aainstitutetext: Universidad Nacional de San Antonio Abad del Cusco, Av. La
Cultura 733, Cusco, Perú.bbinstitutetext: Universidad Técnica Federico Santa
María, Av. España 1680, Valparaíso, Chile.

# Periapsis shift and deflection of light by hairy black holes

David Choque a Willson D. Llerena a Jhiwsell D. Vargas<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the orbit equations and the eikonal equation for light
respectively, under influence of the hairy black holes (asymptotically flat)
in fourth dimensions. We consider two hairy black hole solutions with non-
trivial potentials, and one of these solutions has Schwarzschild case as a
smooth limit. Following to Landau and Lifshitz we use the Hamilton-Jacobi
method, and we show hairy corrections for periapsis shift, where the effect of
the hair is to increase it. In the same way, using the eikonal equation we
show the deflection of the light and the relevant scalar hair corrections.
Interestingly we find that the hair screening the gravitational field,
decreasing the angle of deflection as the hair increases.

##  1 Introduction

The Hamilton-Jacobi formalism was applied in different gravity situations, for
example in the context of AdS/CFT Maldacena:1997re  it was used to find
counterterms Martelli:2002sp ; Batrachenko:2004fd . On the other hand, the
Hamilton-Jacobi equations of canonical gravity are the main tool in the
derivation of renormalization group equations deBoer:2000cz ; Heemskerk:2010hk
.

The scalar field has a fundamental structure in string theory, and the
expectation value controls the string coupling constant $g_{s}=\langle
e^{\phi}\rangle$. In high energy physics the scalar field is the Higgs
particle Aad:2012tfa , and in Cosmology is the $inflaton$ that describes
cosmological perturbations observed in the experiments COBE, WMAP and Planck
Spergel:2006hy ; Planck:2013jfk . In the context of AdS/CFT the scalar field
is a source of the boundary operators111This depends on Dirichlet, Newmann or
mix boundary conditions of the scalar field Anabalon:2015xvl ; Henneaux:2006hk
.. Interestingly the hairy solutions studied here can be obtained from his
asymptotically AdS solution. By suitably adjusting the cosmological constant
or the scalar field potential so that the effective cosmological vanishes, one
can obtain asymptotically flat black holes Anabalon:2013qua . We considered
the boundary condition of a scalar field such that $\phi=0$ (at spatial
infinity)222 In Gibbons:1996af ; Astefanesei:2006sy  considered another
interesting boundary condition in which the moduli have a non-trivial radial
dependence, where the properties of these black holes depend on the values
$\phi_{\infty}$ of moduli at spatial infinity.. In Anabalon:2013qua ;
Anabalon:2012ih ; Gibbons:1987ps  was constructed various exact and regular
solutions of hairy black holes (charged) asymptotically flat, and AdS
Anabalon:2012ta ; Anabalon:2012ih ; Anabalon:2012dw . Is important
understanding that the hair (scalar field) lives behind horizon and boundary,
and does not exist a conserved quantity associated with the scalar field. In
Anabalon:2013qua ; Nunez:1996xv  argument that the back-reaction ensure that
the scalar field can over in a strong gravitational field without collapsing
completely. In the present paper, we considered Einstein-dilaton theories
minimally coupled and with non-trivial potentials, in Nunez:1996xv ;
Anabalon:2012ih ; Anabalon:2012dw  constructed hairy black holes in dilaton-
charged gravity theories without scalar potential, in these cases the coupling
of the dilaton with gauge fields ensures the existence of an effective
potential. In the extremal case the near horizon data depend completely by the
electric and magnetic charges and so the attractor mechanism Ferrara:1995ih ;
Strominger:1996kf ; Ferrara:1996dd  works like the no-hair theorem.

To describe the geodesics (and null geodesic) we need to consider two actions,
one for the gravitational field which describes the geometric of space-time
for a given distribution of matter $T_{\mu\nu}$. The another action gives us
information about how the particles (of mass $m$) moves in that space-time and
is given by

$I=-mc\int_{a}^{b}ds$ (1)

where $ds$ is the infinitesimal line universe
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$. Taking the variation we get the geodesic
equation

$\frac{d^{2}x^{\mu}}{ds^{2}}+\Gamma_{\alpha\beta}^{\mu}\frac{dx^{\alpha}}{ds}\frac{dx^{\beta}}{ds}=0$
(2)

but to describe the null geodesics ($ds=0$) the before equation is not
complete333That problem can solve easily, but we consider another option.. The
Hamilton-Jacobi is a powerful method to obtain the orbit equations. Landau-
Lifshitz showed that the Hamilton-Jacobi equation for a particle in any space-
time is $(\mathcal{M},g)$

$g^{\mu\nu}\frac{\partial I}{\partial x^{\mu}}\frac{\partial I}{\partial
x^{\mu}}-m^{2}=0$ (3)

In literature there is an extensive bibliography about the study of the orbits
of several black holes in different theories Frye:2013xia ; Cruz:2004ts ;
Magnan:2007uw ; Cruz:2011yr ; Wells:2011st ; Hackmann:2008zz ;
Olivares:2013zta . Recently was study Light propagation in a plasma on Kerr
space-time using the Hamilton-Jacobi equation Perlick:2017fio , but in fourth
dimensions is complicated find a rotated hairy black holes, although there are
currently several solutions in three dimensions Correa:2012rc ;
Natsuume:1999at .

For light trajectory (null geodesic) we consider the null module condition for
four-vector wave $k_{\mu}k^{\mu}=0$. So, replacing
$k_{\mu}=\partial\psi/\partial x^{\mu}$ we find eikonal equation in a
gravitational field

$g^{\mu\nu}\frac{\partial\psi}{\partial x^{\mu}}\frac{\partial\psi}{\partial
x^{\mu}}=0$ (4)

The present paper is structured as follows: In section 2, we consider a small
review of Newtonian results and its corrections given by general relativity.
In section 3, we present the two hairy solutions and its properties, in the
section 4 we use the holographic stress tensor method to determine the mass of
hairy black holes. Finally, in section 5, following to intuitive procedure of
section 2, we calculate the periapsis shift and deflection of light for both
hairy black holes, and in the final part, we present the conclusions.

##  2 Hamilton-Jacobi method in Schwarzschild

The Kepler problem in celestial mechanics consist in found and solve the orbit
equations of stellar systems (two body system). Classically is solved
(exactly) considering the Newtonian potential

$U_{N}(r)=-\frac{G_{N}mm^{{}^{\prime}}}{r}$ (5)

And we can be solved alternatively by Hamilton-Jacobi method. Considering the
Lagrangian expression for a particle in the central field force (at the plane
$\theta=\pi/2$)

$L=\frac{m}{2}(\dot{r}^{2}+r^{2}\dot{\varphi}^{2})-U_{N}(r),\qquad
M=\frac{\partial L}{\partial\dot{\varphi}}=mr^{2}\dot{\varphi}$ (6)

The Hamilton-Jacobi equation is

$\frac{1}{2m}\biggl{(}\frac{\partial I}{\partial
r}\biggr{)}^{2}+\frac{1}{2mr^{2}}\biggl{(}\frac{\partial
I}{\partial\varphi}\biggr{)}^{2}+U_{N}(r)=\mathcal{E}^{{}^{\prime}}$ (7)

where the ansatz is $I(r)=-\mathcal{E}^{{}^{\prime}}t+M\varphi+I_{r}^{(0)}$
and its solution is

