# Effects of charge on the interior volume of BTZ black holes

Suraj Maurya  0000-0001-6907-8584<EMAIL_ADDRESS>(S.
Maurya) Sashideep Gutti  0000-0001-7555-8453<EMAIL_ADDRESS>pilani.ac.in (S. Gutti) Rahul Nigam  0000-0002-0497-5898
<EMAIL_ADDRESS>(R. Nigam) Birla Institute of
Technology and Science Pilani (Hyderabad Campus)
Hyderabad 500078, India

(29-Nov-2022)

###### Abstract

In this article we extend the variational technique for maximal volume
estimation of a black hole developed by Christodoulou and Rovelli (CR) [1] to
the case of a charged BTZ black hole in 2+1 dimensions. The technique involves
a study of the equation of motion of a hypothetical particle moving in an
auxiliary manifold defined by spacetime variables. We then compare this
estimation with the volume computed using maximization method and extrinsic
curvature method. The charge Q of the black hole appears as a log term in the
metric and hence an analytical solution for the volume does not exist. So
first we compute the steady state radius and the volume for limiting case when
the charge Q is very small i.e. $Q<<1$ and then carry out a numerical analysis
to solve for the volume for more generic values of the charge. We find that
the volume grows monotonically with the advance time. We further investigate
the functional behaviour of the entropy of a massless scalar field living on
the maximal hypersurface of a near extremal black hole. We show that this
volume entropy exhibits a very different functional form compared to the
horizon entropy.

††preprint: APS/123-QED

##  I Introduction

A black hole boundary is defined by its event horizon. Event horizons have
been studied exhaustively and they play a crucial role in defining the
geometry and more importantly the thermodynamics of a black hole. For a given
black hole the position and the area of its event horizon depend on its mass,
charge and the angular momentum. So given these parameters, the area of event
horizon is well defined and unique. However computing the volume of a black
hole is slightly tricky. In fact there is no unique way of defining the volume
of a black hole. A foliation dependent volume is not an unfamiliar concept and
we come across this idea even in Minkowski space. However, in whichever way
one may define the volume, the boundary area of a black hole which is given by
the area of the event horizon remains the same.

There are many interesting ways of defining the black hole volume. Parikh [2]
describes the black hole volume using an invariant slice of spacetime inside
the horizon. Gibbons et al. [3] have offered a thermodynamical definition of
the volume i.e. $V_{th}$, which is useful in presence of a cosmological
constant $\Lambda$. They define $V_{th}$ as the variable conjugate to the
cosmological constant appearing in the first law of thermodynamics for black
hole, $dE=TdS+\Omega dJ+\Phi dQ+V_{th}d\Lambda$. The volume definition
specific to our interest in this work was provided by Christodoulou and
Rovelli (CR) in [1]. This particular volume grows monotonically with the
advance time and has a maximum contribution from a constant $r$ segment. CR
demonstrated that computing this volume is equivalent to finding the geodesic
equation of a particle in an equivalent spacetime since the analysis involves
extremizing a path length. They showed that this volume for a Schwarzschild
black hole increases as $V\sim 3\sqrt{3}m^{2}v$, in the limit $v>>m$, where
$v$ is the advance time. We used this technique in our earlier work [4] and
had applied it to BTZ black holes. We showed that the constant $r$ segment of
the volume hypersurface can also be derived by demanding a divergence free
extrinsic curvature [7]. In this work we further extend the technique to
charged BTZ black holes. Once we obtain the expression for the volume, we
probe its thermodynamical aspects using semi-classical approach in the near
extremal limit.

##  II General Setup

The Hilbert action $(I_{H})$ coupled to electromagnetism is defined for
charged BTZ black hole as

$\begin{split}I_{H}=\int
d^{3}x\sqrt{-g}\bigg{(}\frac{R-2\Lambda}{2k}-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\bigg{)}\\\
=\frac{1}{2\pi}\int
d^{3}x\sqrt{-g}\bigg{(}R+\frac{2}{l^{2}}-\frac{\pi}{2}F_{\mu\nu}F^{\mu\nu}\bigg{)}\end{split}$
(1)

where, $l$ is the AdS length and $\Lambda$ is the cosmological constant which
satisfies, $\Lambda=-1/l^{2}$ and $k=8\pi G/c^{4}$ is the Einstein
gravitational constant. We use natural units where $G=1/8$ and $\hbar=c=1$.
Hence the factor $1/(16\pi G)=1/(2\pi)$. The metric of a charged BTZ black
hole is defined as

$ds^{2}=-N^{2}(r)dt^{2}+\frac{dr^{2}}{N^{2}(r)}+r^{2}(N^{\phi}dt+d\phi)^{2}$
(2)

where, shift function $N^{\phi}=-J/2r^{2}$ and the lapse function
$N^{2}(r)=-\Lambda
r^{2}-M+\frac{J^{2}}{4r^{2}}-\frac{\pi}{2}Q^{2}ln(\frac{r}{l})$. Here $M$, $J$
and $Q$ are the mass, angular momentum and the charge carried by black hole,
respectively. The metric above exhibits a coordinate singularity at the zeros
of the lapse function so we switch from the Schwarzschild coordinates
$(t,r,\phi)$ to Eddington-Finkelstein coordinates $(v,r,\theta)$ to avoid this
singularity, which are defined as

$v=t+\int^{r}{\frac{dr^{\prime}}{N^{2}(r^{\prime})}}\ and\
\theta=\phi-\int^{r}{\frac{N^{\phi}(r^{\prime})}{N^{2}(r^{\prime})}dr^{\prime}}$
(3)

where, $v$ is the advance time. As the cosmological constant,
$\Lambda=-1/l^{2}$ is negative, the background space is AdS. The metric (2)
can now be written as

$ds^{2}=-\bigg{(}-\Lambda
r^{2}-M-\frac{\pi}{2}Q^{2}ln(\frac{r}{l})\bigg{)}dv^{2}+2dvdr\\\
-Jdvd\theta+r^{2}d\theta^{2}$ (4)

The black hole has an outer and an inner horizon $r_{\pm}^{Q}$. One can define
the volume enclosed between the two horizons by picking a spacelike
hypersurface. As there is a whole family of such hypersurfaces, the volume
enclosed is not unique. We use the CR [1] technique to define the maximal
hypersurface which will enclose maximum volume.

##  III Analytical solution of the maximum volume

###  III.1 The Maximization technique

As discussed earlier, in this work we find the hypersurface which encloses the
maximum volume inside a black hole. CR [1] have shown that the maximum
contribution to such a hypersurface comes from a constant $r$ segment. We
computed such a volume for a BTZ in our earlier work [4]. We now use the
techniques to understand the effects of charge by studying the interior volume
of a charged BTZ black hole. We will be interested only in the constant $r$
segment whose contribution to the volume of the hypersurface is sufficiently
large that we can ignore the remaining segments as justified in [1]. For the
constant $r$ hypersurface, the induced metric (4) can be written as

$ds^{2}=-\bigg{(}-\Lambda
r^{2}-M-\frac{\pi}{2}Q^{2}ln(\frac{r}{l})\bigg{)}dv^{2}-Jdvd\theta+r^{2}d\theta^{2}$
(5)

where, $\Lambda=-1/l^{2}$. Metric tensor $g_{\mu\nu}$ can be written in the
matrix form as

$g_{\mu\nu}=\Bigg{(}\begin{matrix}-f(r)&-J/2\\\
-J/2&r^{2}\end{matrix}\Bigg{)}$ (6)

The determinant of $g_{\mu\nu}$ is,
$g=det(g_{\mu\nu})=-r^{2}f(r)-\frac{J^{2}}{4}$, where
$f(r)=\bigg{(}\frac{r^{2}}{l^{2}}-M-\frac{\pi}{2}Q^{2}ln(\frac{r}{l})\bigg{)}$.
So the volume for constant $r$ hypersurface is

$V_{\Sigma}=2\pi
v\sqrt{-r^{2}\bigg{(}\frac{r^{2}}{l^{2}}-M-\frac{\pi}{2}Q^{2}ln(\frac{r}{l})\bigg{)}-\frac{J^{2}}{4}}$
(7)

This is the volume of charged BTZ black hole interior. We seek the radius
which we call the steady state radius, $r_{ss}^{Q}$, which maximizes this
volume. Such a steady state radius solves the equation $dV_{\Sigma}/dr=0$. So

$\bigg{(}\frac{dV_{\Sigma}}{dr}\bigg{)}_{r=r^{Q}_{ss}}=(r^{Q}_{ss})^{2}-\frac{\pi}{4}Q^{2}l^{2}ln(\frac{r^{Q}_{ss}}{l})-\frac{Ml^{2}}{2}-\frac{\pi
Q^{2}l^{2}}{8}=0$ (8)

To get an analytical solution we work in the limit when the charge $Q<<1$. In
this limit we expect $r^{Q}_{ss}=r_{ss}+\delta$ or
$\frac{r^{Q}_{ss}}{r_{ss}}=1+\frac{\delta}{r_{ss}}$, where $\delta$ is an
infinitesimal parameter $(\frac{\delta}{r_{ss}}\ll 1$ or
$\frac{r^{Q}_{ss}}{r_{ss}}-1\ll 1)$. Substituting the value of $r^{Q}_{ss}$ in
the eq.(8) and using the approximation on logarithmic function i.e.
$\ln(\frac{r^{Q}_{ss}}{l})\approx\ln(\frac{r_{ss}}{l})+\frac{\delta}{r_{ss}}$,
we get

$(r_{ss}+\delta)^{2}-\frac{\pi}{8}Q^{2}l^{2}\bigg{(}1+2ln(\frac{r_{ss}}{l})+\frac{2\delta}{r_{ss}}\bigg{)}-r^{2}_{ss}=0$
(9)

where, $r_{ss}=l\sqrt{\frac{M}{2}}$, is the steady state radius of BTZ black
hole. Solution of equation (9) gives,

$\delta=r^{Q}_{ss}-r_{ss}=\frac{r_{ss}Q^{2}}{2}\Bigg{(}\frac{1+2ln(\frac{r_{ss}}{l})}{\frac{8r^{2}_{ss}}{\pi
l^{2}}-Q^{2}}\Bigg{)}$ (10)

So we obtain the steady state radius for charged BTZ black hole $(r^{Q}_{ss})$
from the eq.(10) as

$r^{Q}_{ss}=r_{ss}\Bigg{[}1+\frac{Q^{2}}{2}\Bigg{(}\frac{1+2ln(\frac{r_{ss}}{l})}{\frac{8r^{2}_{ss}}{\pi
l^{2}}-Q^{2}}\Bigg{)}\Bigg{]}\\\
=l\sqrt{\frac{M}{2}}\Bigg{[}1+\frac{Q^{2}}{2}\Bigg{(}\frac{1+ln(\frac{M}{2})}{\frac{4M}{\pi}-Q^{2}}\Bigg{)}\Bigg{]}$
(11)

###  III.2 Traceless extrinsic curvature technique

One can arrive at the expression for the steady state radius $r_{ss}$ using
another method proposed originally by Bruce L. Reinhart [7]. We noted that the
maximum contribution to the volume is due to the surface r = const. It is
argued in [6, 7] that the trace of the extrinsic curvature of such a surface
should vanish. So we evaluate the trace of extrinsic curvature of the surface
r = cosnt. The unit normal to the surface r = const. for the metric (4) is
given by $n^{\alpha}=(1/\sqrt{f},\sqrt{f},0)$, where
$f(r)=\frac{r^{2}}{l^{2}}-M-\frac{\pi}{2}Q^{2}\ln(\frac{r}{l})$. The condition
for the vanishing trace of extrinsic curvature is same as making the normal
vector divergenceless which is written as $n^{\alpha};_{\alpha}=0$. We note
that the volume element of spacetime with metric given by eq.(4) is
$\sqrt{|g|}=r$. So the condition for the vanishing trace of the extrinsic
curvature reduces to

