# Microscopic State of BHs and an Exact One Body Method for Binary Dynamics in
General Relativity

Ding-fang Zeng<EMAIL_ADDRESS>Beijing University of Technology, Chaoyang,
Bejing 100124, P.R. China,
Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark

###### Abstract

Due to gravitational time delay, in Physical Black Boles formed through
gravitational collapses, the horizon and singularity are only asymptotically
realisable concepts instead of truly realised ones to the outside probe or
observers. Their occurrence in the finite future of the proper time used co-
moving observers happens in the future of infinitely far away future of the
normal time used outside probes. To the latter the horizon and singularity of
a physical black hole are physical realities only in the sense of parallel
universe or statistic ensemble. We provide two exact time dependent solution
families to the dust sourced Einstein equation conceived as physical realities
by two types of observers respectively and show that they form a pair of
complementarity descriptions for the microscopic state of black holes by
showing that the area law featured Bekenstein-Hawking entropy follows properly
from their canonical quantum wave function’s degeneracy. We then develop an
eXact One Body method for the general relativity two-body dynamics whose
conservative part requires no perturbative input from the post newtonian
approximation and applies to the full three stages of Black Hole Binary (BHB)
merger processes. By this method, we analytically calculate the gravitational
wave forms following from the merger of BHBs. In the case the participants
carry exact and apriori horizon and singularity our wave forms agree with
those following from the conventional effective one body method but exhibit
more consistent late time behaviour which is to be compared with the
calculation of black hole perturbation theories. In the case the participants
carry only asympttic horizon and extended inner structure thus experiencing
banana shape deformations during the merger process, our wave forms exhibits
all features especially the late time quasi-normal mode type oscillations
expected in the real observations.

###### Contents

  1. I Introduction
  2. II The outside fixed position observer’s description
  3. III The inside co-moving observer’s description
  4. IV Quantisation
  5. V A working example for the quantisation
  6. VI Black hole complementarity principle
  7. VII The physical reality’s observer dependence
  8. VIII Banana shape deformation
  9. IX An exact one body method
  10. X The full gravitational wave forms
  11. XI Conclusion

##  I Introduction

The existence of the Bekenstein-Hawking entropy hawking1971 ; Bekenstein1972 ;
Bekenstein1974 ; hawking1974 ; Jacobson1996 ; Wald1999  is a strong hint that
black holes are inner-structured Bekenstein1974a ; Mukhanov1986 ;
Bekenstein1995 ; Bekenstein1997a ; Bekenstein1997b ; dfzeng2017 ; dfzeng2018a
; dfzeng2018b ; dfzeng2020 ; dfzeng2021 ; dfzeng2022 ; Allen1987 ;
NambuSasaki1988 ; Nagai1989 ; BFZ1994 ; KMY2013 ; Stojkovic2008b ;
stojkovic2020unp . However, such structures are very seldomly believed by the
string theory strominger1996 ; DasMathur9601 ; CallanMaldacena1996 ;
HorowitzStrominger9602 ; MaldacenaSusskind9604 ; fuzzballsMathur2002 ;
LuninMathur0202 ; fuzzballSkenderis2007 ; LuninMaldacenaMaoz0212 ; MT0103 ;
GMathur0412 ; mathur0502 ; mathur2005 ; Jejjala0504 ; KST0704 ; Mathur0706
and loop quantum gravity community lqgEntropy1996 ; lqgEntropy1997 ;
AshtekarBojowald0509 ; Modesto0509 ; Campiglia0703 ; Modesto0811 ; Modesto0611
; Corichi1506 ; Bohmer0709 ; rovelli2015 ; MMP2021  to have any relevance with
exact solutions to the four dimensional real general relativity. The reason is
simple, most of the researchers believe that in the framework of general
relativity, as long as falling into the horizon, the fate of all matters is
hitting on the singularity and becoming a part of it. So the only existence
form of matters inside the horizon is a singular point characterised by mass,
charge and spin parameters exclusively. So the most general microscopic
structure allowed by general relativity is those exhibited in the Kerr-Neumann
metric.

However, just as all existence questions in philosophies do, the existence
form of matters in physics is also related with the definition of time. In
general relativity, this is especially so. Because of the time delay effects
caused by gravitation, the outside fixed position observers cannot see the
horizon’s formation in a to be black hole collapsing star in any finite future
quantifiable by their physical time definition. These observers are not
constrained to be human beings or any wisdom. They can be any outside fixed or
approximately fixed position probes or detectors. For example, either
participant involved in the merger event of a BHB system works as probes of
its inspiral partner. These probes will not see the horizon’s formation in
their partners’ evolution in any finite future quantifiable by their physical
time definition. So to them, the existence form of matters inside their merger
partners is just those of the frozen star frozenStar1971 ; Buchdahl1959 ;
chand1964a ; chand1964b ; bondi1964 . But the reason for being frozen here is
the infinite gravitational time delaying due to the horizon to form, instead
of any exclusive forces such as those arising from the neutron degeneration
pressure or any other unknown physics BrusteinFrozen2301 ; Brustein2304 ;
Brustein2310 . We will provide exact metric and Penrose-Carter diagram
description for this frozen star in section II.

