\section*{References}


\section{Conclusions}
In conclusion, the possibility of obtaining the scalar and tensor form factors and charges directly from experiment with sufficient precision,
gives an entirely different leverage to neutron beta decay searches.
While lattice QCD provides the only means to calculate quantities that are
unattainable in experiment, for the tensor charge the situation is different. In this case, the hadronic matrix element is  the same which enters the DIS observables measured in precise semi-inclusive and deeply virtual exclusive scattering off polarized targets.
Most importantly the error on the elementary tensor coupling, $\epsilon_T$, depends on both the central value of $g_T$ as well as on the relative error, $\Delta g_T/ g_T$,
therefore, independently from the theoretical accuracy  that can be achieved,
experimental measurements are essential since they simultaneously provide a testing ground for lattice QCD calculations.

We are grateful to H. Avakian, A. Kim, S. Pisano, J. Zhang for details on the experimental extractions at Jefferson Lab,  and to L. Barr\'on Palos, M. Engelhardt, P.Q. Hung,  E. Peinado,  and D. Po\v{c}ani\'c for fruitful discussions. This work was funded by the Belgian Fund F.R.S.-FNRS via the contract of Charg?e de recherches (A.C.), by D.O.E. grant DE-FG02-01ER4120 (S.L.) and by NSF PHY-1205833 (S.B.). M.G.-A. is grateful to the LABEX Lyon Institute of Origins (ANR-10-LABX-0066) of the Universit\'e de Lyon for its financial support within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) of the French government operated by the National Research Agency (ANR).


\section{Introduction}
\label{sec:Introduction}
Relativistic radiation hydrodynamics provides the natural framework
of several high-energy astrophysical processes, essentially all those for
which the interaction of photons with matter takes place
in a strong gravity regime or in the presence of
relativistic motions. One such relativistic system is
represented by geometrically thick disks (tori) around
black holes.
Considered as inviscid purely fluid solutions,
these objects have always attracted much interest,
partly because they allow for relatively simple analytic
or semi--analytic
configurations~\citep{Fishbone76,Abramowicz78,Kozlowski1978,Font02a,Daigne04,Qian2009,Penna2013b}
and partly because they can be adopted to study
various types of fluid instabilities and potentially
observable physical effects
in the vicinity of black holes~\citep{Abramowicz1980,Abramowicz83,Papaloizou84,Abramowicz1998,
  Rezzolla_qpo_03b, Zanotti03, Blaes2006, Montero2010}.
Moreover, a renewed interest for them has been motivated by the
outcome of recent numerical simulations in full general
relativity, showing that
high-density tori are indeed produced after the
merger of unequal--mass neutron star binaries that form a
black hole~\citep{Rezzolla:2010}.

In spite of all these analyses, and although tori
have been extensively studied
in numerical simulations
also for the effects of
magnetic
fields~\citep{DeVilliers03,Komissarov2006a,Fragile2009b,
Barkov2011,Narayan2012,
McKinney2012}, the role of
radiation fields has often been disregarded.
This has certainly been due to the
complexity inherent in the time--dependent
solution of the relativistic
radiation hydrodynamics equations, which, at a rigorous
level, would require the solution of the
Boltzmann equation for the
distribution function of photons.\footnote{See
\citet{McClarren10} and \citet{Radice2013} for some
promising progress in this direction.}
For this reason,
a few more pragmatic approaches have been developed over the
years,  which become useful both from a theoretical and
from a numerical point of view.
Perhaps the most successful one is represented by
the so--called projected symmetric
trace--free (PSTF) moment formalism of
\cite{Thorne1981}, which defines
the moments of the
radiation field similarly to how density, momentum and pressure of a
fluid are defined as moments of the
corresponding distribution function.

In the recent past, by adopting this procedure,
a number of time-dependent numerical codes have been
developed and subsequently applied to
various scenarios related to accretion
flows in a relativistic
context~\citep{Shapiro96,Farris08,Zanotti2011,Fragile2012,
Roedig2012,Sadowski2013,Takahashi2013}.
\citet{Sadowski2013b}, in particular, have studied the
super--critical accretion onto a black hole from a
radiative torus, showing that most of the luminosity
emerges through the funnels and confirmed, broadly
speaking, the physical importance of these objects for
high-energy astrophysics after so many years of active
research.

The goal of this paper is to provide a simple
model for an optically thick radiative
torus around a black hole. Local thermodynamic equilibrium is assumed,
and it is shown that a radiation field can be added while
keeping the underlying hydrodynamic solution unmodified.
The new model can be useful both on theoretical grounds,
for clarifying the physics of radiative tori, and
on numerical grounds, since it offers a test for general
relativistic radiation codes in the optically thick
regime. However, the time--dependent numerical
investigation and the study of the stability properties
of the proposed model are not the focus of this work and will
be considered elsewhere.

In the following, we set the speed of light $c=1$,
and the gravitational constant $G=1$.
We extend the geometric units by setting $m_p/k_B=1$,
where $m_p$ is the mass of the proton, while $k_B$ is the Boltzmann
constant.
However, we have maintained $c$ in an explicit form in expressions of particular physical interest.

\section{Physics of the model}
\label{sec:Physics-of-the-model}

\subsection{Basics assumptions}
We considered a stationary and axisymmetric space-time
around a Kerr black hole, which may be written either in
Boyer-Lindquist or in Kerr-Schild coordinates.
The energy--momentum tensor of the system is given by
\be
T^{\alpha\beta}=T^{\alpha\beta}_{{\rm m}}+T^{\alpha\beta}_{{\rm r}}\,,
\ee
which clearly contains two contribution. The first term
is due to matter and can be written as
\be
T^{\alpha\beta}_{{\rm m}}=\rho
h\,u^{\,\alpha}u^{\beta}+pg^{\,\alpha\beta}\,,
\label{T_matter}
\ee
where $g^{\alpha\beta}$ is the metric of the background space-time, $u^\alpha$ is
the four--vector velocity of the fluid, while $\rho$, $h=1+\epsilon + p/\rho$,
$\epsilon$ and $p$ are the rest--mass density, the specific (i.e. per
unit mass) enthalpy,
the specific internal energy, and the pressure,
respectively. For the time being, we keep the equation of
state unspecified. The fluid is in circular motion around
the black hole and its four--vector velocity is
$u^\alpha=u^t(1,0,0,\Omega)$, where $\Omega=u^\phi/u^t$ is
the angular velocity as observed from infinity.

