because $\sum_{n = N_t(s_t, a_t)}^{N_{T}(s_t, a_t)} \frac{1}{n} \leq \log T$ (harmonic number) and $\lambda_t \leq 1+\frac{1-\gamma}{2}$ (cf. (<ref>)).

(ii) holds due to the condition (<ref>) and our choice of $\alpha$.

In summary with probability at least $1-\delta/6$, (<ref>) holds simultaneously for all $V^{\mathrm{u}} \in \mathbb{R}^{S}$ obeying $V^{\star} \leq V^{\mathrm{u}} \leq \frac{1}{1-\gamma}$.

Step 4: controlling the original quantity of interest.
With the uniform bound over the set of all optimistic vectors in $\mathbb{R}^{S}$ from Step 3, we can finally control the original quantity of interest

\begin{align}
\sum_{t = 1}^T \sum_{n = N_t(s_t, a_t)}^{N_{T}(s_t, a_t)} \frac{\lambda_t}{n}\big(P_{t} - P_{s_t, a_t} \big)\big(V^{\rref}_{T} -V^\star \big).
\end{align}

Substitution into (<ref>) with $V^{\rref}_{T}$ as $V^{\mathrm{u}}$ yields

\begin{align}
&\quad\ \left|\sum_{t = 1}^T \sum_{n = N_t(s_t, a_t)}^{N_{T}(s_t, a_t)} \frac{\lambda_t}{n}\big(P_{t} - P_{s_t, a_t} \big)\big(V^{\rref}_{T} -V^\star \big)\right|\nonumber\\
&\lesssim \sqrt{\frac{SA\log^2 T}{1-\gamma}\sum_{t=1}^{T}P_{s_{i},a_{i}}\left(V^{\rref}_{T}-V^{\star}\right)\log\frac{T}{\delta}}+ \frac{SA}{1-\gamma} \log^2\frac{T}{\delta} \notag\\
&\overset{(\mathrm{i})}{\lesssim} \sqrt{\frac{SA\log^2 T}{1-\gamma}\left\{ T + \frac{SA\log^2\frac{SAT}{\delta}}{(1-\gamma)^{7/2}} + \sqrt{\frac{SAT\log\frac{SAT}{\delta}}{(1-\gamma)^5}}\right\}\log\frac{T}{\delta}}+ \frac{SA}{1-\gamma} \log^2\frac{T}{\delta} \notag\\
&\lesssim \sqrt{\frac{SA\log^2 T}{1-\gamma}\left\{ \sqrt{\frac{SAT}{(1-\gamma)^5}\log\frac{SAT}{\delta}} + T\right\}\log\frac{T}{\delta}}+ \frac{SA}{(1-\gamma)^{9/4}} \log^{5/2}\frac{T}{\delta} \notag\\
&\overset{(\mathrm{ii})}{\lesssim} \sqrt{\frac{SA}{1-\gamma}\left\{ \frac{SA}{(1-\gamma)^5}\log\frac{SAT}{\delta}+T\right\} \log^{3}\frac{SAT}{\delta}} + \frac{SA}{(1-\gamma)^{9/4}} \log^{5/2}\frac{T}{\delta} \notag\\
&\lesssim \frac{SA}{(1-\gamma)^{3}} \log^{5/2}\frac{SAT}{\delta} + \sqrt{\frac{SAT}{1-\gamma}\log^{3}\frac{SAT}{\delta}},
\label{eq:covering-final}
\end{align}

in which (i) is obtained because with probability exceeding $1-\delta/6$,

\begin{align}
\sum_{t = 1}^T P_{s_t, a_t}\big(V^{\rref}_{T} -V^\star \big)
&\le\sum_{t = 1}^T P_{s_t, a_t}\big(V^{\rref}_{t} -V^\star \big) \le T + \frac{SA\log^2\frac{SAT}{\delta}}{(1-\gamma)^{7/2}} + \sqrt{\frac{SAT\log\frac{SAT}{\delta}}{(1-\gamma)^5}},
\end{align}

where the first inequality holds because $V_{t}$ is non-increasing in $t$ and $V^{\rref}_t$ can only be updated from $V_t$ (thus $V^{\rref}_{t}$ is monotonically non-increasing), and the second inequality follows from (<ref>), which we are allowed to use because Lemma <ref> suggests

\begin{align}
V^{\star}(s) \leq V^{\rref}_{T}(s) &\leq \frac{1}{1-\gamma} \quad \text{for any } s\in\cS.
\end{align}

