory of scalar tensors with second-order field equations and second-
order energy-moment tensor Heisenberg:2018vsk ; Horndeski:1974wa ;
Cisterna:2014nua ; Anabalon:2013oea ; Bravo-Gaete:2013dca ; Rinaldi:2012vy .
The Lagrangian for this theory produces second-order equations of motion
Cisterna:2014nua ; Anabalon:2013oea ; Deffayet:2011gz ; VanAcoleyen:2011mj ;
Gomes:2015dhl ; Rinaldi:2016oqp ; Cisterna:2017jmv ; Zumalacarregui:2013pma ;
Feng:2015oea ; Brito:2019ose ; Cisterna:2016vdx . This theory also includes
four arbitrary functions of the scalar field and its kinetic term.

In recent years, the Horndeski gravity Cisterna:2016vdx  has been shown to
support static black hole solutions with asymptotically anti-de Sitter
behavior. In general, on the astrophysical scale, black holes are not expected
to be static and has spherical symmetry, but asymmetric because they have
angular momentum Tattersall:2018nve . However, asymmetric black holes are the
objects expected to be involved in the events detected in the gravitational
wave observatories by the collaborations LIGO and VIRGO Abbott:2016blz ;
Abbott:2016nmj ; Abbott:2017vtc . As discussed by Tattersall:2018nve , the
author performed an extension of the calculation of perturbations from a
Schwarzschild black hole to a Kerr black hole with a slow rotation, showing an
agreement of the almost normal modes analytically with the calculated
numerical frequencies. However, the ’bald’ black holes in Horndeski’s gravity
are identical to their equivalents in General Relativity. They can display a
modified gravitational wave signal during the ringdown, which is characterized
by only one or two observation parameters useful for attempts to restrict
gravity in this new era of gravitational-wave astronomy.

In addition to the results explored by Tattersall:2018nve  to explore the
equivalence between ’bald’ black holes in Horndeski’s gravity and General
Relativity, another scenario explored by Bravo-Gaete:2014haa  showed that due
to a particular truncation of Horndeski’s action, it reduces to
EinsteinHilbert’s Lagrangian with a cosmological constant and a scalar field,
which has a dynamics governed by their usual kinetic term, together with a
non-minimal kinetic coupling. In this sense, the radial component of the
conserved current has to disappear, providing a solution with a geometry of a
BTZ black hole with a radial scalar field that is well defined on the horizon.

Slowly rotating black holes in Horndeski’s gravity context has been presented
by Maselli:2015yva , where first-order rotational corrections were considered
for a wide range of black hole solutions in Horndeski’s gravity. It has been
shown that the drag function, which describes the rotational corrections of
leadership order, is precisely the same as in General Relativity for all known
black hole solutions by Horndeski. However, a very relevant fact for these
black holes with slow rotation in Horndeski’s gravity is that the no-hair
theorem is valid in the static regime and can be extended to apply also to
slow rotation black holes Maselli:2015yva ; Hui:2012qt ; Sotiriou:2015pka . In
this work, we will consider the computational complexity of a rotating black
hole in (2+1)-spacetime in Horndeski’s gravity. For further discussions
regarding the computational complexity of black holes, see the following
references Nagasaki:2017kqe ; Nagasaki:2018csh ; deBoer:2017xdk ;
Banerjee:2019vff ; Brown:2015lvg ; Brown:2015bva . For this black hole,
butterfly effects caused by a small disturbance in an asymptotic region of
this black hole were presented by Reynolds:2016pmi . However, in our study, we
considered the effect of the string moving on this spacetime geometry of the
BTZ black hole. This work is summarized as follows: In Sec. II, we present the
Horndeski gravity. In Sec. III, we address the issue of finding black hole
ansatz, and we explore the effect of the probe string in the black hole. In
the Sec. IV, we investigate the Nambu-Goto (NG) action of the string moving in
the black hole. Finally, in Sec. V, we present our conclusions.