By the freely falling observer’s physical time definition, the matters
consisting of a to be black hole collapsing star would indeed contract into
the mathematical surface defined by $r=2GM$ in finite futures. However, both
the matters being observed and the observer themselves experience not any
exotics as that surface is being walked through, all of them are freely
falling. As local observer and observable, the spacetime regions around them
are completely flatten. The non-zero curvature is physically relevant only to
the global viewpoint holden investigators. This is also the case even when the
$r=0$ point is concerned. The observer and observables coming from the east
world of the central point just normally walk over that point and get to the
west world afterwards. Tidal forces affect only non-local or finite sized
observer and observables. However, general relativity is a time-reverse
invariant theory. No matter how strongly is an extended object tidally
stretched as it gets close to the $r=0$ point, its shape and inner structure
will be retrieved as it walking away from that point just as the world is time
reversed Turok1510 ; Turok1612 ; CPTsymmUniverse2018 ; CPTsymmUniverse2022 ;
Turok2208 . Penrose and Hawking’s singularity theorem Penrose1965 ;
Hawking1976 ; geroch1979 ; SingularityHawking  only requires that all matters
getting into the horizon get onto the singularity (an equal time hyper
surface) in finite proper times. It does not prohibit them from walking across
that epoch. Exact metric description and PC-diagram representation of this
picture will be provided in section III.

With section V provided as a working example, section IV discusses the
quantisation of the inner structure described by the exact metric families of
sections II and III. We decompose the matter contents of the spherically
symmetric physic black holes into all kinds of concentric shells characterised
by each shell’s mass and radial quantum number and write the wave functional
of the whole system as the direct product of all compositing shells. By
considering all possible shell decomposition schemes allowed by the total mass
summation rule and the radial quantum number assigning schemes allowed by the
asymptotic horizon condition, we analytically prove that the degeneracy of the
system’s wave functional is exactly that required by the area law formula of
Bekenstein Hawking entropy, except for some logarithmic type corrections.
Numerically determining these corrections is possible but challenging as the
mass of the black hole becomes large. In contrast with the string theory
calculation Ashoke0504 ; Ashoke0506 ; Ashoke0508 ; Ashoke0611 ; Ashoke0708 ;
Ashoke1005 ; Ashoke1008 ; Ashoke1106 ; Ashoke1108 ; Ashoke1109 ; Ashoke1204 ;
Ashoke1402 ; Ashoke1405 ; Ashoke2306 , our microscopic state definition and
number counting for the black holes are purely general relativity based and
resort not any hyper physic concepts such as supersymmetry or extra spatial
dimension. But our black holes will have similar observational features as the
string theory fuzz balls mayerson2021fuzzball ; mayerson2022microReview  if
their shadow images are taken joao2302 ; GLM2008  or gravitational wave echos
gwEchoCardoso1902 ; gwEchoMaggio1907 ; gwEchoKaloper1912 ; gwEchoCardoso2007 ;
gwEchoStojkovic2011 ; gwEchoMaggio2202  are measured.

The black hole complementarity principle complementarity1990thooft ;
complementarity1993verlinde ; complementarity1993schoutens ;
complementarity1993 ; complementarity1994  believes that both the outside
fixed position observer and the freely falling observer’s description of a
collapsing star’s evolution are equally valid. We will elaborate in section VI
that, for the outside fixed position observers, the uncertainty principle or
the quantum fluctuation’s ubiquity implies the necessity of parallel universe
or statistic ensemble description for the initial status of matters consisting
of the black hole. While for the freely falling co-moving observers, this
necessity can be attributed to the ergodicity feature of the system’s
evolution on the proper time axis. So after the initial status’ uncertainty or
the evolution dynamics’ ergodicity are considered, not only are the two types
of observer’s description both valid, but also are they equivalent. The fire
wall paradoxes fireworksAMPS2012 ; fireworksAMPS2013 ; EREPR  will be analysed
in this section to show that the truly rational resolution to the information
missing puzzle fuzzball2009mathur ; fuzzball2021mathur ; NVW2013a ; NVW2013b ;
giddings2022 ; Traschen ; Stojkovic2007 ; Stojkovic2008a ; Stojkovic2013 ;
Stojkovic2014 ; Stojkovic2015  is to give up the popular vacuum fluctuation
and partial escaping mechanism for the Hawking radiation instead of the idea
of black hole complementarity. This section consists mainly of logic reasoning
instead of mathematical derivations. So we put it in the middle part of the
paper to avoid preaching. The mathematical derivations underlying can be found
in our earlier works dfzeng2021 ; dfzeng2022 .