$n^{\alpha};_{\alpha}=\frac{1}{r}\frac{\partial}{\partial
r}\bigg{(}r\sqrt{\frac{r^{2}}{l^{2}}-M-\frac{\pi}{2}Q^{2}\ln(\frac{r}{l})}\bigg{)}=0$
(12)

the eq.(12) gives

$(r^{Q}_{ss})^{2}-\frac{\pi}{4}Q^{2}l^{2}ln(\frac{r^{Q}_{ss}}{l})-\frac{Ml^{2}}{2}-\frac{\pi
Q^{2}l^{2}}{8}=0$ (13)

which is same as eq.(8). Hence, the solution of the eq.(13) gives same steady
state radius $(r^{Q}_{ss})$ as given in eq.(11) i.e.
$r^{Q}_{ss}=l\sqrt{\frac{M}{2}}\bigg{[}1+\frac{Q^{2}}{2}\bigg{(}\frac{1+ln(\frac{M}{2})}{\frac{4M}{\pi}-Q^{2}}\bigg{)}\bigg{]}$.
Now, substituting the value of $r^{Q}_{ss}$ in the eq.(7) we get the
expression for maximum volume as

$V_{\Sigma}=\int\sqrt{g}dvd\theta=2\pi
v\sqrt{-(r^{Q}_{ss})^{2}\bigg{(}\frac{(r^{Q}_{ss})^{2}}{l^{2}}-M-\frac{\pi}{2}Q^{2}ln(\frac{r^{Q}_{ss}}{l})\bigg{)}-\frac{J^{2}}{4}}$
(14)

or

$V_{\Sigma}=2\pi
v\sqrt{-r^{2}_{ss}\bigg{(}1+\frac{\frac{1}{2}+ln(\frac{r_{ss}}{l})}{\frac{8r^{2}_{ss}}{\pi
Q^{2}l^{2}}-1}\bigg{)}^{2}\Bigg{[}\frac{r^{2}_{ss}}{l^{2}}\bigg{(}1+\frac{\frac{1}{2}+ln(\frac{r_{ss}}{l})}{\frac{8r^{2}_{ss}}{\pi
Q^{2}l^{2}}-1}\bigg{)}^{2}-M-\frac{\pi}{2}Q^{2}ln\frac{r_{ss}}{l}\bigg{(}1+\frac{\frac{1}{2}+ln(\frac{r_{ss}}{l})}{\frac{8r^{2}_{ss}}{\pi
Q^{2}l^{2}}-1}\bigg{)}\Bigg{]}-\frac{J^{2}}{4}}$ (15)

Note here $r_{ss}=l\sqrt{\frac{M}{2}}$, which is the steady state radius in
absence of the charge $Q$ and the expression for volume simplifies to
$V_{\Sigma}=\pi v\sqrt{M^{2}l^{2}-J^{2}}$ when the charge is taken to zero.
The region plot of $V_{\Sigma}$ from eq.(15) as shown below. Here we take the
AdS length $l=4$.

Figure 1:  The graph highlights the region in the parameter space spanned by
the mass $M$, charge $Q$ and angular momentum $J$ of a charged BTZ black hole.
The different shaded regions correspond to angular momentum
$(J=0,0.25,0.50,0.75)$. The volume of hypersurface $V_{\Sigma}$ of black hole
is positive definite and hence well defined inside the shaded regions. In the
region outside, the hypersurface becomes timelike and cannot be used to define
the black hole volume. The boundary of the regions corresponds to lightlike
hypersurface which again cannot be used to define the volume. Here we take the
AdS length $l=4$.

The volume of this hypersurface defining the black hole volume increases
monotonically with the advanced time $v$. As discussed by many [6, 17, 19, 24,
4], such a behaviour can be appropriately used in defining and understanding
the black hole entropy. We will touch on this in latter sections. Note that
the expression for $V_{\Sigma}$ also provides a constraint on the values of
the charge $Q$ and angular momentum $J$ for the $V_{\Sigma}$ to exist. A
negative value for the expression inside the square root would mean the volume
becomes timelike and hence cannot be used to define the black hole volume.
Demanding that the expression inside the square root is positive definite, we
find the region in the parameter space spanned by $Q$ and $J$, which we call
the permitted region for $V_{\Sigma}$ to exist. Note that these graphs in the
figure (1) become less accurate for larger values of $Q$ as we are working
with approximation that $Q<<1$.

###  III.3 Variational method in the auxiliary manifold

In this section we use the formulation developed by Christodoulou et al. [1]
to construct the spacelike hypersurface $\Sigma$ and check if it reconciles
with our results in previous section. We write $\Sigma$, which is a
2-manifold, as a direct product of a 1-sphere defined by $\theta$ and a curve
$\gamma$ embedded in the 2-dimensional space $(v,r)$ paramterized by $\lambda$
as

$\begin{split}\Sigma\sim\gamma\times S^{1}\\\
\gamma\sim(v(\lambda),r(\lambda))\end{split}$ (16)

The line element of the induced metric on the hypersurface $\Sigma$ is

$ds^{2}_{\Sigma}=\big{(}-f(r)\dot{v}^{2}+2\dot{v}\dot{r}\big{)}d\lambda^{2}-J\dot{v}d\lambda
d\theta+r^{2}d\theta^{2}$ (17)

The dot represents derivative with respect to the parameter $\lambda$ and
$f(r)=\frac{r^{2}}{l^{2}}-M-\frac{\pi}{2}Q^{2}\ln(\frac{r}{l})$. For $\Sigma$
to be a spacelike surface, we require

$r^{2}\bigg{(}-f(r)\dot{v}^{2}+2\dot{v}\dot{r}\bigg{)}-\frac{J^{2}\dot{v}^{2}}{4}>0$
(18)

and the proper volume of $\Sigma$ is given by

$V_{\Sigma}[\gamma]=2\pi\int_{0}^{\lambda_{f}}d\lambda\sqrt{r^{2}\bigg{(}-f(r)\dot{v}^{2}+2\dot{v}\dot{r}\bigg{)}-\frac{J^{2}\dot{v}^{2}}{4}}$
(19)

This volume depends on the curve $\gamma$ and we want to find the curve
$\gamma$ which would maximize this volume. This can be viewed as an
extremization problem and our goal then is to find the equations of motion for
the Lagrangian

$L(r,\dot{r},v,\dot{v})=\sqrt{r^{2}\bigg{(}-f(r)\dot{v}^{2}+2\dot{v}\dot{r}\bigg{)}-\frac{J^{2}\dot{v}^{2}}{4}}$
(20)

We will find the equations of motion and then use appropriate normalization to
set

$L(r,\dot{r},v,\dot{v})=1$ (21)

This normalization ensures that $\Sigma$ is spacelike. The curve $\gamma$
parameterized by $\lambda$ has two end points which we choose to be at
$r(\lambda_{i}=0)=r_{+}^{Q}$ and $r(\lambda_{f})=r^{Q}_{-}$, so the curve
stretches between the inner and the outer horizon and $v(\lambda_{i}=0)=v$ and
$v(\lambda_{f})=v_{f}$.

####  III.3.1 Euler-Lagrange equations of motion

The Lagrangian is a function of the variables $v$ and $r$. First we write the
Euler-Lagrangian equation corresponding to $v$ as

$\frac{\partial
L}{\partial\dot{v}}=\frac{1}{2L}\Bigg{(}r^{2}(-2f(r)\dot{v}+2\dot{r})-\frac{J^{2}\dot{v}}{2}\Bigg{)}=a$
(22)

using eqs.(20) and (21) this simplifies to

$\displaystyle\dot{v}\bigg{(}-2r^{2}f(r)-J^{2}/2\bigg{)}=2a-2\dot{r}r^{2}$
$\displaystyle\Rightarrow\dot{v}=\frac{\dot{r}r^{2}-a}{r^{2}f(r)+J^{2}/4}$
(23)

Next using eqs.(22) and (III.3.1), we get

$\dot{v}=\frac{1}{a+\dot{r}r^{2}}$ (24)

and plugging this expression in eq.(20), we find

$\dot{r}=-\frac{1}{r^{2}}\sqrt{r^{2}f(r)+\frac{J^{2}}{4}+a^{2}}$ (25)

The expression for $\dot{v}$ is same as deduced by others for the case of BTZ
black hole and the expression for $\dot{r}$ reduces to the expression for BTZ
black hole as shown by [19] when we put $J$ and $Q=0$.

####  III.3.2 Steady state radius and maximum volume

As we discussed earlier, most of the contribution to the volume of the maximal
hypersurface comes from a steady state segment which corresponds to a constant
$r_{ss}$. We get the expression for $r^{Q}_{ss}$ by taking $\dot{r}=0$ and
equating the right hand side of eq.(25) to zero.

$(r^{Q}_{ss})^{2}\bigg{(}\frac{(r^{Q}_{ss})^{2}}{l^{2}}-M-\frac{\pi}{2}Q^{2}ln(\frac{r^{Q}_{ss}}{l})\bigg{)}+\frac{J^{2}}{4}+a^{2}=0\\\
\Rightarrow
a=\sqrt{-(r^{Q}_{ss})^{2}\bigg{(}\frac{(r^{Q}_{ss})^{2}}{l^{2}}-M-\frac{\pi}{2}Q^{2}ln(\frac{r^{Q}_{ss}}{l})\bigg{)}-\frac{J^{2}}{4}}$
(26)

Note that for $\dot{r}=0$, we have $\dot{v}=1/a$. As $v$ must be a
monotonically increasing function of the parameter $\lambda$, $a$ must be
positive definite. Now the volume of the hypersurface as defined in eq.(19)
can be evaluated as

$V_{\Sigma}=2\pi\int d\lambda=2\pi a\int dv=2\pi av$ (27)

Substituting the expression for $a$ as in eq.(27), we get

$V_{\Sigma}=2\pi
v\sqrt{-(r^{Q}_{ss})^{2}\bigg{(}\frac{(r^{Q}_{ss})^{2}}{l^{2}}-M-\frac{\pi}{2}Q^{2}ln(\frac{r^{Q}_{ss}}{l})\bigg{)}-\frac{J^{2}}{4}}$
(28)

Further substituting the expression for $r^{Q}_{ss}$ obtained from the
traceless extrinsic curvature technique, we obtain

$V_{\Sigma}=2\pi
v\sqrt{-r^{2}_{ss}\bigg{(}1+\frac{\frac{1}{2}+ln(\frac{r_{ss}}{l})}{\frac{8r^{2}_{ss}}{\pi
Q^{2}l^{2}}-1}\bigg{)}^{2}\Bigg{[}\frac{r^{2}_{ss}}{l^{2}}\bigg{(}1+\frac{\frac{1}{2}+ln(\frac{r_{ss}}{l})}{\frac{8r^{2}_{ss}}{\pi
Q^{2}l^{2}}-1}\bigg{)}^{2}-M-\frac{\pi}{2}Q^{2}ln\frac{r_{ss}}{l}\bigg{(}1+\frac{\frac{1}{2}+ln(\frac{r_{ss}}{l})}{\frac{8r^{2}_{ss}}{\pi
Q^{2}l^{2}}-1}\bigg{)}\Bigg{]}-\frac{J^{2}}{4}}$ (29)

This matches with the expression for the volume we derived earlier using the
maximization technique. Hence, the variational method provides a different way
to calculate the volume of a black hole using Euler-Lagrange equation of
motion of a hypothetical particle which is traversing along a geodesic curve
in the auxiliary manifold.

##  IV Numerical solution of the maximum volume

Figure 2: The roots of this graph give the steady state radii $(r^{Q}_{ss})$
of a charged BTZ black hole at different values of black hole parameters
$(M,Q,J)$. Here we take the AdS length $l=1$.

Our analysis till now has been limited to the case when the black hole charge
is very small i.e. $Q<<1$. This was necessary as solving the equation
analytically for the volume with the $ln(r^{Q}_{ss}/l)$ term in eq.(14) is not
possible. However the $Q<<1$ limit provides a good insight into effects of
adding charge on the volume $V_{\Sigma}$ and the steady state radius. Moreover
as we will study a certain thermodynamical aspects of a charged BTZ black hole
in later sections, we will note that such an investigation is possible only in
small charge limit regime. In this section we carry out a numerical analysis
to study the behaviour of the steady state radius $r_{ss}$ and subsequently of
the maximal hypersurface by removing any restrictions on the values of charge
$Q$. The steady state radius $r_{ss}$ is defined by the maximization equation
$dV_{\Sigma}/dr=0$. So we plot $dV_{\Sigma}/dr$ versus $r/l$ seeking the
values of $r$ for which for function has its zeros. The plot is shown in
figure (2), here we take the AdS length $l=1$.