The second contribution to the energy--momentum tensor is
due to the radiation field, and, in terms of its moments,
can be written as \citep{Hsieh1976}
\be
\label{T_rad1}
T_{\rm r}^{\alpha\beta}=(E_{\rm r}+{\cal P}_{\rm r}) u^\alpha u^\beta +
F_{\rm r}^\alpha u^\beta+ u^\alpha F_{\rm r}^\beta + P^{\alpha\beta}_{\rm r}\,,
\ee
where the energy density, the radiation flux and the radiation
stress tensor, all measured in the comoving frame of the fluid, are given by
\bea
&&E_{\rm r}=\frac{1}{c}\int I_\nu d\nu d\Omega \,, \\
&&F_{\rm r}^\alpha=h^\alpha_{\ \beta}\int I_\nu d\nu d\Omega N^\beta
\,, \\
&&P^{\alpha\beta}_{\rm r}=\frac{1}{c}\int I_\nu d\nu d\Omega N^\alpha N^\beta
\,.
\eea
We recall that
$I_\nu=I_\nu(x^\alpha,N^i,\nu)$ is the specific (i.e. an
energy flux per unit time, frequency, and solid angle)
intensity of the radiation,
 $N^\alpha$ is the four--vector defining the photon
propagation direction,
$d\nu$ is the infinitesimal frequency and
$d\Omega$ is the infinitesimal solid angle around the direction of
propagation. Finally,
$h^{\alpha\beta}=u^\alpha u^\beta + g^{\alpha\beta}$
is the projector operator into
the space orthogonal to the four--velocity $u^\alpha$.
In full generality, the four--force density of the radiation field, which
describes the interaction of radiation with matter,
is given by \citep{Mihalas84,Shapiro96}
\be
G^\alpha_{\rm r}=\frac{1}{c}\int(\chi_\nu I_\nu-\eta_\nu)N^\alpha
d\nu d\Omega \,,
\ee
where $\chi_\nu \equiv \chi_\nu^t+\chi_\nu^s$ and $\eta_\nu\equiv
\eta_\nu^t+\eta_\nu^s$ are the total opacity and emissivity
coefficients,
each containing a thermal contribution, indicated with superscript
t, and a scattering contribution, indicated with superscript
s. However, if additionally assume that
the fluid is optically thick and in local
  thermodynamic equilibrium with the radiation,
then the radiation four--force vanishes, and the energy
density is just $E_{\rm r}=4\pi \tilde{B}=a_{\rm
  rad}T^4$,
where $4\pi \tilde{B}=a_{\rm rad}T^4$ is the equilibrium black--body
intensity, $T$ is the local temperature of the fluid, and $a_{\rm rad}$ is
the radiation constant.
Under these conditions, the radiation is isotropic in the
comoving frame of the fluid, the radiation pressure is
${\cal P}_{\rm r}=E_{\rm
  r}/3,$ and the radiation fluxes $F_{\rm r}^\alpha$ also
vanish~\citep{Farris08}.
As a result, the energy--momentum tensor
(\ref{T_rad1}) of
the radiation field reduces to
\be
\label{T_rad2}
T_{\rm r}^{\alpha\beta}=\frac{4}{3}E_{\rm r}u^\alpha u^\beta +
\frac{1}{3}E_{\rm r}g^{\alpha\beta}\,.
\ee

\subsection{Description of the model}
We searched for a stationary, that is, $\partial_t\equiv0$,  and
axisymmetric,  that is, $\partial_\phi\equiv0$,
solution to the
full set of equations describing the dynamics of the
system, that is,
\bea
\label{0eq:mass}
&&\nabla_{\alpha} (\rho u^{\,\alpha})=0, \\
\label{0eq:momentum}
&&\nabla_{\alpha}T^{\alpha\beta}=0, \\
\label{0eq:rad}
&&\nabla_\alpha T_{\rm r}^{\alpha\beta}=-G^\beta_{\rm r} \,.
\eea
The continuity equation is of course trivially satisfied
when the velocity field is purely toroidal.
In addition, since local thermodynamic equilibrium is assumed,
and the radiation four--force vanishes,
Eq.~(\ref{0eq:momentum}) effectively decouples into
\be
\label{1eq:momentum-a}
\nabla_{\alpha}T_{\rm m}^{\alpha\beta}=0
\ee
and
\be
\label{1eq:momentum-b}
\nabla_{\alpha}T_{\rm r}^{\alpha\beta}=0\,,
\ee
which must be satisfied separately.
From (\ref{1eq:momentum-a}), after contracting with the operator
$h^{\alpha\beta}$, we can write the Euler equation in the
form~\citep{Abramowicz78}
\begin{equation}
\label{accel:gas}
u^\alpha\nabla_\alpha u_\beta = -\frac{\nabla_\beta
  p}{h\rho}=\nabla_\beta \ln |u_t| -
\left(\frac{\Omega}{1-\Omega\ell}\right)
\nabla_\beta\ell
\,,
\end{equation}
where the last term is the four--acceleration of the
fluid in circular motion.
This equation was extensively studied in the context of
geometrically thick disks around black holes, and
depending on the specific\footnote{When referred to the
  angular momentum, the adjective {\em specific} means
 ``per unit energy''.} angular momentum
$\ell=-u_\phi/u_t$, different classes of equilibrium
tori can be obtained\footnote{
See \citet{Rezzolla_book:2013} for a pedagogic exposition.}
whose structure and
properties have been discussed in detail
\citep{Abramowicz78,Kozlowski1978,Font02a,Daigne04}.
Here we investigate the possibility of adding an
optically thick radiation field to those solutions
without affecting their hydrodynamics.
This extension is possible, but
only for a limited class of equations of state. First of
all, we contract Eq.~(\ref{1eq:momentum-b})
with  $h^{\alpha\beta}$ to find, after a few
tensor operations, that
\begin{equation}
\label{accel:rad}
u^\alpha\nabla_\alpha u_\beta = -\frac{\nabla_\beta
  E_{\rm r}}{4E_{\rm r}}\,.
\end{equation}
To make (\ref{accel:gas}) and (\ref{accel:rad})
compatible, it must be
\begin{equation}
\label{eq:p-E}
\frac{\nabla_i p}{h\rho}=\frac{\nabla_i
  E_{\rm r}}{4E_{\rm r}}\hspace{1cm}{\rm for}\hspace{0.2cm} i=r,\theta\,.
\end{equation}
We now need to address the thermodynamic
properties of the fluid. We limited our
attention to isentropic fluids, for which $dp=\rho dh$.
This has two important consequences. The first one
is that the specific enthalpy can be computed as
\begin{equation}
\label{eq:h_vs_W}
h = \exp({\mathcal{W}_\mathrm{in}-\mathcal{W}})\,,
\end{equation}
which follows from the second equality of Eq.~(\ref{accel:gas})
after introducing the potential $\mathcal{W}$ as \citep{Kozlowski1978}
\begin{equation}
\label{eq:equilibrium0}
\mathcal{W}-\mathcal{W}_\mathrm{in} = \ln{|u_{t}|}-\ln{|(u_{t})_\mathrm{in}|}-
\int_{\ell_\mathrm{in}}^{\ell}\frac{\Omega\, d\ell'}{1-\Omega \ell'}\,.
\end{equation}
\begin{table*}
\begin{center}
\caption{Main parameters of
a few representative  tori in
local thermodynamic equilibrium with a radiation field.
>From left to
right the columns report the name of the model,
the black hole spin $a$,
 the power--law index $q$ of the angular momentum
distribution (when pertinent),
the constant specific
angular momentum $\ell$ (when pertinent), the inner and outer radii
$r_\mathrm{in}$ and
$r_\mathrm{out}$,
the radial position of the
center $r_\mathrm{c}$,
and the orbital period at the center of the torus
$t_\mathrm{orb}$,
The last column reports the temperature ratio
$T_{\rm c}/T_{\rm in}$.
All models share
the same mass for the black hole, $M=2.5M_{\odot}$,
polytropic exponent
$\Gamma=4/3,$ and disk-to-hole mass
ratio $M_{\rm t}/M=0.1$.
}
\label{tab1}
\begin{tabular}{l|lcccccccc}
\hline
Model   & $a$           &
 $q$& $\ell$  & $r_{\rm in}$     & $r_{\rm out}$
        & $r_{\rm c}$ & $t_{\rm orb}$  & $T_{\rm c}/T_{\rm in}$
          \\
&  & & & & & & ${\rm (ms)}$
          & &\\