Through three concrete examples, section VII will analyse how the belief that
the gravitational collapses will cause observer independent horizon and
singularity makes the mistake of (A) taking the horizon and singularity of the
global viewpoint holders as the horizon and singularity detectible to the
local viewpoint holders and (B) taking the physics law’s general coordinate
invariance as the physical reality’s observer or probe independence. The
physical reality’s observer of probe independence is not the fact at all. Our
three examples are, (i) the frequency of light signals emitted from the
surface of a collapsing star and measured by an outside fixed position
observer and a freely falling observer; (ii) the pressure of a dust type test
mass shell freely falling towards the horizon of a pre-existing Schwarzschild
black hole measured by a far away fixed position observer and a freely falling
observer co-moving with the shell itself and (iii) the horizon of two
cosmological sized super giants separated double hubble radius away in our
real universe. This section is the key part of this work but we burry it here
because it is conceptually so bold that some people may think it non-sense at
all.

Section VIII will show that when the horizon is taken as a probe dependent
physical reality, the black holes involved in the binary merger event will
experience banana shape deformation under the inhomogeneous back reaction
force of the gravitational wave radiation. We will illustrate such a
deformation’s occurrence in the newtonian gravities and argue that the
physical picture would be similar in the fully general relativistic treatments
of the system’s evolution. We then provide a static mechanic analysis for the
banana shape deformed black holes and show that the radiation activity of such
BHBs will be suppressed by a factor of $\frac{\sin 4GMz/a(t)}{4GMz/a(t)}$,
where $z$ is the system’s stretching times relative to the diameter of the
undeformed Schwarzschild horizon. $z$ itself is time dependent whose concrete
form reflects the black hole’s deformation sensitivity or susceptibility under
the back reaction force of gravitational wave radiation.

Section IX will propose an exact one body (XOB) method to calculate the
relative motion orbit of the general relativity two-body systems. This method
will avoid the self-contradicting ingredients of the conventional EOB method
eob1998 ; eob2000 ; eob2009 . Its conservative part involves no post-newtonian
approximation Blanchet9501 ; Damour0005 ; PatiWill0007 ; blanchet0209 ;
pnApproxLivingReview  as inputs. Taking the most simple quadrupole radiation
of gravitational waves as the dissipation source, this method allows us to
trace the relative motion of the binary systems even when their evolution
develops to the ringdown stage. By this method we will calculate the relative
motion orbit and gravitational wave forms for binary systems consisting of
standard Schwarzschild black holes to make comparisons with the results of
conventional EOB method. Our results will coincide with that of EOB at early
times but behaves more rationally as the very late time stage is arrived. By
looking the black hole as an arc of fixed length, this method will yield
gravitational wave forms highly similar with those obtained in the numeric
relativity NmRel0507 ; NmRel0511 ; NmRel0511103 ; tidalDeformationNumeric ;
NR2002 ; NR2004 ; NR2007 ; NR2018  and black hole perturbation theories
BHPTwaveform1970 ; BHPTwaveform1971 ; BHPTwaveform1975 ; QNMcardoso0905 ;
QNMcardoso1602 ; QNMcardoso1806 .

Section X will compare the relative motion orbits and gravitational wave forms
following from our XOB method for three different time variation modes $z(t)$
of banana shape deformation. These three modes describe black holes with three
typical shape deformation sensitivity or susceptibilities under the
inhomogeneous back reaction force of gravitational wave radiations. By
exhibiting the resultant gravitational wave forms’ difference, we will show
that the inner structure and the shape deformation sensitivity of black holes
are observable experimentally instead of pure theoretical arguments. By the
simple energy conservation law, we will derive a universal upper bound for the
late time quasi-normal mode’s frequency as functions of the symmetric mass
ratio of the binary system irrespective of the inner structure detail of black
holes.

We conclude the whole paper in section XI.

##  II The outside fixed position observer’s description

The physical reality’s probe dependence is a very widely accepted concept in
high energy physics. For example, the hadron structures measured in the deep
inelastic scattering experiments depends on the probe particle’s momentum
transfer. One purpose of this work is to show that, the horizon and
singularity’s observer independent is just a counter part of this concept in
general relativity and black hole physics.

By the word of outside fixed position observer, we refer to all out side probe
or detectors that are affected by the gravitation emanate from the to be black
hole collapsing stars. These observers can only probe or detect physics
related with the collapsing stars during their life time, i.e. existence
duration of the probe themselves. Typical examples for these observers
include, either member of a BHB system which are experiencing merger process,
and/or all imaging light signals that are emitted from the background
accretion disk and enter the detector of the event horizon telescope on earth.
These observer or probe’s positions are not truly fixed, but during their
whole existence duration and in the spherical coordinate system of the being
probed black holes, their radial positions are only slowly varying functions
of the time coordinate which is understood as physic times by the infinitely
far away investigators.