Once we have the values of the the steady state radii $r_{ss}$ for different
mass $M$, charge $Q$ and angular momentum $J$ from the graph, we plug the
values into the expression from eq.(15) and evaluate $V_{\Sigma}$. Here we
tabulate the volumes of charged BTZ and BTZ black hole for a comparison. In
the case of charged BTZ black hole, the volume depends on the charge Q and
evidently its numerical value is smaller than the volume of an BTZ black hole.
The complete table is shown below:

Table: Maximum volume $(V_{\Sigma{max}})$ of charged-BTZ black hole and BTZ
black hole at different value of steady state radius $(r^{Q}_{ss})$ is given
in the following table. Here we are taking AdS length $l=1$.  Mass Charge
Angular Momentum charged-BTZ charged-BTZ charged-BTZ BTZ $M$ $Q$ $J$
$r^{Q}_{ss}$ $r^{Q}_{+}$ ${(V_{\Sigma{max}})}/{2\pi v}$
${(V_{\Sigma{max}})}/{2\pi v}$ 0.4 0.2 0.2 0.45 0.47 0.11 0.18 0.5 0.2 0.2
0.47 0.67 0.20 0.23 0.6 0.2 0.2 0.53 0.75 0.26 0.29 0.7 0.2 0.2 0.57 0.82 0.32
0.34 0.8 0.2 0.2 0.62 0.88 0.37 0.39 0.9 0.2 0.2 0.66 0.94 0.42 0.44 From the
above table we note that the event horizon is always larger than the steady
state radius, $r^{Q}_{+}>r^{Q}_{ss}$. This means that the steady state radius
$r^{Q}_{ss}$ is always hidden behind the event horizon $r^{Q}_{+}$. By adding
a small charge $Q$ we find that the volume of charged BTZ black hole decreases
and less than the BTZ black hole.

##  V Analytical solution of the horizons

The horizons of a charged BTZ black hole are the roots of the lapse function
$N^{2}(r)$ which is defined as

$\begin{split}N^{2}(r)=\frac{r^{2}}{l^{2}}-M+\frac{J^{2}}{4r^{2}}-\frac{\pi}{2}Q^{2}ln(\frac{r}{l})\end{split}$
(30)

The outer and the inner horizons are given by the zeros of
$[N^{2}(r)]_{r=r^{Q}_{\pm}}=0$. Again, due to the presence of the log term,
the equation does not have an analytical solution and we rely on perturbative
approach in the limit when charge $Q\rightarrow 0$. In this limit the horizons
are only slightly displaced from the horizons of the BTZ black hole [4] given
by
$r_{\pm}^{2}=\frac{Ml^{2}}{2}\big{(}1\pm\sqrt{1-(\frac{J}{Ml})^{2}}\big{)}$.
Let $r^{Q}_{\pm}=r_{\pm}+\delta$ or
$\frac{r^{Q}_{\pm}}{r_{\pm}}=1+\frac{\delta}{r_{\pm}}$, where $\delta$ is an
infinitesimal parameter such that $\frac{\delta}{r_{\pm}}\ll 1$ or
$\frac{r^{Q}_{\pm}}{r_{\pm}}-1\ll 1$. So
$\log(\frac{r^{Q}_{\pm}}{l})\approx\log(\frac{r_{\pm}}{l})+\frac{\delta}{r_{\pm}}$.
With this approximation, the horizon equation can be written as

$\frac{(r_{\pm}+\delta)^{2}}{l^{2}}-M+\frac{J^{2}}{4(r_{\pm}+\delta)^{2}}-\frac{\pi}{2}Q^{2}\bigg{(}ln(\frac{r_{\pm}}{l})+\frac{\delta}{r_{\pm}}\bigg{)}=0$
(31)

Solving eq.(31) to the first order in $\delta$ gives

$\delta=r^{Q}_{\pm}-r_{\pm}=\frac{\pi
Q^{2}r_{\pm}ln(\frac{r_{\pm}}{l})}{4\bigg{(}\frac{r^{2}_{\pm}}{l^{2}}-\frac{J^{2}}{r^{2}_{\pm}}-\frac{\pi
Q^{2}}{4}\bigg{)}}$ (32)

Hence, the horizons of a charged BTZ black hole $(r^{Q}_{\pm})$ are

$r^{Q}_{\pm}=r_{\pm}\Bigg{[}1+\frac{\pi
Q^{2}ln(\frac{r_{\pm}}{l})}{4\bigg{(}\frac{r^{2}_{\pm}}{l^{2}}-\frac{J^{2}}{r^{2}_{\pm}}-\frac{\pi
Q^{2}}{4}\bigg{)}}\Bigg{]}\\\ =r_{\pm}\Bigg{[}1+\frac{\pi
Q^{2}ln(\frac{r_{\pm}}{l})}{4\bigg{(}M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}}\Bigg{]}$ (33)

As we take the limit $Q<<1$, the denominator in the above expression is
positive definite and the term $ln(r_{\pm}/l)$ in the numerator is negative as
$r_{\pm}<l$. So, the sign of $\delta$ in the eq.(32) is negative and hence
$r_{\pm}^{Q}<r_{\pm}$. We note that the position of the horizons of the
charged BTZ with respect to the neutral one does not depend on the sign but
the magnitude of the charge $Q$. The region plot of event horizon shown in
figure (3).

Figure 3: Graph between Mass $(M)$ and charge $(Q)$ of charged BTZ black hole
for different values of the angular momentum J. The shaded regions represent
the parameters values which allow the existence of horizons and unshaded
regions correspond to the parameter values which forbid horizons to exist.
Here we take the AdS length $l=1$

##  VI Numerical solution of the horizons

Now we investigate the parameter space spanned by mass $M$, charge $Q$ and
angular momentum $J$ to understand how the black hole horizons vary over
different regions. The existence of the horizons is guaranteed only when the
lapse function has real zeros. Generally the lapse function has two roots
which correspond to the two horizons. But, for a certain values of the
charges, the lapse function may have only one root and that would correspond
to an extremal black hole. In this section we analyse these regions by
imposing different conditions on the black hole parameters. We illustrate
these cases in the figure (4) by taking different values of the black hole
parameters to get all the possible outcomes. Let us consider the case of an
extremal black hole which is obtained when the lapse function has only one
root. This is possible only when the minima of the lapse function touches the
$r/l$-axis in $N^{2}(r)-r/l$ plot. We solve the equation,
$\big{(}\frac{d[N^{2}(r)]}{dr}\big{)}_{r=r_{0}}=0$, to get the value of
$r^{2}_{0}$ as

$r^{2}_{0}=\frac{l^{2}}{8}\bigg{(}\pi
Q^{2}+\sqrt{\pi^{2}Q^{4}+\frac{16J^{2}}{l^{2}}}\bigg{)}$ (34)

. We check that at $r=r_{0}$, $\frac{d^{2}[N^{2}(r)]}{dr^{2}}>0$, so
$r=r_{0}=r_{min}$ and the polynomial $N^{2}(r)$ has a minima at $r_{0}$.
Hence, the minimum value of $N^{2}(r)$ is

$N^{2}(r_{min})=\frac{r^{2}_{min}}{l^{2}}-M+\frac{J^{2}}{4r^{2}_{min}}-\frac{\pi}{2}Q^{2}ln(\frac{r_{min}}{l})$
(35)

Plugging the expression for $r_{min}$ from eq.(34), we get

$N^{2}(r_{min})=-M+\frac{\pi
Q^{2}+\sqrt{\pi^{2}Q^{4}+\frac{16J^{2}}{l^{2}}}}{8}\\\
+\frac{2J^{2}}{l^{2}\bigg{(}\sqrt{\pi^{2}Q^{4}+\frac{16J^{2}}{l^{2}}}\bigg{)}}\\\
-\frac{\pi}{4}Q^{2}ln\bigg{[}\frac{1}{8}\bigg{(}\pi
Q^{2}+\sqrt{\pi^{2}Q^{4}+\frac{16J^{2}}{l^{2}}}\bigg{)}\bigg{]}=-M+F(J,Q)$
(36)

where, $F(J,Q)=\frac{\pi
Q^{2}+\sqrt{\pi^{2}Q^{4}+\frac{16J^{2}}{l^{2}}}}{8}+\frac{2J^{2}}{l^{2}\bigg{(}\sqrt{\pi^{2}Q^{4}+\frac{16J^{2}}{l^{2}}}\bigg{)}}-\frac{\pi}{4}Q^{2}ln\bigg{[}\frac{1}{8}\bigg{(}\pi
Q^{2}+\sqrt{\pi^{2}Q^{4}+\frac{16J^{2}}{l^{2}}}\bigg{)}\bigg{]}$
The graph between $N^{2}(r)$ vs. $r/l$ as shown below. Note that we take the
AdS length $l=1$ here.

Figure 4: Lapse function $N^{2}(r)$ versus $r/l$ for different values of black
hole parameters $(M,Q,J)$. Blue curve depicts the existence of two different
horizons, the red curve shows the case when both the horizons overlap and
green curve depicts the case when there is no horizon.

There exist three possible situations which is arising from the eq.(36) and we
investigate them one at a time.
Case (a). When $N^{2}(r_{min})<0$ : In this case both the roots of the
polynomial $N^{2}(r)$ are real and distinct i.e. $r=r^{Q}_{\pm}$. This is the
normal charged BTZ black hole. This is the general condition for the existence
of two horizons of a charged BTZ black hole. If charge Q = 0, then we get
$M>J/l$ or $J<Ml$, which is the normal case of a BTZ black hole. So, from the
eq.(36) for $N^{2}(r_{min})<0$, we obtain an inequality relation between mass
$M$ and polynomial $F(J,Q)$ as

$M-F(J,Q)>0$ (37)

Case (b). When $N^{2}(r_{min})=0$ : In this case both root of the polynomial
$N^{2}(r)$ are real and equal i.e. $r=r_{-}=r_{+}$. This is the extremal case
of a charged BTZ black hole in which both horizons coincide. If charge Q = 0
then, we get $M=J/l$ or $J=Ml$, which is the extremal case for a BTZ black
hole. So, from the eq.(36) for $N^{2}(r_{min})=0$, we get the relation between
mass $M$ and the polynomial $F(J,Q)$ as

$M-F(J,Q)=0$ (38)

Case (c). When $N^{2}(r_{min})>0$ : In this case both the roots of the
polynomial $N^{2}(r)$ are imaginary and hence the horizons do not exist. This
is the case of naked singularity. If charge Q = 0, then we get the inequality
$M<J/l$ or $J>Ml$, which is the case of naked singularity of a BTZ black hole.
So, from the eq.(36) for $N^{2}(r_{min})>0$ we get the inequality relation
between mass $M$ and polynomial $F(J,Q)$ as

$M-F(J,Q)<0$ (39)

The graphical representation of the nature of black hole for all three
possible cases as discussed above is depicted in the figure (5).

Figure 5:  Graph between black hole mass $M$ and charge $Q$ for different
values of the angular momentum $J$. Here the shaded regions represent a normal
charged BTZ black hole with well defined horizons, the unshaded region
represent a naked charged BTZ black hole and the boundary between these two
regions represents the extremal charged BTZ case.