\hline
(a)   & 0.0     & $-$ & 3.7845  &
        4.646 & 14.367& 8.165 & 1.80    & 1.0079\\
(b)   & 0.0      & $-$ & 3.8022  &
        4.566 & 16.122 & 8.378 & 1.87   &  1.0099\\
(c)   & 0.0      & $-$ & 3.8425
        &4.410 & 21.472 & 8.839 &2.03    &  1.0152\\
(d)   & 0.0      & $-$ & 3.8800
        &4.290 &29.539& 9.246 &2.17   &   1.0209\\
\hline
(e)   & 0.7     & 0.1 & $-$
        &3.004 & 11.032 & 5.556 & 1.06  &   1.0132 \\
(f)   & 0.9     & 0.1 & $-$
        &1.971 & 16.535 & 4.124 & 0.71  &   1.0485\\
\end{tabular}
\end{center}
\end{table*}
Note that subscript $\mathrm{in}$ refers to the inner edge of the
disk in the equatorial plane, where $h_\mathrm{in}=1$.
The second consequence of the isentropic assumption
is that
Eq.~(\ref{eq:p-E}) can be integrated to give
\begin{equation}
\label{eq:h-E}
h=\left(\frac{E_{\rm r}}{E_{\rm r,in}}\right)^{1/4}=\frac{T}{T_{\rm in}}\,,
\end{equation}
where $E_{\rm r,in}=a_{\rm rad} T_{\rm in}^4$ is the energy density of the
radiation field at $r=r_{in}$, while $T_{\rm in}$ is the
surface temperature.
Eq.~(\ref{eq:h-E}) poses some restrictions on the
equations of state that can be accepted. For instance, an
ideal fluid equation of state $p=\rho\epsilon(\gamma-1)$,
for which $T=p/\rho$ and $h=1+\gamma/(\gamma-1)T$, would imply
a constant temperature from Eq.~(\ref{eq:h-E}). This is
clearly not
acceptable, since $p/\rho$ is not uniform within the torus.
Hence, we need an equation of state for which the
temperature is a free parameter.
This is the case of the
polytropic equation of state
\begin{equation}
\label{polytrope-Gamma}
p=K\rho^\Gamma\,,
\end{equation}
where $K$ is the polytropic constant and $\Gamma$ is the
adiabatic index of the polytrope.
We stress
that the equation of state of  isentropic  polytropes
can still be written formally like
$p=\rho\epsilon(\Gamma-1)$. However, there are two
important differences with respect to
the equation
of state of an ideal fluid, even when the latter is isentropic.
The first difference is that
the exponent $\Gamma$ of Eq.~(\ref{polytrope-Gamma}) is not
necessarily
given by the ratio of the specific heats at constant
pressure and at constant volume. The second difference is
that for polytropes like (\ref{polytrope-Gamma}),
the temperature $T$ is not
necessarily given by the ratio $p/\rho$ and can therefore
be regarded as a free parameter.\footnote{A famous example is
given by the degenerate Fermi
fluid, whose equation of state  can be written like
Eq.~(\ref{polytrope-Gamma}).
}
It is interesting to note that while a purely
hydrodynamic torus is compatible with the ideal fluid
equation of state, in which case its temperature is given by $p/\rho$,
the radiation hydrodynamic torus that we are describing
is not.

After adopting the equation of state (\ref{polytrope-Gamma}),
Eq.~(\ref{eq:h-E}) can be used to fix the temperature of
the torus as
\begin{equation}
\label{eq:T-h}
T=T_{\rm in} h\,.
\end{equation}
Hence,
the isocurves of the temperature
$T$ and those of the specific enthalpy $h$ of the torus
coincide.

A well-known property of polish doughnuts is that they
possess
an internal Keplerian point, $r_{\rm c}$,
lying on the equatorial
plane, where the rest--mass density reaches a maximum
$\rho_{\rm c}$. This also corresponds to the highest
temperature of the torus, and the ratio of the central to
the surface temperature is given by
\begin{equation}
\label{eq:T-ratio}
\frac{T_{\rm c}}{T_{\rm in}}=h_{\rm c}=
1+\frac{\Gamma}{\Gamma-1}\frac{p_{\rm c}}{\rho_{\rm c}}=
\exp({\mathcal{W}_\mathrm{in}-\mathcal{W}_\mathrm{c}})>1\,.
\end{equation}

We note that the surface
temperature $T_{\rm in}$, or, alternatively, the central
temperature $T_{\rm c}$, is the only missing
parameter that must be specified in addition to those
necessary for constructing the hydrodynamic model.
We also stress that since the temperature scales like
the specific enthalpy, and the specific enthalpy is given
by Eq.~(\ref{eq:h_vs_W}), the temperature does not
depend on the polytropic constant $K$ or on the index
of the polytrope $\Gamma$. This is reminiscent of a
similar result originally proved by \citet{Rezzolla_qpo_03b} for
purely hydrodynamic relativistic tori,
namely that they have ratios $p/\rho,$ which do not depend on the
polytropic constant $K$.