To these observers, the general ansatz for the spacetime metric of the
collapsing star can be written as

$\displaystyle
ds^{2}_{\mathrm{full}}{=}-h{\cdot}B^{-1}dt^{2}{+}h^{-1}dr^{2}{+}r^{2}d\Omega^{2},h{=}1{-}\frac{2GM}{r}$
(II.1) $\displaystyle
B^{-1}=1+h^{-2}\dot{M}^{2}/{M^{\prime}}^{2},M[t,r]^{\mathrm{outside.matt.}}_{occup.region}\equiv
M_{\mathrm{tot}}$ (II.2) $\displaystyle
M[t,r]\xrightarrow[r<2GM_{\mathrm{tot}}]{t\rightarrow\infty}\frac{r}{2G}-\mathrm{small~{}deviation}$
(II.3)

The function form of $B$ is determined by the normalisation of matter sources’
four velocity
$u^{\mu}=\frac{dx^{\mu}}{d\tau}=\\{1,\frac{\dot{m}}{m^{\prime}}\\}\cdot\frac{dt}{d\tau}$.
We choose the scaling of $t$ so that $hdt^{2}={d\tau}^{2}$, where $\tau$ is
the proper time measured by observers fixed on the collapsing matter, up to a
regular $r$-dependent scale factor. In the case of gravity dominating over all
other interactions, neglecting the pressure is a rational doing. In this case,
the Einstein equation $R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi G\rho
u_{\mu}u_{\nu}$ will tell us

$\displaystyle\frac{\dot{m}\dot{m}^{\prime}}{{m^{\prime}}^{2}}{-}\big{[}\frac{2m}{r(r{-}2m)}{+}\frac{m^{\prime\prime}}{{m^{\prime}}}\big{]}\frac{\dot{m^{2}}}{{m^{\prime}}^{2}}{-}\frac{(r{-}2m)m^{\prime}}{r^{2}}{=}0$
(II.4)
$\displaystyle\frac{\ddot{m}}{m^{\prime}}{-}\big{[}\frac{3m}{r(r{-}2m)}{+}\frac{m^{\prime\prime}}{m^{\prime}}\big{]}\frac{\dot{m}^{2}}{{m^{\prime}}^{2}}{-}\frac{(r{-}2m)(m{+}2rm^{\prime})}{r^{3}}{=}0$
(II.5)

where we used short notations $m=GM[t,r]$. Equation (II.4) follows from the
condition that $G^{0}_{0}G^{1}_{1}-G^{0}_{1}G^{1}_{0}=0$. Equation (II.5)
follows from the condition of $G^{\theta}_{\theta}=G^{\phi}_{\phi}=0$
supplemented by (II.4).

Eq(II.4) can be written into the form of a first order differential equation
and integrated formally

$\displaystyle y^{\prime}(t,r)-p(t,r)y(t,r)=q(t,r),y\equiv\dot{m}^{2}$ (II.6)
$\displaystyle
p\equiv\frac{2m^{\prime\prime}}{m^{\prime}}{+}\frac{4m/r}{(r{-}2m)},q\equiv\frac{2(r{-}2m){m^{\prime
3}}}{r^{2}}$ (II.7) $\displaystyle
y=e^{\int^{r}_{\mathrm{r_{1}}}\\!\\!p(t,z)dz}{\cdot}c(t){+}\\!\\!\int^{r}_{\mathrm{r_{2}}}\\!\\!\\!\\!e^{\int^{r}_{x}\\!p(t,z)dz}q(t,x)dx$
(II.8)

where $c(t)$ is an arbitrarily tunable function of $t$-coordinate, while
$r_{1}$ and $r_{2}$ are two arbitrary reference point. Taking
$\frac{m/r}{m^{\prime}{-}1/2}$ as a slowly varying function of $r$, the
integration $\int\\!p(t,z)dz$ in (II.8) can be done approximately so that

$\displaystyle\hskip 14.22636pt\int\\!p(t,x){\approx}[\log{m^{\prime
2}}\\!(x{-}2m)^{\frac{4m/x}{1{-}2m^{\prime}_{x}}}]$ (II.9)
$\displaystyle\frac{\dot{m}^{2}}{m^{\prime
2}}{\approx}{\\!(r{-}2m)^{\\!\frac{4m/r}{1{-}2m^{\prime}_{r}}\\!}}{\cdot}\big{[}c(t){+}\\!\\!\int^{r}_{r_{2}}\\!\\!\\!\frac{(x{-}2m)^{1{-}\frac{4m/x}{1{-}2m^{\prime}_{x}}}2m^{\prime}}{x^{2}}dx\big{]}$
(II.10)