##  VII Entropy of a massless scalar field in the black hole volume

After understanding the geometry of spacelike hypersurfaces inside the charged
BTZ black hole, we now try to gain some insight into the process of state
counting in the interior of a black hole. Since a black hole evaporates and
shrinks with time through Hawking radiation, it is not in an equilibrium
state. Moreover the metric inside a black hole is not static. In general we
cannot do a state counting in such a dynamical background. However we can
avoid this problem by considering a near extremal black hole which has the
horizon temperature close to absolute zero. One can indeed apply the
techniques of quantum statistics to do a state counting in such a quasi-static
background and we work in this limit through the rest of the analysis. Now we
count the accessible states in the phase space [5]. The 4-dimensional phase
space for a charged BTZ black hole is spanned by the conjugate variables
$v,\theta,p_{v}$ and $p_{\theta}$. One quantum state corresponds to a cell of
volume $(2\pi\hbar)^{2}$ in this 4-dimensional phase space. The total number
of quantum states would be given by the integral of the measure

$\frac{dvd\theta dp_{v}dp_{\theta}}{(2\pi\hbar)^{2}}$ (40)

over the accessible region. To find the accessible region, we introduce a
massless scalar field on the hypersurface $\Sigma$ which satisfies the Klein-
Gordon equation,
$\frac{1}{\sqrt{g}}\partial_{\mu}(\sqrt{g}g^{\mu\nu}\partial_{\nu}\phi)=0$. As
we focus on the part of $\Sigma$ which corresponds to $\dot{r}=0$, the line
element as given in eq.(17) would reduce to
$ds^{2}=-f(r^{Q}_{ss})dv^{2}-Jdvd\theta+(r^{Q}_{ss})^{2}d\theta^{2}$. Hence,
the metric tensor $g_{\mu\nu}$ define as

$g_{\mu\nu}=\Bigg{(}\begin{matrix}-f(r^{Q}_{ss})&-\frac{J}{2}\\\
\frac{-J}{2}&(r^{Q}_{ss})^{2}\end{matrix}\Bigg{)}$ (41)

Let $\lambda_{v}$ and $\lambda_{\theta}$ be the eigenvalues of the above
metric $g_{\mu\nu}$. So, we get $\lambda_{v}$$\lambda_{\theta}$ = g =
det$(g_{\mu\nu})$. The Klein-Gordon equation in the momentum space gives the
following relation

$E^{2}=\frac{1}{\lambda_{v}}p_{v}^{2}+\frac{1}{\lambda_{\theta}}p_{\theta}^{2}$
(42)

Now we count the number of states in the phase space with energy less than E.
That would be written as

$g(E)=\frac{1}{(2\pi\hbar)^{2}}\int\int dvd\theta dp_{v}dp_{\theta}\\\
=\frac{1}{(2\pi\hbar)^{2}}\int\
dvd\theta\int_{-E/\sqrt{\lambda_{\theta}}}^{E/\sqrt{\lambda_{\theta}}}dp_{\theta}\bigg{[}\int_{-\sqrt{(E^{2}-\lambda_{\theta}p_{\theta}^{2})/\lambda_{\theta}}}^{\sqrt{(E^{2}-\lambda_{\theta}p_{\theta}^{2})/\lambda_{v}}}dp_{v}\bigg{]}\\\
=\frac{\pi
E^{2}}{(2\pi\hbar)^{2}}\int\sqrt{\lambda_{v}\lambda_{\theta}}dvd\theta=\frac{\pi
E^{2}}{(2\pi\hbar)^{2}}\int\sqrt{g}dvd\theta=\frac{E^{2}V_{\Sigma}}{4\pi}$
(43)

where, we put $\hbar=1$. The limits of both the integrals are
$P_{v}=\pm\sqrt{(E^{2}-\lambda_{\theta}p_{\theta}^{2})/\lambda_{v}}$ and
$p_{\theta}=\pm\sqrt{E/\lambda_{\theta}}$. Now the free energy for the scalar
field at inverse temperature $\beta=(k_{B}T)^{-1}$ is defined as

$F(\beta)=\frac{1}{\beta}\int dg(E)ln(1-e^{-\beta E})\\\
=\frac{1}{\beta}\int\frac{g(E)(-\beta)e^{-\beta E}dE}{1-e^{-\beta
E}}=-\int\frac{g(E)dE}{e^{\beta E}-1}$ (44)

From eqs.(43) and (44), we get

$F(\beta)=-\frac{v}{2}\int\frac{E^{2}dE}{e^{\beta
E}-1}=-\frac{\zeta(3)V_{\Sigma}}{2\pi\beta^{3}}$ (45)

where, $\zeta(3)$ is the Riemann zeta function. We then obtain the entropy of
a scalar field living on the maximal hypersurface $\Sigma$ corresponding to
maximal slicing as

$S_{\Sigma}=-\frac{\partial F}{\partial T}=k_{B}\beta^{2}\frac{\partial
F}{\partial\beta}=\frac{3\zeta(3)V_{\Sigma}}{2\pi\beta^{2}}\propto\frac{v}{\beta^{2}}$
(46)

where, we takes the Boltzmann constant $(k_{B})=1$ in final expression. The
eq.(46) shows that the entropy of the maximal hypersurface $\Sigma$
corresponding to maximal slicing grows linearly with advance time $(v)$. The
inverse temperature of the field here can be well approximated by the horizon
temperature $T_{H}$ in the near extremal limit. Now the horizon temperature of
a black hole is defined as

$T_{H}=\bigg{(}\frac{\partial S_{H}}{\partial M}\bigg{)}_{J,Q}^{-1}$ (47)

where, $S_{H}$ is the horizon entropy of the black hole. We now find the
relation between the advance time, the field temperature and the black hole
parameters as follows.

In 1972, Jacob Bekenstein [9] proposed that black hole should have entropy
which is proportional to the area of the horizon. In 1974 Stephen Hawking [10,
11] used quantum field theoretical calculations in curved space time to show
that black holes emit thermal radiation corresponding to a certain
temperature. Further, using the thermodynamic relationship between energy,
temperature and entropy, Hawking was able to confirm Bekenstein’s proposal and
fix the constant of proportionality at 1/4. The horizon entropy is also known
as Bekenstein-Hawking entropy which is proportional to surface area of the
black hole. For the case of 2+1 dimensional BTZ black hole, the horizon
entropy $S_{H}$ is found to be two times the horizon length $L_{H}$ [12] i.e.

$S_{H}=\frac{2L_{H}}{l_{p}}=\frac{4\pi r^{Q}_{+}}{l_{p}}$ (48)

where, $l_{p}=\sqrt{\hbar G/c^{3}}$ is the Planck length and $r^{Q}_{+}$ is
the event horizon of the charged BTZ black hole. Substituting the value of
$r^{Q}_{+}$ from the eq.(33) we get the horizon entropy as

$S_{H}=\frac{4\pi r_{+}}{l_{p}}\Bigg{[}1+\frac{\pi
Q^{2}ln(\frac{r_{+}}{l})}{4(\frac{r^{2}_{+}}{l^{2}}-\frac{J^{2}}{4r^{2}_{+}}-\frac{\pi
Q^{2}}{4})}\Bigg{]}\\\ =\frac{4\pi r_{+}}{l_{p}}\Bigg{[}1+\frac{\pi
Q^{2}ln(\frac{r_{+}}{l})}{4\bigg{(}M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}}\Bigg{]}$ (49)

We proceed by plugging eq.(48) into the expression for horizon temperature of
charged BTZ black hole as defined in eq.(47) to get

$T_{H}=\bigg{(}\frac{\partial S_{H}}{\partial
M}\bigg{)}_{J,Q}^{-1}=\frac{1}{4\pi}\bigg{(}\frac{\partial r^{Q}_{+}}{\partial
M}\bigg{)}_{J,Q}^{-1}$ (50)

Now differentiating the horizon radius $r_{+}^{Q}$ with respect to the black
hole mass $M$ we get

$\bigg{(}\frac{\partial r^{Q}_{+}}{\partial M}\bigg{)}_{J,Q}=\frac{\partial
r_{+}}{\partial
M}\Bigg{[}\frac{\bigg{(}\frac{r^{2}_{+}}{l^{2}}-\frac{J^{2}}{4r^{2}_{+}}-\frac{\pi
Q^{2}}{4}\bigg{)}^{2}+\frac{\pi
Q^{2}}{4}\Bigg{\\{}\bigg{(}\frac{r^{2}_{+}}{l^{2}}-\frac{J^{2}}{4r^{2}_{+}}-\frac{\pi
Q^{2}}{4}\bigg{)}-\bigg{(}\frac{r^{2}_{+}}{l^{2}}+\frac{3J^{2}}{4r^{2}_{+}}-\frac{\pi
Q^{2}}{4}\bigg{)}ln\bigg{(}\frac{r_{+}}{l}\bigg{)}\Bigg{\\}}}{\bigg{(}\frac{r^{2}_{+}}{l^{2}}-\frac{J^{2}}{4r^{2}_{+}}-\frac{\pi
Q^{2}}{4}\bigg{)}^{2}}\Bigg{]}\\\ =\frac{\partial r_{+}}{\partial
M}\Bigg{[}\frac{\bigg{(}M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}^{2}+\frac{\pi
Q^{2}}{4}\Bigg{\\{}\bigg{(}M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}-\bigg{(}2M-M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}ln(\frac{r_{+}}{l})\Bigg{\\}}}{\bigg{(}M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}^{2}}\Bigg{]}$ (51)

We use the expression for the event horizon $r_{+}$ of a BTZ black hole in 2+1
dimensions [4, 12] to get

$r^{2}_{+}=\frac{Ml^{2}}{2}\Bigg{(}1+\sqrt{1-(\frac{J}{Ml})^{2}}\Bigg{)}\Rightarrow\frac{\partial
r_{+}}{\partial
M}=\frac{r_{+}}{2M\sqrt{1-(\frac{J}{Ml})^{2}}}=\frac{r_{+}l^{2}}{2(r^{2}_{+}-r^{2}_{-})}$
(52)

then using the eqs.(50), (51) and (52), we get the horizon temperature
$(T_{H})$ for charged BTZ black hole as

$\begin{split}T_{H}=\frac{(r^{2}_{+}-r^{2}_{-})}{2\pi
r_{+}l^{2}}\Bigg{[}\frac{\bigg{(}\frac{r^{2}_{+}}{l^{2}}-\frac{J^{2}}{4r^{2}_{+}}-\frac{\pi
Q^{2}}{4}\bigg{)}^{2}}{\bigg{(}\frac{r^{2}_{+}}{l^{2}}-\frac{J^{2}}{4r^{2}_{+}}-\frac{\pi
Q^{2}}{4}\bigg{)}^{2}+\frac{\pi
Q^{2}}{4}\Bigg{\\{}\bigg{(}\frac{r^{2}_{+}}{l^{2}}-\frac{J^{2}}{4r^{2}_{+}}-\frac{\pi
Q^{2}}{4}\bigg{)}-\bigg{(}\frac{r^{2}_{+}}{l^{2}}+\frac{3J^{2}}{4r^{2}_{+}}-\frac{\pi
Q^{2}}{4}\bigg{)}ln\bigg{(}\frac{r_{+}}{l}\bigg{)}\Bigg{\\}}}\Bigg{]}\\\
=\frac{\frac{M}{2\pi
r_{+}}\sqrt{1-(\frac{J}{Ml})^{2}}\bigg{(}M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}^{2}}{\bigg{(}M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}^{2}+\frac{\pi
Q^{2}}{4}\bigg{[}\bigg{(}M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}-\bigg{(}2M-M\sqrt{1-(\frac{J}{Ml})^{2}}-\frac{\pi
Q^{2}}{4}\bigg{)}ln(\frac{r_{+}}{l})\bigg{]}}\end{split}$ (53)

As discussed earlier, we need to approach the extremal limit for the
statistical counting to make sense. In the near extremal limit the Hawking
temperature tends to zero and a quasi static limit is attained. For a BTZ
black hole, this limit is attained as the mass tends to the angular momentum
$(Ml\rightarrow J)$. In our case of a charged BTZ, we further have to make the
charge $Q$ very small to get $T_{H}\rightarrow 0$. So at the near extremal
limit, $Ml-J=\delta=lQ^{2}$, from eq.(53) we get the horizon temperature to be
$T_{H}\propto-\frac{1}{l}\frac{\sqrt{\delta/l}}{ln(r_{+}/l)}\propto\frac{1}{l}\frac{\sqrt{\delta/l}}{|ln(r_{+}/l)|}$.
Note that the horizon $r_{+}$ is smaller than the AdS length $l$ so,
$ln(r_{+}/l)$ is a negative quantity. The expression for the Hawking
temperature is positive definite and tends to zero in the near extremal limit.
Now as the Hawking radiation is thermal in nature, the mass loss of the black
hole per unit horizon area can be written using the Stefan-Boltzmann law for
2+1 dimensions which is defined as

$\frac{1}{r^{Q}_{+}}\frac{dM}{dv}=-\sigma T_{H}^{3}$ (54)

where, $\sigma=5.670374419\times 10^{-8}\frac{W}{m^{2}K^{4}}$ is the Stefan-
Boltzmann constant. We want to write the advance time $v$ as a function of the
black hole parameters so we integrate this equation in the near extremal limit
in which the event horizon of charged BTZ black hole becomes
$r^{Q}_{+}=l\sqrt{\frac{M}{2}}\bigg{[}1+\frac{\pi}{4}\sqrt{\frac{\delta}{2Ml}}ln\bigg{(}\sqrt{\frac{r_{+}}{l}}\bigg{)}\bigg{]}\approx
l\sqrt{\frac{M}{2}}$. Now eq.(54) can be written as