Finally, a consistency check can be performed to show that even
in the presence of a temperature gradient, the
relativistic radiation flux
is zero. In fact,
in the framework of Eckart's formulation of relativistic
standard irreversible thermodynamics~\citep{Eckart40c},
the  flux is given by the relativistic form of Fourier's law, namely
\citep{Israel76}
\be
\label{heat-flow}
F_{\rm r}^\mu=-\lambda T(h^{\mu\nu} \nabla_\nu\ln T + a^\mu)\,,
\ee
where
$\lambda$ is the thermal conductivity.
It is easy to see that if the four--vector acceleration is given
by the right-hand side of Eq.~(\ref{accel:rad}), $F_{\rm rad}^\mu$
as given by Eq.~(\ref{heat-flow}) is indeed zero.

\subsection{A few representative examples}
As a first illustrative example, we consider a
Schwarzschild space-time
with the fluid having a constant (in radius) specific
angular momentum $\ell$.
In these circumstances the rest--mass density distribution can be solved
analytically and is~\citep{Font02a}
\begin{equation}
 \label{eq:rho_vs_W}
\rho  = \left[\left(\frac{\Gamma-1}{K \Gamma}\right)
\left[\exp(\mathcal{W}_\mathrm{in}-\mathcal{W})
-1\right]\right]^{1/(\Gamma-1)}\,,
\end{equation}
where
\begin{equation}
\label{eq:Weq_a=0}
\mathcal{W}(r,\theta) = \ln|u_t| = \frac{1}{2} \ln \left[
\frac{r^2(r-2)\sin^2\theta}{r^3\sin^2\theta - \ell^2(r-2)} \right]
\end{equation}
is the potential.
If we  fix the position of the inner edge of
torus $r_{\rm in}$ at the position of the cusp $r_{\rm
  cusp}$ of the potential $\mathcal{W}(r,\pi/2)$,
the torus will exactly fill its Roche lobe.
Under this assumption, we have considered a few
tori orbiting around a
$M=2.5M_{\odot}$ black hole.
The first four rows of Table~\ref{tab1} report
four models previously considered by \citet{Zanotti03} in
the Schwarzschild space-time, but without the
radiation field.\footnote{
Models (a)--(d) in Table~\ref{tab1}
are the same as models (e)--(h) of \citet{Zanotti03}.
}
All the models have the same disk-to-hole
mass ratio $M_t/M=0.1$, but different sizes.
The last column
reports the ratio $T_{\rm c}/T_{\rm in}$, computed
according to Eq.~(\ref{eq:T-ratio}).
As can be seen, tori that are in local thermodynamic equilibrium
and obey Eq.~(\ref{eq:h-E}) are close to be
isothermal, with deviations from $T={\rm const}$ that
increase as the inner edge of the torus approaches the
black hole.

The property of being almost isothermal remains true also
when power-law distributions of the specific
angular momentum are adopted, namely when
$\ell (r,\theta = \pi/2) = {\cal S} r^q$, although
a richer phenomenology is possible for them.\footnote{
See \citet{Daigne04} for an extended discussion about
tori with a power-law distribution of the specific
angular momentum.}
We recall that in the Kerr metric the radius $r_{\rm  ms}$ of the marginally stable orbit
is a decreasing function of the black-hole spin parameter
$a$. On the other hand, when the
power law-index $q$ is increased, the inner radius of the torus increases as well. Hence,
for low values of $q$, the torus
penetrates deeper into the potential well, placing the cusp below the marginally
stable orbit. In these circumstances the highest
values of $T_{\rm c}/T_{\rm in}$ are found.
The last two rows of Table~\ref{tab1} report
two representative models of
this class,
which were previously considered by \citet{Zanotti05} in Kerr space-time, but without the
radiation field.\footnote{
Models (e)--(f) in Tab.~\ref{tab1}
are the same as models (D2a) and (E1a) of \citet{Zanotti05}.
}

In contrast, when
the exponent $q$  is close to $1/2$,
the rotation law tends to the Keplerian one, with a
torus that flattens
toward the equatorial plane.
Tori of this kind are generally much larger
than $\ell ={\rm const}$ tori \citep{Zanotti2010},
with radii at the maximum rest--mass density
point as high as $r_{\rm c}\sim1000$. Consequently,
they are very mildly
relativistic, with values of the specific enthalpy
$h_{\rm c}$ very close to unity. We have
verified this trend for a few representative cases, not
reported in Table~\ref{tab1}, and found that
$T_{\rm c}/T_{\rm in}\approx 1$ to a
few parts over $10^6$.

\section{Discussion and conclusions}
\label{sec:conclusions}
We have shown that purely hydrodynamical geometrically
thick disks around a black hole,
extensively studied in the past and sometimes referred to
as {\em polish doughnuts},
can be {\em dressed} with a
radiation field in local thermodynamic equilibrium with the fluid.
The equation of
state required for this extension is that of an
isentropic and polytropic fluid, with a temperature that
scales like the specific enthalpy. The resulting models,
computed for a few representative cases of both constant
and varying distributions of the specific angular
momentum, are almost isothermal, with
deviations of only a few percent.

Since the equation of state of the model is nonthermal
and well--suited to describe degenerate matter,
the most natural astrophysical scenario
where such a model can become relevant
is that of a high--density
torus, with densities close to those of neutron stars.
Although still rather simplified,
these models may indeed offer a better description
of high--density tori produced after the merger of
unequal--mass neutron star
binaries collapsing onto a black hole, as reported by
fully relativistic and purely hydrodynamical
numerical simulations~\citep{Rezzolla:2010}.
This marks an important difference with respect to the
radiation torus discussed in \citet{McKinney2013b}, which
adopts a thermal equation of state and is instead well--suited to
describe (much) lower density accretion disks around
black holes.

