# Radiative gravastar with Gibbons-Hawking temperature

Ken-ichi Nakao1,2 Kazumasa Okabayashi1 Tomohiro Harada3 1 Department of
Mathematics and Physics, Graduate School of Science, Osaka City University,
3-3-138 Sugimoto, Sumiyoshi, Osaka 558-8585, Japan
2Nambu Yoichiro Institute of Theoretical and Experimental Physics, Osaka City
University, Sumiyoshi, Osaka City 558-8585, Japann
3Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan

###### Abstract

We study the quantum particle creation in a toy model of spherically symmetric
gravitational collapse whose final product is not a black hole but a
gravastar. Precedent studies revealed that even in the case of the
gravitational collapse to form a horizonless ultra-compact object, thermal
radiation named transient Hawking radiation is generated at the late stage of
the gravitational collapse, and a sudden stop of collapsing motion to form a
horizonless ultra-compact object causes one or two bursts of quantum particle
creation. The very different behavior of the model studied in this paper from
the precedent ones is quantum radiation with a thermal spectrum from the
gravastar between two bursts. The temperature of the radiation is not the same
as the Hawking one determined by the gravitational mass of the system but the
Gibbons-Hawking one of the de Sitter core inside the gravastar.

††preprint: OCU-PHYS-560††preprint: AP-GR-180††preprint: NITEP 133††preprint:
RUP-22-7

##  I Introduction

The gravitational collapse to form black holes is one of the most important
and fascinating subjects in the physics of strong gravity and has gathered
special interests. Recent observational developments have led to a more
activity research field of the black hole physics; the observations through
gravitational waves LIGO:2016 ; LVK:2021  and the imaging of the so-called
black hole shadow EHT:2019  gave us very impressive evidence of the existence
of black hole candidates in our universe.

The black hole is defined as a complement of the causal past of the future
conformal infinity, and its boundary is called the event horizon Penrose:1969
; Hawking:1970 . This definition captures the essence of the black hole and
the event horizon. On the other hand, any observational confirmation of the
black hole defined in such a manner is impossible Cardoso-P ; Nakao-YH .
Exactly speaking, the gravitational waves observed by LIGO and Virgo were
generated by not black hole binaries but collapsing objects; what was taken
through EHT is not a direct image of a black hole but an image of collapsing
matter (see, for example, Appendix A of Ref. OAN ). This is true also for
semi-classical processes which are quantum phenomena in classical spacetime.
The Hawking radiation is a celebrated example which is regarded as a
characteristic of the black hole Hawking:1974 ; Hawking:1975 . However,
Paranjape and Padmanabhan studied a free quantum field in the gravitational
collapse to form a static horizonless spherical ultra-compact object with a
hollow inside and showed that even if no black hole eventually forms,
radiation power whose value is the same as that of the Hawking radiation is
generated at the late stage of the gravitational collapse PP:2009 . The study
by Barceló, Liberati, Sonego and Visser BLSV:2011  revealed that the quantum
radiation studied by Paranjape and Padmanabhan has the thermal spectrum of the
Hawking temperature. Even quantum radiation with thermal spectrum is not an
evidence of a black hole but merely implies it is a black hole candidate.

What we can observationally confirm is that the black hole candidate is a
black hole mimicker. The black hole mimicker is a horizonless ultra-compact
object observationally very similar to a black hole. Various black hole
mimickers have been proposed (see eg. Ref. Cardoso-P ). The situation studied
by Paranjape and Padmanabhan is regarded as a formation process of a black
hole mimicker. One of the present authors, TH, in collaboration with Cardoso
and Miyata, studied the quantum particle creation in the similar situation to
that studied by Paranjape and Padmanabhan Harada-CM . They showed that the two
bursts of the particle creation occur due to the stop of the gravitational
collapse in addition to the thermal radiation revealed by Paranjape and
Padmanabhan. Kokubu and TH studied the quantum field which interacts with the
collapsing object Kokubu-H  in the spacetime similar to that studied in Ref.
Harada-CM . They also studied the case in which the interaction between the
quantum field and the collapsing object is represented as the reflection
boundary condition for the mode function at the surface of the collapsing
object. Their study revealed that, even in this case, the transient Hawking
radiation is generated, and furthermore only one burst of the particle
creation due to the stop of the collapse occurs. In Refs. PP:2009 ; Harada-CM
; Kokubu-H , the gravitational collapse of the spherically symmetric object
with hollow inside, or in other words, an infinitesimally thin spherical shell
were studied. By contrast, the present authors studied the free quantum field
in the gravitational collapse of the spherically symmetric object composed of
homogeneously distributed matter enclosed by an infinitesimally thin crust,
which will be called OHN model Okabayashi-HN . The result obtained in Ref.
Okabayashi-HN  is basically the same as the results in Refs. Harada-CM : the
homogeneous distribution of matter does not affect the time variation of the
radiation power of the quantum particle creation even though the homogeneous
component dominates the total mass of the system.

The final product of OHN model is the spherically symmetric ultra-compact
horizonless object composed of homogeneously distributed matter with
infinitesimally thin crust. The equation of state of the homogeneous matter is
$p=-\rho/3$, where $p$ and $\rho$ are the pressure and the energy density,
respectively. By contrast, in this paper, we consider an OHM type model, but
the final product is a gravastar. The gravastar is one of black hole mimickers
which was proposed by Mazur and Mottola in order to solve problems related to
the black hole (e.g., the information loss problem)MM2004 . Its inside is
occupied by the dark energy of $p=-\rho<0$, or equivalently, the cosmological
constant and hence the geometry of its inside is equivalent to that of the de
Sitter spacetime.

This paper is organized as follows. In Sec. II. we briefly review the quantum
particle creation of the massless scalar field in the spherically symmetric
spacetime and give a formula to calculate the radiation power in the spacetime
with a homogeneous star. In Sec. III, we explain the model of the gravastar
formation through the gravitational collapse of a spherically symmetric dust
star. In Sec. IV, we show an example of quantum particle creation in the
gravastar formation process, and briefly see what happens in the present
model. In Sec. V, we estimate the radiation power in the radiative gravastar
phase and its duration. Sec. VI is devoted to summary. Five Appendices are
given at the end of the paper; Appendices A and B are for the readers
unfamiliar to this topics, whereas Appendices C, D and E show the detail of
the calculations.

We adopt the natural unit $c=\hbar=1$. Newton’s gravitational constant and the
Boltzmann constant are denoted by $G$ and $k_{\rm B}$, respectively. The sign
convention of the metric follows the text book written by Wald Wald .

##  II Quantum particle creation

We study the quantum dynamics of the massless scalar field $\phi$ in the
spherically symmetric spacetime by the so called semi-classical treatment in
which the effect of the quantum field on the classical spacetime geometry is
ignored.

###  II.1 Radiation power

We consider the quantum particle creation in the spherically symmetric
asymptotically flat spacetime. By adopting the double null coordinates, the
infinitesimal world interval can be written in the following form:

$ds^{2}=-h^{2}(u,v)dudv+r^{2}(u,v)d\Omega^{2},$ (1)

where $u$ and $v$ are the retarded and the advanced time coordinates,
respectively, whereas $d\Omega^{2}$ is the round metric. In the case of the
Minkowski spacetime, $h=1$ and the areal radius $r$ agrees with $(v-u)/2$, and
hence $u=v$ at the symmetry center $r=0$. By contrast, in general dynamical
cases, the symmetry center $r=0$ is not $v=u$ but

$v=F(u).$ (2)

In order to study the quantum effect in this spacetime, we consider the
massless free scalar field which is the simplest model but is sufficient for
the present purpose. The Lagrangian density is given as

${\cal
L}=-\frac{1}{2}\sqrt{-g}g^{\mu\nu}(\partial_{\mu}\phi)\partial_{\nu}\phi,$ (3)

where $g_{\mu\nu}$, $g^{\mu\nu}$ and $g$ are the metric tensor, its inverse
and its determinant, respectively. The average of radiation power of the
massless scalar field due to the quantum effect caused by the spacetime
curvature is estimated through the expectation value of the stress-energy-
momentum tensor $T_{\mu\nu}$ as

$\displaystyle P(u)=\oint_{r\rightarrow\infty}\langle
0|(\hat{T}_{uu}-\hat{T}_{vv})|0\rangle
r^{2}d\Omega=\frac{1}{48\pi}\kappa^{2}(u),$ (4)

where, being a prime to be a derivative with respect to $u$,

$\displaystyle\kappa(u):=-\frac{F^{\prime\prime}(u)}{F^{\prime}(u)},$ (5)

and we have ignored the total derivative term in $P$, since it does not
contribute the total radiated energy. This expression is derived by invoking
the S-wave approximation whose sketch is given in Appendix A. It is known that
if the adiabatic condition $|\kappa^{\prime}|/\kappa^{2}\ll 1$ is satisfied,
the spectrum of the radiation is thermal with the temperature BLSV:2011

$k_{\rm B}T=\frac{\kappa}{2\pi}.$ (6)

###  II.2 How to calculate radiation power in the case of a homogeneous star

The outside of the star is assumed to be vacuum. By Birkhoff’s theorem, the
outside domain is described by the Schwarzschild geometry whose metric is
given as

$\displaystyle ds^{2}$
$\displaystyle=-f(r)dt^{2}+\frac{dr}{f(r)}+r^{2}d\Omega^{2}=-f(r)\left(dt^{2}-dr_{*}^{2}\right)+r^{2}d\Omega^{2},$
(7)

where

$f(r)=1-\frac{2GM}{r},$ (8)

and

$r_{*}=\int\frac{dr}{f(r)}=r+2GM\ln\left|r-2GM\right|.$ (9)

The null coordinates are defined as

$u=t-r_{*}~{}~{}~{}~{}~{}{\rm and}~{}~{}~{}~{}~{}v=t+r_{*}.$ (10)

As mentioned, the inside of the spherical star is assumed to be homogeneous
and so described by the Robertson-Walker geometry whose metric is given as

$\displaystyle
ds^{2}=a^{2}(\eta)\left[-d\eta^{2}+d\chi^{2}+\varSigma^{2}(\chi)d\Omega^{2}\right],$
(11)

where

$\varSigma(\chi)=\begin{cases}\sin\chi&\text{for positive curvature space,}\\\
\chi&\text{for flat space,}\\\ \sinh\chi&\text{for negative curvature
space.}\end{cases}$ (12)

The null coordinates are defined as

$U:=\eta-\chi~{}~{}~{}~{}{\rm and}~{}~{}~{}~{}V:=\eta+\chi.$ (13)

We denote the proper time along the world line of the surface of the star by
$\tau$. Then by using the coordinates outside the star, the world line of the
surface of the star, which is a curve with constant round coordinates, is
represented in the form

$\displaystyle u$ $\displaystyle=u_{\rm s}(\tau)=t_{\rm s}(\tau)-r_{\rm
s}(\tau)-2M\ln\left|r_{\rm s}(\tau)-2GM\right|,$ (14) $\displaystyle v$
$\displaystyle=v_{\rm s}(\tau)=t_{\rm s}(\tau)+r_{\rm
s}(\tau)+2M\ln\left|r_{\rm s}(\tau)-2GM\right|,$ (15)

whereas, by using the coordinates inside the star, the world line of the
surface of the star is represented as

$\displaystyle U$ $\displaystyle=U_{\rm s}(\tau)=\eta_{\rm s}(\tau)-\chi_{\rm
s}(\tau),$ (16) $\displaystyle V$ $\displaystyle=V_{\rm s}(\tau)=\eta_{\rm
s}(\tau)+\chi_{\rm s}(\tau).$ (17)

Since the coordinate system we adopt here is different from that of Eq. (1),
we show how to derive $F^{\prime\prime}/F^{\prime}$ in detail. The ingoing
radial null with $v=v_{\rm in}=$ constant and $V=V_{\rm in}=$ constant hits
the surface of the star at $\tau=\tau_{\rm in}$. After the radial null arrives
at the origin $\chi=0$, it becomes the outgoing null with $U=U_{\rm
out}=V_{\rm in}$ and $u=u_{\rm out}$ and again hits the surface of the star at
$\tau=\tau_{\rm out}$. Then, we denote

$\dot{U}_{\rm out}=\dot{U_{\rm s}}(\tau_{\rm out}),~{}~{}\dot{V}_{\rm
in}=\dot{V_{\rm s}}(\tau_{\rm in}),~{}~{}\dot{u}_{\rm out}=\dot{u_{\rm
s}}(\tau_{\rm out})~{}~{}{\rm and}~{}~{}\dot{v}_{\rm in}=\dot{v_{\rm
s}}(\tau_{\rm in}),$ (18)

where a dot represents a derivative with respect to the proper time $\tau$.
Since we have

$1=\frac{dV_{\rm in}}{dU_{\rm out}}=\frac{\dot{V}_{\rm in}d\tau_{\rm
in}}{\dot{U}_{\rm out}d\tau_{\rm out}},$ (19)

we obtain

$\frac{\dot{V}_{\rm in}}{\dot{U}_{\rm out}}=\frac{d\tau_{\rm out}}{d\tau_{\rm
in}}.$ (20)

Then, we have

$\frac{dv_{\rm in}}{du_{\rm out}}=\frac{\dot{v}_{\rm in}}{\dot{u}_{\rm
out}}\frac{d\tau_{\rm in}}{d\tau_{\rm out}}=\frac{\dot{v}_{\rm
in}}{\dot{u}_{\rm out}}\frac{\dot{U}_{\rm out}}{\dot{V}_{\rm in}}.$ (21)

We define $A_{\rm out}$ and $B_{\rm in}$ as

$A_{\rm out}:=\frac{\dot{U}_{\rm out}}{\dot{u}_{\rm out}}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}B_{\rm in}:=\frac{\dot{V}_{\rm in}}{\dot{v}_{\rm in}}.$ (22)

Then, we have

$F^{\prime}(u_{\rm out})=\frac{dv_{\rm in}}{du_{\rm out}}=\frac{A_{\rm
out}}{B_{\rm in}}.$ (23)