$I(r)=-\mathcal{E}^{{}^{\prime}}t+M\varphi+\int{\sqrt{2m(\mathcal{E}^{{}^{\prime}}-U_{N})-\frac{M^{2}}{r^{2}}}dr}$
(8)

The trajectory is given by equation $\frac{\partial I}{\partial M}=cte$

$\varphi(r)=\int\frac{Mdr}{r^{2}\sqrt{2m(\mathcal{E}^{{}^{\prime}}-U_{N})-\frac{M^{2}}{r^{2}}}}+\varphi_{0}\Rightarrow\frac{p}{r}=1+e\cos{(\varphi-\varphi_{0})}$
(9)

Where $p=M^{2}/m^{2}m^{{}^{\prime}}G_{N}$ and the eccentricity is
$e=\sqrt{1+2\mathcal{E}^{{}^{\prime}}p/G_{N}mm^{{}^{\prime}}}$. Is easy to
show that for elliptical orbit 444In this case we have
$\mathcal{E}^{{}^{\prime}}<0$ and $r_{min}=p/(1+e)=a(1-e)$,
$r_{max}=p/(1-e)=a(1+e)$. $e<1$

$\Delta\varphi^{(0)}=-\frac{\partial\Delta I_{r}^{(0)}}{\partial
M}=2\int_{r_{min}}^{r_{max}}\frac{Mdr}{r^{2}\sqrt{2m(\mathcal{E}^{{}^{\prime}}-U_{N})-\frac{M^{2}}{r^{2}}}}=2\pi$
(10)

This is the angle that vector position tour when $r$ changed from $r_{max}$ to
$r_{min}$ and return to $r_{max}$555Another interesting result is
$\mathcal{E}^{{}^{\prime}}=-G_{N}mm^{{}^{\prime}}/2a$, where $m^{{}^{\prime}}$
is the mass of the star, black hole or another big gravitational source. And
$a$ is the length of the semi-major axis of the elliptical orbit.. The
astronomical observations showed that perihelion of Mercury has deviations
$\Delta\varphi^{(0)}\rightarrow\Delta\varphi^{(0)}+\delta\varphi$ 666The
closest point of the celestial object that orbits (closed orbit) another is
known as $periapsis$, for a particular case of the solar system we use
$perihelion$.. Exist different forms to deal with this problem, classically we
can consider perturbations to Newtonian potential $U_{N}\rightarrow
U_{N}+\delta U(r)$ Landau ; LandauL ; Wells:2011st . Another option is to
consider the corrections of general relativity. The usual method consist in to
solve the geodesic equation Magnan:2007uw , but here we focus on the Hamilton-
Jacobi method Vasudevan:2005js .

###  2.1 General relativistic corrections

In this section, we describe step by step the procedure proposed by Landau and
Lifshitz Landau . This gives us a correct intuition when we face the hairy
case. We will consider the very know Schwarzschild asymptotically flat
solution, where the action is

$S[g_{\mu\nu}]=\frac{1}{2\kappa}\int_{\mathcal{M}}{d^{4}x\sqrt{-g}R}+\frac{1}{\kappa}\int_{\partial\mathcal{M}}{d^{3}xK\sqrt{-h}}$
(11)

Here $\kappa=8\pi G_{N}$, and the last term is the Gibbons-Hawking boundary
term. Where $h_{ab}$ is the boundary metric and $K$ is the trace of the
extrinsic curvature. The solution is

$ds^{2}=-c^{2}N(r)dt^{2}+\frac{dr^{2}}{N(r)}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\varphi^{2})$
(12)

where, $N(r)=1-r_{g}/r$ and $r_{g}=2G_{N}m^{{}^{\prime}}/c^{2}$. Here
$m^{{}^{\prime}}$ is the gravitational mass. Considering geodesics in the
plane $\theta=\pi/2$. The Hamilton-Jacobi equation of the trajectory is

$\frac{1}{c^{2}N(r)}\biggl{(}\frac{\partial I}{\partial
t}\biggr{)}^{2}-N(r)\biggl{(}\frac{\partial I}{\partial
r}\biggr{)}^{2}-\frac{1}{r^{2}}\biggl{(}\frac{\partial
I}{\partial\varphi}\biggr{)}^{2}-m^{2}c^{2}=0$ (13)

The ansatz is $I(t,r,\varphi)=-\mathcal{E}_{0}t+M\varphi+I_{r}(r)$, where the
energy and mass of particle is $\mathcal{E}_{0}$, $m$, and its angular
momentum is $M$. Replacing the expression $I(t,r,\varphi)$ in (13) we can
solve $I_{r}$

$I_{r}(r)=\int{\biggl{[}\frac{r^{2}(\mathcal{E}_{0}^{2}-m^{2}c^{4})+m^{2}c^{4}rr_{g}}{c^{2}(r-r_{g})^{2}}-\frac{M^{2}}{r(r-r_{g})}\biggr{]}}^{1/2}dr$
(14)

The trajectory equation can be determined by $\varphi(r)=-\frac{\partial
I_{r}}{\partial M}+cte$, and is easy to show

$\varphi(r)=\int\biggl{[}r^{2}\sqrt{\frac{\mathcal{E}_{0}^{2}}{c^{2}}-\biggl{(}m^{2}c^{2}+\frac{M^{2}}{r^{2}}\biggr{)}\biggl{(}1-\frac{r_{g}}{r}\biggr{)}}\biggr{]}^{-1}Mdr+cte$
(15)

The classical limit given in (9) can be obtained considering
$c\rightarrow\infty$, $r_{g}=0$ and
$\mathcal{E}_{0}=mc^{2}+\mathcal{E}^{{}^{\prime}}$. Here
$\mathcal{E}^{{}^{\prime}}$ is the non-relativistic energy. To obtain the
relevant corrections of general relativity we consider small velocities of the
planets respect to light777For example, the star S2 that orbit to Sagittarius
$A^{*}$ (super massive black hole) in the periapsis the velocity is five
percent of the light velocity Schodel:2002vg ., this means $r_{g}/r<<1$. The
angular change of closed orbits can be calculated in the similar form to (10)

$\Delta\varphi(r)=-\frac{\partial\Delta I_{r}}{\partial M}$ (16)

For that purpose, we need expand (14) and check the Newtonian term and its
relativistic correction. Comparing $M^{2}/r^{2}$ part of (8) and (14) is easy
concluded that we need to consider the change variable

$r(r-r_{g})=r^{{}^{\prime}2}\Rightarrow r-\frac{r_{g}}{2}\approx
r^{{}^{\prime}}$ (17)

and introducing the non-relativistic energy $\mathcal{E}^{{}^{\prime}}$.
Expanding around $r_{g}/r<<1$ keeping fix $M^{2}/r^{2}$ (regardless of the
apostrophe) we have

$I_{r}=\int\sqrt{2m(E-U)-\frac{M^{2}}{r^{2}}+\frac{3m^{2}c^{2}r_{g}^{2}}{2r^{2}}+O\biggl{(}\frac{r_{g}^{3}}{r^{3}}\biggr{)}}dr$
(18)

where

$E=\mathcal{E}^{{}^{\prime}}+\frac{\mathcal{E}^{{}^{\prime}2}}{c^{2}},\qquad
U=U_{N}\biggl{(}1+\frac{4\mathcal{E}^{{}^{\prime}}}{mc^{2}}\biggr{)},\qquad
U_{N}=-\frac{mm^{{}^{\prime}}G_{N}}{r}$ (19)