This means that given initial mass function $m(0,r)$, we can always tune the
form of $c(t)$ so that on the boundary of the matter occupation region
$\frac{\dot{m}}{m^{\prime}}|^{\mathrm{m.o.r}}_{\mathrm{bndry}}=\frac{dr}{dt}|^{\mathrm{m.o.r}}_{\mathrm{bndry}}$
matches with the geodesic motion of test particles freely released in the
Schwarzschild metric caused by the total mass of the collapsing star. But
according to equation (II.4), the initial value of $\dot{m}^{2}(0,r)$ is
completely determined by the form of $m(0,r)$ thus allows no artificial tuning
to implement a given
$\frac{\dot{m}}{m^{\prime}}(0,r^{\mathrm{m.o.r}}_{\mathrm{bndry}})$. This is
because we neglect all pressure effects as well as the external
force/potentials exerted on the matter contents so that their initial speed
equals zero. In our numerical examples, we will simply set $c(t)=0$ and take
$r_{2}$ as a small parameter to assure integral convergence and accuracy. In
this case equation (II.10) implies that as $m\rightarrow\frac{r}{2}$,
$\frac{\dot{m}^{2}}{m^{\prime 2}}\sim(r{-}2m)\rightarrow 0$. In this limit,
equation (II.4) leads to

$\displaystyle\dot{m}\sim(1{-}\frac{2m}{r})^{-1/2},{m^{\prime}}\sim(1{-}\frac{2m}{r})^{-1},$
(II.11)

This explains the origin of asymptotic expression (II.3), which is in fact a
result independent with respect to the matter source’s equation of state. The
Ricci scalar of spacetime metric (II.1)-(II.2) when simplified using (II.4)
and (II.5) has the following form

$R=\frac{2(r-2m)^{2}{m^{\prime}}^{3}}{r^{2}[(r-2m)^{2}{m^{\prime}}^{2}+r^{2}\dot{m}^{2}]}$
(II.12)

This is regular as $t\rightarrow\infty$, because the asymptotic behaviour of
(II.11). Divergent happens only when we try to go beyond the Future of
infinite Far Future (FiFF), that is, when we use alternative time coordinate
which covers range outside that $t$ does.

Eqs(II.4)-(II.5) allows us to express all the higher order time derivatives of
$m$ in terms of its spatial derivatives, e.g.

$\ddot{m}{=}\big{[}\frac{3m}{r(r{-}2m)}{+}\frac{m^{\prime\prime}}{m^{\prime}}\big{]}\frac{\dot{m}^{2}}{m^{\prime}}{+}\frac{(r{-}2m)(m{+}2rm^{\prime})m^{\prime}}{r^{3}}$
(II.13)
$\displaystyle\dddot{m}{=}\dot{m}^{3}\big{[}\frac{m^{\prime\prime\prime}}{{m^{\prime}}^{3}}{+}\frac{9mm^{\prime\prime}}{r(r{-}2m){m^{\prime}}^{3}}{+}\frac{4r^{2}m^{\prime}{-}6rm{+}24m^{2}}{r^{2}(r-2m)^{2}{m^{\prime}}^{2}}\big{]}$
(II.14)
$\displaystyle{+}\dot{m}\big{[}\frac{2[6m^{2}{-}rm(1{-}8m^{\prime}){+}3r^{2}{m^{\prime}}^{2}]}{r^{4}}{+}$
$\displaystyle\frac{3(r{-}2m)(m{+}3rm^{\prime})m^{\prime\prime}}{r^{3}m^{\prime}}\big{]}+\frac{(r{-}2m)^{2}{m^{\prime}}^{3}(2m{+}3rm^{\prime})}{r^{5}\dot{m}}$

These expressions tell us that, as $t\rightarrow\infty$ the $n$-th order time
derivative of $m$ are proportional to $(r-2m)^{-n}\sim\infty$. This makes
numeric integration of equations (II.4) or (II.8) more and more difficult as
we approach the configuration $m(t,r)\approx\frac{r}{2}$. However, the basic
feature of
$\frac{dr}{dt}|_{\mathrm{fix.mss.pt}}=\frac{\dot{m}}{m^{\prime}}\sim(r-2m)^{\frac{1}{2}}\rightarrow
0$ is very robust. This causes the outside probes not being able to probe
$m(t,r)$’ evolution into profiles of $r-2m<0$ in any finite $t$ time.
Physically this is just the fact that, as outside observers or probes, they
cannot see the formation of horizons in any finite $t$ durations. Of course,
if the proper time description is adopted, the co-moving observers can see
matter’s collapse into the horizon in finite durations, just as was
illustrated in references dfzeng2017 . However, since those observers or
detectors are freely falling, they will experience not any exotics as they
pass through the horizon surface.