By differentiating this expression with respect to $u$, we have

$\displaystyle F^{\prime\prime}(u_{\rm out})$
$\displaystyle=\frac{dF^{\prime}(u_{\rm out})}{du_{\rm out}}=\frac{A_{\rm
out}}{B_{\rm in}\dot{u}_{\rm out}}\left(\frac{d\ln A_{\rm out}}{d\tau_{\rm
out}}-\frac{\dot{U}_{\rm out}}{\dot{V}_{\rm in}}\frac{d\ln B_{\rm
in}}{d\tau_{\rm in}}\right).$ (24)

Hence, by defining $C_{\rm out}$ and $D_{\rm in}$ as

$\displaystyle C_{\rm out}$ $\displaystyle:=-\frac{1}{\dot{u}_{\rm
out}}\frac{d\ln A_{\rm out}}{d\tau_{\rm out}},$ (25) $\displaystyle D_{\rm
in}$ $\displaystyle:=-\frac{1}{\dot{v}_{\rm in}}\frac{d\ln B_{\rm
in}}{d\tau_{\rm in}},$ (26)

we obtain

$\displaystyle\kappa(u_{\rm out})$ $\displaystyle:=-(\ln
F^{\prime})^{\prime}(u_{\rm out})=C_{\rm out}-\frac{A_{\rm out}}{B_{\rm
in}}D_{\rm in}.$ (27)

The world line of the surface of the star is represented as

$\displaystyle\eta$ $\displaystyle=\eta_{\rm s}(\tau),$ (28)
$\displaystyle\chi$ $\displaystyle=\chi_{\rm s}(\tau)$ (29)

in the inside coordinates, and

$\displaystyle t$ $\displaystyle=t_{\rm s}(\tau),$ (30) $\displaystyle r$
$\displaystyle=r_{\rm s}(\tau),$ (31)

in the outside coordinates. Denoting the scale factor at the surface
$a(\eta_{\rm s})$ by $a_{\rm s}$, we have

$\displaystyle\frac{d\eta_{\rm s}}{d\tau}$
$\displaystyle=\sqrt{\dot{\chi}_{\rm s}^{2}+\frac{1}{a_{\rm s}^{2}}},$ (32)
$\displaystyle\frac{dt_{\rm s}}{d\tau}$
$\displaystyle=\frac{\sqrt{\dot{r}_{\rm s}^{2}+f(r_{\rm s})}}{f(r_{\rm s})},$
(33)

and hence we obtain

$\displaystyle\dot{U}_{\rm s}$ $\displaystyle=\sqrt{\dot{\chi}_{\rm
s}^{2}+\frac{1}{a_{\rm s}^{2}}}-\dot{\chi}_{\rm s},$ (34)
$\displaystyle\dot{V}_{\rm s}$ $\displaystyle=\sqrt{\dot{\chi}_{\rm
s}^{2}+\frac{1}{a_{\rm s}^{2}}}+\dot{\chi}_{\rm s},$ (35)
$\displaystyle\dot{u}_{\rm s}$ $\displaystyle=\frac{\sqrt{\dot{r}_{\rm
s}^{2}+f(r_{\rm s})}-\dot{r}_{\rm s}}{f(r_{\rm s})},$ (36)
$\displaystyle\dot{v}_{\rm s}$ $\displaystyle=\frac{\sqrt{\dot{r}_{\rm
s}^{2}+f(r_{\rm s})}+\dot{r}_{\rm s}}{f(r_{\rm s})}.$ (37)

We introduce the following quantities:

$\displaystyle A$ $\displaystyle:=\frac{\dot{U}_{\rm s}}{\dot{u}_{\rm
s}}=\left(\sqrt{\dot{r}_{\rm s}^{2}+f(r_{\rm s})}+\dot{r}_{\rm
s}\right)\left(\sqrt{\dot{\chi}_{\rm s}^{2}+\frac{1}{a_{\rm
s}^{2}}}-\dot{\chi}_{\rm s}\right),$ (38) $\displaystyle B$
$\displaystyle:=\frac{\dot{V}_{\rm s}}{\dot{v}_{\rm
s}}=\left(\sqrt{\dot{r}_{\rm s}^{2}+f(r_{\rm s})}-\dot{r}_{\rm
s}\right)\left(\sqrt{\dot{\chi}_{\rm s}^{2}+\frac{1}{a_{\rm
s}^{2}}}+\dot{\chi}_{\rm s}\right),$ (39) $\displaystyle C$
$\displaystyle:=-\frac{1}{\dot{u}_{\rm s}}\frac{d\ln
A}{d\tau}=-\frac{1}{\dot{U}_{\rm s}}\frac{dA}{d\tau}$
$\displaystyle=-\left(\frac{\ddot{r}_{\rm s}}{\sqrt{\dot{r}_{\rm
s}^{2}+f(r_{\rm s})}}-\frac{\ddot{\chi}_{\rm s}}{\sqrt{\dot{\chi}_{\rm
s}^{2}+\dfrac{1}{a_{\rm s}^{2}}}}\right)\left(\sqrt{\dot{r}_{\rm
s}^{2}+f(r_{\rm s})}+\dot{r}_{\rm s}\right)$
$\displaystyle-\frac{GM\dot{r}_{\rm s}}{r_{\rm s}^{2}\sqrt{\dot{r}_{\rm
s}^{2}+f(r_{\rm s})}}+\frac{1}{a_{\rm s}^{3}}\frac{da_{\rm s}}{d\eta_{\rm
s}}\frac{\sqrt{\dot{r}_{\rm s}^{2}+f(r_{\rm s})}+\dot{r}_{\rm
s}}{\sqrt{\dot{\chi}_{\rm s}^{2}+\dfrac{1}{a_{\rm s}^{2}}}-\dot{\chi}_{\rm
s}},$ (40) $\displaystyle D$ $\displaystyle:=-\frac{1}{\dot{v}_{\rm
s}}\frac{d\ln B}{d\tau}=-\frac{1}{\dot{V}_{\rm s}}\frac{dB}{d\tau}$
$\displaystyle=\left(\frac{\ddot{r}_{\rm s}}{\sqrt{\dot{r}_{\rm
s}^{2}+f(r_{\rm s})}}-\frac{\ddot{\chi}_{\rm s}}{\sqrt{\dot{\chi}_{\rm
s}^{2}+\dfrac{1}{a_{\rm s}(\eta_{\rm s})}}}\right)\left(\sqrt{\dot{r}_{\rm
s}^{2}+f(r_{\rm s})}-\dot{r}_{\rm s}\right)$
$\displaystyle-\frac{GM\dot{r}_{\rm s}}{r_{\rm s}^{2}\sqrt{\dot{r}_{\rm
s}^{2}+f(r_{\rm s})}}+\frac{1}{a_{\rm s}^{3}}\frac{da_{\rm s}}{d\eta_{\rm
s}}\frac{\sqrt{\dot{r}_{\rm s}^{2}+f(r_{\rm s})}-\dot{r}_{\rm
s}}{\sqrt{\dot{\chi}_{\rm s}^{2}+\dfrac{1}{a_{\rm s}^{2}}}+\dot{\chi}_{\rm
s}}.$ (41)

Then, we have

$A_{\rm out}=A(\tau_{\rm out}),~{}~{}B_{\rm in}=B(\tau_{\rm in}),~{}~{}C_{\rm
out}=C(\tau_{\rm out}),~{}~{}D_{\rm in}=D(\tau_{\rm in}).$ (42)

##  III Gravastar formation through dust collapse

As already mentioned, we consider the gravastar formation through the
gravitational collapse of a spherically symmetric star. The time evolution of
the star is divided into three phases. In the first phase, the star is
composed of homogeneously distributed dust and begins collapsing from the
momentarily static configuration. We call this period the collapsing phase. In
the second phase, the speed of the gravitational collapse slows down. This
period is called the slowing-down phase. The ingredient of the star is not the
dust in this phase. Finally, the star stops collapsing and becomes a static
gravastar. We call this period the gravastar phase. We explain the details of
the model below.

###  III.1 Collapsing phase

The gravitational collapse of a spherically symmetric star composed of
homogeneous dust is described by the Oppenheimer-Snyder solution. The metric
inside the star is given by Eq. (11) with $\varSigma(\chi)=\sin\chi$. The
surface of the dust sphere is $\chi_{\rm s}=\chi_{\rm i}=$ a positive
constant, and we have

$\displaystyle a$ $\displaystyle=\frac{GM}{\sin^{3}\chi_{\rm
i}}\left(1+\cos\eta\right),$ (43) $\displaystyle\tau$
$\displaystyle=\frac{GM}{\sin^{3}\chi_{\rm i}}(\eta+\sin\eta),$ (44)

where $M$ is a positive constant representing the gravitational mass of the
star, and the domain of $\eta$ is restricted to $\eta\geq 0$. In this model, a
black hole forms when $r_{\rm s}=a_{\rm s}\sin\chi_{\rm i}=2GM$ is satisfied,
or equivalently,

$\eta=\eta_{\rm bh}:=\pi-2\chi_{\rm i}$ (45)

is satisfied. Note that $\chi_{\rm i}$ should be less than $\pi/2$ so that
$\eta_{\rm bh}$ is positive.

###  III.2 Parameters to determine the duration and characteristic radius of
each phase

As mentioned, we assume that the gravitational collapse stops before the
formation of a black hole at $\eta=\eta_{\rm bh}$, and the star eventually
becomes a gravastar. We represent the conformal time $\eta_{\rm G}$ at which
the star becomes the static gravastar by introducing a small positive
parameter $\epsilon$ as

$\eta_{\rm G}=\eta_{\rm bh}-\epsilon^{2}.$ (46)

By introducing a positive parameter $\beta$, the conformal time $\eta_{\rm S}$
at which the collapsing motion begins slowing down denotes

$\eta_{\rm S}=\eta_{\rm G}-\epsilon^{2\beta}=\eta_{\rm
bh}-\epsilon^{2}-\epsilon^{2\beta}$ (47)

so that $\eta_{\rm S}<\eta_{\rm G}$. Thus, the collapsing phase, the slowing-
down phase and the gravastar phase are $0\leq\eta<\eta_{\rm S}$, $\eta_{\rm
S}\leq\eta<\eta_{\rm G}$, and $\eta_{\rm G}\leq\eta$, respectively.

The areal radius of the star at the beginning of the slowing-down phase,
$r_{\rm s}=R_{\rm S}$, is given as

$R_{\rm S}=\frac{GM}{\sin^{2}\chi_{\rm i}}(1+\cos\eta_{\rm S})$ (48)

Note that the areal radius of the gravastar, $R_{\rm G}$, is also a free
parameter of the model. By introducing a parameter $\alpha$ restricted to
$0<\alpha<1$, we represent it as

$\displaystyle R_{\rm G}$ $\displaystyle=\frac{GM}{\sin^{2}\chi_{\rm
i}}\left\\{1+\cos\left[\eta_{\rm
S}+(1-\alpha)\epsilon^{2\beta}\right]\right\\}$
$\displaystyle=\frac{GM}{\sin^{2}\chi_{\rm i}}\left[1+\cos(\eta_{\rm
G}-\alpha\epsilon^{2\beta})\right]$ (49)

so that $R_{\rm S}>R_{\rm G}$. Note that both $R_{\rm S}$ and $R_{\rm G}$ are
larger than the gravitational radius $2GM$ and approach $2GM$ in the limit of
$\epsilon\rightarrow 0$.

###  III.3 Gravastar phase

Before mentioning the detail of the slowing-down phase, we describe the
gravastar as a final product in the present model. The gravastar is a
horizonless ultra-compact object composed of the dark energy with the equation
of state $p=-\rho<0$, or equivalently, the positive cosmological constant,
which is enclosed by an infinitesimally thin crustMM2004 ; VW2004 . In the
present model, the equation of state of the crust is assumed to be determined
through Darmois-Israel junction condition (see Appendix B)Israel:1966 . Due to
the assumption of the spherical symmetry, the inside of the gravastar is
described by the de Sitter geometry. The scale factor of the de Sitter
spacetime in the coordinate system of Eq. (11) with $\varSigma=\sin\chi$ is
given as

$a(\eta)=\frac{1}{H\sin\left(\eta-\eta_{\rm c}\right)},$ (50)

where $\eta_{\rm c}<\eta<\pi+\eta_{\rm c}$, $\eta_{\rm c}$ is a constant
determined later, and $H$ is the so called Hubble parameter of the de Sitter
spacetime which is related to the positive cosmological constant $\Lambda$ by

$H=\sqrt{\frac{\Lambda}{3}}.$ (51)

The final state of the gravastar is static, and its areal radius is equal to
$R_{\rm G}$ expressed as Eq. (49). Due to the relation $r_{\rm s}=a_{\rm
s}\sin\chi_{\rm s}$, the relation $R_{\rm G}=a_{\rm s}\sin\chi_{\rm s}$ should
be satisfied for the static gravastar. Hence, we have

$\chi_{\rm s}=\arcsin\left[HR_{\rm G}\sin(\eta_{\rm s}-\eta_{\rm c})\right]$
(52)

in the final state. Note that $\chi_{\rm s}$ is not constant. It should also
be noted that an inequality

$R_{\rm G}<H^{-1}$ (53)

should hold so that the timelike Killing vector in the de Sitter spacetime is
tangent to the surface of the gravastar.

In the original scenario of the gravastar, $H^{-1}$ is almost equal to
$2GM$MM2004 . Thus, for simplicity, we assume here

$H^{2}R_{\rm G}^{2}=\frac{2GM}{R_{\rm G}}.$ (54)

By using formulae given in Appendix B, we find that this condition leads to
the equation of state of the crust whose surface energy density $S_{(0)(0)}$
vanishes and tangential stresses $S_{(2)(2)}$ and $S_{(3)(3)}$ is given as

$S_{(2)(2)}=S_{(3)(3)}=\frac{3M}{8\pi R_{\rm G}^{2}\sqrt{f(R_{\rm G})}}.$ (55)

The crust of the gravastar does not satisfy reasonable energy conditions,
e.g., the dominant energy condition(see Ref. Wald ). However, the possibility
of such a gravastar model may not be excluded, since the gravastar model is a
proposal by assuming some unknown framework beyond Einstein’s theory, in which
the effective stress-energy-momentum tensor does not necessarily satisfy the
reasonable energy conditions.