We can see the fundamental correction to $M^{2}/r^{2}$ is given by the term
$r_{g}^{2}/r^{2}$, and this term can describe the periapsis shift of the
orbit. We consider $E\approx\mathcal{E}^{{}^{\prime}}$, $U\approx U_{N}$ and
only expanding around $r_{g}/r<<1$ keeping fix $M^{2}/r^{2}$

$I_{r}\approx\int\sqrt{2m(\mathcal{E}^{{}^{\prime}}-U_{N})-\frac{M^{2}}{r^{2}}}~{}dr+\frac{3m^{2}c^{2}r_{g}^{2}}{4}\int\frac{1}{r^{2}}\biggl{[}\sqrt{2m(\mathcal{E}^{{}^{\prime}}-U_{N})-\frac{M^{2}}{r^{2}}}\biggr{]}^{-1}dr+\ldots$
(20)

Considering the limits of integration $(r_{min},r_{max})$, we have
$I_{r}\rightarrow\Delta I_{r}$

$\Delta
I^{(0)}_{r}=\int_{r_{min}}^{r_{max}}\sqrt{2m(\mathcal{E}^{{}^{\prime}}-U_{N})-\frac{M^{2}}{r^{2}}}~{}dr$
(21)

So 888You can see that (21) is the same expression given in (8).

$\Delta I_{r}\approx\Delta
I_{r}^{(0)}-\frac{3m^{2}c^{2}r_{g}^{2}}{4M}~{}\frac{\partial\Delta
I_{r}^{(0)}}{\partial M}$ (22)

then, using (16) is easy to show the correction given for general
relativity999For perihelion shift of Mercury this give
$43^{{}^{\prime\prime}}$ and astronomical observations give
$43,1^{{}^{\prime\prime}}\pm 0,4^{{}^{\prime\prime}}$.

$\Delta\varphi\approx 2\pi+\frac{6\pi
G_{N}^{2}m^{2}m^{{}^{\prime}2}}{c^{2}M^{2}}$ (23)

To determine the trajectory of light ray (null geodesics) we need the eikonal
equation (4). This equation is similar to (13) with $m^{2}=0$ and
$I\rightarrow\psi$, and in this case, instead of energy $\mathcal{E}=-\partial
I/\partial t$ of the particle, we consider the light frequency
$\omega_{0}=-\partial\psi/\partial t$, then

$\psi(r)=-\omega_{0}t+M\varphi+\psi_{r}(r)$ (24)

From eikonal equation we have

$\psi_{r}(r)=\frac{\omega_{0}}{c}\int\biggl{[}\frac{r^{2}}{(r-r_{g})^{2}}-\frac{\varrho^{2}}{r(r-r_{g})}\biggr{]}^{1/2}dr,\qquad\varrho=\frac{Mc}{\omega_{0}}$
(25)

where $\varrho$ is the impact parameter. One more time we need consider the
before transformations given in (17)

$\psi_{r}(r)\approx\frac{\omega_{0}}{c}\int\biggl{(}1+\frac{2r_{g}}{r}-\frac{\varrho^{2}}{r^{2}}\biggr{)}^{1/2}dr$
(26)

the equation for the trajectory of a light ray is
$\varphi(r)=-\frac{\partial\psi_{r}}{\partial M}+cte$, then

$\varphi(r)=\int\frac{1}{r^{2}}\biggl{[}\sqrt{\frac{1}{\varrho^{2}}-\frac{1}{r^{2}}\biggl{(}1-\frac{r_{g}}{r}\biggr{)}}\biggr{]}^{-1}dr+cte$
(27)

the gravitational correction is given by $r_{g}$, if we take $r_{g}=0$ the
integral is $r=\varrho/\cos{\varphi}$. Namely a line that passing a distance
$\varrho$ from the origin, integrating (26) when $r_{g}=0$

$\Delta\psi_{r}^{(0)}=\frac{\omega_{0}}{c}\int_{0}^{\pi}\sqrt{1-\frac{\varrho^{2}}{r^{2}}}dr=-M\pi$
(28)

The angle formed by the asymptotes of ray light that comes from a very great
distance $(r\rightarrow\infty,\varphi=-\pi/2)$, approaches the nearest point
$(r=\varrho,\varphi=0)$ and moves away a great distance
$(r\rightarrow\infty,\varphi=\pi/2)$, is
$\Delta\varphi^{(0)}=-\partial\Delta\psi_{r}^{(0)}/\partial M=\pi$. The
relativistic corrections are given by

$\Delta\varphi=-\frac{\partial\Delta\psi_{r}}{\partial M}$ (29)

and for that purpose we need expand (26) around $r_{g}/r<<1$ (keeping constant
$\varrho^{2}/r^{2}$)

$\psi_{r}(r)\approx\psi^{(0)}_{r}+\frac{r_{g}\omega_{0}}{c}\ln{\varrho}+\frac{r_{g}\omega_{0}}{c}\cosh^{-1}\biggl{(}\frac{r}{\varrho}\biggr{)}$
(30)

The term $\psi^{(0)}_{r}$ corresponds to the classical ray rectilinear
$\psi_{r}(r_{g}=0)=\psi^{(0)}_{r}$ in (26). The total changing of $\psi_{r}$
during propagation of the light, from one large distance $R$ at the point
$r=\varrho$ close to the center and back again to the distance $R$ is

$\Delta\psi_{r}=\Delta\psi^{(0)}_{r}+\frac{2r_{g}\omega_{0}}{c}\cosh^{-1}\biggl{(}\frac{r}{\varrho}\biggr{)}$
(31)

replacing in (29)

$\Delta\varphi=-\frac{\partial\Delta\psi^{(0)}_{r}}{\partial
M}+\frac{2r_{r}R}{\varrho\sqrt{R^{2}-\varrho^{2}}}=\Delta\varphi^{(0)}+\frac{2r_{r}R}{\varrho\sqrt{R^{2}-\varrho^{2}}}$
(32)

taking the limit $R\rightarrow\infty$ and remember that
$\Delta\varphi^{(0)}=\pi$ for the rectilinear ray, we obtain

$\Delta\varphi=\pi+\frac{2r_{g}}{\varrho}$ (33)

##  3 Hairy black hole solutions

We will consider two hairy solutions one of which is the smooth limit of the
first when the hairy parameter $\nu\rightarrow\infty$. We are interested in
asymptotically flat hairy black hole solutions with a spherical horizon
Anabalon:2013sra ; Anabalon:2013eaa . The action is

$S[g_{\mu\nu},\phi]=\frac{1}{2\kappa}\int_{\mathcal{M}}{d^{4}x\sqrt{-g}\biggl{[}R-\frac{(\partial\phi)^{2}}{2}-V(\phi)\biggr{]}}+\frac{1}{\kappa}\int_{\partial\mathcal{M}}{d^{3}xK\sqrt{-h}}+S^{ct}$
(34)

where $V(\phi)$ is the scalar potential, $\kappa=8\pi G_{N}$ and $S^{ct}$ is
the boundary counterterm which we use for constructed the renormalized quasi-
local stress tensor $\tau_{ab}$. The equations of motion for the dilaton and
metric are

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi\right)-\frac{\partial
V}{\partial\phi}=0$
$E_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-\frac{1}{2}T_{\mu\nu}^{\phi}$

where the stress tensor of the scalar field is

$T_{\mu\nu}^{\phi}=\partial_{\mu}\phi\partial_{\nu}\phi-
g_{\mu\nu}\left[\frac{1}{2}\left(\partial\phi\right)^{2}+V(\phi)\right]$

When we consider convex or positive semi-definite potential the no-hair
theorems ensure that does not exist regular black holes solutions
Israel:1967wq ; Bekenstein:1995un ; Sudarsky:1995zg . In Martinez:2006an ;
Acena:2013jya ; Anabalon:2013sra ; Anabalon:2013qua ; Acena:2012mr ;
Anabalon:2012dw  they were able to relax some of those conditions and find a
large (exact) family of hairy black holes. In the present article, we will
work with following potentials, where
$l_{\nu}^{-1}=\sqrt{(\nu^{2}-1)/2\kappa}$ and $j^{-1}=1/\sqrt{2\kappa}$
101010In Anabalon:2013eaa ; DallAgata:2012mfj ; deWit:2013ija  was showed that
these scalar potentials (35), (36) for asymptotically-AdS space-times, fixing
some particular values of the parameters it becomes the one of a truncation of
$\omega$-deformed gauged $\mathcal{N}=8$ supergravity.