##  II Horndeski gravity

In this section, we present the model of John Lagrangian Bruneton:2012zk ,
which is a specific model related to F4 theories for further discussions on F4
theories see Charmousis:2011bf ; Charmousis:2011ea , they are a particular
subclass of Horndeski’s theory. This subclass has attracted attention in
recent years when carrying out investigations around the standard kinetic term
for the scalar Starobinsky:2016kua , providing a model that is sometimes
called Fab Five (F5), which gained high visibility in the study of the
cosmological scenarios to know Santos:2019ljs . The following action gives
John Lagrangian Bruneton:2012zk  model

$S[g_{\mu\nu},\phi]=\int{d^{4}x\sqrt{-g}\left[\kappa(R-2\Lambda)-\frac{1}{2}(\alpha
g_{\mu\nu}-\gamma G_{\mu\nu})\nabla^{\mu}\phi\nabla^{\nu}\phi\right]}+S_{m}.$
(1)

Where $\kappa=(16\pi G)^{-1}$, we can define a new field
$\phi^{{}^{\prime}}\equiv\psi$, and $S_{m}$ describes an ordinary matter that
is supposed to be a perfect fluid. We can see that in action the field has a
dimension of $(mass)^{2}$ with the parameters $\alpha$ and $\gamma$
controlling the strength of the kinetic couplings, note that $\alpha$ is
dimensionless and $\gamma$ has a dimension of $(mass)^{-2}$. The Einstein-
Horndeski field equations for the action (1) can be written formally by
varying this action, that is, $\delta S[g_{\mu\nu},\phi]$ and assuming
$S_{m}=$constant, we can write the equations of motion as in the form

$\displaystyle E_{\mu\nu}[g_{\mu\nu},\phi]$ $\displaystyle=$ $\displaystyle
G_{\mu\nu}+\Lambda
g_{\mu\nu}-\frac{\alpha}{2\kappa}\left(\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\lambda}\phi\nabla^{\lambda}\phi\right)$
$\displaystyle-$
$\displaystyle\frac{\gamma}{2\kappa}\left(\frac{1}{2}\nabla_{\mu}\phi\nabla_{\nu}\phi
R-2\nabla_{\lambda}\phi\nabla_{(\mu}\phi
R^{\lambda}_{\nu)}-\nabla^{\lambda}\phi\nabla^{\rho}\phi
R_{\mu\lambda\nu\rho}\right)$ $\displaystyle-$
$\displaystyle\frac{\gamma}{2\kappa}\left(-(\nabla_{\mu}\nabla^{\lambda}\phi)(\nabla_{\nu}\nabla_{\lambda}\phi)+(\nabla_{\mu}\nabla_{\nu}\phi)\Box\phi+\frac{1}{2}G_{\mu\nu}(\nabla\phi)^{2}\right)$
$\displaystyle-$
$\displaystyle\frac{\gamma}{2\kappa}\left[-g_{\mu\nu}\left(-\frac{1}{2}(\nabla^{\lambda}\nabla^{\rho}\phi)(\nabla_{\lambda}\nabla_{\rho}\phi)+\frac{1}{2}(\Box\phi)^{2}-(\nabla_{\lambda}\phi\nabla_{\rho}\phi)R^{\lambda\rho}\right)\right],$
$\displaystyle E_{\phi}[g_{\mu\nu},\phi]$ $\displaystyle=$
$\displaystyle\nabla_{\mu}J^{\mu};\quad J^{\mu}=[(\alpha g^{\mu\nu}-\gamma
G^{\mu\nu})\nabla_{\nu}\phi].$ (3)

Using the fact that $E_{\mu\nu}[g_{\mu\nu},\phi]=0$ and
$E_{\phi}[g_{\mu\nu},\phi]=0$, we can write

$G_{\mu\nu}+\Lambda g_{\mu\nu}=\frac{1}{2\kappa}T_{\mu\nu},$ (4)

where $T_{\mu\nu}=\alpha T^{(1)}_{\mu\nu}+\gamma T^{(2)}_{\mu\nu}$ and the
energy-momentum tensors $T^{(1)}_{\mu\nu}$ and $T^{(2)}_{\mu\nu}$ take the
following format