$v=-\frac{1}{\sigma}\int\frac{dM}{r^{Q}_{+}T^{3}_{H}}\approx\frac{\sqrt{2}l^{2}}{\sigma}\bigg{(}\sqrt{\frac{l}{\delta}}\bigg{)}^{3}\int\frac{[ln(\sqrt{M/2})]^{3}}{\sqrt{M}}dM$
(55)

after solving the integral (55) we get

$v\approx\frac{6\sqrt{2}l^{2}\sqrt{M}[ln(\frac{r_{+}}{l})]^{2}}{\sigma(\sqrt{\delta/l})^{3}}\bigg{[}1-\frac{1}{3}ln(\frac{r_{+}}{l})-2\bigg{(}\frac{1-ln(\frac{r_{+}}{l})}{[ln(\frac{r_{+}}{l})]^{2}}\bigg{)}\bigg{]}$
(56)

Here, $\beta=(k_{B}T)^{-1}=(k_{B}T_{H})^{-1}\propto
l\frac{|ln(\frac{r_{+}}{l})|}{k_{B}\sqrt{\delta/l}}$. Now dividing the eq.(56)
by $\beta^{2}$, we get

$\frac{v}{\beta^{2}}\approx\frac{6\sqrt{2}\sqrt{M}k^{2}_{B}}{\sigma}\sqrt{\frac{l}{\delta}}\bigg{[}1-\frac{1}{3}ln(\frac{r_{+}}{l})-2\bigg{(}\frac{1-ln(\frac{r_{+}}{l})}{[ln(\frac{r_{+}}{l})]^{2}}\bigg{)}\bigg{]}$
(57)

Hence, from the eqs.(46) and (57), we get the entropy of the scalar field
living on the maximal hypersurface $\Sigma$ to be

$S_{\Sigma}\propto\frac{r_{+}}{l}\bigg{[}1-\frac{1}{3}ln(\frac{r_{+}}{l})-2\bigg{(}\frac{1-ln(\frac{r_{+}}{l})}{[ln(\frac{r_{+}}{l})]^{2}}\bigg{)}\bigg{]}$
(58)

In this section we calculated the volume entropy using a semi-classical
approach. As the horizon entropy of the black hole is proportional to the
horizon length i.e. $S_{H}\propto r_{+}$, evidently the two expressions do not
match. The second and the third terms inside the bracket in eq.(58) are not
small compared to 1 and hence cannot be thought of as corrections. In fact if
we try to make the second term small by choosing an appropriate ratio
$r_{+}/l$, the third term becomes large and vice versa. We note that the two
terms together are less than $0.5$ only if $r_{+}/l<0.1$ and $S_{H}$ and
$S_{\Sigma}$ are of same order. However, in general the formula for the
internal entropy $S_{\Sigma}$ exhibits quite a different functional behavior
compare to the horizon entropy $S_{H}$. This is a different result from our
earlier work [4], where we found that both $S_{\Sigma}$ and $S_{H}$ were
proportional to each other. Other earlier works [6, 17, 19, 24, 25] have also
found the two entropies to be proportional to each other. But all these
results were restricted to neutral black holes. Evidently the different
functional form of entropy is a result of introducing charge $Q$ in the
metric. It will be interesting to understand the exact physical reason for
this difference which we plan to investigate in our future work.

##  VIII Conclusions

In this work we have extended the idea of maximal slicing proposed by Bruce L.
Reinhart [7] and indendently by F. Estabrook [8] and the variational method to
setup the Euler-Lagrange equation of motion by Christodoulou and Rovelli [1]
for Schwarzschild black hole, to the case of a charged BTZ black hole. It is
also an extension of our earlier work in which we had probed a BTZ black hole
[4]. The charge appears as a logarithmic term in the metric which makes it
difficult to find an analytical solution. So a part of our work is done in the
small charge limit. We have computed the steady state radius and the volume of
the maximal spacelike hypersurface in this limit. The expression for the
volume allowed us to impose constraints on the black hole parameters mass $M$,
Charge $Q$ and angular momentum $J$. We further investigated the system
numerically and also analysed the locations of the black hole horizons with
respect to the steady state radius. We found the conditions on the black hole
parameters for which the horizons exist which is necessary for the maximal
slicing technique to be valid. We then investigated a certain thermodynamical
aspects of the maximal volume. We do this by introducing a massless scalar
field on the maximal hypersurface and computing the entropy in the near
extremal limit of black hole. The near extremal limit was necessary to justify
a static background in which quantum statics could be applied. We found the
expression for the volume entropy $S_{\Sigma}$ as a function of horizon
length. This volume entropy $S_{\Sigma}$ is found to exhibit a very different
functional behaviour compared to the Bekenstein-Hawking horizon entropy
$S_{H}$. It will be interesting to study this different nature of the two
entropy functions.

###### Acknowledgements.

We would like to thank our institute BITS Pilani Hyderabad campus for
providing the required infrastructure to carry out this research work. Suraj
Maurya would further like to thank Government of India for providing CSIR
(NET-JRF) fellowship to support this research work.

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# Possible Universal Limit for Valence Parton Distributions

Christopher Leon, Misak M. Sargsian  Department of Physics, Florida
International University, Miami, FL 33199 USA

###### Abstract

We report the observation of the existence of a possible universal limit for
valence parton distributions that should exist once partonic degrees of
freedom are relevant for high energy scattering from strongly interacting
bound systems like a nucleon, meson or a few nucleon system at very short
distances. Our observation is based on the notion that the Bjorken x weighted
valence parton distribution function has a peak, $x_{p}$, that characterizes
the average momentum fraction carried out by the valence quarks in the system.
Within the residual mean-field model of the valence quark distribution we
found that $x_{p}$ has an upper limit: $x_{p}\leq{1\over 2(n_{V}-1)}$, where
$n_{V}$ is the number of valence quarks which can be considered in the cluster
embedded in the strongly interacting environment of the bound system. The
existence of such a limit imposes a new constraint on choosing the starting
resolution scale $Q_{0}$ for PDFs. Our prediction for the nucleon is that
$x_{p}\mid_{Q\to Q_{0}}\leq{1\over 4}$, which is in agreement with all the
available valence PDFs that employ the standard approach for selecting
starting $Q_{0}$. We also demonstrate how the existence of this limit can be
used to check the onset of quark-clusters in short range nucleon correlations
in nuclei.

##  I Introduction

Valence quarks are one of the most important constituents of hadrons, defining
their baryonic number and representing “effective” fermions interacting
mutually and with the hadronic interior. These interactions being non-
perturbative pose significant challenges in their description within Quantum
Chromodynamics (QCD). One of the quantities representing a testing ground for
a non-perturbative QCD description of valence quarks are the partonic
distribution functions (PDFs). During the last several decades there have been
extensive efforts in the phenomenological extraction of valence PDFs from the
analysis of deep inelastic scattering (DIS) from hadronic targets and Drell-
Yan processescteq18 ; cj15 ; cteq14 . There are few empirical constraints on
the behavior of valence PDFs and possible new constraints are important for
the precision of the extraction of PDFs in global analyses. Theoretical
modeling of valence PDFs and comparing them with phenomenological PDFs is one
direction that allows the introduction of additional constraints, progressing
the understanding of the valence structure of hadrons.

Even though the shapes of the valence PDFs are not observables, many
theoretical models however predict specific shapes at different ranges of
Bjorken x. The feedback of such modeling is that their predictions are used in
constructing the ansatzes for PDF parameterizations. For example, a
$(1-x)^{N}$ asymptote at large x is predicted from the quark counting rule,
according to which $N=2n-3+2|\lambda_{h}-\lambda_{i}|$, where $n$ is the
number of valence quarks and $\lambda_{h}$ and $\lambda_{i}$ are the
helicities of the hadron and struck quark, respectively. Similarly, the
$x^{s}$ behavior for valence quarks is expected at small x following from
dominance of Regge dynamics with exchanged spin $s=0.5$. Both of these
predictions are used in the ansatzes of phenomenological PDFs.

Figure 1: The valence and sea PDFs for nucleon using the CJ15nlocj15  and
NNPDF31nnloNNPDF:2017mvq  sets evaluated at $Q_{0}^{2}=4\>GeV^{2}$. Error
bands show the Hessian 68 % confidence level.

In the present work we focus on another unique characteristic of valence PDFs,
that is the x-weighted distribution has a distinguishable peaking property for
all approximation orders, factorization, and renormalization scales (Fig.1).
The QCD evolution shifts the peak position of the distribution at smaller x
values resulting in an interesting correlation between $x_{p}$ and the height
of the peak in the formLeon:2020nfb :

$x_{p}(Q^{2})q_{V}(x_{p},Q^{2})=h(x_{p})=Ce^{Dx_{p}(Q^{2})},\vspace{-0.2cm}$
(1)

where $C$ and $D$ are constants. At $Q^{2}$ large enough that quark degrees of
freedom are relevant, and small enough such that the QCD evolution does not
shift the peak position substantially towards smaller x, the peak position
within the partonic picture characterizes the average momentum fraction
carried by the interacting parton. For the nucleon, the naive expectation
(e.g. Ref.Halzen:1984mc ) is that $x_{p}={1\over 3}$, indicating that each
parton carries one third of the overall longitudinal momentum of the nucleon.
Such a result is also expected for a simple non-relativistic constituent quark
model. However, a relativistic description of a three-quark system on the
Light-Front (LF) deals with more complicated dynamics of momentum sharing
between quarks, resulting in, for example Brodsky:1981jv , a position of the
peak at $x_{p}=0.2$ (see Fig.2).

Figure 2: Peak position of $xu_{V}(x)$ distribution compared with prediction
of different models calculating valence PDFs.

Moreover, as Fig.1 shows, the position of the peak for valence $d$ quarks is
systematically lower than that of the $u$-quarks. To the best of our knowledge
this inequality first was addressed within the di-quark modelClose:1988br ;
Roberts:2013mja ; Lu:2022cjx  in which it was observed that the magnitude of
$x_{p}$ is related to the mass of the recoil di-quark, $m_{d}$, in the form:
$x_{p}\approx(1-{m_{d}\over m_{N}})$. In this picture the empirical
observation that $x_{p}^{d}<x_{p}^{u}$ follows from the expectation that the
vector di-quark being a spectator in the valence d-quark distribution is
heavier than the scalar di-quark for the case of valence u-quark distribution.
In the di-quark model, however, one needs a dynamical model generating masses
for di-quarks $\sim 500-700$ MeV/c. In our spectral function approach
described in the text, we give a different explanation of this difference
based on the difference in masses of the residual systems of hadrons.

This article is arranged as follows: in Sec. II we present a spectral function
approach to modeling hadrons, outline the assumptions and derive an expression
for the valence PDF in the residual mean-field model. In Sec. III, we present
predictions, specifically for $x_{p}$ and bound it with an upper limit for all
parameterizations. This limit is tested using existing phenomenologically
extracted PDFs. Finally, Sec IV contains a summary and outlook. The appendices
contain technical calculations and a numerical investigation of $x_{p}$ with
different parameterizations.

##  II Spectral Function Approach in Description of Valence Quark
Distributions

Recently we developed a new approachLeon:2020bvt  in which the dynamics of the
valence quarks in the nucleon is described through a spectral function
framework that distinguishes between the massless valence quark cluster and a
residual nucleon system, the latter being characterized by the residual mass
distribution $m_{R}$. The model resulted in a prediction that the above
discussed peak position of the valence PDFs are defined according to:

$x_{p}={1\over 4}(1-{m_{R}\over M_{N}}),\vspace{-0.0cm}$ (2)

where $M_{N}$ is the mass of the nucleon. Due to the expected different masses
for the residual system, depending on whether the u- or d- valence quark is
being considered, the model predicted $x_{p}^{d}<x_{p}^{u}$ in accordance with
empirical data (Fig.1). Another interesting consequence of relation (2) is
that it predicts an upper limit of $x^{max}_{p}={1\over 4}$.

To generalize the above relation for any hadron (including also nuclei in the
extreme condition of quark-cluster correlations), we present the derivation of
partonic distribution function in leading-order approximation for a system
consisting of n- massless valence quark cluster embedded in a strongly
interacting bound system.

###  II.1 The valence PDF of a “hadron” containing of n-valence quarks

In the spectral function approach we separate a “hadron” containing of
n-valence quarks into a cluster of n-valence quarks and a residual system
consisting of sea-quarks and gluons (see Fig. 3). In this scenario the valence
quarks are embedded in the residual field, thus we refer it as residual mean-
field (RMF) model, which was first applied to the case of the nucleon
Leon:2020cev ; Leon:2020bvt . In the current generalization to the n-valence
quarks the final result, in addition to the nucleon, can be applied to mesons
as well as a six-quark system that can be formed due to the strong overlap of
two nucleons in the deuteron. Furthermore, by hadron we will also mean
strongly overlapped two-nucleon system.