$\displaystyle V(\phi)$
$\displaystyle=\frac{2\alpha}{\nu^{2}}\biggl{[}\frac{\nu-1}{\nu+2}\sinh{\phi
l_{\nu}(\nu+1)}-\frac{\nu+1}{\nu-2}\sinh{\phi
l_{\nu}(\nu-1)}+4\frac{\nu^{2}-1}{\nu^{2}-4}\sinh{\phi l_{\nu}}\biggr{]}$ (35)
$V(\phi)=\frac{\alpha
j\phi}{\kappa}[2+\cosh(j\phi)]-\frac{3\alpha}{\kappa}\sinh(j\phi)$ (36)

Along the present paper, we centered in the negative branch:
$\phi\in(-\infty,0]$ for which $\alpha>0$

Figure 1:  The left hand side graphic (a) describe the potential (35) at
$\nu=5$. There are two families: $\phi\in(-\infty,0]$ for which $\alpha>0$ and
$\phi\in[0,+\infty)$ for which $\alpha<0$.
The right hand side graphic (b) describe the potential (36). There are two
families: $\phi\in(-\infty,0]$ for which $\alpha>0$ and $\phi\in[0,+\infty)$
for which $\alpha<0$.

###  3.1 Black hole solution

Considering the action (34) with scalar potential (35) and the metric ansatz

$ds^{2}=\Omega(x)\left[-c^{2}f(x)dt^{2}+\frac{\eta^{2}dx^{2}}{f(x)}+d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right]$ (37)

The equations of motion can be integrated for the conformal factor
Anabalon:2013sra ; Anabalon:2013qua ; Acena:2012mr ; Acena:2013jya :

$\Omega(x)=\frac{\nu^{2}x^{\nu-1}}{\eta^{2}(x^{\nu}-1)^{2}}$ (38)

where the parameters that characterize the hairy solutions are $\alpha$ and
$\nu$. The solutions for scalar field

$\phi(x)=l_{\nu}^{-1}\ln{x}$ (39)

and metric function

$f(x)=\alpha\biggl{[}\frac{1}{\nu^{2}-4}-\frac{x^{2}}{\nu^{2}}\biggl{(}1+\frac{x^{-\nu}}{\nu-2}-\frac{x^{\nu}}{\nu+2}\biggr{)}\biggr{]}+\frac{x}{\Omega(x)}$
(40)

where $\eta$ is the only integration constant and
$l_{\nu}^{-1}=\sqrt{(\nu^{2}-1)/2\kappa}$.

The scalar potential given in (35) and the solution for
$\Omega(x),\phi(x),f(x)$ are invariant under the transformation
$\nu\rightarrow-\nu$. The boundary where $\Omega(x)$ is blowing-up correspond
to $x=1$ and the theory has a standard flat vacuum $V(\phi=0)=0$. The hairy
parameter vary in the range $\nu\in[1,+\infty)$111111Since the potential and
the solution is symmetric about $\nu\rightarrow-\nu$, then the behavior is the
same if we consider the range $\nu\in(-\infty,-1]$., and in the limit $\nu=1$
one gets $l_{\nu}\rightarrow\infty$ and $\phi\rightarrow 0$ so that the
Schwarzschild black hole (asymptotically flat) is smoothly obtained.
There are two distinct branches, one that corresponds to $x\in(0,1]$ and the
other one to $x\in[1,\infty)$ — the curvature singularities are at $x=0$ for
the first branch and $x\rightarrow\infty$ for the second one (these are the
locations where the scalar field is also blowing up). Considering the change
of coordinates121212For the negative branch $x<1$. The procedure for obtaining
the coordinate transformation (41) can be found in Anabalon:2015xvl .

$x=1-\frac{1}{\eta
r}+\frac{(\nu^{2}-1)}{24\eta^{3}r^{3}}\biggl{[}1+\frac{1}{\eta
r}-\frac{9(\nu^{2}-9)}{80\eta^{2}r^{2}}\biggr{]}+O(r^{-6})$ (41)

we can read off the mass from the sub-leading term of $g_{tt}$:

$-g_{tt}=f(x)\Omega(x)=1-\frac{\alpha+3\eta^{2}}{3\eta^{3}r}+O(r^{-3})$ (42)

In the section (4) we use the quasilocal stress tensor to show that the mass
of this black hole is

$m_{1}^{{}^{\prime}}=\frac{4\pi
c^{2}}{\kappa}\biggl{(}\frac{\alpha+3\eta^{2}}{3\eta^{3}}\biggr{)}$ (43)

where the gravitational radio (or mass parameter) is
$r_{g}=2m_{1}^{{}^{\prime}}G_{N}/c^{2}$ In the section (4) we verified that
this is the gravitational mass of the black hole

###  3.2 Black hole solution $\nu=\infty$

In Anabalon:2013qua , was studied the following solution, but here we consider
the neutral case.
One more time, considering the action (34) with scalar potential (36) and the
metric ansatz

$ds^{2}=\Omega(x)\left[-c^{2}f(x)dt^{2}+\frac{\eta^{2}dx^{2}}{x^{2}f(x)}+d\theta^{2}+\sin^{2}{\theta}d\phi^{2}\right]$
(44)

The equations of motion can be integrated for the conformal factor
Anabalon:2013sra ; Anabalon:2013qua :

$\Omega(x)=\frac{x}{\eta^{2}\left(x-1\right)^{2}}$ (45)

Where $\alpha$ is the parameter that characterizes the hairy solution. With
this choice of the conformal factor is straightforward to obtain the
expressions for the scalar field

$\phi(x)=j^{-1}\ln{x},\qquad j^{-1}=\frac{1}{\sqrt{2\kappa}}$ (46)

and the metric function

$f(x)=\alpha\left[\frac{(x^{2}-1)}{2x}-\ln(x)\right]+\frac{1}{\Omega(x)}$ (47)

At the boundary $\Omega(x)$ is blowing-up, this correspond to $x=1$. The
theory has a standard flat vacuum $V(\phi=0)=0$. There are two distinct
branches, one that corresponds to $x\in(0,1]$ and the other one to
$x\in[1,\infty)$ — the curvature singularities are at $x=0$ for the first
branch and $x\rightarrow\infty$ for the second one (these are the locations
where the scalar field is also blowing up). But the difference with the before
hairy solution is that does not exist a hairy parameter, and we can not obtain
the Schwarzschild case.
Here, $\eta$ is the constant of integration and considering the horizon
equation we can solve it

$f(\eta,x_{h})=0\rightarrow\eta^{2}=\frac{x_{h}}{(x_{h}-1)^{2}}\biggl{[}\frac{\alpha(2x_{h}\ln{x_{h}}-x_{h}^{2}+1)}{2x_{h}}\biggr{]}$
(48)

Considering the following asymptotic coordinate transformation131313For the
negative branch.