The model discussed here may also offer
a stationary test for
numerical  codes that solve the equations of general
relativistic radiation hydrodynamics.
However, a few words of caution need to be said
in this respect.
The first warning
is that since the torus is isentropic,
the evolution should preferably be performed
without integrating the energy density of
the fluid, which can be computed algebraically
from the remaining quantities by exploiting
Eq.~(\ref{polytrope-Gamma}). Secondly, since the opacity
is typically a function of the rest--mass density, which drops at the surface of the torus,
the
assumption of rigorous optical thickness is likely
to fail in a thin shell at the surface of the
torus. However, this problem is mitigated by the
possibility of building models for which such a shell
is arbitrarily thin. Thirdly, any modern numerical code for
the solution of the general
relativistic radiation hydrodynamics equations will require
the treatment of a tenuous ``atmosphere'', that is, of a
low--density region filling the supposedly vacuum space
around the torus. The atmosphere is
intrinsically optically thin and this requires a code
capable of handling both the optically thick and the optically thin
regimes. Significant progress in this respect has recently been
obtained by \citet{Sadowski2013,Takahashi2013,McKinney2013b}.

Finally, the stability properties of these new models
and their response to perturbations remain an open question,
which may deserve a dedicated
investigation, both analytical and numerical.

\begin{acknowledgements}
I thank John C.~Miller and the
referee Jonathan McKinney for very useful suggestions that
helped me to improve the quality of the manuscript.
This work  has been financed in parts by the European Research Council under
the European Union's Seventh Framework Programme (FP7/2007-2013) in the frame of the research
project \textit{STiMulUs}, ERC Grant agreement no. 278267.
\end{acknowledgements}

\bibliographystyle{aa}


\section{Introduction and result}

Let $k$ be a knot in $S^3$. In \cite{HS}, H.N. Howards and J. Schultens introduced a method to construct a manifold decomposition of double branched cover (abbreviate it as \textit{DBC}, and call the method \textit{the H-S method}) of $(S^3, k)$ (see section \ref{section-DBC}) and they proved that for $2$-bridge knots and $3$-bridge knots in thin position DBC inherits thin manifold decomposition (note that a knot in a thin position may not induce a thin manifold decomposition by the H-S method in general, see \cite{HS} and \cite{HRS}.) Indeed, if $k$ is a non-prime $3$-bridge knot, then $k=k_1\#k_2$ for $2$-bridge knots $k_1$ and $k_2$, and thin position of $k$ is the sum of $k_1$ and $k_2$ by stacking a thin position of one of the knots on top of a thin position of the other (see Corollary 2.5 of \cite{HS}.) So $w(k)=w(k_1)+w(k_2)-2$, i.e. $k$ holds the \textit{nearly additivity} of knot width (for $k=k_1\#k_2$, $w(k)=w(k_1)+w(k_2)-2$, see \cite{ST2} for more details on ``the nearly additivity of knot width''.)

So we get a question whether the property that a thin position of a knot induces a thin manifold decomposition of DBC by the H-N method implies the nearly additivity of knot width in general.
If a thin position of $k$ is the sum of an ordered stack of prime summands of $k$ where each summand is in a thin position (``\textit{a sum of an ordered stack of prime summands}'' means like the left of Figure \ref{figure-thin}, where the bottom summand is a Montesinos knot $M(0; (2,1), (3,1), (3, 1), (5, 1))$ in a thin position (this figure is borrowed from Figure 5.2.(c) of \cite{HK}) and the top summands are trefoils in thin position,)
then the sum of an ordered stack like that in a different order also determines a thin position of $k$. So $k$ must hold the nearly additivity of knot width by the uniqueness of prime factorization of $k$.
In \cite{RS}, Y. Rieck and E. Sedgwick proved that thin position of the sum of small knots is the sum of an ordered stack in that manner, i.e. it holds the nearly additivity of knot width. But M. Scharlemann and A. Thompson proposed a way to construct a example to contradict the nearly additivity of knot width (see \cite{ST2}.) Although R. Blair and M. Tomova proved that most of Scharlemann and Thompson's constructions do not produce counterexamples for the nearly additivity of knot width (see \cite{BT},) the question seems not obvious.

In this article, we will prove that the question is true.

\begin{theorem}\label{theorem-1}
	If a thin position of a knot $k$ induces a thin manifold decomposition of double branched cover of $(S^3,k)$ by the Howards and Schultens' method, then the thin position is the sum of prime summands by stacking  a thin position of one of prime summands of $k$ on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width in this case.
\end{theorem}

In section \ref{section-critical}, we will generalize Theorem 1.1 by using the concept \textit{critical surface} originated from D. Bachman (see \cite{Bachman1} and \cite{Bachman3} for the original definition and the recently modified definition of ``critical surface''.) So we will get the corollary.\\

\Corollaryno{5.3}\textit{
If a thin position of a knot $k$ induces a manifold decomposition of double branched cover $M$ of $(S^3,k)$ by the Howards and Schultens' method where each thick surface $H_+$ of the manifold decomposition of $M$ is strongly irreducible or critical in $M(H_+)$, then the thin position of $k$ is the sum of prime summands by stacking  a thin position of one of prime summands on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width in this case.}

\section{Generalized Heegaard splittings}

In this section, we will introduce some definitions about generalized Heegaard splittings. We use the notations and definitions by D. Bachman in \cite{Bachman3} through this section for convenience.

\begin{definition}
A \textit{compression body} (a \textit{punctured compression body} resp.) is a $3$-manifold which can be obtained by starting with some closed, orientable, connected surface, $H$, forming the product $H\times I$, attaching some number of $2$-handles to $H\times\{1\}$ and capping off all resulting $2$-sphere boundary components (some $2$-sphere boundaries resp.) that are not contained in $H\times\{0\}$ with $3$-balls. The boundary component $H\times\{0\}$ is referred to as $\partial_+$. The rest of the boundary is referred to as $\partial_-$.
\end{definition}

\begin{definition}
A \textit{Heegaard splitting} of a $3$-manifold $M$ is an expression of $M$ as a union $V\cup_H W$, where $V$ and $W$ are compression bodies that intersect in a transversally oriented surface $H=\partial_+V=\partial_+ W$. If $V\cup_H W$ is a Heegaard splitting of $M$ then we say $H$ is a \textit{Heegaard surface}.
\end{definition}