Here, we determine $\eta_{\rm c}$. From Eqs. (45) and (46), the formation time
of the static gravastar is equal to $\eta_{\rm G}=\pi-2\chi_{\rm
i}-\epsilon^{2}$. Equation (50) implies that $a_{\rm s}$ diverges in the limit
of $\eta-\eta_{\rm c}\rightarrow\pi$, and hence the static gravastar appears
in $\eta_{\rm G}<\eta<\pi+\eta_{\rm c}$, or equivalently, $\pi-(2\chi_{\rm
i}+\epsilon^{2})<\eta<\pi+\eta_{\rm c}$. As a result, we find that the
inequality

$\eta_{\rm c}>-(2\chi_{\rm i}+\epsilon^{2})$ (56)

should hold. The variation of the areal radius of the star in the slowing-down
phase, $R_{\rm G}-R_{\rm S}$, is approximately given as

$R_{\rm G}-R_{\rm S}\simeq-\frac{GM\sin(2\chi_{\rm
i}+\epsilon^{2})}{\sin^{2}\chi_{\rm s}}(1-\alpha)\epsilon^{2\beta}.$ (57)

We impose that the variation of $\chi_{\rm s}$ in the slowing-down phase is
the order of max$\left[\epsilon^{2},\epsilon^{2\beta}\right]$:

$\sin\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm G}}-\sin\chi_{\rm i}=HR_{\rm
G}\sin(\eta_{\rm G}-\eta_{\rm c})-\sin\chi_{\rm i}={\cal
O}\left(\rm{max}\left[\epsilon^{2},\epsilon^{2\beta}\right]\right)<0,$ (58)

where, since $\chi_{\rm s}$ is a decreasing function of $\eta_{\rm s}$ in the
gravastar phase, we have also imposed the last inequality, i.e., the variation
of $\chi_{\rm s}$ is negative. Thus, for example, we put $\eta_{\rm c}$ as

$\eta_{\rm c}=-\left(\chi_{\rm i}+\frac{3}{2}\epsilon^{2}\right),$ (59)

so that Eq. (58) holds (see Eq. (188) in the case of $0<\beta<1$), where we
assume $\chi_{\rm i}>\epsilon^{2}$.

###  III.4 Slowing-down phase before gravastar formation

We determine the behavior of the star for the slowing-down phase, $\eta_{\rm
S}<\eta<\eta_{\rm G}$. We assume both the radial coordinates $\chi_{\rm s}$
and $r_{\rm s}$ of the surface of the star are $C^{3}$ functions of $\eta_{\rm
s}$. Then, we represent $\chi_{\rm s}$ as

$\sin\chi_{\rm s}=\begin{cases}\sin\chi_{\rm i}&\text{for $0<\eta_{\rm
s}\leq\eta_{\rm S}$}\\\ \sin\chi_{\rm
i}+\displaystyle{\sum_{n=0}^{3}}c_{n}(\eta_{\rm s}-\eta_{\rm
S})^{n+4}&\text{for $\eta_{\rm S}<\eta_{\rm s}<\eta_{\rm G}$}\\\ HR_{\rm
G}\sin(\eta_{\rm s}-\eta_{\rm c})&\text{for $\eta_{\rm G}\leq\eta_{\rm
s}<\eta_{\rm c}+\pi$}\end{cases}~{}.$ (60)

where putting

$\displaystyle S_{\chi}$ $\displaystyle:=HR_{\rm G}\sin(\eta_{\rm G}-\eta_{\rm
c}),$ (61) $\displaystyle C_{\chi}$ $\displaystyle:=HR_{\rm G}\cos(\eta_{\rm
G}-\eta_{\rm c}),$ (62)

the coefficients $c_{n}$ are given as

$\displaystyle c_{0}$
$\displaystyle=\frac{1}{6\epsilon^{8\beta}}\left[210(S_{\chi}-\sin\chi_{\rm
i})-90C_{\chi}\epsilon^{2\beta}-15S_{\chi}\epsilon^{4\beta}+C_{\chi}\epsilon^{6\beta}\right],$
(63) $\displaystyle c_{1}$
$\displaystyle=\frac{1}{2\epsilon^{10\beta}}\left[-168(S_{\chi}-\sin\chi_{\rm
i})+78C_{\chi}\epsilon^{2\beta}+14S_{\chi}\epsilon^{4\beta}-C_{\chi}\epsilon^{6\beta}\right],$
(64) $\displaystyle c_{2}$
$\displaystyle=\frac{1}{2\epsilon^{12\beta}}\left[140(S_{\chi}-\sin\chi_{\rm
i})-68C_{\chi}\epsilon^{2\beta}-13S_{\chi}\epsilon^{4\beta}+C_{\chi}\epsilon^{6\beta}\right],$
(65) $\displaystyle c_{3}$
$\displaystyle=\frac{1}{6\epsilon^{14\beta}}\left[-120(S_{\chi}-\sin\chi_{\rm
i})+60C_{\chi}\epsilon^{2\beta}+12S_{\chi}\epsilon^{4\beta}-C_{\chi}\epsilon^{6\beta}\right].$
(66)

Since the world line of the surface of the star should be timelike,

$\left|\frac{d\chi_{\rm s}}{d\eta_{\rm s}}\right|<1$ (67)

should hold. As shown below, this condition constrains the parameter $\beta$.
Since the period $\eta_{\rm S}<\eta_{\rm s}<\eta_{\rm G}=\eta_{\rm
S}+\epsilon^{2\beta}$ is the slowing-down phase, we introduce a normalized
time coordinate $\gamma$ for this phase, which is defined as

$\eta_{\rm s}=\eta_{\rm S}+\gamma\epsilon^{2\beta},$ (68)

where $\gamma$ increases from $0$ to $1$. Then, by differentiating Eq. (60),
we have

$\displaystyle\cos\chi_{\rm s}\frac{d\chi_{\rm s}}{d\eta_{\rm s}}$
$\displaystyle=\gamma^{3}\biggl{\\{}140(1-\gamma)^{3}(S_{\chi}-\sin\chi_{\rm
i})\epsilon^{-2\beta}+\left(70\gamma^{3}-204\gamma^{2}+195\gamma-60\right)C_{\chi}$
$\displaystyle-(1-\gamma)\left(14\gamma^{2}-25\gamma+10\right)S_{\chi}\epsilon^{2\beta}+\frac{1}{6}(1-\gamma)\left(7\gamma^{2}-11\gamma+4\right)C_{\chi}\epsilon^{4\beta}\biggr{\\}}$
(69)

We are only interested in the case of $0<\epsilon\ll 1$. In the limit of
$\epsilon\rightarrow 0+$, we have

$\displaystyle\cos\chi_{\rm s}$ $\displaystyle\longrightarrow\cos\chi_{\rm
i},$ (70) $\displaystyle C_{\chi}$ $\displaystyle\longrightarrow-\cos\chi_{\rm
i},$ (71) $\displaystyle S_{\chi}$ $\displaystyle\longrightarrow\sin\chi_{\rm
i},$ (72)

and

$(S_{\chi}-\sin\chi)\epsilon^{-2\beta}\longrightarrow\begin{cases}-\dfrac{1}{2}\alpha\cos\chi_{\rm
i}&\text{for $0<\beta<1$}\\\ -\left(1+\dfrac{1}{2}\alpha\right)\cos\chi_{\rm
i}&\text{for $\beta=1$}\\\ -\cos\chi_{\rm
i}~{}\displaystyle{\lim_{\epsilon\rightarrow
0}}\epsilon^{-2(\beta-1)}=-\infty&\text{for $\beta>1$}\end{cases}$ (73)

Hence, $0<\beta\leq 1$ should be imposed so that Eq. (67) holds for very small
positive $\epsilon$. Further careful investigation shows that $\beta=1$ should
be excluded (see Appendix C). Hence, hereafter, we assume

$0<\beta<1.$ (74)

We represent $r_{\rm s}$ as

$r_{\rm s}=\begin{cases}\dfrac{GM}{\sin^{2}\chi_{\rm i}}(1+\cos\eta)&\text{for
$0<\eta_{\rm s}\leq\eta_{\rm S}$}\\\ R_{\rm
G}+\displaystyle{\sum_{n=0}^{3}}C_{n}(\eta_{\rm s}-\eta_{\rm
G})^{n+4}&\text{for $\eta_{\rm S}<\eta_{\rm s}<\eta_{\rm G}$}\\\ R_{\rm
G}&\text{for $\eta_{\rm G}\leq\eta_{\rm s}<\eta_{\rm c}+\pi$}\end{cases}~{}.$
(75)

where putting

$\displaystyle S_{r}$ $\displaystyle:=\frac{GM}{\sin^{2}\chi_{\rm
i}}\sin\eta_{\rm S},$ (76) $\displaystyle C_{r}$
$\displaystyle:=\frac{GM}{\sin^{2}\chi_{\rm i}}\cos\eta_{\rm S},$ (77)

the coefficients $C_{n}$ are given as

$\displaystyle C_{0}$
$\displaystyle=\frac{1}{6\epsilon^{8\beta}}\left[210(R_{\rm S}-R_{\rm
G})-90S_{r}\epsilon^{2\beta}-15C_{r}\epsilon^{4\beta}+S_{r}\epsilon^{6\beta}\right],$
(78) $\displaystyle C_{1}$
$\displaystyle=\frac{1}{2\epsilon^{10\beta}}\left[168(R_{\rm S}-R_{\rm
G})-78S_{r}\epsilon^{2\beta}-14C_{r}\epsilon^{4\beta}+S_{r}\epsilon^{6\beta}\right],$
(79) $\displaystyle C_{2}$
$\displaystyle=\frac{1}{2\epsilon^{12\beta}}\left[140(R_{\rm S}-R_{\rm
G})-68S_{r}\epsilon^{2\beta}-13C_{r}\epsilon^{4\beta}+S_{r}\epsilon^{6\beta}\right],$
(80) $\displaystyle C_{3}$
$\displaystyle=\frac{1}{6\epsilon^{14\beta}}\left[120(R_{\rm S}-R_{\rm
G})-60S_{r}\epsilon^{2\beta}-12C_{r}\epsilon^{4\beta}+S_{r}\epsilon^{6\beta}\right].$
(81)

The scale factor $a_{\rm s}$ is given by

$a_{\rm s}=\frac{r_{\rm s}}{\sin\chi_{\rm s}}.$ (82)

Since the radial coordinates $\chi_{\rm s}$ and $r_{\rm s}$ and the scale
factor $a$ are explicitly expressed as functions of $\eta_{\rm s}$ as in Eqs.
(60), (75) and (82), we represent the derivatives of them with respect to
$\tau$ by using their derivatives with respect to $\eta_{\rm s}$. From the
normalization condition $a^{2}\left(\dot{\eta}_{\rm s}^{2}-\dot{\chi}_{\rm
s}^{2}\right)=1$, we have

$\dot{\eta}_{\rm s}=\frac{1}{a_{\rm s}\sqrt{1-\left(\dfrac{d\chi_{\rm
s}}{d\eta_{\rm s}}\right)^{2}}}.$ (83)

By using this relation, we obtain

$\displaystyle\dot{\chi}_{\rm s}$ $\displaystyle=\dot{\eta}_{\rm
s}\frac{d\chi_{\rm s}}{d\eta_{\rm s}}=\frac{1}{a_{\rm
s}\sqrt{1-\left(\dfrac{d\chi_{\rm s}}{d\eta_{\rm
s}}\right)^{2}}}\dfrac{d\chi_{\rm s}}{d\eta_{\rm s}},$ (84)
$\displaystyle\ddot{\chi}_{\rm s}$ $\displaystyle=\frac{1}{a_{\rm
s}^{2}\left[1-\left(\dfrac{d\chi_{\rm s}}{d\eta_{\rm
s}}\right)^{2}\right]^{2}}\dfrac{d^{2}\chi_{\rm s}}{d\eta_{\rm
s}^{2}}-\frac{1}{a_{\rm s}^{3}\left[1-\left(\dfrac{d\chi_{\rm s}}{d\eta_{\rm
s}}\right)^{2}\right]}\frac{da_{\rm s}}{d\eta_{\rm s}}\frac{d\chi_{\rm
s}}{d\eta_{\rm s}},$ (85) $\displaystyle\dot{r}_{\rm s}$
$\displaystyle=\dot{\eta}_{\rm s}\frac{dr_{\rm s}}{d\eta_{\rm
s}}=\frac{1}{a_{\rm s}\sqrt{1-\left(\dfrac{d\chi_{\rm s}}{d\eta_{\rm
s}}\right)^{2}}}\dfrac{dr_{\rm s}}{d\eta_{\rm s}},$ (86)
$\displaystyle\ddot{r}_{\rm s}$ $\displaystyle=\frac{1}{a_{\rm
s}^{2}\left[1-\left(\dfrac{d\chi_{\rm s}}{d\eta_{\rm
s}}\right)^{2}\right]^{2}}\left[\dfrac{d^{2}r_{\rm s}}{d\eta_{\rm
s}^{2}}+\dfrac{d\chi_{\rm s}}{d\eta_{\rm s}}\left(\dfrac{dr_{\rm
s}}{d\eta_{\rm s}}\dfrac{d^{2}\chi_{\rm s}}{d\eta_{\rm
s}^{2}}-\dfrac{d^{2}r_{\rm s}}{d\eta_{\rm s}^{2}}\dfrac{d\chi_{\rm
s}}{d\eta_{\rm s}}\right)\right]$ $\displaystyle-\frac{1}{a_{\rm
s}^{3}\left[1-\left(\dfrac{d\chi_{\rm s}}{d\eta_{\rm
s}}\right)^{2}\right]}\frac{da_{\rm s}}{d\eta_{\rm s}}\frac{dr_{\rm
s}}{d\eta_{\rm s}}.$ (87)

From Eq. (82), we have

$\displaystyle\frac{da_{\rm s}}{d\eta_{\rm s}}$ $\displaystyle=a_{\rm
s}\left(\frac{1}{r_{\rm s}}\frac{dr_{\rm s}}{d\eta_{\rm s}}-\cot\chi_{\rm
s}\frac{d\chi_{\rm s}}{d\eta_{\rm s}}\right),$ (88)

As shown in Appendix E, the dominant energy condition does not hold in this
phase for $0<\epsilon\ll 1$.