$x=1-\frac{1}{\eta
r}+\frac{1}{2\eta^{2}r^{2}}-\frac{1}{8\eta^{3}r^{3}}+O(r^{-5})$ (49)

the gravitational radio $r_{g}$ (or mass parameter) can be read-off $-g_{tt}$
component in r-coordinate

$\Omega(x)f(x)=1-\frac{\alpha}{6\eta^{3}r}+O(r^{-3})$ (50)

then $r_{g}=\alpha/6\eta^{3}$, and the mass of the hairy black hole is

$m_{2}^{{}^{\prime}}=\frac{4\pi
c^{2}}{\kappa}r_{g}=\frac{c^{2}}{2G_{N}}\biggl{(}\frac{\alpha}{6\eta^{3}}\biggr{)},\qquad$
(51)

In the next section (4) we verified that this is the gravitational mass of the
black hole

##  4 Mass of hairy black holes

Similar to holographic formalism for asymptotic AdS space-times Myers:1999psa
; Anabalon:2015ija ; Anabalon:2015xvl ; Anabalon:2016izw ;
Balasubramanian:1999re , for asymptotically-flat exist an identical proposal
given in Astefanesei:2005ad ; Astefanesei:2009wi ; Astefanesei:2006zd ;
Astefanesei:2010bm . If the boundary has the following topology $S^{2}\times
R\times S^{1}$ the gravitational counterterm is

$S^{ct}=-\frac{1}{\kappa}\int_{\partial\mathcal{M}}d^{3}x\sqrt{2\mathcal{R}}~{}\sqrt{-h}$
(52)

The (quasilocal) stress tensor was defined in Brown:1992br  like

$\tau_{ab}\equiv\frac{2}{\sqrt{-h}}\frac{\delta S}{\delta h^{ab}}$ (53)

where for the total action (that including the boundary terms) given in (34)
we have Astefanesei:2005ad

$\tau_{ab}=-\frac{1}{\kappa}\left[K_{ab}-h_{ab}K-\Psi(\mathcal{R}_{ab}-\mathcal{R}h_{ab})-h_{ab}\Box\Psi+\Psi_{;ab}\right]$
(54)

Considering the foliation ($x=constant$) of the metrics (37) and (44) (both
foliations are similar)

$h_{ab}dx^{a}dx^{b}=\Omega(x)[-c^{2}f(x)dt^{2}+d\theta^{2}+\sin^{2}{\theta}d\varphi^{2}]$
(55)

Where $\mathcal{R}_{ab}$, $\mathcal{R}$ are the Ricci tensor and Ricci scalar
of the foliation (55). The expression for $\Psi$ was given in
Astefanesei:2006zd , this is $\Psi=\sqrt{2/\mathcal{R}}$, then

$\mathcal{R}_{00}=0\quad,\quad\mathcal{R}_{\theta\theta}=1\quad,\quad\mathcal{R}_{\varphi\varphi}=\sin^{2}\theta\quad,\quad\mathcal{R}=\frac{2}{\Omega(x)}\quad,\quad\Psi=\sqrt{\Omega(x)}$
(56)

Here $K_{ab}$, $K$ and $n_{a}$ are the extrinsic curvature, its trace and the
normal of time-like hypersurface defined by induced metric (55). We use the
following very useful expressions141414These expressions are correct only when
the metric $g_{\mu\nu}$ and the induced metric $h_{ab}$ are diagonal.

$K_{ab}=\frac{\sqrt{g^{xx}}}{2}\partial_{x}h_{ab},\qquad
n_{a}=\frac{\delta_{a}^{x}}{\sqrt{g^{xx}}},\qquad
K\sqrt{-h}=n^{a}\partial_{a}\sqrt{-h}$ (57)

And is easy to show $T^{\nu}_{tt}$ for solution (37) and $T^{\infty}_{tt}$ for
(44)

$T^{\nu}_{tt}=-\frac{c^{2}}{\kappa}\biggl{[}-\frac{(\Omega
f)^{{}^{\prime}}}{2\eta}\sqrt{\frac{f}{\Omega}}+\frac{\sqrt{\Omega
f}}{2\eta}\biggl{(}\frac{3\Omega^{{}^{\prime}}f}{\Omega}+f^{{}^{\prime}}\biggr{)}-2\sqrt{\Omega}f\biggr{]}$
(58) $T^{\infty}_{tt}=-\frac{c^{2}}{\kappa}\biggl{[}-\frac{x(\Omega
f)^{{}^{\prime}}}{2\eta}\sqrt{\frac{f}{\Omega}}+\frac{x\sqrt{\Omega
f}}{2\eta}\biggl{(}\frac{3\Omega^{{}^{\prime}}f}{\Omega}+f^{{}^{\prime}}\biggr{)}-2\sqrt{\Omega}f\biggr{]}$
(59)

The conserved quantity associated with the generator of time translations
symmetry $\xi=\partial/c\partial t$ is the energy. For both metrics (37) and
(44) we have

$E=c^{4}\int d^{2}\Sigma\sqrt{\sigma}m^{a}\xi^{b}T_{ab}=4\pi
c^{2}T_{tt}\sqrt{\frac{\Omega}{f}}~{}\biggr{|}_{x\rightarrow 1}$ (60)

Where $\sigma_{ab}$ is the metric of the transversal section (spherical) where
$\sqrt{\sigma}=\Omega\sin{\theta}$. The normal to foliation $(t=constant)$ is
$m_{a}=\delta_{a}^{t}c\sqrt{\Omega f}$. The energy or mass of the hairy black
holes (37), (44) are respectively ($E=m^{{}^{\prime}}c^{2}$)

$m^{{}^{\prime}}_{1}=\frac{4\pi
c^{2}}{\kappa}\biggl{[}\frac{\alpha+3\eta^{2}}{3\eta^{3}}+O(r^{-1})\biggr{]}_{r=\infty}$
(61) $m^{{}^{\prime}}_{2}=\frac{4\pi
c^{2}}{\kappa}\biggl{[}\frac{\alpha}{6\eta^{3}}+O(r^{-1})\biggr{]}_{r=\infty}$
(62)

##  5 Hamilton-Jacobi method

In this section, we use the Hamilton-Jacobi method to obtain the correction to
periapsis shift and deflection of light by hairy solutions given in (3.1) and
(3.2). We considered the intuitive procedure given in the section (2.1)

###  5.1 Corrections of hairy solution

We study the geodesics and null geodesic of the metric given in (37). Setting
$\theta=\pi/2$, we have

$ds^{2}=\Omega(x)\biggl{[}-c^{2}f(x)dt^{2}+\frac{\eta^{2}dx^{2}}{f(x)}+d\varphi^{2}\biggr{]}$
(63)

The Hamilton-Jacobi equation for geodesic of some celestial body of mass $m$,
that orbit around a hairy black hole is