$\begin{array}[]{rclrcl}T^{(1)}_{\mu\nu}&=&\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\nabla_{\lambda}\phi\nabla^{\lambda}\phi\\\
T^{(2)}_{\mu\nu}&=&\frac{1}{2}\nabla_{\mu}\phi\nabla_{\nu}\phi
R-2\nabla_{\lambda}\phi\nabla_{(\mu}\phi
R^{\lambda}_{\nu)}-\nabla^{\lambda}\phi\nabla^{\rho}\phi
R_{\mu\lambda\nu\rho}\\\
&-&(\nabla_{\mu}\nabla^{\lambda}\phi)(\nabla_{\nu}\nabla_{\lambda}\phi)+(\nabla_{\mu}\nabla_{\nu}\phi)\Box\phi+\frac{1}{2}G_{\mu\nu}(\nabla\phi)^{2}\\\
&-&g_{\mu\nu}\left[-\frac{1}{2}(\nabla^{\lambda}\nabla^{\rho}\phi)(\nabla_{\lambda}\nabla_{\rho}\phi)+\frac{1}{2}(\Box\phi)^{2}-(\nabla_{\lambda}\phi\nabla_{\rho}\phi)R^{\lambda\rho}\right].\end{array}$
(5)

And the scalar field equation is given by

$\nabla_{\mu}[(\alpha g^{\mu\nu}-\gamma G^{\mu\nu})\nabla_{\nu}\phi]=0.$ (6)

##  III Black hole solutions and probe string

Let us consider for Horndeski’s gravity that the string is moving in the
spacetime of the BTZ black hole Reynolds:2016pmi . The BTZ black hole at
($2+1$)-dimensions is given by

$ds^{2}=-f(r)dt^{2}+r^{2}\left(d\chi-\frac{J}{r^{2}}dt\right)^{2}+\frac{dr^{2}}{f(r)}.$
(7)

where $J$ is the angular momentum, to escape the no-hair theorem, which has
been well discussed in Bravo-Gaete:2013dca , we impose that the conserved
current’s radial component disappears identically without restricting the
radial dependence of the scalar field:

$\alpha g_{rr}-\gamma G_{rr}=0.$ (8)

Mind that $\phi^{{}^{\prime}}(r)\equiv\psi(r)$, we can easily see that this
condition annihilates $\psi^{2}(r)$, regardless of its behavior on the
horizon. It is possible to find the function $f(r)$ using the equation (8).
Thus, equation (6) is satisfied with the following solution

$\displaystyle f(r)$ $\displaystyle=$ $\displaystyle-M+\frac{\alpha
r^{2}}{\gamma}+\frac{J^{2}}{r^{2}},$ (9) $\displaystyle\psi^{2}(r)$
$\displaystyle=$
$\displaystyle-\frac{2\kappa(\alpha+\gamma\Lambda)}{\alpha\gamma f(r)}.$ (10)

The Einstein-Horndeski field equations (4) and (6) are satisfied by these
equations. Besides, as discussed by Anabalon:2013oea , we have that
$\alpha/(\gamma)=l^{-2}_{AdS}$ is defined as an effective radius of AdS
$l_{AdS}$. For the case in which we have limitations in the storage of
information, we have that the area of the black hole delimits the information
Brown:2015lvg ; Brown:2015bva . The black hole entropy, which can be obtained
by applying the first law of black hole thermodynamics $dM=TdS$, can be
written as

$\displaystyle
S=\frac{1}{2G}\int^{r_{h}}_{0}{\frac{1}{T(r_{h})}\frac{dM}{dr_{h}}dr_{h}}=\frac{2\pi
r_{h}}{G}=\frac{A}{4G}$ (11) $\displaystyle M(r_{h})=\left(\frac{\alpha
r^{2}_{h}}{\gamma}+\frac{J^{2}}{r^{2}_{h}}\right)$ (12) $\displaystyle
T(r_{h})=\frac{1}{2\pi}\left(\frac{\alpha
r_{h}}{\gamma}-\frac{J^{2}}{r^{3}_{h}}\right)$ (13)