###  III.5 Time coordinate outside the star

In order to see the $\eta$-dependence of $t_{\rm s}$, we need to solve the
following differential equation:

$\frac{dt_{\rm s}}{d\eta_{\rm s}}=\frac{d\tau}{d\eta_{\rm s}}\dot{t}_{\rm
s}=\frac{1}{f(r_{\rm s})}\sqrt{\left(\frac{dr_{\rm s}}{d\eta_{\rm
s}}\right)^{2}+a_{\rm s}^{2}f(r_{\rm s})\left[1-\left(\frac{d\chi_{\rm
s}}{d\eta_{\rm s}}\right)^{2}\right]}.$ (89)

In the dust collapsing phase, $0<\eta_{\rm s}<\eta_{\rm S}$, this equation
becomes

$\frac{dt_{\rm s}}{d\eta_{\rm s}}=\frac{GM\cos\chi_{\rm i}}{\sin^{3}\chi_{\rm
i}}\left(1+2\sin^{2}\chi_{\rm i}+\cos\eta_{\rm s}+\frac{4\sin^{4}\chi_{\rm
i}}{\cos\eta_{\rm s}+\cos 2\chi_{\rm i}}\right).$ (90)

This differential equation can be analytically solved, and we have

$t_{\rm s}=\frac{GM\cos\chi_{\rm i}}{\sin^{3}\chi_{\rm
i}}\left[(1+2\sin^{2}\chi_{\rm i})\eta_{\rm s}+\sin\eta_{\rm
s}+\frac{2\sin^{3}\chi_{\rm i}}{\cos\chi_{\rm i}}\ln\left|\frac{\cot\chi_{\rm
i}+\tan\dfrac{\eta_{\rm s}}{2}}{\cot\chi_{\rm i}-\tan\dfrac{\eta_{\rm
s}}{2}}\right|\right],$ (91)

where we have chosen the integration constant so that $t_{\rm s}$ vanishes at
$\eta_{\rm s}=0$.

In the static gravastar phase, $\eta_{\rm s}>\eta_{\rm G}$, Eq. (89) becomes

$\frac{dt_{\rm s}}{d\eta_{\rm s}}=\frac{1}{H\sin\left(\eta_{\rm s}-\eta_{\rm
c}\right)\sqrt{1-H^{2}R_{\rm G}^{2}\sin^{2}\left(\eta_{\rm s}-\eta_{\rm
c}\right)}}.$ (92)

This differential equation can also be analytically integrated and we obtain

$\displaystyle t_{\rm s}$ $\displaystyle=\frac{1}{2H}\ln\left|\frac{HR_{\rm
G}\sin^{2}(\eta_{\rm s}-\eta_{\rm c})+1-\cos(\eta_{\rm s}-\eta_{\rm
c})\sqrt{1-H^{2}R_{\rm G}^{2}\sin^{2}\left(\eta_{\rm s}-\eta_{\rm
c}\right)}}{HR_{\rm G}\sin^{2}(\eta_{\rm s}-\eta_{\rm c})-1-\cos(\eta_{\rm
s}-\eta_{\rm c})\sqrt{1-H^{2}R_{\rm G}^{2}\sin^{2}\left(\eta_{\rm s}-\eta_{\rm
c}\right)}}\right|+t_{\rm c},$ (93)

where $t_{\rm c}$ is an integration constant. For the slowing-down phase,
$\eta_{\rm S}<\eta_{\rm s}<\eta_{\rm G}$, we need to numerically integrate Eq.
(89). In order that the time coordinate $t_{\rm s}$ of the surface of the star
is continuous at $\eta_{\rm s}=\eta_{\rm G}$, the integration constant $t_{\rm
c}$ should satisfy

$\displaystyle t_{\rm c}$ $\displaystyle=\int_{\eta_{\rm S}}^{\eta_{\rm
G}}\frac{1}{f(r_{\rm s})}\sqrt{\left(\frac{dr_{\rm s}}{d\eta_{\rm
s}}\right)^{2}+a_{\rm s}^{2}f(r_{\rm s})\left[1-\left(\frac{d\chi_{\rm
s}}{d\eta_{\rm s}}\right)^{2}\right]}d\eta_{\rm s}$
$\displaystyle+\frac{GM\cos\chi_{\rm i}}{\sin^{3}\chi_{\rm
i}}\left[(1+2\sin^{2}\chi_{\rm i})\eta_{\rm S}+\sin\eta_{\rm
S}+\frac{2\sin^{3}\chi_{\rm i}}{\cos\chi_{\rm i}}\ln\left|\frac{\cot\chi_{\rm
i}+\tan\dfrac{\eta_{\rm S}}{2}}{\cot\chi_{\rm i}-\tan\dfrac{\eta_{\rm
S}}{2}}\right|\right]$ $\displaystyle-\frac{1}{2H}\ln\left|\frac{HR_{\rm
G}\sin^{2}(\eta_{\rm G}-\eta_{\rm c})+1-\cos(\eta_{\rm G}-\eta_{\rm
c})\sqrt{1-H^{2}R_{\rm G}^{2}\sin^{2}(\eta_{\rm G}-\eta_{\rm c})}}{HR_{\rm
G}\sin^{2}(\eta_{\rm G}-\eta_{\rm c})-1-\cos(\eta_{\rm G}-\eta_{\rm
c})\sqrt{1-H^{2}R_{\rm G}^{2}\sin^{2}(\eta_{\rm G}-\eta_{\rm c})}}\right|.$
(94)

##  IV An example and classification of radial null geodesics

Figure 1:  The radiation power $P$ in the case of $\chi_{\rm i}=0.1$,
$\alpha=\beta=0.5$, and $\epsilon=10^{-4}$ is depicted as a function of time
$t$. $P_{\rm H}$ and $P_{\rm GH}$ are the values of the radiation power of the
Hawking radiation with the gravitational mass $M$ and that of the Gibbons-
Hawking radiation with the Hubble constant $H$, respectively.

In Fig. 1, we show an example of the the radiation power detected in the
asymptotic region $r\gg 2GM$ as a function of time $t$; $\chi_{\rm i}=0.1$,
$\alpha=\beta=0.5$, and $\epsilon=10^{-4}$ are assumed. We numerically checked
that the world line of the surface of the collapsing object is timelike in
this case. The initial radius at $\eta_{\rm s}=0$ is equal to
$2GM/\sin^{2}\chi_{\rm i}\simeq 200GM$. From this figure, we find that there
are three characteristic periods divided by two bursts of the quantum particle
creation. In the first period, the radiation power $P$ increases from almost
zero and eventually becomes the value of the so-called Hawking radiation

$P_{\rm H}=\frac{1}{48\pi(4GM)^{2}}.$ (95)

This value is kept until the 1st burst. Since the adiabatic condition
$|\kappa^{\prime}|\ll\kappa^{2}$ holds at this stage, the spectrum is thermal
one with the so-called Hawking temperature

$k_{\rm B}T_{\rm H}=\frac{1}{8\pi GM}.$ (96)

Thus, following Refs. Harada-CM ; Kokubu-H , we call this radiation the
transient Hawking radiation. In the period between the first and second
bursts, the radiation power $P$ becomes small but does not vanish; later, we
will discuss this non-vanishing radiation power in detail. After the second
burst, the radiation power $P$ vanishes completely.

Figure 2:  The schematic spacetime diagram by using the coordinates
$(\eta,\chi)$ inside the star is depicted. The thick solid black curve is the
world line of the surface of the star. Hence, only the left-hand-side domain
of this curve is covered by this coordinates. Blue lines represent the radial
null geodesics, each of which is a typical one of Class I, II, III, IV and V.
The two red lines are the first and last radial null geodesics in Class III.
Note that the areal radius $r_{\rm s}$ of the surface of the star is equal to
the constant $R_{\rm G}$ in the gravastar phase.

As mentioned, we adopt the $S$-wave approximation in order to estimate the
radiation power of quantum particle creation. In this approximation scheme,
the flux of particles detected at the asymptotic region comes from the
nontrivial deformation of the phase of the spherically symmetric mode function
propagating from the past null infinity to the future null infinity. Figure 2
depicts the spacetime diagram by using the coordinates $(\eta,\chi)$. The
black solid curve is the world line of the surface of the star. Blue lines
represent the radial null geodesics along which the spherically symmetric mode
function propagate: each null geodesic is categorized into the following five
classes:

  * •

Class I: null geodesics categorized into this class enter and come out from
the star in the collapsing phase.

  * •

Class II: null geodesics categorized into this class enter the star in the
collapsing phase and come out from the star in the slowing-down phase.

  * •

Class III: null geodesics categorized into this class enter the star in the
collapsing phase and come out from the star in the gravastar phase.

  * •

Class IV: null geodesics categorized into this class enter the star in the
slowing-down phase and come out from the star in the gravastar phase.

  * •

Class V: null geodesics categorized into this class enter and come out from
the star in the gravastar phase.

The mode function along the null geodesic in Class I causes the transient
Hawking radiation if the null comes out from the star in the very late stage
of the collapsing phase. The mode function propagating along the null
geodesics in Class II causes the 1st burst, i.e, the post-Hawking burst named
in Ref. Harada-CM . The mode function along the null geodesics in Class III
causes the non-vanishing radiation. The mode function along the null geodesics
in Class IV causes the 2nd burst. The mode function along the null geodesics
in Class V causes no particle creation, since these null geodesics only go
through the static domain.

The occurrence of the two bursts of the quantum particle creation is a common
feature to both the present study and precedent onesHarada-CM ; Okabayashi-HN
. By contrast, the non-vanishing radiation in the period between two bursts is
the characteristic of the present model, since there is no radiation in this
period in the models investigated in the precedent studies. The gravastar has
already formed in the causal past of the asymptotic observer after the 1st
burst, and therefore this radiation can be attributed to the gravastar
formation. We call this period the radiative gravastar phase. Details of this
phase will be discussed in the next subsection.

The complicated temporal variation of the radiation power $P$ during the two
bursts is also characteristic in the present model. This temporal variation
merely comes from a bit complicated behavior of the radial coordinate of the
surface of the star, $\chi_{\rm s}$ and $r_{\rm s}$, especially non-
monotonicity of the dependence of $\chi_{\rm s}$ on the conformal time
$\eta_{\rm s}$. By differentiating Eqs. (60) and (75) with respect to
$\eta_{\rm s}$ once or twice and taking a limit of $\epsilon\rightarrow 0$, we
have

$\displaystyle\frac{d\chi_{\rm s}}{d\eta_{\rm s}}\longrightarrow$
$\displaystyle-\gamma^{3}\left[1+3(1-\gamma)+6(1-\gamma)^{2}-70(1-\alpha)(1-\gamma)^{3}\right],$
(97) $\displaystyle\frac{dr_{\rm s}}{d\eta_{\rm s}}\longrightarrow$
$\displaystyle-2GM(1-\gamma)^{3}\left[70(1-2\alpha)\gamma^{3}+6\gamma^{2}+3\gamma+1\right]\cot\chi_{\rm
i},$ (98) $\displaystyle\epsilon^{2\beta}\frac{d^{2}\chi_{\rm s}}{d\eta_{\rm
s}^{2}}\longrightarrow$ $\displaystyle
5\gamma^{2}\left[\alpha-6(2-7\alpha)(1-\gamma)^{2}+84(1-\alpha)(1-\gamma)^{3}\right],$
(99) $\displaystyle\epsilon^{2\beta}\frac{d^{2}r_{\rm s}}{d\eta_{\rm
s}^{2}}\longrightarrow$ $\displaystyle
60GM\gamma^{2}(1-\gamma)^{2}\left[7(1-2\alpha)\gamma+7\alpha-3\right]\cot\chi_{\rm
i}.$ (100)

for the slowing-down phase $\eta_{\rm S}<\eta_{\rm s}<\eta_{\rm G}$.
Substituting these results into Eqs. (84)–(88) and using Eqs. (27), (38)–(42),
we find that the amplitude of the radiation power $P$ of the burst is
proportional to $\epsilon^{-4\beta}$ for $1-\gamma\gg\epsilon^{2\beta}$, in
the case that the mode function propagates along the null in Class II or Class
IV.

##  V The radiative gravastar phase

As mentioned, in the radiative gravastar phase, the mode function propagates
along radial null geodesics categorized into Class III. We investigate the
quantity $\kappa$ associated to the null geodesics in Class III in this
subsection.