In the current approach rather than taking a full or truncated Fock expansion
of the hadron, $h$, we model its ket state via:

$\displaystyle|h\rangle=\psi_{n_{V}q}|n_{V}q\rangle\otimes\psi_{VR}|VR\rangle,$
(3)

where $\psi_{n_{V}q}$ and $\psi_{VR}$ are the light-front wave functions (LF)
of the n-valence cluster ($n_{V}q$) and the cluster-residual $VR$ system,
respectively.

Within our framework the LF wave functions can be related to valence PDFs if
one calculates the $F_{2}(x,Q^{2})$ structure function that enters in the
cross section of deep inelastic scattering (DIS), $eh\rightarrow e^{\prime}X$.
We consider this reaction in the Drell-Yan-West reference frame

$\displaystyle p^{\mu}$ $\displaystyle=$
$\displaystyle\left(p^{+},p^{-},\mathbf{p}^{\perp}\right)=\left(p^{+},\frac{m_{H}^{2}}{p^{+}},\mathbf{0}_{\perp}\right),$
$\displaystyle q^{\mu}$ $\displaystyle=$
$\displaystyle\left(q^{+},q^{-},\mathbf{q}^{\perp}\right)=\left(0,\frac{2p\cdot
q}{p^{+}},\mathbf{q}_{\perp}\right),$ (4)

where $p^{\mu}$ and $q^{\mu}$ are four-momenta of the incoming hadron and
virtual photon in light cone (LC) co-ordinates, with
$Q^{2}=-q^{2}=|\mathbf{q}_{\perp}|^{2}$, $x=\frac{Q^{2}}{2p\cdot q}$ and
$m_{H}$ being the mass of the hadron.

The hadronic tensor is given by:

$\displaystyle W^{\mu\nu}(x,Q^{2})=\frac{1}{4\pi
m_{H}}\sum_{q,h_{i}}\int\delta(k_{R}^{2}-m_{R}^{2})\frac{d^{4}k_{R}}{(2\pi)^{3}}$
$\displaystyle\times\delta(k^{\prime,2}_{1}-m_{1}^{2})\frac{d^{4}k^{\prime}_{1}}{(2\pi)^{3}}\prod_{i=2}^{n_{V}}\delta(k_{i}^{2}-m_{i}^{2})\frac{d^{4}k_{i}}{(2\pi)^{3}}$
$\displaystyle\times(2\pi)^{4}\delta^{(4)}(P+q-k^{\prime}_{1}-\sum_{i=2}^{n_{V}}k_{i}-k_{R})A^{\mu\dagger}A^{\nu},$
(5)

where the amplitude, $A^{\mu}$, is that of the scattering process depicted in
Fig. 3. Here, $k_{i}^{\mu}$, $m_{i}$ and $h_{i}$ are 4-momentum, mass, and
helicity respectively, with the $i$ index denoting the valence quarks from
$i=1,2,\ldots,n_{V}$, with $i=1$ being the struck quark before interacting
with the photon and $i=1^{\prime}$ denoting it after. The index $i=R,V$ label
the residual and valence subsystem, respectively.

The Lorentz invariant phase space for a single particle in light front co-
ordinates is:

$\displaystyle\delta(k^{2}-m^{2})d^{4}k=\left.\frac{dxd^{2}\mathbf{k}_{\perp}}{2x}\right|_{k^{-}=\frac{k_{\perp}^{2}+m^{2}}{k^{+}}}.$
(6)

In the Bjorken limit, $Q^{2}\rightarrow\infty$ with $x=\frac{Q^{2}}{2p\cdot
q}$ fixed:

$\displaystyle\delta^{(4)}(P+q-k^{\prime}_{1}-\sum_{i=2}^{n_{V}}k_{i}-k_{R})$
$\displaystyle=\frac{x_{1}}{p_{N}\cdot
q}\delta(x_{1}-x)\delta\left(1-x_{1}-\sum_{i=2}^{n_{V}}x_{i}-x_{R}\right)$
$\displaystyle\times\delta^{(2)}\left(\sum_{i=1}^{n_{V}}\mathbf{k}_{i,\perp}+\mathbf{k}_{R,\perp}\right).$
(7)

.

Figure 3: Light cone time ordered diagram of DIS. Intermediate states are
identified by the vertical dashed lines.

For the calculation of the amplitude $A^{\mu}$ in RefLeon:2020bvt  we
developed effective light-front diagrammatic rules in which we introduced
LFWFs through the phenomenological vertices and light-front energy
denominators in the same form which enters in LF equation of bound states. For
the n-valence cluster ($n_{V}q$) and cluster-residual system (VR) LFWFs are
defined as:

$\displaystyle\psi_{VR}(x_{V},\mathbf{k}_{R,\perp},x_{R},\mathbf{k}_{V,\perp})=\frac{\bar{\chi_{V}}\bar{\chi_{R}}\Gamma^{h\rightarrow
VR}\chi_{h}}{m_{h}^{2}-\frac{k_{V,\perp}^{2}+m_{V}^{2}}{x_{V}}-\frac{k_{R,\perp}^{2}+m_{R}^{2}}{x_{R}}}$
$\displaystyle\psi_{n_{V}q}(\\{\beta_{i},\mathbf{k}_{i,\perp},h_{i}\\}_{i=1}^{n_{V}})=\frac{\prod\limits_{i=1}^{n_{V}}\bar{u}(k_{i},h_{i})\Gamma^{V\rightarrow
n_{V}q}\chi_{V}}{m_{V}^{2}-\sum\limits_{i=1}^{n_{V}}\frac{k_{i,\perp}^{2}+m_{i}^{2}}{\beta_{i}}},$
(8)

where $(x_{V},k_{V,\perp})$ and $(x_{R},k_{R,\perp})$ characterize LC momenta
of the cluster and the residual system and
$\\{\beta_{i},\mathbf{k}_{i,\perp},h_{i}\\}_{i=1}^{nV}$ denotes the LC momenta
and helicities of the n-valence quarks in the cluster. In the above defined
LFWFs $\Gamma^{h\rightarrow VR}$ and $\Gamma^{V\rightarrow n_{V}q}$ are
phenomenological vertices describing the transition of the hadron to the
cluster-residual system and the valence cluster to n-valence quarks. These
vertices absorb all the complexities of LFWS (for example null-modes) that in
principle could be calculated in an almost infinite number coupled channel
equation for LF Fock state wave functions (see e.g. Ref.Brodsky:1997de ). In
our approach these wave functions are found phenomenologically by
parameterizing them and then verifying their parameters in different QCD
processes in which the same LFWFs enter.

In leading order we can relate the amplitude to the LFWFs as follows:

$\displaystyle A^{\mu}$
$\displaystyle=\sum_{h_{1},h_{V}}\bar{u}(k_{1}^{\prime},h_{1}^{\prime})(ie_{1}\gamma^{\mu})u(k_{1},h_{1})\frac{\psi_{VR}}{x_{V}}\frac{\psi_{n_{V}q}}{\beta_{1}}.$
(9)

Here $e_{1}$ is the charge of quark $i=1$.The LFWFs have dependencies of:

$\displaystyle\psi_{VR}=\psi_{VR}(x_{V},\mathbf{k}_{R,\perp},x_{R},\mathbf{k}_{V,\perp})$
(10)
$\displaystyle\psi_{n_{V}q}=\psi_{n_{V}q}(\\{\beta_{i},\mathbf{\tilde{k}}_{i,\perp},h_{i}\\}_{i=1}^{n_{V}})$
(11)

where

$\displaystyle\mathbf{\tilde{k}}_{i,\perp}$
$\displaystyle=\mathbf{k}_{i,\perp}-\frac{x_{i}}{x_{V}}\mathbf{k}_{V,\perp},$
(12) $\displaystyle\beta_{i}$ $\displaystyle=\frac{x_{i}}{x_{V}}\ \ \mbox{(i =
$1$,...$n_{V}$.)}.$ (13)

Eq. 9 simplifies for $\mu=+$ , resulting in

$\displaystyle A^{+}$
$\displaystyle=2ie_{1}\sum_{h_{1},h_{V}}\psi_{VR}\psi_{n_{V}q}.$ (14)

In the considered reference frame we can relate the second structure function
to the hadronic tensor:

$\displaystyle F_{2}(x,Q^{2})$ $\displaystyle={m_{H}(p\cdot
q)\over(p^{+})^{2}}W^{++}={m_{H}Q^{2}\over 2x(p^{+})^{2}}W^{++}.$ (15)

The non-negativity condition on the ‘+’ momentum in Eq.(4) eliminates the
vacuum contribution where the photon fluctuates into a quark/anti-quark pair,
$\gamma^{*}\rightarrow q\bar{q}$. Substituting Eq.’s (6), (II.1) and (14) into
$W^{++}$ in Eq. (5), from Eq. (15) one obtains:

$\displaystyle
F_{2}(x)=\sum_{q}\sum_{h_{i}}\int^{Q^{2}}[dx]_{n_{V}+R}[d^{2}\mathbf{k}_{\perp}]_{n_{V}+R}$
$\displaystyle\times
e_{q}^{2}x_{1}\delta(x_{1}-x)|\psi_{3q}|^{2}|\psi_{V}|^{2},$ (16)

where

$\displaystyle[dx]_{n_{V}+R}$
$\displaystyle=\prod_{i=1,...n_{V},R}\frac{dx_{i}}{x_{i}}\delta\left(1-\sum_{i=1,...n_{V},R}x_{i}\right)$
(17) $\displaystyle[d^{2}\mathbf{k}_{\perp}]_{n_{V}+R}$
$\displaystyle=\prod_{i=1,...n_{V},R}\frac{d^{2}\mathbf{k}_{i,\perp}}{16\pi^{3}}16\pi^{3}\delta^{(2)}\left(\sum_{i=1,...n_{V},R}\mathbf{k}_{i,\perp}\right).$
(18)

Using the leading order relation of
$F_{2}(x,Q^{)}=\sum_{i}e_{i}^{2}xf_{i}(x,Q^{2})$, one can then relate the
valence PDF through the LFWFs as follows:

$\displaystyle
f_{q}(x)=\sum_{h_{i}}\int^{Q^{2}}[dx]_{n_{V}+R}[d^{2}\mathbf{k}_{\perp}]_{n_{V}+R}$
$\displaystyle\times\delta(x_{1}-x)|\psi_{n_{V}q}|^{2}|\psi_{V}|^{2}.$ (19)

Using $\mathbf{\tilde{k}}_{i,\perp}$ for the valence quarks allows us to
factorize the transverse integral:

$\displaystyle f(x)=\int[dx]_{n_{V}+R}\delta(x_{1}-x)$
$\displaystyle\times\int^{\tilde{Q}^{2}_{max}}[d^{2}\tilde{\mathbf{k}}_{\perp}]_{n_{V}}\left|\psi_{n_{V}q}(\\{\beta_{i},\mathbf{\tilde{k}}_{i,\perp},h_{i}\\}_{i=1}^{n_{V}})\right|^{2}$
$\displaystyle\times\int^{Q_{VR,max}^{2}}\frac{d^{2}\mathbf{k}_{R,\perp}}{16\pi^{3}}|\psi_{VR}(x_{V},\mathbf{k}_{R,\perp},x_{R},\mathbf{k}_{V,\perp})|^{2},$
(20)

where:

$\displaystyle[d^{2}\mathbf{\tilde{k}}_{\perp}]_{n_{V}}=16\pi^{3}\delta^{(2)}\left(\sum_{i=1}^{n_{V}}\mathbf{\tilde{k}}_{i,\perp}\right)\prod_{i=1}^{n_{V}}\frac{d^{2}\mathbf{\tilde{k}}_{i,\perp}}{16\pi^{3}}$
$\displaystyle\mathbf{\tilde{k}}_{i,\perp}=\mathbf{k}_{i,\perp}-\frac{x_{i}}{x_{V}}\mathbf{k}_{V,\perp},\
\ \beta_{i}=\frac{x_{i}}{x_{V}}\ \ \mbox{(i = $1$,...$n_{V}$)}.$ (21)

###  II.2 Valence PDF with Light Front Harmonic Oscillator LFWFs

From the fact that the hadron is a bound, relativistic object we model
$\psi_{n_{V}q}$ through the mutually coupled relativistic harmonic light-front
wave functionsLeon:2020bvt , to account the effect of confinement. For
$\psi_{VR}$, we use a Gaussian function with non-relativistic kinematics used
to estimate the $z$ component of momentum.