Figure 1: The $t$-time evolution of the mass function of two collapsing stars.
The upper has central sparser initial configuration, the downer has a central
denser one. Different colour curves family denotes the mass function evolution
inside three concentric spheres of different radius. Figure 2: The red line is
the asymptotic mass function of all collapsing stars after infinite $t$-time
evolution. $t$ is the time of the outside fixed position observers. The other
three curves are the mass function of collapsing star after finite $t$-time
evolution. The matters live in the region
$0{<}r{<}2GM_{\mathrm{tot}}{+}\epsilon$. Figure 3: The Penrose-Carter diagram
of the three inner mass profiles of FIG.2. The horizon and singularity will be
realised only when $\infty<t$. The two blue hyperbolas in the leftmost sub-
figure correspond to the $r=2GM_{\mathrm{tot}}$ and the equal-$r$ sphere which
are mostly close to the $M=r/2G$ line. The two green hyperbolas in the middle
and the three maganta hyperbolas in the rightmost are similar.

Consider two simple initial mass function (i) $m(0,r<1)=\frac{1}{4}r^{3}$,
(ii) $m(0,r<1)=\frac{1}{4}r^{1}$ and $m(0,1<r)=1$. The former has central
sparser initial mass density profile $\rho\sim(1-\frac{r^{2}}{4})^{-1}$, the
latter has a central denser and singular profile $\rho\sim r^{-1}$ but no
horizon apriori. For these two initial mass functions the $t=0$ integration
(II.8) can be done exactly as well as analytically

$\displaystyle\mathrm{(i)}~{}\dot{m}^{2}(0,r)=\frac{r^{4}c(t)}{(2{-}r^{2})}{+}\frac{9r^{4}}{128}\big{[}\frac{(2-r_{2}^{2})^{3}}{2-r^{2}}{-}(2-r^{2})^{2}\big{]}$
(II.15)
$\displaystyle\mathrm{(ii)}~{}\dot{m}^{2}(0,r)=r^{2}c(t)+\frac{r^{2}}{128}\big{[}\frac{1}{r_{2}^{2}}-\frac{1}{r^{2}}\big{]}$
(II.16)

FIG.1 displays their latter time evolution following from equations
(II.4)-(II.5). From the figure we can easily see that different radius
concentric shells need different $t$-duration to reach their horizon size. The
duration difference $\Delta$ is finite. This means that to the outside
detectors, the horizon of a collapsing star is not a single equal-r surface of
$r=2GM_{\mathrm{tot}}$, but the collection of all concentric spheres of
different sizes $\\{r_{i}=2GM^{i}\\}$, where $M^{i}$ means the mass of matters
inside the sphere of radius $r_{i}$ and $i$ varies continuously on classical
levels. Each one of those horizons is never reached but only infinitely
closely reachable by the corresponding shells at different $t$-time epochs
depending on the initial mass function of the star. As outside observer or
probes, one can use the initial mass function to represent the microscopic
state of the collapsing star with approximate horizon of radius
$r_{h}=2GM_{\mathrm{tot}}+\epsilon$. FIG.2 gives an illustration for three
typical examples, while FIG.3 is their Penrose-Carter diagram representation.

The physical black holes form through gravitational collapses are defined by
their asymptotic horizon. The asymptotic horizon is only mathematical surface
which is infinitely approachable but never reached in any finite physic time.
It provides us a boundary to look the physical black holes as statistic
systems and introduce parameters such as internal energy and entropy to
describe their macroscopic features. Just as all normal statistic systems do,
the microscopic state of the physical black holes can be simply attributed to
the mechanical motion of their matter contents. At here, this motion is the
freely falling of concentric matter shells consisting of the collapsing star
under randomly specified initial radius and radial speed. The randomness of
initial conditions arise from the uncertainty principle. We will discuss the
quantisation of this mechanical motion and show that its wave functionals are
indeed degenerated in the number of ways required by the area law formula of
Bekenstein Hawking entropy. But before this doing, let us consider this
motion’s description from the inside co-moving observer’s aspect.

##  III The inside co-moving observer’s description

By the term inside co-moving observer, we refer to all local detectors or
probes which are freely falling together with the matters consisting of the
collapsing star and use proper time as time coordinate to describe physic
evolutions accessible to them. As local probes, these observers are not
necessarily point particles. As long as their size is much smaller than the
space-time region’s scale they are detecting, things will be okay. Their
detection are just interactions between them and the to be detected spacetime
region itself. The interaction between different parts of the system is also a
detection which has the potential of putting the microscopic state of the
whole system onto an ergodic evolution orbit. To these observers, the most
general metric ansatz for an isotropic but inhomogeneous collapsing star can
be written as