###  V.1 Calculation of $\kappa$

The first radial null geodesic of Class III comes out from the star at
$\eta_{\rm s}=\eta_{\rm G}$, and hence is described by $\eta-\eta_{\rm
G}=\chi-\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm G}}$. It intersects the origin
$\chi=0$ at $\eta=\eta_{\rm G}-\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm G}}$ and
is the ingoing null $\eta-\eta_{\rm G}+\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm
G}}=-\chi$ before this moment. The conformal time at which the first radial
null geodesic enters the star is denoted by $\eta_{\rm b}$. Since the star is
in the collapsing phase with $\chi_{\rm s}=\chi_{\rm i}$ at this moment, we
have

$\eta_{\rm b}=\eta_{\rm G}-\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm G}}-\chi_{\rm
i}\simeq\pi-4\chi_{\rm i}.$ (101)

By contrast, the last radial null geodesic in Class III enters the star at
$\eta_{\rm s}=\eta_{\rm S}$. The ingoing radial null arriving at the surface
of the star at $\eta=\eta_{\rm S}$ is represented as $\eta-\eta_{\rm
S}=-\chi+\chi_{\rm i}$ since $\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm
S}}=\chi_{\rm i}$ holds. It will arrive at the center $\chi=0$ at

$\eta=\eta_{\rm rf}:=\eta_{\rm S}+\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm
S}}\simeq\pi-\chi_{\rm i}-\epsilon^{2\beta},$ (102)

and then becomes the outgoing radial null represented as $\eta-\eta_{\rm
rf}=\chi$. The radial null which has become outgoing again crosses the surface
of the star, and we denote the conformal time at this moment by $\eta_{\rm
e}$. Since $\eta_{\rm e}-\eta_{\rm rf}=\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm
e}}$ holds, Eq. (52) leads to,

$\sin(\eta_{\rm e}-\eta_{\rm rf})=HR_{\rm G}\sin(\eta_{\rm e}-\eta_{\rm c}).$
(103)

Rewriting the left hand side of this equation as

$\sin\left[\eta_{\rm e}-\eta_{\rm c}+(\eta_{\rm c}-\eta_{\rm
rf})\right]=\sin(\eta_{\rm e}-\eta_{\rm c})\cos(\eta_{\rm c}-\eta_{\rm
rf})+\sin(\eta_{\rm c}-\eta_{\rm rf})\cos(\eta_{\rm e}-\eta_{\rm c}),$ (104)

we obtain

$\left[\cos(\eta_{\rm e}-\eta_{\rm c})-HR_{\rm G}\right]\sin(\eta_{\rm
e}-\eta_{\rm c})=-\sin(\eta_{\rm c}-\eta_{\rm rf})\cos(\eta_{\rm e}-\eta_{\rm
c}).$ (105)

By taking the square of both sides of this equation, we have

$\sin^{2}(\eta_{\rm e}-\eta_{\rm c})=\frac{\sin^{2}(\eta_{\rm c}-\eta_{\rm
rf})}{1-2HR_{\rm G}\cos(\eta_{\rm c}-\eta_{\rm rf})+H^{2}R_{\rm G}^{2}}.$
(106)

Since $\sin(\eta_{\rm e}-\eta_{\rm c})>0$ holds and, as shown below,
$\sin(\eta_{\rm c}-\eta_{\rm rf})<0$ is satisfied, we have

$\sin(\eta_{\rm e}-\eta_{\rm c})=-\frac{\sin(\eta_{\rm c}-\eta_{\rm
rf})}{\sqrt{1-2HR_{\rm G}\cos(\eta_{\rm c}-\eta_{\rm rf})+H^{2}R_{\rm
G}^{2}}}.$ (107)

By using Eqs. (59) and (102), we obtain

$\eta_{\rm c}-\eta_{\rm rf}\simeq-\left(\pi-\epsilon^{2\beta}\right),$ (108)

and hence we have

$\sin(\eta_{\rm c}-\eta_{\rm rf})\simeq-\epsilon^{2\beta}~{}~{}~{}{\rm
and}~{}~{}~{}\cos(\eta_{\rm c}-\eta_{\rm
rf})\simeq-1+\frac{1}{2}\epsilon^{4\beta}.$ (109)

By these equations and Eq. (107), we have

$\sin(\eta_{\rm e}-\eta_{\rm c})\simeq\frac{1}{2}\epsilon^{2\beta}.$ (110)

Since $\eta_{\rm e}-\eta_{\rm c}\simeq\pi$ holds,

$\eta_{\rm e}-\eta_{\rm c}\simeq\pi-\frac{1}{2}\epsilon^{2\beta}$ (111)

is obtained.

It is easy to obtain the following results for the null which enters the star
in the collapsing phase. Since $\dot{\chi}_{\rm s}=0=\ddot{\chi}_{\rm s}$
holds in this phase, we have

$\displaystyle\frac{D_{\rm in}}{B_{\rm in}}$ $\displaystyle=-\frac{GM}{r_{\rm
s}\sin\chi_{\rm i}(\cos\chi_{\rm i}-\dot{r}_{\rm s})}+\frac{\dot{r}_{\rm
s}}{\sin\chi_{\rm i}}$ $\displaystyle=-\frac{1}{2}\left[\frac{1-f(r_{\rm
s})}{f(r_{\rm s})}\left(1-\sqrt{1-\frac{f(r_{\rm s})}{\cos^{2}\chi_{\rm
i}}}\right)+2\sqrt{1-\frac{f(r_{\rm s})}{\cos^{2}\chi_{\rm
i}}}\right]\cot\chi_{\rm i}$ (112)

where we have used the fact that

$\dot{r}_{\rm s}^{2}=\cos^{2}\chi_{\rm i}-f(r_{\rm s})$ (113)

holds in the collapsing phase. The first radial null geodesic enters the star
with $r_{\rm s}=r_{\rm s}|_{\eta_{\rm s}=\eta_{\rm b}}\simeq
8GM\cos^{2}\chi_{\rm i}$, which is assumed to be in the collapsing phase,
i.e., $0<\eta_{\rm b}<\eta_{\rm S}$. Because of $0<\eta_{\rm
b}\simeq\pi-4\chi_{\rm i}+\alpha\epsilon^{2\beta}/2$ (see Eqs. (101) and
(188)), we impose a stringent restriction on $\chi_{\rm i}$ more than
$0<\chi_{\rm i}<\pi/2$ mentioned below Eq. (45) as

$0<\chi_{\rm i}\leq\pi/4.$ (114)

Hence we have

$\left.\frac{D_{\rm in}}{B_{\rm in}}\right|_{\rm
1st}\simeq-\frac{4\cos^{2}2\chi_{\rm i}+2\cos 2\chi_{\rm i}+1}{2\sin
2\chi_{\rm i}\left(2\cos 2\chi_{\rm i}+1\right)}.$ (115)

By contrast, since the last radial null geodesic enters the star with $r_{\rm
s}=R_{\rm S}\simeq 2GM$ (see Eq. (185)), we have

$\left.\frac{D_{\rm in}}{B_{\rm in}}\right|_{\rm last}\simeq-\frac{2\cos
2\chi_{\rm i}+3}{2\sin 2\chi_{\rm i}}.$ (116)

In the gravastar phase, $\dot{r}_{\rm s}=0=\ddot{r}_{\rm s}$ holds, whereas we
have

$\displaystyle\dot{\chi}_{\rm s}$ $\displaystyle=-\frac{R_{\rm G}}{a_{\rm
s}\sqrt{1-H^{2}R_{\rm G}^{2}}}\sqrt{H^{2}-\frac{1}{a_{\rm s}^{2}}},$ (117)
$\displaystyle\ddot{\chi}_{\rm s}$ $\displaystyle=\frac{R_{\rm G}}{a_{\rm
s}(1-HR_{\rm G})}\left(H^{2}-\frac{2}{a_{\rm
s}^{2}}\right)\sqrt{1-\frac{R_{\rm G}^{2}}{a_{\rm s}^{2}}}.$ (118)

By using these results, we obtain

$\displaystyle A_{\rm out}$ $\displaystyle=\frac{1}{a_{\rm
s}}\left(\sqrt{1-\frac{R_{\rm G}^{2}}{a_{\rm s}^{2}}}+HR_{\rm
G}\sqrt{1-\frac{1}{H^{2}a_{\rm s}^{2}}}\right)=\frac{\sin\chi_{\rm s}}{R_{\rm
G}}\left(\cos\chi_{\rm s}+\sqrt{\cos^{2}\chi_{\rm s}-f(R_{\rm G})}\right),$
(119) $\displaystyle C_{\rm out}$ $\displaystyle=H\sqrt{1-\frac{1}{H^{2}a_{\rm
s}^{2}}}\left(\sqrt{1-\frac{R_{\rm G}^{2}}{a_{\rm s}^{2}}}-HR_{\rm
G}\sqrt{1-\frac{1}{H^{2}a_{\rm s}^{2}}}\right)+H^{2}R_{\rm
G}\left(1-\frac{2}{H^{2}a_{\rm s}^{2}}\right)$ $\displaystyle=\frac{1}{R_{\rm
G}}\sqrt{\cos^{2}\chi_{\rm s}-f(R_{\rm G})}\left(\cos\chi_{\rm
s}-\sqrt{\cos^{2}\chi_{\rm s}-f(R_{\rm G})}\right)+\frac{\cos 2\chi_{\rm
s}-f(R_{\rm G})}{R_{\rm G}}.$ (120)

where we have used $a_{\rm s}=R_{\rm G}/\sin\chi_{\rm s}$. Since $0<f(R_{\rm
G})=1-H^{2}R_{\rm G}^{2}\ll 1$ is assumed, we obtain

$\displaystyle A_{\rm out}$ $\displaystyle\simeq H\sin 2\chi_{\rm s},$ (121)
$\displaystyle C_{\rm out}$ $\displaystyle\simeq H\cos 2\chi_{\rm s}.$ (122)

Since the first radial null geodesic comes out from the star with $\chi_{\rm
s}=\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm G}}\simeq\chi_{\rm i}$, we have

$\displaystyle A_{\rm out}|_{\rm 1st}$ $\displaystyle\simeq H\sin 2\chi_{\rm
i},$ (123) $\displaystyle C_{\rm out}|_{\rm 1st}$ $\displaystyle\simeq H\cos
2\chi_{\rm i},$ (124)

and

$\kappa|_{\rm 1st}\simeq H\left(\frac{8\cos^{2}(2\chi_{\rm
i})+4\cos(2\chi_{\rm i})+1}{4\cos(2\chi_{\rm i})+2}\right).$ (125)

For $0<\chi_{\rm i}\ll 1$, we have $\kappa|_{\rm 1st}\simeq 13H/6$ and hence

$P|_{\rm 1st}\simeq\left(\frac{13}{6}\right)^{2}P_{\rm GH},$ (126)

where $P_{\rm GH}$ is the radiation power of the Gibbons-Hawking radiation in
the de Sitter spacetime with the Hubble constant $H$;

$P_{\rm GH}=\frac{H^{2}}{48\pi}.$ (127)

By using Eqs. (52) and (111), we have

$\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm e}}\simeq\frac{1}{2}\epsilon^{2\beta}.$
(128)

From this equation and Eqs. (116), (121) and (122), we obtain

$\displaystyle A_{\rm out}|_{\rm last}$ $\displaystyle\simeq
H\epsilon^{2\beta},$ (129) $\displaystyle C_{\rm out}$ $\displaystyle\simeq H$
(130)

and hence

$\kappa|_{\rm last}\simeq H$ (131)

and

$P|_{\rm last}=\frac{1}{48\pi}\kappa^{2}\simeq P_{\rm GH}.$ (132)

In the late stage of the radiative gravastar phase, $|\chi_{\rm s}|$ becomes
much less than unity. For $|\chi_{\rm s}|\ll 1$, $|A_{\rm out}|\ll H$ and
$C_{\rm out}\simeq H$ hold and hence $\kappa\simeq H$. Thus, in this stage,
the adiabatic condition $|\kappa^{\prime}|\ll\kappa^{2}$ holds, which implies
the thermal spectrum with the Gibbons-Hawking temperature

$k_{\rm B}T_{\rm GH}=\frac{H}{2\pi}.$ (133)

###  V.2 Duration of radiative gravastar phase

We estimate the duration of the radiative gravastar phase with respect to the
time of the asymptotic observer. From Eqs. (111) and (187), we have

$\displaystyle HR_{\rm G}\sin^{2}(\eta_{\rm e}-\eta_{\rm c})-1-\cos(\eta_{\rm
e}-\eta_{\rm c})\sqrt{1-H^{2}R_{\rm G}^{2}\sin^{2}(\eta_{\rm e}-\eta_{\rm
c})}$ $\displaystyle=$ $\displaystyle\frac{(1-HR_{\rm
G})^{2}\sin^{2}(\eta_{\rm e}-\eta_{\rm c})}{HR_{\rm G}\sin^{2}(\eta_{\rm
e}-\eta_{\rm c})-1+\cos(\eta_{\rm e}-\eta_{\rm c})\sqrt{1-H^{2}R_{\rm
G}^{2}\sin^{2}(\eta_{\rm e}-\eta_{\rm c})}}$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{16}\alpha^{2}\epsilon^{8\beta}\cot^{2}\chi_{\rm i},$
(134)

and

$HR_{\rm G}\sin^{2}(\eta_{\rm e}-\eta_{\rm c})+1-\cos(\eta_{\rm e}-\eta_{\rm
c})\sqrt{1-H^{2}R_{\rm G}^{2}\sin^{2}(\eta_{\rm e}-\eta_{\rm c})}\simeq 2.$
(135)

Thus, we have, from Eq. (93), the time coordinate $t=t_{\rm s}$ outside the
star at $\eta=\eta_{\rm e}$ as

$t_{\rm s}|_{\eta_{\rm s}=\eta_{\rm
e}}\simeq\frac{1}{2H}\ln\left(\frac{32\tan^{2}\chi_{\rm
i}}{\alpha^{2}\epsilon^{8\beta}}\right)+t_{\rm c}.$ (136)

This is the time just before the 2nd burst occurs. By contrast, since

$\displaystyle\sin(\eta_{\rm G}-\eta_{\rm c})$
$\displaystyle=\sin\left(\pi-\chi_{\rm
i}+\frac{1}{2}\epsilon^{2}\right)=\sin\chi_{\rm i}+{\cal O}(\epsilon^{2}),$
(137) $\displaystyle\cos(\eta_{\rm G}-\eta_{\rm c})$
$\displaystyle=-\cos\chi_{\rm i}+{\cal O}(\epsilon^{2})$ (138)

hold, we have

$\displaystyle HR_{\rm G}\sin^{2}(\eta_{\rm G}-\eta_{\rm c})-1-\cos(\eta_{\rm
G}-\eta_{\rm c})\sqrt{1-H^{2}R_{\rm G}^{2}\sin^{2}(\eta_{\rm G}-\eta_{\rm
c})}$ $\displaystyle=$ $\displaystyle\frac{(1-HR_{\rm
G})^{2}\sin^{2}(\eta_{\rm G}-\eta_{\rm c})}{HR_{\rm G}\sin^{2}(\eta_{\rm
G}-\eta_{\rm c})-1+\cos(\eta_{\rm G}-\eta_{\rm c})\sqrt{1-H^{2}R_{\rm
G}^{2}\sin^{2}(\eta_{\rm G}-\eta_{\rm c})}}$ $\displaystyle\simeq$
$\displaystyle\frac{\alpha^{2}\epsilon^{4\beta}\cos^{2}\chi_{\rm
i}}{4\left[HR_{\rm G}\sin^{2}\chi_{\rm i}-1-\cos\chi_{\rm
i}\sqrt{1-H^{2}R_{\rm G}^{2}\sin^{2}\chi_{\rm i}}\right]},$ (139)