$\displaystyle\psi_{n_{V}q}$
$\displaystyle=16\pi^{3}m_{H}A_{R}\exp{\left[-\frac{B_{V}}{8}\sum_{i=1}^{n_{V}}\frac{\tilde{k}^{2}_{i,\perp}+m_{i}^{2}}{\beta_{i}}\right]}$
$\displaystyle\times e^{\frac{n^{2}m^{2}B_{V}}{8}}\sqrt{x_{2}....x_{n_{V}}}$
(22) $\displaystyle\psi_{VR}$ $\displaystyle=\sqrt{16\pi^{3}m_{H}}A_{V}s$
$\displaystyle\times\exp{\left[-\frac{B_{R}}{2}\left(\left(m_{H}x_{R}-m_{R}\right)^{2}k_{R,\perp}^{2}\right)\right]}\sqrt{x_{R}}.$
(23)

Inserting the above wave functions into Eq.(20) one obtains :

$\displaystyle f(x)$
$\displaystyle=(16\pi^{3}m_{H})^{5}|A_{R}A_{V}|^{2}e^{\frac{n^{2}m^{2}B_{V}}{8}}$
$\displaystyle\times\int[dx]_{n_{V}+R}\delta(x_{1}-x)x_{2}....x_{n}$
$\displaystyle\times\int^{\tilde{Q}^{2}_{max}}[d^{2}\tilde{\mathbf{k}}_{\perp}]_{n_{V}}e^{-\frac{B_{V}}{4}\sum_{i}\frac{\tilde{k}^{2}_{\perp,i}+m_{i}^{2}}{\beta_{i}}}$
$\displaystyle\times\int^{Q_{VR,max}^{2}}\frac{d^{2}\mathbf{k}_{R,\perp}}{16\pi^{3}}|\psi_{VR}|^{2}.$
(24)

Taking the $Q^{2}\rightarrow\infty$ limit (see Appendix A) results in :

$\displaystyle\int[d^{2}\tilde{\mathbf{k}}]_{n_{V}}e^{-\frac{B_{V}}{4}\sum_{i}\frac{\tilde{k}^{2}_{\perp,i}+m_{i}^{2}}{\beta_{i}}}$
$\displaystyle=\frac{e^{-\frac{B_{V}}{4}\sum_{i}\frac{m_{i}^{2}}{\beta_{i}}}}{(16\pi^{2})^{n_{V}-1}(B_{V}/4)^{n_{V}-1}}\frac{x_{1}x_{2}...x_{n_{V}}}{x^{n_{V}}_{V}}.$
(25)

Similarly, for the residual transverse integral:

$\displaystyle\int\frac{d^{2}\mathbf{k}_{R,\perp}}{16\pi^{3}}|\psi_{VR}|^{2}$
(26) $\displaystyle=\frac{\pi
e^{-B_{R}m_{H}^{2}(x_{R}-\mu_{R})^{2}}x_{R}}{16\pi^{3}}\int
dk^{2}_{R,\perp}e^{-B_{R}k^{2}_{R,\perp}}$
$\displaystyle=\frac{e^{-B_{R}m_{H}^{2}(x_{R}-\mu_{R})^{2}}x_{R}}{16\pi^{2}B_{R}},$
(27)

where $\mu_{R}=\frac{m_{R}}{m_{H}}$. Inserting Eq. (25) and (27) into Eq.
(II.2) for the valence PDF one obtains:

$\displaystyle f(x)$ $\displaystyle=\mathcal{N}_{n_{V}}\int[dx]_{n_{V}+R}$
$\displaystyle\times\delta(x_{1}-x)\frac{x_{1}x^{2}_{2}...x^{2}_{n_{V}}}{x^{n_{V}}_{V}}e^{-B_{R}m_{H}^{2}(x_{R}-\mu_{R})^{2}}x_{R},$
(28)

where

$\displaystyle\mathcal{N}_{n_{V}}$
$\displaystyle=\frac{(16\pi^{3}m_{H})^{5}4^{n_{V}-1}|A_{R}A_{V}|^{2}}{(16\pi^{2})^{n_{V}}B_{V}^{n_{V}-1}B_{R}}e^{\frac{B_{V}n_{V}^{2}m^{2}}{4}}.$
(29)

This can be simplified (see Appendix B) to obtain:

$\displaystyle f(x)$ $\displaystyle=\frac{\mathcal{N}_{n_{V}}}{(2n_{V}-3)!}$
$\displaystyle\times\int\limits_{0}^{1-x}dx_{R}\frac{(1-x_{R}-x)^{2n-3}}{(1-x_{R})^{n}}e^{-B_{R}m_{H}^{2}(x_{R}-{m_{R}\over
m_{H}})^{2}}.$ (30)

##  III Analytic Predictions of the Model

Obtained in Eq.(30) expression uses a coupled $n_{V}$-dimensional harmonic
oscillator on the Light-Front which does not contain the hard component, which
can be generated through the hard gluon exchanges between valence quarks. With
such a “soft” wave function one expects that the model has more validity at
moderate $x$ relevant to the peak position of the $x$ weighted valence PDF
distribution (see Fig.1 and Ref.Leon:2020nfb ). We expect our model to
underestimate the high $x$ part of the valence PDFs if they are dominated by
hard interactions between valence quarksBrodsky:1974vy .

Our procedure for the case of the nucleonLeon:2020nfb  was to fit the height
and the position of the x-weighted valence PDF peak and thereby to evaluate
parameters entering the LFWFs of Eq.(8). Once these parameters are fixed then
one can use the same wave functions in the calculation of different QCD
“objects” like form-factors, generalized partonic distributions or
transversities that can be accessed experimentally. This allows us to verify
the universalities of LFWFs for multitude of processes at moderate range of
$x\sim 0.1-0.4$.

In the present work we focus on analytic features of the model that does not
require a fitting. For this, one observes that for moderate $x$, using the
exponent in the integrand in Eq.(30) we can evaluate it at its minimum,
$x_{R}\sim 1-{m_{R}\over m_{H}}$, resulting in:

$f(x)\approx\frac{\mathcal{N}_{n_{V}}}{(2n_{V}-3)!(1-\mu_{R})^{n_{V}}}(1-x-\mu_{R})^{2n_{V}-3}.$
(31)

The derivation of the approximation in Eq. (31) utilizes the saddle point
approximation, which has a correction of
$\mathcal{O}(\frac{1}{B_{R}m_{H}^{2}})$. Values of $B_{R}m_{H}^{2}$ extracted
from fits to phenomenologically derived proton valence PDFs are typically
$\sim 10-50$ Leon:2020bvt . Using Eq. (31) one can calculate the position of
the peak of $xf(x)$ to obtain:

$x_{p}=\frac{1}{2(n_{V}-1)}\left(1-{m_{R}\over m_{H}}\right).\vspace{-0.2cm}$
(32)

By the evolution of the DGLAP equations, there is a dependence on $x_{p}$ on
$Q^{2}$. While there is no exact prescription as to what minimal $Q_{0}^{2}$
to use, one expects it to be large enough for factorization and pQCD to be
valid and small enough so that the exact characteristics of the residual
structure are not important. However, once fixed at a particular $Q^{2}$ their
renormalization flow is uniquely determined.

Since the mass of the residual system, $m_{R}$, is positive the above equation
results in a universal upper limit of the peak position of valence quark PDFs
of the form:

$x_{p}\leq\frac{1}{2(n_{V}-1)}.\vspace{-0.2cm}$ (33)

This upper bound is valid for all $Q^{2}$. A numerical investigation into the
validity of Eq. (32) and (33) can be found in Appendix D, where both are found
to generally hold for the parameters explored.

For the nucleon this results in Eq.(2) and an upper limit of
$x_{p}^{max}={1\over 4}$. This allows us to state that if the partonic degrees
are resolved in the nucleon then the position of the peak of the x weighted
valence PDFs should not exceed ${1\over 4}$. To check this observation, in
Fig.4 we compare the results from several valence PDFs at starting $Q_{0}$.
Note that only those PDFs are chosen that apply the standard approach in
defining starting $Q_{0}$, that is, it assumes the onset of partonic degrees
of freedom in the nucleon. The figure shows that at the lowest possible
starting $Q_{0}=1$ GeV (for which partonic degrees of freedom can only start
to be relevant) the position of the peak is in agreement with the predicted
upper limit of ${1\over 4}$. For all $Q>Q_{0}$ the peak position is less than
${1\over 4}$, thus satisfying inequity (33). Note that this result is
independent on the form of the ansatz in empirical PDF parameterizations.

Figure 4: The up valence peak for various PDFs sets at their starting $Q_{0}$
cteq18 ; cj15 ; cteq14 ; NNPDF:2017mvq ; Owens:2012bv ; Harland-Lang:2014zoa ;
Martin:2009iq ; NNPDF:2014otw .

One may think that $x_{p}^{max}={1\over 4}$ can be a simple reflection that,
in $m_{R}\to 0$ limit the model becomes like a four-massless—body problem. If
this is the case, then mesons will correspond to three-massless-body problem
expecting $x_{p}^{max}={1\over 3}$. However, as Eq.(33) shows for mesons the
model predicts $x_{p}<{1\over 2}$, which is in agreement with available pion
PDFsBarry:2018ort ; Barry:2021osv . Thus, the result is due to a nontrivial
light-cone momentum sharing between valence quarks and residual system.

It is interesting that for the case of 6q-system the model predicts
$x_{p}<{1\over 10}$, which should be compared to ${1\over 8}$ that follows
from the convolution model in which two nucleons in the deuteron did not merge
into a 6q state. Thus, the shift of the peak position towards lower $x$
reaching to ${1\over 10}$ for the deuteron valence PDFs at fixed $Q^{2}$ with
increase of internal momentum in the deuteron will indicate the transition
from NN to 6q state.

Furthermore, without fixing parameters of the LF wave functions we can
calculate the analytic form of the valence PDFs at $x\to 1$. Substituting
$x\to 1-\epsilon$ we calculate the integral in Eq.(30) in the small $\epsilon$
limit, resulting in:

$f(x)={\mathcal{N}_{n_{V}}\over(2n_{V}-3)!}e^{-B_{R}m_{H}^{2}(1-\mu_{R})^{2}}(1-x)^{2n_{V}-2}.$
(34)

The above relation predicts $(1-x)^{4}$ behavior for nucleon valence PDFs at
$x\to 1$, which should be compared with $(1-x)^{3}$ that follows from
perturbative QCD arguments that require two hard gluon exchanges between three
valence quarksBrodsky:1974vy . It is interesting that for mesons one ovbtains
$(1-x)^{2}$, which is the same as the prediction from pQCD. This indicates
that the observation of $(1-x)^{2}$ behavior of the pion valence PDF may not
necessarily indicate the dominance of the hard component in the pion wave
function. This observation is in agreement with that of Efremov:2009dx . The
same is true for the deuteron in which case Eq.(34) predicts $(1-x)^{10}$ the
same behavior as predicted within pQCDBrodsky:1974vy .

##  IV Summary and Outlook

We introduced a residual field approach in the calculation of valence PDFs of
a hadron containing $n_{V}$-valence quarks. In the approach we describe a
hadron as consisting of an $n_{V}$-valence cluster embedded in the residual
system of the hadron. Phenomenological LF wave functions are introduced, which
do not contain a hard component and the calculation is done within the
effective light-front diagrammatic method. Analyzing analytic features of the
calculated valence PDFs we predict that there is a universal upper limit for
the peak position of the $x$-weighted valence PDFs. Namely, for the nucleon we
predict an upper limit for $x_{p}^{max}={1\over 4}$, which is in agreement
with the all available phenomenological PDFs. This result in our view is non
trivial considering that only three-valence quarks exist in the nucleon. One
obtains also $x_{p}^{max}={1\over 2}$ and $x_{p}^{max}={1\over 10}$ for meson
and six-quark system respectively.

In discussing the high $x$ behavior of the PDFs one observes that soft LFWFs
predict a faster drop-off for the nucleon PDFs than that of pQCD, while the
prediction for pion and the deuteron PDFs are the same as the pQCD prediction.