Which obeys the celebrated Hawking’s law. If an object can be forced to
undergo a gravitational collapse by adding mass, the second law of
thermodynamics insists that it must have less entropy than the resulting black
hole. Now, we will present the effect of the string moving in this spacetime
geometry. For this, we will calculate the induced metric using the parameters
$\tau$ and $\sigma$ in the world-sheet of the fundamental string. These
parameters are given as follows:

$\displaystyle t=\tau,\quad r=\sigma\quad,\chi=v\tau+\xi(\sigma),$ (14)

Where $v$ is a constant velocity, and $\xi(\sigma)$ is a function that
determines string’s shape. However, the metric induced in the world-sheet is
given by

$\displaystyle
ds^{2}_{ind}=H(\sigma)d\tau^{2}+G(\sigma)d\sigma^{2}+2F(\sigma)d\tau d\sigma$
(15) $\displaystyle
H(\sigma)=-f(\sigma)+\left(v\sigma-\frac{J}{\sigma}\right)^{2}$ $\displaystyle
G(\sigma)=\frac{1}{f(\sigma)}+\sigma^{2}\xi^{{}^{\prime}2}(\sigma)$
$\displaystyle F(\sigma)=\xi^{{}^{\prime}}(\sigma)(v\sigma^{2}-J)$

Through the equation (8) we can provide that the solution for (15), which is
given by

$\displaystyle f(\sigma)=\left(v\sigma-\frac{J}{\sigma}\right)^{2}$ (16)
$\displaystyle\psi^{2}(\sigma)=\frac{4\kappa\Lambda
G(\sigma)(-F^{2}(\sigma)+G(\sigma)H(\sigma))}{\alpha(-2F^{2}(\sigma)+G(\sigma)H(\sigma))}$
(17)

Where this solution satisfies all equations (4) and (6). Now, integrating the
equation (6), we can write that

$\displaystyle\psi(\sigma)=\frac{-F^{2}(\sigma)+G(\sigma)H(\sigma)}{\alpha
H(\sigma)}$ (18)

##  IV Evaluating of the NG action

We are now going to carry out the NG action evaluation process, which was
recently presented by Nagasaki:2017kqe ; Nagasaki:2018csh ; deBoer:2017xdk ;
Banerjee:2019vff . Let us address the action of the Nambu-Goto (NG) term,
which is given by

$\displaystyle S_{NG}=-T_{s}\int{d\sigma^{2}\sqrt{-detg_{ind}}},$ (19)

where $T_{s}$ is the fundamental string tension and the horizon is determined
by $f(r)=0$. Adding the Wilson loop implies the insertion of a fundamental
string whose worldsheet has a limit in the Wilson loop. In this sense, we can
calculate the NG action of this fundamental string. Besides, we can calculate
the time derivative of the NG action, which is obtained by integrating the
square root of the determinant of the induced metric, and write that

$\displaystyle\frac{dS_{NG}}{dt}=T_{s}\int^{r_{+}}_{r_{-}}{d\sigma\sqrt{\xi^{{}^{\prime}2}(\sigma)(v\sigma^{2}-J)^{2}}},$
(20)

The Lagrangian is given by

$\displaystyle\mathcal{L}=T_{s}\xi^{{}^{\prime}}(\sigma)(v\sigma^{2}-J),$ (21)

which has the following equations of motion

$\displaystyle\frac{d}{d\sigma}\frac{\partial\mathcal{L}}{\partial\xi^{{}^{\prime}}(\sigma)}-\frac{\partial\mathcal{L}}{\partial\xi(\sigma)}=0$
(22)

Where through the equation of motion (22), we can easily show that $v=0$, this
fact characterizes a stationary string. So, by the equation (20), we have

$\displaystyle\frac{dS_{NG}}{dt}=\left.T_{s}J\xi(\sigma)\right|^{r_{+}}_{r_{-}},$
(23)

We can see that through the equation (16) combined with $H(\sigma)$, we can
draw from the equation (18) that $\xi(\sigma)=c_{\xi}/J$ where $c_{\xi}$ is an
integration constant that we will take to be $c_{\xi}>0$ and $c_{\xi}<0$ in
our analysis. So, we can write for equation (23):