(ii) holds since

\begin{equation*}
\sqrt{\frac{SAT}{(1-\gamma)^5}\log\frac{SAT}{\delta}} = \sqrt{\frac{SA}{(1-\gamma)^5}\log\frac{SAT}{\delta}}\sqrt{T}\lesssim \frac{SA}{(1-\gamma)^5}\log\frac{SAT}{\delta}+T.
\end{equation*}


# Lower bound on the radii of light rings in traceless black-hole spacetimes

Shahar Hod  The Ruppin Academic Center, Emeq Hefer 40250, Israel  The Hadassah
Institute, Jerusalem 91010, Israel

###### Abstract

Photonspheres, curved hypersurfaces on which massless particles can perform
closed geodesic motions around highly compact objects, are an integral part of
generic black-hole spacetimes. In the present compact paper we prove, using
analytical techniques, that the innermost light rings of spherically symmetric
hairy black-hole spacetimes whose external matter fields are characterized by
a traceless energy-momentum tensor cannot be located arbitrarily close to the
central black hole. In particular, we reveal the physically interesting fact
that the non-linearly coupled Einstein-matter field equations set the lower
bound $r_{\gamma}\geq{6\over 5}r_{\text{H}}$ on the radii of traceless black-
hole photonspheres, where $r_{\text{H}}$ is the radius of the outermost black-
hole horizon.

##  I Introduction

Theoretical Bar ; Chan ; Shap ; Hodub ; Herne ; Hodrec ; YY  as well as
observational Aki  studies have recently established the fact that closed
light rings exist in the external spacetime regions of generic black holes. It
has long been known that the presence of null circular geodesics in highly
curved spacetimes has many implications on the physical and mathematical
properties of the corresponding central black holes Bar ; Chan ; Shap ; Herne
; Hodns ; Hodrec ; YY ; Aki ; Mash ; Goeb ; Hod1 ; Dec ; Hodhair ; Hodfast ;
YP ; Hodm ; Hodub ; Lu1 ; Hodlwd ; Pod ; Ame ; Ste .

For instance, the unstable circular motions of massless fields along closed
null rings determine the characteristic relaxation timescale of a perturbed
black-hole spacetime in the short wavelength (eikonal) regime Mash ; Goeb ;
Hod1 ; Dec . In addition, the optical appearance of a black hole to far away
asymptotic observers is influenced by the presence of a light ring in the
highly curved near-horizon region Pod ; Ame ; Ste . Moreover, as measured by
asymptotic observers, the equatorial null circular geodesic determines the
shortest possible orbital period around a central non-vacuum black hole
Hodfast ; YP .

Intriguingly, it has also been proved Hodhair ; Hodub ; Hodlwd ; Hod1  that
the innermost light ring of a non-trivial (non-vacuum) black-hole spacetime
determines the non-linear spatial behavior of the supported hair. In
particular, it has been revealed, using the non-linearly coupled Einstein-
matter field equations, that the non-linear behavior of external hairy
configurations which have a non-positive energy-momentum trace must extend
beyond the null circular geodesic that characterizes the curved black-hole
spacetime Hodhair ; Hodub ; Hodlwd ; Hod1 .

Motivated by the well established fact that null circular geodesics (closed
light rings) are an important ingredient of generic black-hole spacetimes Bar
; Chan ; Shap ; Hodub ; Herne ; Hodrec ; YY ; Aki , in the present paper we
raise the following physically intriguing question: How close can the
innermost light ring of a central black hole be to its outer horizon?

This is a seemingly simple question but, to the best of our knowledge, in the
physics literature there is no general (model-independent) answer to it which
is rigorously based on the Einstein equations.