Thus, we get, from Eq. (93), the time coordinate $t=t_{\rm s}$ outside the
star at $\eta=\eta_{\rm G}$ as

$\displaystyle t_{\rm s}|_{\eta_{\rm s}=\eta_{\rm G}}$
$\displaystyle\simeq\frac{1}{2H}\ln\left|\frac{4\left[(1+H^{2}R_{\rm
G}^{2})\sin^{2}\chi_{\rm i}-2-2\cos\chi_{\rm i}\sqrt{1-H^{2}R_{\rm
G}^{2}\sin^{2}\chi_{\rm
i}}\right]}{\alpha^{2}\epsilon^{4\beta}\cos^{2}\chi_{\rm i}}\right|+t_{\rm
c}.$ (140)

At this moment, the 1st burst ceases. The duration $\Delta t$ between 1st and
2nd bursts, i.e., of the radiative gravastar phase is then given as

$\Delta t=t_{\rm s}|_{\eta=\eta_{\rm e}}-t_{\rm s}|_{\eta=\eta_{\rm
G}}=\frac{1}{2H}\ln\epsilon^{-4\beta}+{\rm
constant}\simeq\frac{1}{H}\ln\epsilon^{-2\beta}.$ (141)

By definition, the value of $\epsilon$ determines when the slowing-down phase
begins and ceases. Smaller the value of $\epsilon$ is, later the 1st burst
occurs and the radiative gravastar phase starts. Furthermore, as shown in the
above, the value of $\epsilon$ also determines the duration of the radiative
gravastar phase.

From Eq. (91), the duration $\delta t$ with respect to the outside time
coordinate $t$ from the beginning of the gravitational collapse, $\eta_{\rm
s}=0$, to a moment before the slowing-down phase, $0<\eta_{\rm s}=\eta_{\rm
bh}-2\varepsilon<\eta_{\rm S}$, is given as

$\displaystyle\delta t$ $\displaystyle=\frac{GM\cos\chi_{\rm
i}}{\sin^{3}\chi_{\rm i}}\biggl{[}\left(1+2\sin^{2}\chi_{\rm
i}\right)(\pi-2\chi_{\rm i}-2\varepsilon)+\sin(\pi-2\chi_{\rm
i}-2\varepsilon)$ $\displaystyle+\frac{2\sin^{3}\chi_{\rm i}}{\cos\chi_{\rm
i}}\ln\left|\frac{\cot\chi_{\rm i}+\cot(\chi_{\rm
i}+\varepsilon)}{\cot\chi_{\rm i}-\cot(\chi_{\rm
i}+\varepsilon)}\right|\biggr{]}.$ (142)

Here we assume

${\max}\left[\epsilon^{2\beta},~{}\exp\left(-\frac{\pi}{2}\chi_{\rm
i}^{-3}\right)\right]\ll\varepsilon\ll\chi_{\rm i}\ll 1.$ (143)

Then, we have

$\delta t\simeq\pi GM\chi_{\rm
i}^{-3}\simeq\sqrt{\frac{3\pi}{32G\rho|_{\eta_{\rm s}=0}}},$ (144)

where $\rho$ is the energy density of the dust. This result implies that the
dust star shrinks to $r_{\rm s}=2GM(1+2\chi_{\rm i}^{-1}\varepsilon)\simeq
2GM$ for the free-fall time.

After the star collapsed to $r_{\rm s}\simeq 2GM$, the areal radius of the
surface of the star is determined by

$\frac{dr_{\rm s}}{dt_{\rm s}}=-\frac{f(r_{\rm s})}{\cos\chi_{\rm
i}}\sqrt{\cos^{2}\chi_{\rm i}-f(r_{\rm s})}\simeq-f(r_{\rm s})$ (145)

and thus, for $t_{\rm s}>\delta t$, we have

$r_{\rm s}-2GM\simeq 4GM\chi_{\rm i}^{-1}\varepsilon\exp\left(-\frac{t_{\rm
s}-\delta t}{2GM}\right).$ (146)

Denoting the time $t_{\rm s}$ at the beginning of the slowing-down phase by
$t_{\rm S}$, we have

$R_{\rm S}\simeq 2GM\left[1+2\chi^{-1}\varepsilon\exp\left(-\frac{t_{\rm
S}-\delta t}{2GM}\right)\right].$ (147)

By comparing this equation with Eq. (185), we have

$\epsilon^{2\beta}\simeq 2\varepsilon\exp\left(-\frac{t_{\rm S}-\delta
t}{2GM}\right).$ (148)

Substituting this result into Eq. (141), we obtain

$\Delta t\simeq H^{-1}\left(\frac{t_{\rm S}-\delta
t}{2GM}+\ln\varepsilon^{-1}\right).$ (149)

Here note that Eqs. (143) and (144) imply $\delta t\gg GM\ln\varepsilon^{-1}$.
Thus, since $H^{-1}\simeq 2GM$ holds in the present model, if the period from
the start of the gravitational collapse to the beginning time of the slowing-
down phase is much longer than the free-fall time of the system, the duration
of the radiative gravastar phase is nearly equal to $t_{\rm S}$.

##  VI Summary

We studied the quantum particle creation in the gravastar formation process
through the gravitational collapse of a spherically symmetric star. The star
is assumed to be initially composed of homogeneously distributed dust and
collapses in the freely falling manner. Just before the formation of the event
horizon, the gravitational collapse stops due to the change of the equation of
state of the star, and then the star eventually becomes the gravastar, in the
present model.

At the late stage of the gravitational collapse of the dust, the thermal
radiation is generated by the quantum effect as pointed out by Hawking
Hawking:1974 ; Hawking:1975 . The sudden stop of the gravitational collapse
causes two bursts of the particle creation. The occurrence of two bursts was
revealed by precedent researches in which a star with hollow inside and that
occupied homogeneous matter enclosed by an infinitesimally thin crust were
studied Harada-CM ; Okabayashi-HN . The characteristic behavior of the present
gravastar formation model is that non-vanishing radiation is released in the
period between 1st and 2nd bursts. Since this radiation is attributed to the
gravastar, we have called this period the radiative gravastar phase. Its
spectrum approaches the thermal one with the Gibbons-Hawking temperature of
the de Sitter spacetime inside the gravastar. Then, after the 2nd burst, no
radiation is generated. The duration of the radiative gravastar phase is
almost equal to $t_{\rm S}$, i.e., the period from the start of the
gravitational collapse to the beginning of the slowing-down phase, if $t_{\rm
S}$ is much longer than the free-fall time of the system. We should note that
there is no black hole horizon and no cosmological horizon, but the transient
thermal radiation appears.

###### Acknowledgements.

KN is grateful to Hideki Ishihara and colleagues at the research groups of
elementary particle physics and astrophysics in Osaka City University for
useful discussions at colloquium. This work was supported by JSPS KAKENHI
Grant Number JP21K03557(KN), JP19K03876, JP19H01895, JP20H05853 (TH), and
JP21J15676 (KO).

##  Appendix A Derivation of radiation power due to quantum effects

In this section, we review the formulation given by Ford and Parker to derive
the radiation power of quantum particle creation in general spherically
symmetric gravitational collapseFord-Parker .

The field operator $\hat{\phi}$ is represented as

$\hat{\phi}=\sum_{l,m}\int d\omega\left[a_{\omega lm}{\cal G}_{\omega
lm}(t,r_{*},\Omega)+a^{\dagger}_{\omega lm}{\cal G}^{*}_{\omega
lm}(t,r_{*},\Omega)\right],$ (150)

where the mode function ${\cal G}_{\omega lm}$ satisfies the equation of
motion for the scalar field

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\nu}\partial_{\nu}{\cal
G}_{\omega lm}\right)=0,$ (151)

and the orthonormal conditions with respect to the Klein-Gordon norm

$\displaystyle({\cal G}_{\omega lm},{\cal
G}_{\omega^{\prime},l^{\prime},m^{\prime}})$ $\displaystyle:=-i\int\left[{\cal
G}_{\omega lm}\left(\partial_{\mu}{\cal
G}^{*}_{\omega^{\prime},l^{\prime},m^{\prime}}\right)+{\cal
G}^{*}_{\omega^{\prime},l^{\prime},m^{\prime}}\left(\partial_{\mu}{\cal
G}_{\omega lm}\right)\right]\sqrt{-g}~{}n^{\mu}d\Sigma$
$\displaystyle=\delta(\omega-\omega^{\prime})\delta_{ll^{\prime}}\delta_{mm^{\prime}},$
(152)

where the integral is taken over a Cauchy surface and $n^{\mu}$ is the unit
normal to this Cauchy surface. From the canonical commutation relation, we
have

$[a_{\omega
lm},a_{\omega^{\prime},l^{\prime},m^{\prime}}]=0,~{}~{}[a^{\dagger}_{\omega
lm},a^{\dagger}_{\omega^{\prime},l^{\prime},m^{\prime}}]=0,~{}~{}[a_{\omega
lm},a^{\dagger}_{\omega^{\prime},l^{\prime},m^{\prime}}]=\delta(\omega-\omega^{\prime})\delta_{ll^{\prime}}\delta_{mm^{\prime}}$
(153)

The vacuum state $|0\rangle$ is defined as the state which satisfies

$a_{\omega lm}|0\rangle=0.$ (154)

We adopt the mode functions which agree with those of the Minkowski spacetime
in the past null infinity $u\rightarrow-\infty$. Hence, the mode functions in
the asymptotic region $R\rightarrow\infty$ take the following form:

${\cal G}_{\omega lm}=\frac{1}{\sqrt{4\pi\omega}r}\left(e^{-i\omega
F_{lm}(u)}+e^{-i\omega v}\right)Y_{lm}(\Omega),$ (155)

where $Y_{lm}$ is the spherical harmonics. Here note that $F_{00}(u)$ is equal
to $F(u)$ in Eq. (2). Then, the state $|0\rangle$ defined as Eq. (154) is
regarded as the Minkowski vacuum in the past null infinity.

The stress-energy-momentum tensor operator of the scalar field is given as

$\hat{T}_{\mu\nu}=(\partial_{\mu}\hat{\phi})\partial_{\nu}\hat{\phi}-\frac{1}{2}g_{\mu\nu}g^{\alpha\beta}(\partial_{\alpha}\hat{\phi})\partial_{\beta}\hat{\phi}.$
(156)

The average energy flux is the expectation value of the following components
of the stress-energy-momentum tensor:

$\displaystyle\langle 0|(\hat{T}_{uu}-\hat{T}_{vv})|0\rangle$
$\displaystyle=\langle
0|\left[(\partial_{u}\hat{\phi})^{2}-(\partial_{v}\hat{\phi})^{2}\right]|0\rangle$
$\displaystyle=\sum_{lm}\int_{0}^{\infty}d\omega\left[\left(\partial_{u}{\cal
G}_{\omega lm}\right)\partial_{u}{\cal G}^{*}_{\omega
lm}-\left(\partial_{v}{\cal G}_{\omega lm}\right)\partial_{v}{\cal
G}^{*}_{\omega lm}\right].$ (157)

In order to obtain the average energy flux detected in the asymptotic region,
we substitute Eq. (155) into this equation, and replacing $u$ and $v$ by
$u+\epsilon$ and $v+\epsilon$ in ${\cal G}^{*}_{\omega lm}$, we evaluate the
integral with respect to $\omega$ as

$\displaystyle\langle 0|(\hat{T}_{uu}-\hat{T}_{vv})|0\rangle$
$\displaystyle=\lim_{\epsilon\rightarrow 0}\frac{1}{4\pi
r^{2}}\sum_{lm}|Y_{lm}|^{2}\int_{0}^{\infty}d\omega\omega\left[F_{lm}^{\prime}(u)F_{lm}^{\prime}(u+\epsilon)e^{i\omega\left[F_{lm}(u+\epsilon)-F_{lm}(u)\right]}-e^{i\omega\epsilon}\right]$
$\displaystyle\simeq\frac{1}{48\pi
r^{2}}\left[\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{2}-2\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{\prime}\right]|Y_{00}|^{2},$
(158)

where a prime represents a derivative with respect to $u$, and we have ignored
the contributions from non-vanishing $l$ modes since those will not suffer the
dynamical effect in the neighborhood of the center $r=0$ due to the
centrifugal potential so much, and hence
$|F^{\prime\prime}_{lm}|\ll|F_{00}^{\prime\prime}|=|F^{\prime\prime}|$ holds
for $l>0$. Hereafter, we ignored the total derivative term
$(F^{\prime\prime}/F^{\prime})^{\prime}$ in the last equality, which does not
contribute the total emitted energy. The radiation power $P$ detected in the
asymptotic region is obtained as

$\displaystyle P=\oint_{r\rightarrow\infty}\langle
0|(\hat{T}_{uu}-\hat{T}_{vv})|0\rangle
r^{2}d\Omega=\frac{1}{48\pi}\left(\frac{F^{\prime\prime}}{F^{\prime}}\right)^{2}.$
(159)

##  Appendix B Stress-energy-momentum of infinitesimally thin shell

The Darmois-Israel junction condition specifies the stress-energy-momentum
tensor of the infinitesimally thin shell on the surface of the starIsrael:1966
.