It will be interesting if the future studies of valence PDFs, both
experimentally and on the lattice, could verify the existence of the universal
upper limit for the nucleon $x_{p}^{max}={1\over 4}$. Such a position of the
peak will be an indicator of the onset of partonic degrees of freedom in the
nucleon.

Acknowledgment: This work is supported by United States Department of Energy
grant under contract DE-FG02-01ER41172

## References

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##  Appendix A General Case for Transverse Integral

LF relativistic harmonic oscillators were used in modelling the LFWFs. In
integrating over the transverse momentum we will have integrals of the form,

$\displaystyle
I_{n_{V}}=\int\prod_{j=1}^{n_{V}}[d^{2}\mathbf{\tilde{k}}_{j,\perp}]\exp\bigg{[}-\sum_{j=1}^{{n_{V}}}B_{j}\frac{\tilde{k}_{j,\perp}^{2}}{x_{j}}\bigg{]}.$
(35)

We now take $Q^{2}\rightarrow\infty$ in the integral bounds. The
$\tilde{k}_{x}$ and $\tilde{k}_{y}$ integrals are identical, thus we can take
it as the square of an ${n_{V}}$-dimensional integral:

$\displaystyle
I_{n_{V}}=\int\prod_{j=1}^{n_{V}}\frac{d^{2}\tilde{k}_{j,\perp}}{16\pi^{3}}16\pi^{3}\delta^{(2)}\left(\sum_{j=1}^{{n_{V}}}\mathbf{\tilde{k}}_{j,\perp}\right)$
$\displaystyle\times\exp\bigg{[}-\sum_{j=1}^{{n_{V}}}B_{j}\frac{\tilde{k}_{j,\perp}^{2}}{x_{j}}\bigg{]}$
$\displaystyle=\frac{1}{(16\pi^{3})^{n_{V}-1}}\left(\int\prod_{j=1}^{n_{V}}d\tilde{k}_{j}\delta(\sum_{j=1}^{{n_{V}}}\tilde{k}_{j})\exp\bigg{[}-\sum_{j=1}^{{n_{V}}}B_{j}\frac{\tilde{k}_{j}}{x_{j}}\bigg{]}\right)^{2}.$
(36)

Now, we make use of the fact that we can take a Fourier decomposition of Dirac
delta:
$\delta(\sum_{j=1}^{N}\tilde{k}_{i})=\frac{1}{2\pi}\int_{-\infty}^{\infty}dze^{iz(\sum_{j=1}^{N}\tilde{k}_{i})}$
in the above equation obtaining,

$\displaystyle I_{n_{V}}=\frac{1}{(16\pi^{3})^{n_{V}-1}}$
$\displaystyle\times\left(\frac{1}{2\pi}\int_{-\infty}^{\infty}dz\prod_{j=1}^{n_{V}}\int_{-\infty}^{\infty}dk_{j}\exp\bigg{[}izk_{j}\bigg{]}\exp\bigg{[}-B_{j}\frac{k_{j}^{2}}{x_{j}}\bigg{]}\right)^{2}.$

The integral in the product is just a Fourier transform of a Gaussian. Using
the relation $\int_{-\infty}^{\infty}dxe^{-\alpha x^{2}}e^{i\omega
x}=\sqrt{\frac{\pi}{\alpha}}e^{-\frac{\omega^{2}}{4\alpha}}$ in the above
equation, one obtains:

$\displaystyle I_{n_{V}}$ $\displaystyle=\frac{1}{(16\pi^{3})^{n_{V}-1}}$
$\displaystyle\times\bigg{(}\frac{1}{2\pi}\int_{-\infty}^{\infty}dze^{-z^{2}(\sum_{j=1}^{n_{V}}\frac{x_{j}}{4B_{j}})}\prod_{j=1}^{n_{V}}\sqrt{\frac{\pi
x_{j}}{B_{j}}}\bigg{)}^{2}$
$\displaystyle=\frac{1}{(16\pi^{3})^{n_{V}-1}}\bigg{(}\frac{1}{2\pi}\sqrt{\frac{\pi}{\sum_{j=1}^{n_{V}}\frac{x_{j}}{4B_{j}}}}\prod_{j=1}^{n_{V}}\sqrt{\frac{\pi
x_{j}}{B_{j}}}\bigg{)}^{2}.$

. Thus,

$\displaystyle I_{n_{V}}$
$\displaystyle=\frac{1}{(16\pi^{3})^{n_{V}-1}}\frac{(\pi)^{n_{V}-1}}{\sum_{j=1}^{n_{V}}\frac{x_{j}}{B_{j}}}\prod_{j=1}^{n_{V}}\frac{x_{j}}{B_{j}}.$
(37)

In Eq. (25) the fact the $B_{j}$’s were the same for all $i$ and that
$\sum_{i}\beta_{i}=1$ was used.

##  Appendix B The x Integration of PDFs

Starting at Eq. (II.2) we use $x_{V}=1-x_{R}$ and making the $[dx]_{n_{V}+R}$
term explicit:

$\displaystyle f(x)$ $\displaystyle=\mathcal{N}_{n_{V}}\int dx_{1}\ldots
dx_{n_{V}}dx_{R}\delta\Big{(}1-x_{R}-\sum^{n_{V}}_{i=1}x_{i}\Big{)}$
$\displaystyle\times\delta(x_{1}-x)\frac{x_{2}...x_{n_{V}}}{(1-x_{R})^{n_{V}}}e^{-B_{R}m_{H}^{2}(x_{R}-\mu_{R})^{2}}.$
(38)

The Dirac delta is taken to set $x_{1}$ to $x$:

$\displaystyle
f(x)=\mathcal{N}_{n_{V}}\int^{1-x}_{0}dx_{R}\int^{1-x-x_{R}}_{0}dx_{2}...$
$\displaystyle\times\int^{1-x-x_{R}-\sum_{i=2}^{n_{V}-1}x_{i}}dx_{n_{V}}\frac{x_{2}...x_{n}}{(1-x_{R})^{n_{V}}}e^{-B_{R}m_{H}^{2}(x_{R}-\mu_{R})^{2}}$
$\displaystyle\times\delta\Big{(}1-x_{R}-x-\sum^{n_{V}}_{i=2}x_{i}\Big{)}.$
(39)

Now, let $y_{i}$ be the relative momentum fraction for the $i=2,\ldots,n_{V}$
subsystem:

$\displaystyle
y_{i}=\frac{x_{i}}{x_{2}+x_{3}+\ldots+x_{n_{V}}}=\frac{x_{i}}{1-x_{R}-x}$ (40)

Doing so, the integral factorizes,

$\displaystyle
f(x)=\mathcal{N}_{n_{V}}\int^{1-x}_{0}dx_{R}\frac{e^{-B_{R}m_{H}^{2}(x_{R}-\mu_{R})^{2}}}{(1-x_{R})^{n_{V}}}\int^{1}_{0}dy_{2}y_{2}\ldots$
$\displaystyle\int_{0}^{1-\sum_{i=2}^{n_{V}-1}y_{i}}dy_{n_{V}}y_{n_{V}}\frac{(1-x_{R}-x)^{2(n_{V}-1)}}{(1-x_{R})^{n_{V}}}$
$\displaystyle\times\delta\Big{(}1-x_{R}-x-\sum^{n_{V}}_{i=2}y_{i}(1-x-x_{R})\Big{)}$
$\displaystyle=\mathcal{N}_{n_{V}}J_{n_{V}}\int^{1-x}_{0}dx_{R}\frac{(1-x_{R}-x)^{2n_{V}-3}}{(1-x_{R})^{n}}e^{-B_{R}m_{H}^{2}(x_{R}-\mu_{R})^{2}},$
(41)

where $J_{n_{V}}$ is a constant independent of $x_{R}$ and $x$. One can show
$J_{n_{V}}=\frac{1}{(2n_{V}-3)!}$ (see Appendix C). Thus,

$\displaystyle f(x)$ $\displaystyle=\frac{\mathcal{N}_{n_{V}}}{(2n_{V}-3)!}$
$\displaystyle\times\int^{1-x}_{0}dx_{R}\frac{(1-x_{R}-x)^{2n_{V}-3}}{(1-x_{R})^{n_{V}}}e^{-B_{R}m_{H}^{2}(x_{R}-\mu_{R})^{2}}$
(42)

##  Appendix C Valence Spectator’s x Integral

In integrating over all momentum fractions of the spectator system we get the
following integral as a constant:

$\displaystyle J_{n_{V}}$
$\displaystyle=\int^{1}_{0}dy_{2}y_{2}\ldots\int_{0}^{1-\sum_{i=2}^{n_{V}-1}y_{i}}dy_{n_{V}}y_{n_{V}}\delta\Big{(}1-\sum^{n_{V}}_{i=2}y_{i}\Big{)}$
$\displaystyle=\frac{1}{(2n_{V}-3)!}$ (43)

The last line of Eq. (C) can be proven with mathematical induction. For the
starting case of $n_{V}=2$, $I_{2}=1=\frac{1}{1!}$, Now, do the induction step
by assuming Eq. (C) is true up to $n_{V}$. Then,

$\displaystyle J_{n_{V}+1}$ $\displaystyle=\int^{1}_{0}dy_{2}y_{2}\ldots$
$\displaystyle\times\int_{0}^{1-\sum_{i=2}^{n_{V}}y_{i}}dy_{n_{V}+1}y_{n_{V}+1}\delta\Big{(}1-\sum_{i=2}^{n_{V}+1}y_{i}\Big{)}$
(44)

For $i=3,\ldots,n_{V}+1$ we change the variables to the relative momentum
fraction in the $3,\ldots,n_{V}+1$ system:

$z_{i}=\frac{y_{i}}{\sum_{j=3}^{n_{V}+1}y_{j}}=\frac{y_{i}}{1-y_{2}}$ (45)

Then, using the relation in Eq. C

$\displaystyle J_{n_{V}+1}$
$\displaystyle=\int_{0}^{1}dy_{2}y_{2}(1-y_{2})^{2(n_{V}-2)}\>J_{n_{V}}$
$\displaystyle=\frac{1}{(2n_{V}-3)!}\int_{0}^{1}dy_{2}y_{2}(1-y_{2})^{2(n_{V}-2)}.$
(46)

The integral can be taken in terms of the Beta function, $B(m,n)$, which in
turn can be expressed in terms of factorials since its arguments are integers,

$\displaystyle\int_{0}^{1}dy_{2}y_{2}(1-y_{2})^{2(n_{V}-2)}$
$\displaystyle=B(2,2n_{V}-3)$
$\displaystyle=\frac{1!(2n_{V}-2-1)!}{(2n_{V}-1)!}.$ (47)

Thus, using Eq. (47) in Eq. (46), one obtains:

$\displaystyle J_{n_{V}+1}$
$\displaystyle=\frac{1}{(2n_{V}-3)!}\frac{1!(2n_{V}-2-1)!}{(2n_{V}-1)!}$
$\displaystyle=\frac{1}{(2n_{V}-1)!}$ (48)

and Eq. (C) holds.

Figure 5: A comparison of $x_{p}$ with the approximation in Eq. 32 for various
configurations and with a mass ratio range of $0<m_{R}/m_{H}<0.6$. The
$\epsilon$ displayed represents the largest relative error of the
approximation found for the given configuration.

##  Appendix D Numerical Investigation of x Peaking Approximation and Upper
Limit

Using the saddle point approximation in Eq. (30), an approximation to the x
peak, $x_{p}$, of the PDF was found to be:

$x_{p}^{(sp)}=\frac{1}{2(n_{V}-1)}\left(1-\frac{m_{R}}{m_{H}}\right).$ (49)

From this we inferred that:

$x_{p}\leq\frac{1}{2(n_{V}-1)}.\vspace{-0.2cm}$ (50)

Here we explore the validity of the above approximation numerically by looking
at the cases of $n_{V}=2,3,4$ and $B_{R}m_{H}^{2}=10,45,80$, the latter values
being typical when the model was applied to the case of the proton
Leon:2020bvt .

Fig. 5 shows how the peak varies with the mass ratio, $\frac{m_{R}}{m_{H}}$.
The $x_{p}$ value was found numerically, using the scipy.optimize Python
library. It is interesting that for the above plots $x_{p}^{(sp)}$
qualitatively matches $x_{p}$ well, monotonically decreasing with
$m_{R}/m_{H}$ and being nearly linear. Also, in all cases investigated the
upper limit $x_{p}≤\frac{1}{2(n_{V}-1)})$ held.