$ds^{2}_{\mathrm{in}}{=}{-}d\tau^{2}{+}\frac{\big{[}1{-}\big{(}\frac{2GM}{\varrho^{3}}\big{)}\\!^{\frac{1}{2}}\\!\frac{M^{\prime}\\!\varrho}{2M}\tau\big{]}^{2}\\!d\varrho^{2}}{a[\tau,\varrho]}{+}a[\tau,\varrho]^{2}\varrho^{2}d\Omega^{2}_{2}$
(III.1)
$ds^{2}_{\mathrm{out}}=-d\tau^{2}{+}\frac{r_{s}^{2/3}d\varrho^{2}}{[\frac{3}{2}(\varrho{-}\tau)^{\frac{2}{3}}]}{+}[\frac{3}{2}(\varrho{-}\tau)^{\frac{2}{3}}]^{2}r_{s}^{\frac{2}{3}}d\Omega_{2}^{2}$
(III.2)

where as was done in the previous section, we have neglected the pressures so
that the inner structures of the system are characterised by a single function
$M[\varrho]$ exclusively. This is reasonable when the gravitation dominates
all non-gravitational exclusive interactions; $\tau$ and $\varrho$ are so
called Lemaitre coordinate and $r_{s}\equiv 2GM_{\mathrm{tot}}$. By the
conventional Boyer-Linderquist coordinate, $ds^{2}_{\mathrm{out}}$ can be
equivalently written as

$ds^{2}_{\mathrm{out}}{=}{-}hdt^{2}{+}h^{-1}dr^{2}{+}r^{2}d\Omega^{2},h{=}1{-}\frac{2GM_{\mathrm{tot}}}{r}$
(III.3)

The advantage of Lemaitre coordinate is that, we can more directly see the
smoothness of the inside-outside metric’s connection. The advantage of Boyer-
Linderquist coordinate is that, we can more easily understand what we are
describing is a simple neutral and spherical symmetry black hole in the
standard general relativity. Since we neglect all exclusive pressures, the
concrete form of $a[\tau,\varrho]$ can be obtained exactly from the dust
sourced Einstein equation $R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}=8\pi
G\\{\rho,0,0,0\\}$

$\displaystyle
a[\tau\\!\in\\!|_{0}^{\\!\frac{p\;\\!\\!^{\varrho}}{4}},\varrho]\\!=\\!\big{(}1\\!-\\!\frac{4\tau}{p^{\varrho}}\big{)}^{\\!\frac{2}{3}},~{}a[\tau|_{\frac{p\;\\!\\!^{\varrho}}{4}}^{\frac{p\;\\!\\!^{\varrho}}{2}},\varrho]{=}{-}a[\frac{p\;\\!\\!^{\varrho}}{2}{-}\tau,\varrho]\rule{5.69054pt}{0.0pt}$
(III.4) $\displaystyle
a[\tau|_{\frac{p\;\\!\\!^{\varrho}}{2}}^{\frac{p\;\\!\\!^{\varrho}}{~{}}},\varrho]{=}{-}a[p^{\varrho}{-}\tau,\varrho],~{}a[\tau|_{p\;\\!\\!^{\varrho}}^{p\;\\!\\!^{\varrho}{\\!\scriptscriptstyle+}},\varrho]=a[\tau{-}p^{\varrho},\varrho]\rule{5.69054pt}{0.0pt}$
(III.5) $\displaystyle
p^{\varrho}\equiv\frac{8}{3}\big{(}\frac{\varrho^{3}}{2GM[\varrho]}\big{)}\\!^{\frac{1}{2}},~{}M[\varrho\\!\geqslant\\!\varrho_{\mathrm{max}}]{=}M_{\mathrm{tot}}\rule{36.98857pt}{0.0pt}$
(III.6)

Since $a(\tau,\varrho)$ is negative allowable, the sign of the coefficient of
the $d\varrho^{2}$ term in (III.1) must be defined in such a way that the
whole term is positive definitely.

For each given $\varrho$, the behavior of $a[\tau,\varrho]$ involved in
(III.1) or (III.4) is just the oscillation driven by a square inverse force or
linearly inverse potential,

$m\ddot{x}+\frac{k}{x^{2}}=0~{}\mathrm{or}~{}\frac{1}{2}m\dot{x}^{2}{-}\frac{k}{x}{=}\varepsilon$
(III.7)

The full period of this oscillation is
$T{=}\frac{k\sqrt{m}}{\sqrt{2}(-\varepsilon)^{3/2}}\frac{\pi}{2}$. During each
first quater period, the oscillation equation can be integrated implicitly

$\displaystyle
t(x)=\frac{k\sqrt{m}\arcsin[(1+\frac{x\varepsilon}{k})^{\frac{1}{2}}]}{\sqrt{2}(-\varepsilon)^{3/2}}-\frac{\sqrt{mx(k+x\varepsilon)}}{\sqrt{2}\varepsilon}$
(III.8)

and the full oscillation function can be written as

$\displaystyle\\{t,x\\}\equiv\left\\{\begin{array}[]{ll}\\{~{}0+t(x),+x\\},&0<t\leqslant\frac{T}{4}\\\
\\{\frac{T}{2}-t(x),-x\\},&\frac{T}{4}<t\leqslant\frac{T}{2}\\\
\\{\frac{T}{2}+t(x),-x\\},&\frac{T}{2}<t\leqslant\frac{3T}{4}\\\
\\{T-t(x),+x\\},&\frac{3T}{4}<t\leqslant T\end{array}\right.$ (III.13)