We adopt the orthonormal basis $\\{\bm{e}_{(\alpha)}\\}$ on the surface of the
star; $\bm{e}_{(0)}$ is the unit tangent to the world line of an observer at
rest on the stellar surface. Their components with respect to the coordinates
just inside the star are given as

$\displaystyle e_{(0)}{}^{\mu}$ $\displaystyle=\left(\dot{\eta}_{\rm
s},\dot{\chi}_{\rm s},0,0\right)=\left(\sqrt{\dot{\chi}_{\rm
s}^{2}+\frac{1}{a_{\rm s}^{2}}},\dot{\chi}_{\rm s},0,0\right)$ (160)
$\displaystyle e_{(1)}{}^{\mu}$ $\displaystyle=\left(\dot{\chi}_{\rm
s},\sqrt{\dot{\chi}_{\rm s}^{2}+\frac{1}{a_{\rm s}^{2}}},0,0\right),$ (161)
$\displaystyle e_{(2)}{}^{\mu}$ $\displaystyle=\left(0,0,\frac{1}{a_{\rm
s}\sin\chi_{\rm s}},0\right)=\left(0,0,\frac{1}{r_{\rm s}},0\right),$ (162)
$\displaystyle e_{(3)}{}^{\mu}$ $\displaystyle=\left(0,0,0,\frac{1}{a_{\rm
s}\sin\chi_{\rm s}\sin\theta}\right)=\left(0,0,0,\frac{1}{r_{\rm
s}\sin\theta}\right),$ (163)

whereas, with respect to the coordinates just outside the star, as

$\displaystyle e_{(0)}{}^{\mu^{\prime}}$ $\displaystyle=\left(\dot{t}_{\rm
s},\dot{r}_{\rm s},0,0\right)=\left(\frac{\sqrt{\dot{r}_{\rm s}^{2}+f(r_{\rm
s})}}{f(r_{\rm s})},\dot{r}_{\rm s},0,0\right)$ (164) $\displaystyle
e_{(1)}{}^{\mu^{\prime}}$ $\displaystyle=\left(\frac{\dot{r}_{\rm s}}{f(r_{\rm
s})},\sqrt{\dot{r}_{\rm s}^{2}+f(r_{\rm s})},0,0\right),$ (165) $\displaystyle
e_{(2)}{}^{\mu^{\prime}}$ $\displaystyle=\left(0,0,\frac{1}{r_{\rm
s}},0\right),$ (166) $\displaystyle e_{(3)}{}^{\mu^{\prime}}$
$\displaystyle=\left(0,0,0,\frac{1}{r_{\rm s}\sin\theta}\right).$ (167)

We define

$e^{(\alpha)\mu}=\eta^{(\alpha)(\beta)}e_{(\beta)}{}^{\mu}~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}~{}\eta_{(\alpha)(\beta)}e^{(\beta)\mu}=e_{(\alpha)}{}^{\mu}$
(168)

where $\eta^{(\alpha)(\beta)}={\rm diag}[-1,1,1,1]=\eta_{(\alpha)(\beta)}$.

The components of the second fundamental form $K^{\rm(in)}_{\mu\nu}$ of the
surface of the star with respect to the coordinates inside the star are given
as

$K^{\rm(in)}_{\mu\nu}=\left(e^{(0)}{}_{\mu}e_{(0)}{}^{\alpha}+e^{(2)}{}_{\mu}e_{(2)}{}_{\alpha}+e^{(3)}{}_{\mu}e_{(3)}{}^{\alpha}\right)\nabla_{\alpha}e_{(1)\nu}.$
(169)

By using this expression, we have the tetrad components of the second
fundamental form in the form;

$\displaystyle K^{\rm(in)}_{(0)(0)}$ $\displaystyle=-\frac{1}{a_{\rm
s}\dot{\eta}_{\rm s}}\left(\ddot{\chi}_{\rm s}+\frac{2}{a_{\rm
s}}\frac{da_{\rm s}}{d\eta_{\rm s}}\dot{\eta}_{\rm s}\dot{\chi}_{\rm
s}\right),$ (170) $\displaystyle K^{\rm(in)}_{(2)(2)}$
$\displaystyle=\frac{1}{a_{\rm s}}\frac{da_{\rm s}}{d\eta_{\rm
s}}\dot{\chi}_{\rm s}+\cot\chi_{\rm s}\dot{\eta}_{\rm s}=K^{\rm(in)}_{(3)(3)}$
(171)

and all the other components vanish. By contrast, the components of the second
fundamental form $K^{\rm(out)}_{\mu\nu}$ of the surface of the star with
respect to the coordinates outside the star is given as

$K^{\rm(out)}_{\mu^{\prime}\nu^{\prime}}=\left(e^{(0)}{}_{\mu^{\prime}}e_{(0)}{}^{\alpha^{\prime}}+e^{(2)}{}_{\mu^{\prime}}e_{(2)}{}_{\alpha^{\prime}}+e^{(3)}{}_{\mu^{\prime}}e_{(3)}{}^{\alpha^{\prime}}\right)\nabla_{\alpha^{\prime}}e_{(1)\nu^{\prime}}.$
(172)

Then, we obtain the tetrad components of the second fundamental form in the
form

$\displaystyle K^{\rm(out)}_{(0)(0)}$ $\displaystyle=-\frac{1}{f(r_{\rm
s})\dot{t}_{\rm s}}\left(\ddot{r}_{\rm s}+\frac{GM}{r_{\rm s}^{2}}\right),$
(173) $\displaystyle K^{\rm(out)}_{(2)(2)}$ $\displaystyle=\frac{f(r_{\rm
s})}{r_{\rm s}}\dot{t}_{\rm s}=\frac{1}{r_{\rm s}}\sqrt{\dot{r}_{\rm
s}^{2}+f(r_{\rm s})}=K^{\rm(out)}_{(3)(3)}$ (174)

and all the other components vanish. By using Eqs. (83)–(87), we concretely
obtain $K_{(\alpha)(\beta)}^{\rm(in)}$ and $K_{(\alpha)(\beta)}^{\rm(out)}$.

The Darmoise-Israel junction condition is

$K^{\rm(out)}_{(\alpha)(\beta)}-K^{\rm(in)}_{(\alpha)(\beta)}=-8\pi
G\left(S_{(\alpha)(\beta)}-\frac{1}{2}h_{(\alpha)(\beta)}{\rm tr}S\right),$
(175)

where $h_{(\alpha)(\beta)}={\rm diag}[-1,0,1,1]$. Then, the tetrad components,
$S_{(\alpha)(\beta)}$, are given as

$\displaystyle S_{(0)(0)}$ $\displaystyle=-\frac{1}{4\pi
G}\left(K^{\rm(out)}_{(2)(2)}-K^{\rm(in)}_{(2)(2)}\right),$ (176)
$\displaystyle S_{(2)(2)}$ $\displaystyle=-\frac{1}{8\pi
G}\left(K^{\rm(out)}_{(0)(0)}-K^{\rm(out)}_{(2)(2)}-K^{\rm(in)}_{(0)(0)}+K^{\rm(in)}_{(2)(2)}\right)=S_{(3)(3)}$
(177)

and the other components vanish. $S_{(0)(0)}$ is the energy per unit area,
whereas $S_{(2)(2)}=S_{(3)(3)}$ is the tangential pressure.

##  Appendix C On the case of $\beta=1$

We examine $|d\chi_{\rm s}/d\eta_{\rm s}|$ in the case of $\beta=1$. Here, we
regard $\chi_{\rm s}$ as a function of $\eta_{\rm s}$ and the parameter
$\alpha$. Then, from Eqs. (69) to (73), we have

$\frac{\partial\chi_{\rm s}}{\partial\eta_{\rm
s}}\xrightarrow[\epsilon\rightarrow
0]{}-\gamma^{3}\left[70\alpha(1-\gamma)^{3}-70\gamma^{3}+216\gamma^{2}-225\gamma+80\right].$
(178)

Because of $0<\gamma<1$, we have

$\frac{\partial}{\partial\alpha}\left.\frac{\partial\chi_{\rm
s}}{\partial\eta_{\rm s}}\right|_{\epsilon=0}=-70\gamma^{3}(1-\gamma)^{3}<0.$
(179)

Because of $0<\alpha<1$, the following inequality should hold:

$-\gamma^{3}\left[70(1-\gamma)^{3}-70\gamma^{3}+216\gamma^{2}-225\gamma+80\right]<\frac{\partial\chi_{\rm
s}}{\partial\eta_{\rm
s}}<-\gamma^{3}\left[-70\gamma^{3}+216\gamma^{2}-225\gamma+80\right].$ (180)

The rightmost side (RMS) of this inequality is rewritten in the form

${\rm RMS}=-\gamma^{3}\left[(1-\gamma)(70\gamma^{2}-146\gamma+79)+1\right].$
(181)

It is easy to see that $70\gamma^{2}-146\gamma+79>0$ holds for real $\gamma$.
Hence we have RMS$<0$ and

$\left|\frac{\partial\chi_{\rm s}}{\partial\eta_{\rm
s}}\right|_{\epsilon=0}>|{\rm RMS}|,$ (182)

for $0<\gamma<1$. It is easy to get

$\frac{d}{d\gamma}|{\rm
RMS}|=420\gamma^{2}(1-\gamma)^{2}\left(\frac{4}{7}-\gamma\right).$ (183)

This equation implies that $|$RMS$|$ has a maximum at $\gamma=4/7$ in
$0<\gamma<1$, and the maximum value of $|$RMS$|$ is

${\rm max}[|{\rm RMS}|]=\left|{\rm
RMS}\right|_{\gamma=4/7}=\frac{27904}{16807}>1.$ (184)

This result implies that $|\partial\chi_{\rm s}/\partial\eta_{\rm
s}|_{\epsilon=0}$ exceeds unity in the neighborhood of $\gamma=4/7$. Thus, the
world line of the surface of the star cannot be kept timelike in the case of
$\beta=1$ for any $0<\alpha<1$.

##  Appendix D Often used formulae

Under the assumption of $0<\beta<1$, we often use the following
approximations:

$\displaystyle R_{\rm S}$ $\displaystyle=\frac{GM}{\sin^{2}\chi_{\rm
i}}\left(1+\cos\eta_{\rm S}\right)\simeq
2GM\left(1+\epsilon^{2\beta}\cot\chi_{\rm i}\right),$ (185) $\displaystyle
R_{\rm G}$ $\displaystyle=\frac{GM}{\sin^{2}\chi_{\rm
i}}\left[1+\cos\left(\eta_{\rm G}-\alpha\epsilon^{2\beta}\right)\right]\simeq
2GM\left(1+\alpha\epsilon^{2\beta}\cot\chi_{\rm i}\right),$ (186)
$\displaystyle HR_{\rm G}$ $\displaystyle\simeq
1-\frac{1}{2}\alpha\epsilon^{2\beta}\cot\chi_{\rm i},$ (187)
$\displaystyle\chi_{\rm s}|_{\eta_{\rm s}=\eta_{\rm G}}$
$\displaystyle\simeq\chi_{\rm i}-\frac{1}{2}\alpha\epsilon^{2\beta}.$ (188)

##  Appendix E Equation of state

The equation of state of the star is nontrivial in the slowing-down phase. We
write the equation of state in the form $p=w\rho$, where $p$ and $\rho$ are
the pressure and the energy density, respectively. The Einstein equations
imply

$\frac{1}{a}\frac{d^{2}a}{d\tau^{2}}=-\frac{4\pi
G}{3}(\rho+3p)~{}~{}~{}\Longrightarrow~{}~{}~{}w=\frac{p}{\rho}=-\frac{1}{4\pi
G\rho}\frac{1}{a}\frac{d^{2}a}{d\tau^{2}}-\frac{1}{3}$ (189)

and

$\left(\frac{1}{a}\frac{da}{d\tau}\right)^{2}=\frac{8\pi
G}{3}\rho-\frac{1}{a^{2}}~{}~{}~{}\Longrightarrow~{}~{}~{}4\pi
G\rho=\frac{3}{2a^{2}}\left[\left(\frac{da}{d\tau}\right)^{2}+1\right].$ (190)

Combining these equations, we have

$\displaystyle w$
$\displaystyle=-\frac{2a}{3}\left[\left(\frac{da}{d\tau}\right)^{2}+1\right]^{-1}\frac{d^{2}a}{d\tau^{2}}-\frac{1}{3}$
$\displaystyle=-\frac{2}{3a}\left[\left(\frac{1}{a}\frac{da}{d\eta}\right)^{2}+1\right]^{-1}\left[\frac{d^{2}a}{d\eta^{2}}-\frac{1}{a}\left(\frac{da}{d\eta}\right)^{2}\right]-\frac{1}{3}.$
(191)

In order to see $w$ in the slowing-down phase, we may invoke Eq. (88) and

$\frac{d^{2}a_{\rm s}}{d\eta_{\rm s}^{2}}=a_{\rm s}\left[\frac{1}{r_{\rm
s}}\frac{d^{2}r_{\rm s}}{d\eta_{\rm s}^{2}}-\cot\chi_{\rm
s}\frac{d^{2}\chi_{\rm s}}{d\eta_{\rm s}^{2}}-\frac{2}{r_{\rm s}}\cot\chi_{\rm
s}\frac{dr_{\rm s}}{d\eta_{\rm s}}\frac{d\chi_{\rm s}}{d\eta_{\rm
s}}+\frac{1+\cos^{2}\chi_{\rm s}}{\sin^{2}\chi_{\rm s}}\left(\frac{d\chi_{\rm
s}}{d\eta_{\rm s}}\right)^{2}\right].$ (192)

By Eqs. (97)–(100), we find

$\frac{da_{\rm s}}{d\eta_{\rm s}}=O(\epsilon^{0})~{}~{}~{}~{}~{}{\rm
and}~{}~{}~{}~{}\frac{d^{2}a_{\rm s}}{d\eta_{\rm
s}^{2}}=O(\epsilon^{-2\beta}),$ (193)

and hence

$w=O(\epsilon^{-2\beta}).$ (194)

In Fig. 3, we show the numerical result for $w$ in the slowing-down phase of
the example given in Sec. IV. The dominant energy condition will not be
satisfied so that the gravitational collapse slows down just before the event
horizon forms.