In the present compact paper we shall reveal the fact that, for spherically
symmetric hairy black-hole spacetimes whose supported field configurations are
characterized by a traceless energy-momentum tensor, the non-linearly coupled
Einstein-matter field equations provide an explicit quantitative answer to
this physically important question. In particular, we shall explicitly prove
that the radii of light rings in spherically symmetric traceless hairy black-
hole spacetimes are bounded from below by the functional relation

$r_{\gamma}\geq{6\over 5}r_{\text{H}}\ ,$ (1)

where $r_{\text{H}}$ is the radius of the outermost horizon.

It is worth noting that our theorem, to be presented below, is valid for the
canonical family of colored black-hole spacetimes that characterize the non-
linearly coupled Einstein-Yang-Mills (EYM) field theory (see Volk ; Notesta
and references therein). In particular, it is worth emphasizing the fact that
the highly non-linear character of the coupled Einstein-Yang-Mills field
equations has restricted most former studies of this physically important
field theory to the numerical regime. It is therefore of physical interest to
reveal, using purely analytical techniques, some of the generic physical
characteristics of this highly non-linear field theory. This is one of the
main goals of the present paper.

##  II Description of the system

We shall study, using analytical techniques, the radial locations of compact
photonspheres (closed light rings) in spherically symmetric hairy black-hole
spacetimes which are described by the curved line element Hodfast ; Hodm ;
Noteunit

$ds^{2}=-e^{-2\delta}\mu
dt^{2}+\mu^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2})\ ,$ (2)

where $\\{t,r,\theta,\phi\\}$ are the Schwarzschild-like coordinates of the
spacetime.

The radial functional behaviors of the matter-dependent metric functions
$\mu=\mu(r)$ and $\delta=\delta(r)$ are determined by the non-linearly coupled
Einstein-matter field equations $G^{\mu}_{\nu}=8\pi T^{\mu}_{\nu}$ Hodfast ;
Hodm :

${{d\mu}\over{dr}}=-8\pi r\rho+{{1-\mu}\over{r}}\ $ (3)

and

${{d\delta}\over{dr}}=-{{4\pi r(\rho+p)}\over{\mu}}\ ,$ (4)

where the radially-dependent matter functions Bond1

$\rho\equiv-T^{t}_{t}\ \ \ \ ,\ \ \ \ p\equiv T^{r}_{r}\ \ \ \ ,\ \ \ \
p_{T}\equiv T^{\theta}_{\theta}=T^{\phi}_{\phi}\ $ (5)

in the differential equations (3) and (4) are respectively the energy density,
the radial pressure, and the tangential pressure of the external matter
configurations in the non-trivial (non-vacuum) black-hole spacetime (2).

The radial metric functions $\\{\mu,\delta\\}$ of the black-hole spacetime are
characterized by the horizon boundary relations Bekreg

$\mu(r=r_{\text{H}})=0\ $ (6)

and

$\delta(r=r_{\text{H}})<\infty\ \ \ \ ;\ \ \ \
[d\delta/dr]_{r=r_{\text{H}}}<\infty\ .$ (7)

In addition, the asymptotic functional relations Bekreg

$\mu(r\to\infty)\to 1\ $ (8)

and

$\delta(r\to\infty)\to 0\ $ (9)

characterize the metric functions of asymptotically flat black-hole
spacetimes.

Our theorem, to be presented below, is based on the assumption that the
external matter fields respect the dominant energy condition, which implies
that the energy density is positive semi-definite Bekreg ,

$\rho\geq 0\ ,$ (10)

and that it bounds from above the absolute values of the pressure components
of the matter fields Bekreg :

$|p|,|p_{T}|\leq\rho\ .$ (11)

In addition, we shall assume that the external matter fields are characterized
by a traceless energy-momentum tensor:

$T=0\ ,$ (12)

where $T=-\rho+p+2p_{T}$. In particular, the analytically derived lower bound
on the characteristic radii of compact photonspheres [see Eq. (31) below]
would be valid for the well-known colored black-hole spacetimes that
characterize the composed Einstein-Yang-Mills field theory Volk .

Taking cognizance of the Einstein field equation (3), one finds the functional
relation

$\mu(r)=1-{{2m(r)}\over{r}}\ $ (13)

for the dimensionless metric function $\mu(r)$, where the radially-dependent
physical parameter

$m(r)=m(r_{\text{H}})+\int_{r_{\text{H}}}^{r}4\pi r^{2}\rho(r)dr\ $ (14)

is the gravitational mass which is contained within an external sphere of
radius $r\geq r_{\text{H}}$. Here $m(r_{\text{H}})$, which is characterized by
the simple relation

$m(r=r_{\text{H}})={{r_{\text{H}}}\over{2}}\ ,$ (15)

is the horizon mass (the mass contained within the black hole).