$-\frac{1}{c^{2}\Omega f}\biggl{(}\frac{\partial I}{\partial
t}\biggr{)}^{2}+\frac{f}{\Omega\eta^{2}}\biggl{(}\frac{\partial I}{\partial
x}\biggr{)}^{2}+\frac{1}{\Omega}\biggl{(}\frac{\partial
I}{\partial\varphi}\biggr{)}^{2}+m^{2}c^{2}=0$ (64)

The ansatz $I=-\mathcal{E}_{0}t+M\varphi+I_{x}(x)$ and its solution is

$I_{x}(x)=\int\sqrt{\frac{\eta^{2}}{f^{2}}\biggl{(}\frac{\mathcal{E}_{0}^{2}}{c^{2}}-m^{2}c^{2}\Omega
f\biggr{)}-\frac{\eta^{2}M^{2}}{f}}~{}dx$ (65)

Considering the coordinate transformation (41) and similar to Schwarzschild
case we need consider $r\rightarrow r+\frac{r_{g}}{2}$ and
$\mathcal{E}_{0}=mc^{2}+\mathcal{E}^{{}^{\prime}}$, then

$x=1-\frac{1}{\eta(r+\frac{r_{g}}{2})}+\frac{(\nu^{2}-1)}{24\eta^{3}(r+\frac{r_{g}}{2})^{3}}\biggl{[}1+\frac{1}{\eta(r+\frac{r_{g}}{2})}-\frac{9(\nu^{2}-9)}{80\eta^{2}(r+\frac{r_{g}}{2})^{2}}\biggr{]}+O(r^{-6})$
(66)

We can write $I_{x}$ like $I_{r}$ when $\frac{r_{g}}{r}<<1$

$I_{r}=\int\sqrt{2m(E-U)+\frac{3\mathcal{A}m^{2}c^{2}r_{g}^{2}}{2r^{2}}-\frac{M^{2}}{r^{2}}+O\biggl{(}\frac{r_{g}^{3}}{r^{3}}\biggr{)}}dr,\qquad\mathcal{A}=1-\frac{\mathcal{E}^{{}^{\prime}}(\nu^{2}-1)}{3mc^{2}\eta^{2}r_{g}^{2}}$
(67)

Where the expressions for $E$, $U$ was given in (19). The new term
$\mathcal{A}$ depend on hairy parameter $\nu$ and constant integration $\eta$.
We can show that $\mathcal{A}$ give a hairy correction to periapsis. For this
purpose, we need expand (67) around $r_{g}/r<<1$ (keeping fix $M^{2}/r^{2}$)
and $E\approx\mathcal{E}^{{}^{\prime}}$, $U\approx U_{N}$

$\Delta
I_{r}=\int_{r_{min}}^{r_{max}}\sqrt{2m(\mathcal{E}^{{}^{\prime}}-U_{N})-\frac{M^{2}}{r^{2}}}+\frac{3m^{2}c^{2}r_{g}^{2}\mathcal{A}}{4}\int_{r_{min}}^{r_{max}}\frac{1}{r^{2}}\biggl{[}\sqrt{2m(\mathcal{E}^{{}^{\prime}}-U_{N})-\frac{M^{2}}{r^{2}}}\biggr{]}^{-1}dr+O\biggl{(}\frac{r_{g}^{4}}{r^{4}}\biggr{)}$
(68) $\Delta I_{r}=\Delta
I_{r}^{(0)}-\frac{3m^{2}c^{2}r_{g}^{2}\mathcal{A}}{4M}\frac{\partial\Delta
I_{r}^{(0)}}{\partial M}+O\biggl{(}\frac{r_{g}^{4}}{r^{4}}\biggr{)}$ (69)

And is easily to show,

$\Delta\varphi\approx 2\pi+\frac{3\pi
m^{2}c^{2}r_{g}^{2}\mathcal{A}}{2M^{2}},\qquad\mathcal{A}=1+\frac{(\nu^{2}-1)}{4a}\biggl{(}\frac{\eta}{\alpha+3\eta^{2}}\biggr{)}$
(70)

when $\nu=1$, we have $\mathcal{A}=1$, e.i. the Schwarzschild case given in
(23). Where the orbital parameter $a$ is the semi-major axis.151515According
to astronomical observations Eisenhauer:2005cv ; Ghez:2003qj , nine stars are
currently known that orbit around of Sagittarius $A^{*}$, including $S1$,
$S2$, $S8$. And its semi-major axis oscillate between $a\sim 900-3500~{}AU$.
The astronomical unit $(AU)$ or radius of Earth’s orbit, and it is equivalent
to $1.495\times 10^{11}~{}m$.
According to Ghez:1998ph ; Ghez:2003qj ; Schodel:2002vg  we can show
$r_{g}=\frac{2m^{*}G_{N}}{c^{2}}\approx 0.1AU$, where $m^{*}$ is the mass of
Sagittarius $A^{*}$, then $r_{g}<<a$. If we consider the horizon equation
$f(x_{h},\eta,\alpha)=0$ and solve $\eta=\eta(x_{h},\alpha)$, we can write the
hairy correction $\mathcal{A}$ as a function
$\mathcal{A}=\mathcal{A}(a,\nu,x_{h},\alpha)$

Figure 2:  The left hand side graphic (c) describe (70) versus $x_{h}$ for the
negative branch, we fix $a=2000~{}AU$ and $\alpha=2~{}AU^{-2}$. And according
to our approximation $r_{g}/r<<1$ which means that $x_{h}<<1$. That is, in the
graphic all these hairy black holes ($\nu>1$) have the same corrective factor,
which is $\mathcal{A}>1$.
The right hand side graphic (d) describe (70) versus $x_{h}$ for the negative
branch, we fix $a=2000~{}AU$ and $\alpha=2~{}AU^{-2}$. And according to our
approximation $r_{g}/r<<1$ which means that $x_{h}<<1$. That is, in the
graphic all these hairy black holes ($\nu>1$) have the same corrective factor,
which is $\mathcal{A}>1$. But here the correction is larger than the previous
case, because the hair is bigger.

Similar to Schwarzschild case, we can construct the eikonal equation, whose
solution is

$\psi=-\omega_{0}t+M\varphi+\psi_{x}$ (71)
$\psi_{x}=\frac{\omega_{0}}{c}\int{\sqrt{1-\varrho^{2}f}}\frac{\eta}{f}dx,\qquad\varrho=\frac{Mc}{\omega_{0}}$
(72)

The equation of light trajectory is given by161616Is easy to show that the
Schwarzschild case ($\nu=1$) given in (27) can be obtained when
$x=1-\frac{1}{\eta r},\qquad\Omega
f=1-\frac{r_{g}}{r},\qquad\Omega=r^{2},\qquad\eta dx=\frac{dr}{r^{2}}$ (73)

$\varphi(x)=\int\frac{\eta dx}{\sqrt{\frac{1}{\varrho^{2}}-f}}+cte$ (74)

We need to consider the hairy-gravity corrections, considering one more time
the coordinate transformations given in (66) we can show

$\psi_{r}\approx\frac{\omega_{0}}{c}\int\sqrt{1+\frac{2r_{g}}{r}-\frac{\varrho_{\nu}^{2}}{r^{2}}}dr$
(75)

where the impact parameter $\varrho$ has a new hairy correction

$\varrho^{2}_{\nu}=\varrho^{2}+\frac{\nu^{2}-1}{4\eta^{2}}$ (76)