Figure 3:  The coefficient $w$ in the equation of state in the slowing-down
phase of the example given in Sec. IV is depicted as a function of normalized
$\epsilon^{-2\beta}(\eta_{\rm s}-\eta_{\rm S})$. It highly exceeds unity, and
hence the dominant energy condition is not satisfied.

## References

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]www.polyphys.mat.ethz.ch

# Quantum entanglement without superposition

Hans Christian Öttinger<EMAIL_ADDRESS>[ ETH Zürich, Department of
Materials, CH-8093 Zürich, Switzerland

###### Abstract

Superposition states are at the origin of many paradoxes in quantum mechanics.
By unraveling the von Neumann equation for density matrices, we develop a
superposition-free formulation of quantum mechanics. Stochastic quantum jumps
are a key feature of this approach, in blatant contrast with the continuity of
the deterministic Schrödinger equation. We explain how quantum entanglement
arises. Our superposition-free formulation results offers a new perspective on
quantum mechanics.

## Introduction.

Many of the famous paradoxes in quantum mechanics begin with the assumption of
superpositions of states. The usual discussion of the Einstein-Podolsky-Rosen
(EPR) _Gedankenexperiment_ [1] in Bohm’s spin version is based on the singlet
state of a pair of spin-$\frac{1}{2}$ particles [2, 3, 4]. In Schrödinger’s
cat version, the superposition state is considerably more abstract by invoking
complex systems. In the tradition of “Wigner’s friend” [5], Frauchiger and
Renner [6] assume the applicability of quantum mechanics to even more complex
superposition states that include agents who are themselves using quantum
theory, and they then reveal inconsistencies in conventional quantum
mechanics.

Whereas entanglement is an essential feature of quantum physics, superposition
is not. It is the purpose of this letter to propose an alternative,
superposition-free implementation of quantum entanglement, thus eliminating a
prolific source of paradoxes in conventional quantum mechanics. The
mathematical details of the alternative approach have recently been developed
in the context of a stochastic simulation technique for reversible quantum
dynamics [7]. We here elaborate on the implications of that simulation
technique for the foundations and interpretation of quantum mechanics.

## The von Neumann equation.

The stochastic nature of quantum mechanics calls for an appropriate setting to
describe randomness. The most natural choice is given by a density matrix
$\rho$ on a Hilbert space for a quantum system of interest, by which the
average of any observable $A$ can be obtained as a trace, ${\rm tr}(\rho A)$.
In the Schrödinger picture, the evolution of the density matrix $\rho_{t}$
describing the time-dependent state of a quantum system is given by the von
Neumann equation

$i\hbar\frac{d\rho_{t}}{dt}=[H,\rho_{t}],$ (1)

where $\hbar$ is the reduced Planck constant, $H$ is the Hamiltonian of the
system, and the square brackets denote the commutator. Except for its greater
flexibility in choosing initial conditions, the von Neumann equation is
equivalent to the Schrödinger equation. Quantum master equations for density
matrices have the additional advantage that they are perfectly suited not only
for describing reversible dynamics, but also for dissipative quantum systems
[8, 9]. A class of robust quantum master equations for dissipative systems has
been obtained by quantizing the geometric structures behind classical
nonequilibrium thermodynamics [10, 11].

Schrödinger’s complex wave function $\psi_{t}$ is less directly associated
with the randomness of quantum mechanics than the density matrix $\rho_{t}$.
One needs the additional rule that the squared modulus of the wave function,
$|\psi_{t}|^{2}$, should be interpreted as a probability density, whereas Eq.
(1) is linear in the primary probabilistic quantity of interest, $\rho_{t}$.
When one solves the continuous Schrödinger equation for an interacting system,
superposition of states seems to occur inevitably. For quantum master
equations, an alternative option is available, which we discuss next.

## Two-process unravelings.

The general strategy of so-called unravelings is to introduce stochastic
processes in Hilbert space such that a density matrix evolving according to
the von Neumann equation (1) can be extracted in terms of suitable averages.
This idea of passing from probabilistic concepts to equivalent stochastic
processes was originally motivated by computer simulations for dissipative
quantum systems [8] but, as we show here, it is useful also for reversible
systems and conceptual clarifications. Unravelings can even offer an
alternative interpretation of quantum mechanics. The passage from
probabilities to stochastic objects is particularly relevant to ontology [12].
We here focus on the opportunity to obtain entanglement without superposition,
with the underlying goal of avoiding paradoxes in quantum mechanics.

For dissipative systems, unravelings employ continuous evolution for
reversible dynamics and a combination of continuous evolution and stochastic
jump processes for dissipative dynamics [8]. In the context of dissipative
quantum field theory [13], it has been proposed to treat also reversible
interactions by jumps. This idea has been elaborated in detail for simulating
purely reversible quantum dynamics in [7].

The interpretation of interactions as jumps requires a splitting of the full
Hamiltonian into free and interacting parts, $H=H^{\rm free}+H^{\rm int}$, and
a distinguished basis $\cal B$ of orthonormal eigenstates of the free
Hamiltonian $H^{\rm free}$. For unravelings in terms of two stochastic
processes $\left|\phi\right\rangle_{t}$ and $\left|\psi\right\rangle_{t}$ in
Hilbert space, one wishes to reproduce the density matrix $\rho_{t}$ by the
following expectation,

$\rho_{t}=E\big{(}\left|\phi\right\rangle_{t}\left\langle\psi\right|_{t}\big{)},$
(2)

where we have used Dirac’s bra-ket notation. In this setting we can impose the
strict superselection rule that the states $\left|\phi\right\rangle_{t}$ and
$\left|\psi\right\rangle_{t}$ can only be multiples of basis vectors from
$\cal B$. Reasons for believing in the existence of such a distinguished basis
will be offered below.

The strict superselection rule has the pleasing side-effect of reducing the
enormous number of possibilities for constructing unravelings. It naturally
guides us to the following construction of piecewise continuous trajectories
with interspersed jumps occurring at a rate $r$:
Continuous evolution results from solving separate Schrödinger equations for
$\left|\phi\right\rangle_{t}$ and $\left|\psi\right\rangle_{t}$ for the free
Hamiltonian $H^{\rm free}$. The basis vectors remain unchanged. Their
prefactors are modified by exponential phase factors that are determined by
the corresponding eigenvalues of $H^{\rm free}$.
Quantum jumps occurring at rate $r$ reproduce the effect of the interaction
Hamiltonian $H^{\rm int}$. With probability $1/2$ a jump occurs for one of the
vectors $\left|\phi\right\rangle_{t}$ or $\left|\psi\right\rangle_{t}$. The
transition from an initial base vector $b\in\cal B$ to a particular final base
vector $b^{\prime}\in\cal B$ then occurs with a probability proportional to
$|\left\langle b^{\prime}\right|H^{\rm int}\left|b\right\rangle|$. The
prefactor inherited by $b^{\prime}$ from $b$ needs to be modified by a proper
normalization factor.
This level of detail in introducing unravelings should be sufficient for the
conceptual issues discussed in this letter. For more details, curious readers
could look at the actual implementation of unravelings in computer simulations
of reversible quantum dynamics [7].

The continuous evolution shows the importance of allowing for scalar
multiplication. The jumps eliminate the need for any addition of states.

## Triplet unravelings.

Two-process unravelings might work perfectly well on the analog quantum
computer called “Nature,” which seems to possess unlimited computational
resources. For man-made computers, however, efficiency is a major issue, in
particular in view of the infamous sign problem plaguing many quantum
simulations [14].

Actual simulations of reversible quantum dynamics based on superposition-free
unravelings have been performed in [7]. For reasons of efficiency, these
simulations are performed in Laplace-transformed time. Importance sampling and
approximations have been introduced to alleviate the sign problem. In
practice, the sign problem can be addressed more efficiently by switching from
two-process unravelings to triplet unravelings. In the triplets
$(c_{t},\left|\phi\right\rangle_{t},\left|\psi\right\rangle_{t})$, the states
$\left|\phi\right\rangle_{t}$ and $\left|\psi\right\rangle_{t}$ are elements
of the discrete set of distinguished basis vectors without any prefactors, and
the products of previously allowed prefactors are combined into the complex
numbers $c_{t}$. Ensembles can then be compressed efficiently by adding up all
the factors $c_{t}$ for a given pair of base vectors.

## Initial conditions.

We have focussed on reproducing the dynamics of the von Neumann equation by
unravelings. How about initial conditions?

Let us consider a two-dimensional Hilbert space with orthonormal basis ${\cal
B}=\\{b_{1},b_{2}\\}$ for illustration. Let us assume that the initial density
matrix $\rho_{0}$ is associated with the superposition state
$(b_{1}+b_{2})/\sqrt{2}$. Can such a density matrix for a pure state be
unraveled without violating the strict superposition principle? To obtain a
positive answer, assume that the random variable $\left|\phi\right\rangle_{0}$
takes the values $\sqrt{2}b_{1}$ and $\sqrt{2}b_{2}$ with probability $1/2$,
and assume the same independently for $\left|\psi\right\rangle_{0}$; then, the
desired density matrix $\rho_{0}$ of rank $1$ is recovered from Eq. (2). In
conclusion, we have shown how to represent entangled initial conditions and
their time evolution without any superpositions.

## Limitations of quantum mechanics.

In conventional quantum mechanics one is free to do basis changes, which
typically involve superpositions. Therefore, we need to justify our assumption
of the existence of a distinguished basis $\cal B$ in superposition-free
quantum mechanics. To this end one needs to recognize the limitations and the
approximate nature of quantum mechanics and to look at its deeper origin.

Dirac’s fascinating quantization recipe formally suggests that we can quantize
any classical Hamiltonian system, preferably in canonical coordinates, by
replacing classical Poisson brackets by quantum commutators. However, the
quantization of say planetary motion can hardly be particularly meaningful. A
restriction of quantum theory to fundamental particles, which would be the
other extreme, is unpractical because we can never be sure that we know any
fundamental particles. For example, nobody hesitates to apply quantum
mechanics to protons.

A nice textbook example for the application of quantum mechanics is the
hydrogen atom, which leads to the famous prediction for the line spectrum of
atomic hydrogen. Note that the Hamiltonian for this problem contains the $1/r$
Coulomb potential of classical electrostatics. In that sense, quantum
mechanics has a semi-classical character. Particularly questionable is the
occurrence of electron-proton interactions at a distance.

In a more rigorous treatment of the hydrogen atom, the Coulomb potential
should result from an exchange of photons between the proton, or the quarks in
a proton, and an electron. We are then in the domain of quantum field theory,
allowing for the creation and annihilation of photons. Quantum mechanics can
only be an approximate theory that arises in the low-energy limit of quantum
field theory, when particle creation and annihilation are suppressed. Quantum
mechanics should emerge from the quantum field theory of fundamental particles
and interactions after making approximations.

In quantum field theory, Fock space comes with a natural basis $\cal F$ [15,
16, 13]. The Fock basis vectors describe states with well-defined (free)
particle content. The strict superselection rule implies that a system can
never be in a superposition state of different particle contents. In making
approximations for passing from quantum field theory to quantum mechanics one
should make sure that a distinguished basis $\cal B$ emerges from the natural
Fock basis $\cal F$. For example, the treatment of a proton as a fundamental
particle on the atomic length scale (or low energies) in quantum mechanics
relies on color confinement.

## EPR experiment.

As an example of getting guidance from quantum field theory in quantum
mechanics, we revisit the EPR experiment in the version for photons [2]. Pairs
of photons moving with the same circular polarization in opposite directions
can be created in the decay of properly excited calcium atoms. Whereas the
actual decay occurs through an intermediate state within some $5\,{\rm ns}$,
we here use the idealization of a single collision in which a calcium
excitation is annihilated and two photons are created.

We start the unraveling with a calcium excitation, from which a photon pair is
created eventually, rather than from the usually assumed superposition state.
By evolving the initial state via the two-process unraveling, a pair of left-
or right-handed photons can be created in the $\left|\phi\right\rangle$ and
$\left|\psi\right\rangle$ states. If re-excitations of calcium atoms can be
neglected, this situation corresponds exactly to the example considered in the
above discussion of initial conditions.

## Measurements.

In the context of linear quantum master equations, a general theory of
measurable multitime correlation functions has been developed in [8]. For the
semilinear quantum master equations of thermodynamic origin, which are linear
for scalar multiplication but, in general, nonlinear for addition, this theory
of measurable correlations has been generalized in [13]. Note that this kind
of semilinearity goes nicely together with the strict superposition rule. A
detailed discussion of measurements is beyond te scope of this letter.

The idea of eigenstates of an observable $A$ is generalized to the set of
density matrices $\rho$ that commute with $A$. Such density matrices
characterize the superposition-free mixtures of eigenstates. If, for such a
density matrix, $A\rho=a\rho$ holds with $a\in\mathbb{R}$, then $a$ is an
eigenvalue.

One can ask the question whether density matrices can be measured. This
question can, for example, be addressed by quantum-state tomography [17, 18].

## Conclusions.

We have shown how to eliminate superposition from quantum mechanics by
imposing the strongest possible superselection rule. In our approach to
reversible quantum dynamics based on two-process unravelings, we need to solve
the two free Schrödinger equations,
$i\hbar\partial\left|\phi\right\rangle_{t}/\partial t=H^{\rm
free}\left|\phi\right\rangle_{t}$ and
$i\hbar\partial\left|\psi\right\rangle_{t}/\partial t=H^{\rm
free}\left|\psi\right\rangle_{t}$ (with trivial solutions), interrupted by
stochastic quantum jumps between multiples of base vectors due to
interactions. The density matrix is then given by Eq. (2). In such a
stochastic unraveling, the interrupted Schrödinger dynamics is more elementary
than in the GRW approach [19].

Unravelings provide an opportunity of a new interpretation of quantum
mechanics. In the context of dissipative quantum field theory, this
opportunity has been explored in [13, 12].

In the proposed superposition-free approach, entanglement arises from the
interplay of two stochastic processes in Hilbert space. Complete knowledge of
a quantum state requires two state vectors. Tow “semi-worlds” play together to
characterize the quantum state of the ”full world,” including entanglements.

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