##  III Lower bound on the radii of light rings in spherically symmetric
traceless black-hole spacetimes

In the present section we shall address the following question: How close can
a black-hole photonsphere be to its outer horizon? Intriguingly, below we
shall prove that an explicit answer to this physically important question,
which is based on the non-linearly coupled Einstein-matter field equations,
can be given for non-trivial (non-vacuum) hairy black-hole spacetimes whose
external matter fields are characterized by a traceless energy-momentum
tensor. In particular, we shall reveal the fact that the innermost light rings
cannot be located arbitrarily close to the outer horizons of the central black
holes.

The radial locations of null circular geodesics (closed light rings) in
spherically symmetric hairy black-hole spacetimes are determined by the roots
of the dimensionless function Hodhair

${\cal N}(r)\equiv 3\mu-1-8\pi r^{2}p\ .$ (16)

Taking cognizance of the fact that non-extremal black holes are characterized
by the dimensionless horizon relations Bekreg

$0\leq 8\pi r^{2}_{\text{H}}\rho(r_{\text{H}})=-8\pi
r^{2}_{\text{H}}p(r_{\text{H}})<1\ ,$ (17)

one finds that the function (16) is characterized by the horizon boundary
condition [see Eq. (6)]

${\cal N}(r=r_{\text{H}})<0\ .$ (18)

In addition, from Eqs. (8), (11), (13), and (14) one deduces the asymptotic
functional behavior

$r^{2}p\to 0\ \ \ \ \text{for}\ \ \ \ r\to\infty\ ,$ (19)

which implies the simple radial behavior

${\cal N}(r\to\infty)\to 2\ .$ (20)

The characteristic properties (18) and (20) of the dimensionless radial
function (16) guarantee the existence of an external compact sphere with the
property $r=r_{\gamma}>r_{\text{H}}$ for which

${\cal N}(r=r_{\gamma})=0\ $ (21)

and

$\Big{[}{{d{\cal N}}\over{dr}}\Big{]}_{r=r_{\gamma}}\geq 0\ .$ (22)

The functional relations (21) and (22) determine the radial location of the
innermost light ring which characterizes the spherically symmetric non-vacuum
(hairy) black-hole spacetime (2).

Before proceeding, it is worth emphasizing that it has recently been proved
Hodrec , using the non-linearly coupled Einstein-matter field equations, that
extermal black-hole spacetimes are characterized by the horizon relations
${\cal N}(r=r_{\text{H}})=0$ and $[{{d{\cal N}}/{dr}}]_{r=r_{\text{H}}}<0$
which, together with the asymptotic radial behavior ${\cal N}(r\to\infty)\to
2$ [see Eq. (20)] of the dimensionless function (16), guarantee that extremal
black holes, like non-extremal ones, possess external light rings (with
$r=r_{\gamma}>r_{\text{H}}$) which are characterized by the functional
properties (21) and (22). Thus, our analysis is also valid for spherically
symmetric extremal black-hole spacetimes.

Taking cognizance of the Einstein equations (3) and (4) together with the
characteristic conservation equation

$T^{\mu}_{r;\mu}=0\ ,$ (23)

one finds the gradient relation

${{d}\over{dr}}(r^{2}p)={{r}\over{2\mu}}\Big{[}(3\mu-1-8\pi
r^{2}p)(\rho+p)+2\mu(-\rho-p+2p_{T})\Big{]}\ ,$ (24)

which yields the functional relation [see Eqs. (3) and (16)] Hodrole

$\Big{[}{{d{\cal
N}}\over{dr}}\Big{]}_{r=r_{\gamma}}={{2}\over{r_{\gamma}}}\big{[}1-8\pi
r^{2}_{\gamma}(\rho+p_{T})\big{]}\ .$ (25)

Substituting Eq. (25) into (22) and using the trace relation (12) for the
external matter fields, one obtains the relation