Now, expanding around $r_{g}/r<<1$, keeping fix $\varrho_{\nu}^{2}/r^{2}$

$\psi_{r}\approx\frac{\omega_{0}}{c}\int\sqrt{1-\frac{\varrho^{2}}{r^{2}}}dr+\frac{\omega_{0}r_{g}}{c}\int\frac{dr}{\sqrt{r^{2}-\varrho_{\nu}^{2}}}$
(77)

In the first term of the expansion of $\psi_{r}$ we have
$\sqrt{1-\frac{\varrho^{2}}{r^{2}}-\frac{\nu^{2}-1}{4\eta^{2}r^{2}}}\approx\sqrt{1-\frac{\varrho^{2}}{r^{2}}}$,
this is because if we consider the horizon equation we have
$\eta=\eta(x_{h},\alpha,\nu)$, and you can prove that for $x_{h}<<1$
($r_{g}<<r$) we have $\eta^{2}$ large, then $\frac{3}{4r^{2}\eta^{2}}\approx
0$. But for the second term we have only
$\sqrt{r^{2}-\varrho^{2}-\frac{\nu^{2}-1}{4\eta^{2}}}$, from which
$\frac{\nu^{2}-1}{4\eta^{2}}$ is relevant. This is precisely the relevant
hairy correction

$\Delta\varphi=-\frac{\partial\Delta\psi_{r}^{(0)}}{\partial
M}+\frac{2r_{g}Mc}{\omega_{0}\varrho_{\nu}^{2}}\frac{1}{\sqrt{1-\frac{\varrho_{\nu}^{2}}{R^{2}}}}$
(78)

when $R\rightarrow\infty$

$\Delta\varphi\approx\pi+\frac{2r_{g}}{\varrho}~{}\frac{\varrho^{2}}{\varrho_{\nu}^{2}}$
(79)

And one more time when $\nu=1$ we obtain the Schwarzschild171717For a ray of
light passing near the edge of the sun we obtain $1.75^{{}^{\prime\prime}}$.
case giving in (33). Using the horizon equation $f(\eta,x_{h},\nu,\alpha)=0$
we have $\eta=\eta(x_{h},\nu,\alpha)$

$\frac{\varrho_{\nu}^{2}}{\varrho^{2}}\biggr{|}_{(x_{h},\nu,\alpha)}=1+\frac{(\nu^{2}-1)}{4\eta^{2}\varrho^{2}}$
(80)

To graph $\varrho^{2}/\varrho_{\nu}^{2}$ vs $x_{h}$, we consider $\varrho$
like multiples of $R_{\odot}$ (sun radius). For example the stars early-B
hyper-giants (BHGs), has radios in the range of
$\varrho=63.5R_{\odot}-246R_{\odot}$ Clark:2012ne . The figures in (3) show
that the scalar field (hair) screening the effect of gravitational field,
causing the deviation of light to be smaller respect to Schwarzschild case
($\nu=1$)

Figure 3:  The left hand side graphic (e) describe (80) versus $x_{h}$ for the
negative branch, we fix $\varrho=200R_{\odot}$ and $\alpha=10R_{\odot}^{-2}$.
And according to our approximation $r_{g}/r<<1$ which means that $x_{h}<<1$,
(see the graphic) all these hairy black holes ($\nu>1$) have the same
screening factor, which is $\frac{\varrho^{2}}{\varrho_{\nu}^{2}}<1$.
The right hand side graphic (f) describe (80) versus $x_{h}$ for the negative
branch, we fix $\varrho=200R_{\odot}$ and $\alpha=10R_{\odot}^{-2}$. And
according to our approximation $r_{g}/r<<1$ which means that $x_{h}<<1$, (see
the graphic) all these hairy black holes ($\nu>1$) have the same screening
factor, which is $\frac{\varrho^{2}}{\varrho_{\nu}^{2}}<1$. But here the
screening is larger than the previous case, because the hair is bigger.

###  5.2 Corrections of hairy solution $\nu=\infty$

We study the geodesics and null geodesics of the metric given in (44). Setting
$\theta=\pi/2$, we have

$ds^{2}=\Omega(x)\biggl{[}-c^{2}f(x)dt^{2}+\frac{\eta^{2}dx^{2}}{x^{2}f(x)}+d\varphi^{2}\biggr{]}$
(81)

The Hamilton-Jacobi equation is

$-\frac{1}{c^{2}\Omega f}\biggl{(}\frac{\partial I}{\partial
t}\biggr{)}^{2}+\frac{x^{2}f}{\Omega\eta^{2}}\biggl{(}\frac{\partial
I}{\partial x}\biggr{)}^{2}+\frac{1}{\Omega}\biggl{(}\frac{\partial
I}{\partial\varphi}\biggr{)}^{2}+m^{2}c^{2}=0$ (82)

the ansatz $I=-\mathcal{E}_{0}t+M\varphi+I_{x}(x)$, and the solution for
$I_{x}(x)$ is

$I_{x}=\int\sqrt{\frac{\eta^{2}}{x^{2}f^{2}}\biggl{(}\frac{\mathcal{E}_{0}}{c^{2}}-m^{2}c^{2}\Omega
f\biggr{)}-\frac{\eta^{2}M^{2}}{x^{2}f}}~{}dx$ (83)

Considering the coordinate transformation (49), and in a similar form to
Schwarzschild case, we need to consider $r\rightarrow r+\frac{r_{g}}{2}$ and
$\mathcal{E}_{0}=mc^{2}+\mathcal{E}^{{}^{\prime}}$, then, we can write $I_{x}$
like $I_{r}$ when $\frac{r_{g}}{r}<<1$

$I_{r}=\int\sqrt{2m(E-U)+\frac{3\mathcal{A}m^{2}c^{2}r_{g}^{2}}{2r^{2}}-\frac{M^{2}}{r^{2}}+O\biggl{(}\frac{r_{g}^{3}}{r^{3}}\biggr{)}}~{}dr,\qquad\mathcal{A}=1-\frac{\mathcal{E}^{{}^{\prime}}}{mc^{2}\eta^{2}r_{g}^{2}}$
(84)

Where the expressions for $E$, $U$ was given in (19). The new term
$\mathcal{A}$ depend on constant integration $\eta$. We show that
$\mathcal{A}$ give a hairy correction to periapsis shift. For these purpose we
need expand (84) around $r_{g}/r<<1$ (keeping fix $M^{2}/r^{2}$) and
$E\approx\mathcal{E}^{{}^{\prime}}$, $U\approx U_{N}$. And we have,

$\Delta\varphi\approx 2\pi+\frac{3\pi m^{2}c^{2}r_{g}^{2}\mathcal{A}}{2M^{2}}$
(85)

where the orbital parameter $a$ is the semi-major axis181818Here we use the
horizon equation given in (48).,

$\mathcal{A}(a,\alpha,x_{h})=1+\frac{3\sqrt{\alpha}}{2a}~{}\sqrt{\frac{2x_{h}\ln{x_{h}}-x_{h}^{2}+1}{2(x_{h}-1)^{2}}}$
(86)

In this case, there are not a hairy parameter such that $\mathcal{A}=1$, the
unique form to obtain $\mathcal{A}=1$ is when $\alpha=0$, but according to
(40) and (47), these case correspond to a naked singularity. Even if we
consider black holes asymptotically AdS 191919The hairy black holes
(asymptotically AdS) constructed in Anabalon:2012dw ; Anabalon:2013qua ;
Acena:2012mr ; Acena:2013jya , has more general scalar potentials which
consist of two parts $V(\phi)\sim\alpha(\ldots)+\Lambda(\ldots)$, where
$\Lambda=-3/l^{2}$ is the cosmological constant, and when $\alpha=0$ has a new
interesting interpretations, like domain wall. Where its structure is ensured
for the existence of potential $V(\phi)\sim\Lambda(\ldots)$ Anabalon:2013eaa
., the special case $\alpha=0$ is a naked singularity. Is clear that $\alpha$
has an important role to existence of regular horizon. The condition
$\alpha\neq 0$ ensure the existence of scalar potentials given in (35), (36),
and this means that if $\alpha=0$, there are not self-interaction of the
scalar field that ensures its stability and at the same time, we have a naked
singularity.