\begin{definition}
Let $V\cup_H W$ be a Heegaard splitting of a $3$-manifold $M$. Then we say the pair $(V,W)$ is a \textit{weak reducing pair} for $H$ if $V$ and $W$ are disjoint compressing disks on opposite sides of $H$. A Heegaard surface is \textit{strongly irreducible} if it is compressible to both sides but has no weak reducing pairs.
\end{definition}

\begin{definition} \label{definition-GHS}
A \textit{generalized Heegaard splitting} (GHS)\footnote{Note that D. Bachman did not allow thin surfaces in a GHS to be $2$-spheres in \cite{Bachman3}. In particular, he introduced more generalized concept ``pseudo-GHS'' in \cite{Bachman3} to deal with thin spheres and trivial compression bodies.} H of a $3$-manifold $M$ is a pair of sets of pairwise disjoint, transversally oriented, connected surfaces, $\operatorname{Thick}(H)$ and $\operatorname{Thin}(H)$ (in this article, we will call the elements of each of both \textit{thick surfaces} and \textit{thin surfaces}, resp.), which satisfies the following conditions.
	\begin{enumerate}
		\item Each component $M'$ of $M-\operatorname{Thin}(H)$ meets a unique element $H_+$ of $\operatorname{Thick}(H)$ and $H_+$ is a Heegaard surface in $M'$. Henceforth we will denote the closure of the component of $M-\operatorname{Thin}(H)$ that contains an element $H_+\in\operatorname{Thick}(H)$ as $M(H_+)$.
		\item As each Heegaard surface $H_+\subset M(H_+)$ is transversally oriented, we can consistently talk about the points of $M(H_+)$ that are ``above'' $H_+$ or ``below'' $H_+$. Suppose $H_-\in \operatorname{Thin}(H)$. Let $M(H_+)$ and $M(H'_+)$ be the submanifolds on each side of $H_-$. Then $H_-$ is below $H_+$ if and only if it is above $H'_+$.
		\item There is a partial ordering on the elements of $\operatorname{Thin}(H)$ which satisfies the following: Suppose $H_+$ is an element of $\operatorname{Thick}(H)$, $H_-$ is a component of $\partial M(H_+)$ above $H_+$ and $H'_-$ is a component of $\partial M(H_+)$ below $H_+$. Then $H_->H'_-$.
	\end{enumerate}
\end{definition}

\begin{definition}
Suppose $H$ is a GHS of a $3$-manifold $M$ with no $S^3$ components. Then $H$ is \textit{strongly irreducible} if each element $H_+\in \operatorname{Thick}(H)$ is strongly irreducible in $M(H_+)$.
\end{definition}

\section{The manifold decomposition of double branched cover of $(S^3,k)$ from a thin position of $k$\label{section-DBC}}

In this section, we will describe the method to construct a manifold decomposition of DBC from a thin position of a knot originated from H.N. Howards and J. Schultens in \cite{HS} (see section 3 of \cite{HS} for more details.) We borrow the notions and definitions directly from \cite{HS}.

\begin{definition}
Let $h:\{S^3 - (\text{two points})\} \to [0, 1]$ be a height function on $S^3$ that restricts to a Morse function on $k$. Choose a regular value $t_i$ between each pair of adjacent critical values of $h|k$. The \textit{width} of $k$ with respect to $h$ is $\sum_i \#|k\cap h^{-1}(t_i)|$.
Define the \textit{width} of $k$ to be the minimum width of $k$ with respect to $h$ over all $h$. A \textit{thin position} of $k$ is the presentation of $k$ with respect to a height function that realizes the width of $k$.
\end{definition}

\begin{definition}
A \textit{thin level} for $k$ is a $2$-sphere $S$ such that the following hold:
	\begin{enumerate}
		\item $S = h^{-1}(t_0)$ for some regular value $t_0$;
		\item $t_0$ lies between adjacent critical values $x$ and $y$ of $h$, where $x$ is a minimum of $k$ lying above $t_0$ and $y$ is a
maximum of $k$ lying below $t_0$.
	\end{enumerate}
A \textit{thick level} is a $2$-sphere $S$ such that the following hold:
	\begin{enumerate}
		\item $S = h^{-1}(t_0)$ for some regular value $t_0$;
		\item $t_0$ lies between adjacent critical values $x$ and $y$ of $h$, where $x$ is a maximum of $k$ lying above $t_0$ and $y$ is a
minimum of $k$ lying below $t_0$.
	\end{enumerate}
\end{definition}

\begin{definition}
A \textit{manifold decomposition} is a generalized version of GHS in Definition \ref{definition-GHS}\footnote{Many authors use the term ``generalized Heegaard splitting'' to denote ``manifold decomposition'' in \cite{ST} by M. Scharlemann and A. Thompson, but the author distinguished the terms ``manifold decomposition'' and ``generalized Heegaard splitting'' to use two different definitions at the same time.}, where we permit thin surfaces to be 2-spheres (this ``manifold decomposition'' is originated from \cite{ST} by M. Scharlemann and A. Thompson.)
So adjacent thick and thin surfaces in a manifold decomposition cobound a (possibly, punctured-) compression body.
\end{definition}

\begin{definition}
Let the \textit{complexity} of a connected surface $S$ be $c(S)= 1-\chi(S) = 2 \operatorname{genus}(S)-1$ for $S$ of positive genus. Define $c(S^2) = 0$. For $S$ not necessarily connected define $c(S) = \sum\{c(S')| S'\text{ a connected component of }S\}$.
\end{definition}

\begin{definition}
Let the \textit{width} of the manifold decomposition $H$ of $M$ be the set of integers $\{c(S_i)| 1\leq i \leq k, \text{ each }S_i\text{ is the thick surface of }H \}$.
\end{definition}

Order these integers in monotonically non-increasing order. Compare the ordered multi-sets lexicographically.

\begin{definition}
Define the \textit{width} $w(M)$ of $M$ to be the minimal width over all manifold decompositions using the above ordering of the sets of integers.
\end{definition}

\begin{definition}
A given manifold decomposition of $M$ is \textit{thin} if the width of the manifold decomposition is the width of $M$.
\end{definition}