$0\leq[1-8\pi r^{2}(\rho+p_{T})]_{r=r_{\gamma}}=[1-12\pi r^{2}\rho+4\pi
r^{2}p]_{r=r_{\gamma}}\ $ (26)

which, using the dominant energy condition (11), yields the characteristic
dimensionless inequality

$0\leq[1+16\pi r^{2}p]_{r=r_{\gamma}}\ $ (27)

at the radial location of the black-hole innermost photonsphere. Furthermore,
substituting into (27) the relation (21), which characterizes the null
circular geodesics of the black-hole spacetime (2), one obtains the inequality

$[6\mu(r)-1]_{r=r_{\gamma}}\geq 0\ $ (28)

which, using the functional relation (13), can be written in the form

$\Big{[}{{m(r)}\over{r}}\Big{]}_{r=r_{\gamma}}\leq{{5}\over{12}}\ .$ (29)

Finally, taking cognizance of Eqs. (10), (14), (15), and (29), one obtains the
series of inequalities

$r_{\gamma}\geq{{12}\over{5}}m(r_{\gamma})\geq{{12}\over{5}}m(r_{\text{H}})={{6}\over{5}}r_{\text{H}}\
.$ (30)

It is interesting to point out that the canonical family of electrically
charged Reissner-Nordström black-hole spacetimes are characterized by the
relations $r_{\text{H}}=M+(M^{2}-Q^{2})^{1/2}$ and $r_{\gamma}={1\over
2}[3M+(9M^{2}-8Q^{2})^{1/2}]$ Chan ; NoteMQ , in which case one finds that the
dimensionless ratio $r_{\gamma}/r_{\text{H}}$ is a monotonically increasing
function of the dimensionless charge-to-mass ratio $|Q|/M$ of the black hole
from the value $r_{\gamma}/r_{\text{H}}=3/2$ for $Q=0$ to the value
$r_{\gamma}/r_{\text{H}}=2$ for the extremal black hole with $|Q|=M$. Thus,
charged Reissner-Nordström black-hole spacetimes respect the analytically
derived lower bound (30).

##  IV Summary

The non-linearly coupled Einstein-matter field equations of general relativity
predict the existence of compact photonspheres in the external regions of
curved black-hole spacetimes. In particular, it is well established in the
physics literature that closed light rings (null circular geodesics on which
photons and gravitons can perform closed orbital motions around highly compact
astrophysical objects) are of central importance in determining the physical,
mathematical, and observational properties of generic (non-vacuum) black-hole
spacetimes Bar ; Chan ; Shap ; Herne ; Hodns ; Hodrec ; YY ; Aki ; Mash ; Goeb
; Hod1 ; Dec ; Hodhair ; Hodfast ; YP ; Hodm ; Hodub ; Lu1 ; Hodlwd ; Pod ;
Ame ; Ste .

Motivated by the important roles that photonspheres play in the physics of
black holes, in the present paper we have addressed the following question:
How close can the black-hole innermost light ring be to the outer horizon of
the corresponding central black hole? Perhaps somewhat surprisingly, to the
best of our knowledge there is no general answer to this intriguing question
in the physics literature.

Interestingly, in the present compact paper we have proved, using analytical
techniques, that an explicit answer to this physically important question can
be given for spherically symmetric black-hole spacetimes whose external hairy
configurations are characterized by a traceless energy-momentum tensor [It is
worth noting that our main focus here is on the canonical family of colored
black-hole spacetimes that characterize the non-linearly coupled Einstein-
Yang-Mills field equations Volk . However, it should be emphasized that our
analytically derived results are also valid for any Einstein-matter field
theory for which the external matter fields satisfy the traceless energy-
momentum condition (12)].

In particular, we have presented a remarkably compact theorem that reveals the
physically interesting fact that the non-linearly coupled Einstein-matter
field equations set the dimensionless lower bound [see Eq. (30)]

${{r_{\gamma}-r_{\text{H}}}\over{r_{\text{H}}}}\geq{{1}\over{5}}\ $ (31)

on the radii of photonspheres (closed light rings) in spherically symmetric
Notens  traceless hairy black-hole spacetimes.

ACKNOWLEDGMENTS

This research is supported by the Carmel Science Foundation. I thank Yael
Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating
discussions.

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