Figure 4:  The left hand side graphic (g) describe (86) versus $x_{h}$ for the
negative branch, we fix $a=2000~{}AU$ and $\alpha=(100-500)~{}AU^{-2}$. We put
the case $Schw$ (Schwarzschild) to compare. And according to our approximation
$r_{g}/r<<1$ which means that $x_{h}<<1$. That is, in the graphic all these
hairy black holes have different (small) corrective factors, which is
approximate $\mathcal{A}\sim 1$.
The right hand side graphic (h) describe (86) versus $x_{h}$ for the negative
branch, we fix $a=2000~{}AU$ and $\alpha=(0.5-2)\times 10^{4}~{}AU^{-2}$. We
put the case $Schw$ (Schwarzschild) to compare. And according to our
approximation $r_{g}/r<<1$ which means that $x_{h}<<1$. That is, in the
graphic, all these hairy black holes have different corrective factors. But
here the correction is larger than the previous case because $\alpha$ is
bigger.

To describe the light trajectory we use the eikonal equation. The ansatz is
$\psi=-\omega_{0}t+M\varphi+\psi_{x}$, and the solution for $\psi_{x}$ is

$\psi_{x}=\frac{\omega_{0}}{c}\int{\sqrt{1-\varrho^{2}f}}\frac{\eta}{xf}dx,\qquad\varrho=\frac{Mc}{\omega_{0}}$
(87)

The equation of light trajectory is given by

$\varphi(x)=\int\frac{\eta dx}{x\sqrt{1/\varrho^{2}-f}}+cte$ (88)

But in here, we can not obtain the Schwarzschild as a smooth limit. To get the
hairy-gravity corrections, we need to consider one more time the coordinate
transformations given in (49) we can show, for $\frac{r_{g}}{r}<<1$

$\psi_{r}\approx\frac{\omega_{0}}{c}\int\sqrt{1+\frac{2r_{g}}{r}-\frac{\varrho_{\alpha}^{2}}{r^{2}}}dr$
(89)

where the impact parameter $\varrho$ has a new hairy term

$\varrho^{2}_{\alpha}=\varrho^{2}+\frac{3}{4\eta^{2}}$ (90)

Now, expanding around $r_{g}/r<<1$, keeping fix $\varrho_{\nu}^{2}/r^{2}$, we
have

$\psi_{r}\approx\frac{\omega_{0}}{c}\int\sqrt{1-\frac{\varrho^{2}}{r^{2}}}dr+\frac{\omega_{0}r_{g}}{c}\int\frac{dr}{\sqrt{r^{2}-\varrho_{\alpha}^{2}}}$
(91)

In the first term of the expansion of $\psi_{r}$ we have
$\sqrt{1-\frac{\varrho^{2}}{r^{2}}-\frac{3}{4\eta^{2}r^{2}}}\approx\sqrt{1-\frac{\varrho^{2}}{r^{2}}}$,
this is because if we consider $\eta=\eta(x_{h},\alpha)$ given in (48), you
can prove that for $x_{h}<<1$ ($r_{g}<<r$) we have $\eta^{2}$ large, then
$\frac{3}{4r^{2}\eta^{2}}\approx 0$. But for the second term, we have only
$\sqrt{r^{2}-\varrho^{2}-\frac{3}{4\eta^{2}}}$, from which
$\frac{3}{4\eta^{2}}$ is not small enough. This is precisely the relevant
hairy correction. Working in the same way as in the previous section, we can
show that

$\Delta\varphi=\pi+\frac{2r_{g}}{\varrho}~{}\frac{\varrho^{2}}{\varrho_{\alpha}^{2}};\qquad\frac{\varrho_{\alpha}^{2}}{\varrho^{2}}\biggr{|}_{(x_{h},\alpha)}=1+\frac{3}{2\alpha\varrho^{2}}~{}\frac{(x_{h}-1)^{2}}{2x_{h}\ln{x_{h}-x_{h}^{2}+1}}$
(92)

Figure 5:  The left hand side graphic (i) describe (92) versus $x_{h}$ for the
negative branch, we fix the impact parameter like $\varrho=2000R_{\odot}$. And
according to our approximation $r_{g}/r<<1$ which means that $x_{h}<<1$, (see
the graphic) these hairy black holes has different screening factors
$\frac{\varrho^{2}}{\varrho_{\nu}^{2}}<1$, such that $\alpha\approx 0$.
The right-hand side graphic (k) describe (92) versus $x_{h}$ for the negative
branch, we fix $\varrho=2000R_{\odot}$. And according to our approximation,
$r_{g}/r<<1$ which means that $x_{h}<<1$, (see the graphic) these hairy black
holes has small screening factor. Comparing with the before case, the
screening is small, then
$\frac{\varrho^{2}}{\varrho_{\nu}^{2}}\sim\frac{1}{\alpha}$ at fixed
$\varrho$.

##  6 Conclusions

Sagittarius $A^{*}$ is a stellar system located in the center of our galaxy,
it consists of a supermassive black hole, where until now the orbits of nine
stars were studied Eisenhauer:2005cv ; Ghez:2003qj ; Ghez:1998ph ;
Schodel:2002vg ; Clark:2012ne . This black hole has a disk accretion, and
there are indications that dark matter influences the gravitational dynamics
of these stars Ghez:1998ph . In 2004, source IRS 13 black hole was discovered
(and other stars that orbit it) that orbit around of Sagittarius $A^{*}$. The
principal idea of the present paper is that these hairy solutions (3.1) and
(3.2) could be an effective model to describe a more complicated gravitational
system. Our idea is to consider this gravitational system (for example
Sagittarius $A^{*}$) as a black hole with scalar hair, specially the solution
(3.1) has a hairy parameter $\nu$, it can be adjusted in such a way as to
describe for example the Sagittarius $A^{*}$. We propose (like an elemental
model) to use the results for periapsis shift given in (70), (85), and made
astronomical observations to the stars that orbit Sagittarius $A^{*}$
202020There is no concrete evidence of the rotation Sagittarius $A^{*}$ but in
general, all observed the black holes rotate. We consider the hairy black hole
as a simple static model of the stationary system of Sagittarius $A^{*}$,
assuming that its rotation is slow enough., estimate its shift periapsis and
thereby set the parameter of hair $\nu$ (and $\alpha$). In a similar form
maybe is possible use the equation for deflection of light given in (80),
(92). An interesting future direction is, study and classify the different
orbits of these hairy black holes.

##  7 Acknowledgments

D. Choque would like to thank Dumitru Astefanesei for interesting discussions.
D. Choque acknowledge the hospitality of the Universidad Nacional de San
Antonio Abad del Cusco (UNSAAC) during the stages of this research. This work
has been done with support from the 047-2017-FONDECYT-DE. The UNSAAC is funded
by the Peruvian Government through Financing Program of CONCYTEC.

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