Let $k$ be a knot and denote DBC of $(S^3,k)$ by $M$.
If $k$ is in a thin position, then $M$ inherits a manifold decomposition as follows: Denote the thick levels of $k$ by $S_1,\cdots,S_n$ and the thin levels by $L_1,\cdots,L_{n-1}$. Each $S_i$ and each $L_i$ is a sphere that meets the knot some (even) number of times. More specifically, each $S_i$ meets $k$ at least $4$ times and each $L_i$ meets $k$ at least $2$ times. Denote the surface in $M$ corresponding to $S_i$ by $\tilde{S}_i$ and the surface in $M$ corresponding to $L_i$ by $\tilde{L}_i$ . Each $\tilde{S}_i$ is a closed orientable surface of genus at least $1$ and each $\tilde{L}_i$ is a closed orientable surface. More specifically, if $S_i$ meets $k$ exactly $2l$ times, then $\tilde{S}_i$ is a closed orientable surface of genus $l-1$. And if $L_i$ meets $k$ exactly $2l$ times, then $\tilde{L}_i$ is a closed orientable surface of genus $l-1$. See Figure \ref{figure-thin}.

\begin{figure}
\includegraphics[bb=48 170 503 572,width=8.5cm]{fig-thin.pdf}
\caption{Thin and thick levels of $k$ and corresponding surfaces in $M$\label{figure-thin}}
\end{figure}

The $3$-ball bounded by $S_1$ ($S_n$ resp.) in $S^3$ corresponds to a handlebody $H_1$ ($H_n$ resp.) in $M$, where $H_1$ ($H_n$ resp.) is bounded by $\tilde{S}_1$ ($\tilde{S}_n$ resp.) in $M$.
Moreover, the submanifold between $S_i$ and $L_{i-1}$ in $S^3$ (between $L_i$ and $S_{i}$ resp.) corresponds to a (possibly punctured) compression body $C_i^1$ ($C_i^2$ resp.) in $M$, where $\partial_+ C_i^1 = \tilde{S}_i$ ($\partial_+C_i^2=\tilde{S}_{i}$ resp.) and $\partial_- C_i^1 = \tilde{L}_{i-1}$ ($\partial_-C_i^2=\tilde{L}_{i}$ resp.) See Figure \ref{figure-md}. Moreover, $C_i^1$ ($C_i^2$ resp.) is not a trivial compression body, i.e. homeomorphic to $F\times I$ for a closed surface $F$.

\begin{figure}
\includegraphics[bb=176 83 448 550,width=4cm]{fig-md.pdf}
\caption{The manifold decomposition of $M$\label{figure-md}}
\end{figure}

Therefore, the manifold decomposition of $M$ by Howards and Schultens can be written as an ordered set of thin and thick surfaces,
$$\{\tilde{L}_0=\emptyset, \tilde{S}_1, \tilde{L}_1,\tilde{S}_2,\cdots, \tilde{L}_{n-1}, \tilde{S}_n,\tilde{L}_n=\emptyset\},$$
where we denote the empty negative boundaries of $H_1$ and $H_n$ as $\tilde{L}_0$ and $\tilde{L}_n$ for convenience.

\section{Proof of Theorem \ref{theorem-1}\label{section-main-theorem}}

Let us consider the assumption that the given thin position of $k$ induces a thin manifold decomposition of DBC $M$ of $(S^3,k)$. In Rule 6 of \cite{ST}, Scharlemann and Thompson proved that each thick surface $S$ in a thin manifold decomposition is \textit{weakly incompressible}, i.e. any two compressing disks for $S$ on opposite sides of $S$ intersect along their boundary, so every thick surface in a thin manifold decomposition is strongly irreducible in $M(S)$ (we can also check it from Proposition 4.2.3 of \cite{SSS}.)

Let $H=\{\tilde{L}_0=\emptyset, \tilde{S}_1, \tilde{L}_1,\tilde{S}_2,\cdots, \tilde{L}_{n-1}, \tilde{S}_n,\tilde{L}_n=\emptyset\}$ be the manifold decomposition of DBC of $(S^3, k)$ by the H-S method as in the end of section \ref{section-DBC}, where each $\tilde{S}_i$ for $i=1,\cdots,n$ ($\tilde{L}_i$'s for $i=1,\cdots, n-1$ resp.) comes from a thick level (thin level resp.) of the given thin position of $k$.

If there exist $m$-thin levels which intersect $k$ in two points, then we get $m$-thin spheres in $H$. Since the construction of the manifold decomposition allows only one surface for each thick or thin level, these $m$-thin spheres must be $\tilde{L}_{j_1}$, $\cdots$, $\tilde{L}_{j_m}$ for some $1\leq j_1< \cdots< j_m\leq {n-1}$. Let $\tilde{L}_{j_0}$ be $\tilde{L}_0$  and $\tilde{L}_{j_{m+1}}$ be $\tilde{L}_{n}$ for convenience (so they are empty sets.)

Now we cut DBC along the $m$-thin spheres, and cap off all $S^2$ boundaries with $3$-balls. Then we get $(m+1)$-manifolds, $M_0$, $M_1$, $\cdots$, $M_m$, where each $M_i$ comes from the submanifold of DBC bounded by $\tilde{L}_{j_{i}}$ and $\tilde{L}_{j_{i+1}}$ for $i=0,\cdots, m$. In addition, we can induce the canonical manifold decomposition $H_i$ of $M_i$ from $H$, where $H_i=\{\, \emptyset, \tilde{S}_{j_i+1}, \tilde{L}_{j_i+1},\cdots,\tilde{L}_{j_{i+1}-1},\tilde{S}_{j_{i+1}},\emptyset\}$. Now every thin surface of $H_i$ is not homeomorphic to $S^2$ for $i=0,\cdots,m$, so we can say that the manifold decomposition $H_i$ of $M_i$ is a GHS in Definition of \ref{definition-GHS}. Moreover, it is obvious that the strongly irreducibility of each $\tilde{S}_j$ in $M(\tilde{S}_j)$ for $j=1,\cdots,n$ does not change after it becomes a Heegaard surface in $M_i(\tilde{S}_j)$ for some $i$. So either $H_i$ is a strongly irreducible GHS or $M_i$ is homeomorphic to $S^3$ for $i=0,\cdots,m$.

By Lemma 4.7 of \cite{Bachman3}, $M_i$ is irreducible or homeomorphic to $S^3$ (obviously, it is also irreducible) for $i=0,\cdots,m$ , i.e. no one is $S^2\times S^1$. Since the thin spheres $\tilde{L}_{j_1},\cdots,\tilde{L}_{j_m}$ are separating in $M$, no one of them is an essential sphere in a $S^2\times S^1$ piece in the prime decomposition of $M$. So we can assume that there is no $S^2\times S^1$ piece in the prime decomposition of $M$. But it is not obvious whether the sum $M_0\#_{\tilde{L}_{j_1}}M_1\#_{\tilde{L}_{j_2}}\#\cdots\#_{\tilde{L}_{j_m}}M_m$ does not have any trivial summand. So we need the following claim.\\

\Claim{No $M_i$ is homeomorphic to $S^3$ for $i=0,\cdots,m$.}

\begin{proofc}
Suppose that $M_l$ is homeomorphic to $S^3$ for some $l$.