$\displaystyle\frac{dS_{NG}}{dt}=T_{s}c_{\xi}\left(\sqrt{\frac{\gamma
M}{2\alpha}\left(1+\sqrt{1-\frac{4\alpha J^{2}}{\gamma
M^{2}}}\right)}+\sqrt{\frac{\gamma M}{2\alpha}\left(1-\sqrt{1-\frac{4\alpha
J^{2}}{\gamma M^{2}}}\right)}\right),$ (24)

Figure 1: BTZ: Action growth-Black hole angular momentum/Mass for $c_{\xi}=-1$
with the values $M=1$-$\alpha=0.1$-$\gamma=0.5$ (blue curve),
$M=2$-$\alpha=0.5$-$\gamma=1$ (curve red), and $M=3$-$\alpha=1$-$\gamma=1.5$
(green curve). Action growth-Black hole mass (small mass region) for $q=-1$
with the values $J=0.2$-$\alpha=0.1$-$\gamma=0.5$ (blue curve),
$J=0.4$-$\alpha=0.5$-$\gamma=1$ (curve red), and
$J=0.6$-$\alpha=1$-$\gamma=1.5$ (green curve).

Figure 2: BTZ: Action growth-Black hole angular momentum/Mass for $c_{\xi}=1$
with the values $M=1$-$\alpha=0.2$-$\gamma=0.5$ (blue curve),
$M=2$-$\alpha=0.5$-$\gamma=1$ (curve red), and $M=3$-$\alpha=1$-$\gamma=1.5$
(green curve). Action growth-Black hole mass for $q=1$ with the values
$J=0.2$-$\alpha=0.2$-$\gamma=0.5$ (blue curve),
$J=0.4$-$\alpha=0.5$-$\gamma=1$ (curve red), and
$J=0.6$-$\alpha=1$-$\gamma=1.5$ (green curve).

We can observe, according to figure 1, the dependence of the relationship
between the growth of the action and the angular momentum/mass and mass of the
black hole along with Horndeski’s parameters. In this way, the increase in
complexity reaches its maximum when the string is in a steady-state.
Therefore, we can observe that the effect of complexity reaches its maximum
when the relative speed is zero for a rotating black hole. As expected, this
effect is greater for larger black holes, as in our case, where we consider
the AdS4 case. In figure 2, we have a dependence on the angular momentum/mass
where we can see that the string rotates on a different axis, the relative
speed never becomes zero. We can also notice that in figure 2 the dependence
of the growth of the action with the black hole mass, the parameters of
Horndeski’s theory also influence this behavior. When the angular momentum is
small, we have a monotonically increasing function of the mass. As the angular
momentum becomes larger, it stops increasing rapidly, and the extreme appears
around $J=0.6$.

##  V Conclusion

In this work, we show the effects of the probe string on a BTZ black hole, a
solution of Horndeski’s theory without any restrictions on the parameters. We
have seen that although the string is stationary, it can provide maximum
growth in complexity, which is obtained for $c_{\xi}<0$ along with the
parameters of Horndeski’s theory, see figure 1. However, for $c_{\xi}>0$ as
shown in figure 2 the growth of the action with the black hole mass, which has
its behavior influenced by the parameters of Horndeski’s theory where the
complexity increases with the black hole mass giving us an idea of how complex
the physical system. As we showed in the calculation of NG’s action on the
remaining flame, the peak is located at $v=0$, where this behavior derives
from the expansion of time. We can perceive a decrease in complexity, as shown
in figure 1, with the growth of the action with the the black hole mass, which
is a monotonically decreasing function of the black hole mass. One might think
that this is because complexity defines how complex the physical system is.
So, if complexity decreases, we have less information from the physical
system. So, this is a bigger system and has bigger information.

We would like to thank CNPq and CAPES for partial financial support. I would
like to thank Moises Bravo Gaete and Jackson Levi Said for fruitful
discussions at the end of this work.

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