Let the thin levels in the thin position of $k$ corresponding to $\tilde{L}_{j_l}$ and $\tilde{L}_{j_{l+1}}$ be $L'$ and $L''$ (one of both may be empty level if $l=0$ or $m$), and the thick level in the thin position of $k$ corresponding to $\tilde{S}_{j_l+1}$ be $S$. The thin levels $L'$ and $L''$ intersect $k$ in two points, i.e. each of both $L'$ and $L''$ realizes a connected sum of $k$, i.e. $k=k_1 \#_{L'} k' \#_{L''} k_2$.
Moreover, the thick level $S$ must intersect $k$ in $4$ or more points.
Since $M_l\cong S^3$ is DBC of $(k',S^3)$ and $S^3$ has the unique representation as DBC of $S^3$ branched along a knot or a link (see \cite{Wa2}, and this is also true for the other lens spaces, see \cite{HR} or Problem 3.26 of \cite{K},) $k'$ is an unknot.
Since $k$ is in a thin position, the unknot summand $k'$ in $k=k_1 \#_{L'} k' \#_{L''} k_2$ must  intersect all levels between $L'$ and $L''$ in two points, this contradicts the existence of the thick level $S$. So we get a contradiction. In the cases of $l=0$ and $m$, we get a contradiction by similar arguments for each case. This completes the proof of Claim.
\end{proofc}

Now we can say that the thin spheres $\tilde{L}_{j_1}$, $\cdots$, $\tilde{L}_{j_m}$ determine the prime decomposition of $M$. Let us consider the thin levels $L_{j_1}$, $\cdots$, $L_{j_m}$ corresponding to $\tilde{L}_{j_1}$, $\cdots$,$\tilde{L}_{j_m}$. Then they cut $k$ into $m+1$ summands, i.e $k=k_0\#\cdots\#k_m$. In particular, each $M_i$ is DBC of $(S^3,k_i)$ for $i=0,\cdots,m$ and it is also a prime manifold as already proved. Moreover, each $k_i$ must be prime by Corollary 4 of \cite{KT}. Since $k$ is in a thin position, each summand $k_i$ must be in a thin position. This complete the proof of Theorem \ref{theorem-1}.

\section{A generalization of Theorem \ref{theorem-1}\label{section-critical}}

In this section, we will generalize Theorem \ref{theorem-1} using the concept \textit{critical surface}. D. Bachman introduced a concept ``critical surface'' in \cite{Bachman1} and prove several properties about critical surface and minimal common stabilization. Also he proved Gordon's conjecture (see Problem 3.91 of \cite{K}) using critical surface theory in \cite{Bachman3}. In particular, he used the term \textit{topological minimal surface} to denote the class of surfaces such that they are incompressible, strongly irreducible or critical in \cite{Bachman4}.

\begin{definition}[D. Bachman, Definition 3.3 of \cite{Bachman3}]\label{definition-critical} Let $H$ be a Heegaard surface in some $3$-manifold which is compressible to both sides. The surface $H$ is \textit{critical}
if the set of all compressing disks for $H$ can be partitioned into subsets $C_0$ and $C_1$ such that the following hold.
	\begin{enumerate}
		\item For each $i=0,1$ there is at least one weak reducing pair $(V_i,W_i)$, where $V_i$, $W_i\in C_i$.
		\item If $V\in C_0$ and $W\in C_1$ then $(V,W)$ is not a weak reducing pair.
	\end{enumerate}
\end{definition}

\begin{definition}
Suppose $H$ is a GHS of a $3$-manifold $M$ with no $S^3$ components. Then $H$ is \textit{psudo-critical} if each thick surface $H_+\in \operatorname{Thick}(H)$ is strongly irreducible or critical in $M(H_+)$.\footnote{This definition is weaker than the definition ``\textit{critical GHS}'' in \cite{Bachman3}.}
\end{definition}

Now we introduce a generalization of Theorem \ref{theorem-1}.

\begin{corollary}\label{corollary-1}
If a thin position of a knot $k$ induces a manifold decomposition of double branched cover $M$ of $(S^3,k)$ by the Howards and Schultens' method where each thick surface $H_+$ of the manifold decomposition is strongly irreducible or critical in $M(H_+)$, then the thin position of $k$ is the sum of prime summands by stacking  a thin position of one of prime summands on top of a thin position of another prime summand, and so on. Therefore, $k$ holds the nearly additivity of knot width in this case.
\end{corollary}

\begin{proof}
Assume $H=\{\tilde{L}_0=\emptyset, \tilde{S}_1, \tilde{L}_1,\tilde{S}_2,\cdots, \tilde{L}_{n-1}, \tilde{S}_n,\tilde{L}_n=\emptyset\}$ be the manifold decomposition of DBC of $(S^3, k)$ by the H-S method, and $m$-thin spheres $\tilde{L}_{j_1}$, $\cdots$, $\tilde{L}_{j_m}$ cut $M$ into $M_0,\cdots,M_m$ with the canonical GHS $H_0,\cdots,H_m$ as in section \ref{section-main-theorem}. In particular, every thick surface $H_+\in \operatorname{Thick}{H}$ is strongly irreducible or critical in $M(H_+)$ by the hypothesis.

It is obvious that the strongly irreducibility or criticality of each $\tilde{S}_j$ in $M(\tilde{S}_j)$ for $j=1,\cdots,n$ does not change after it becomes a Heegaard surface in $M_i(\tilde{S}_j)$ for some $i$. So either $H_i$ is a psudo-critical GHS or $M_i$ is homeomorphic to $S^3$ for $i=0,\cdots,m$.

Since the proofs of Lemma 4.6 and Lemma 4.7 of \cite{Bachman3} do not depend on the number of critical thick levels, and only depend on the property that each thick surface of the GHS is strongly irreducible or critical, we can extend both lemmas to psudo-critical GHS. So we get each $M_i$ is irreducible or homeomorphic to $S^3$.

The remaining arguments of the proof of Corollary \ref{corollary-1} are the same as those of Theorem \ref{theorem-1}. This completes the proof of Corollary \ref{corollary-1}
\end{proof}

\bibliographystyle{plain}