FIG.4 displays the potential, phase diagram and $x\mathrm{vs}.t$ curve of this
oscillation in details. By the standard Boyer-Lindquist to Lemaitre coordinate
transformation

$\displaystyle
dt+\sqrt{\frac{r_{s}}{x}}\big{(}1{-}\frac{r_{s}}{x}\big{)}^{-1}dx\equiv d\tau$
(III.14) $\displaystyle
dt+\sqrt{\frac{x}{r_{s}}}\big{(}1{-}\frac{r_{s}}{x}\big{)}^{-1}dx\equiv
d\varrho$ (III.15) $\displaystyle
x=\big{[}\frac{3}{2}(\varrho{-}\tau)\big{]}^{\frac{2}{3}}r_{s}^{\frac{1}{3}}$
(III.16)

the oscillation function (III.8)-(III.13) will become the function
(III.4)-(III.6) with fixed $\varrho$ routinely.

Figure 4: The upper part is the potential and phase diagram of oscillators
driven by a linearly inverse potential or square inverse force, the lower part
is its oscillation function. Figure 5: The evolution of three representative
dust stars as the $\tau$-time varies. The magenta arrows denote the co-moving
grid’s contraction as a whole. The left has uniform initial mass density. The
cyan dot displays the matters’ fixing on the co-moving grid. The middle has a
central-sparser initial mass density
$\rho[0,\varrho]\sim\frac{dM[\varrho]}{4\pi\varrho^{2}d\varrho}=\varrho^{1}$.
The matters’ migration on the co-moving grid, marked by cyan arrows, is from
the superficial to central region so that the co-moving $\varrho$-coordinate
covers smaller and smaller regions. The right has a central-denser initial
mass density $\rho[0,\varrho]\sim\varrho^{-1}$. The matters migration on the
co-moving grid in this case is from the central to the outer region, marked by
cyan-arrows.

The mass function $M[\varrho]$ involved in (III.1) and (III.4)-(III.6) are
completely arbitrary at the classic level. Its physics meaning is the initial
mass function of the collapsing star on the co-moving coordinate $\varrho$
line. FIG.5 displays the $\tau$-time evolution of three representative
collapsing stars characterised by different $M[\varrho]$. In the left most
uniform initial density case, the evolution of the system is completely
determined by the contraction and expansion of the co-moving grid as a whole.
But in the other two cases, the co-moving grid’s contraction and the matter’s
migration on the grid are asynchronous. In the central sparser case of the
middle column, the matters’ migration on the co-moving grid is from the
superficial layers to the central ones, so that the matters are distributed on
smaller and smaller co-moving grids. While in the central-denser case of right
most column, the matters’ migration on the co-moving grid is from the central
region to the superficial layers, so that they cover larger and larger co-
moving grids. In both latter cases, the matters’ migration on the co-moving
grid is towards a more and more homogeneous configure until they reach an
exactly uniform distribution and over-cross each other. At the over-cross
epoch, the physical mass density $dM/(a^{3}\varrho^{2}d\varrho)$ is divergent,
but the density $dM/\varrho^{2}d\varrho$ on the co-moving grid is finite.
After that epoch, the mass-energy distribution will retrieve its initial
configuration but with the anti-podal point interchanged. So the matters’
gathering onto the central point and causing divergence there is not the
terminal of their motion inside the horizon but a normal epoch of a periodical
over-cross oscillation. Of course this oscillation happens outside the domain
of time definition of the outside observers. To them this oscillation
especially the matter shells’ expansion away from the gravitational central
happens only in a quantum parallel universe or statistic ensemble spacetime
which is causally in-connected with the one they own live in and is
understandable only as the results of the quantum fluctuation implied by the
uncertainty principle.

Relative to other mono-directional collapsing star solutions to the sourceful
Einstein equation such as Oppenhiemer-Snyder oppenheimer1939 ; MTWbook ,
Yodzis-Seifert-Muller Yodzis1973 ; Yodzis1974  and Vaidya Vaidya1950 ;
Vaidya1970  in the history, the most important feature of our oscillation
solutions family above is its not taking the dusts’ collapse to
$a[\tau,\varrho]=0$ and hitting on the singularity as the terminal of their
motion inside the horizon. Instead it takes such event only as an intermediate
epoch of the dusts’ motion inside the horizon. The next step and following on
motion is their passing through each other and periodic oscillation across the
central point. Their motion over cross the central point is an analogue of two
classic wave’s superposition and independent propagation after encountering
each other. This over cross oscillation mechanism is firstly noted and
quantised with a functional Schrodinger equation to resolve the Schwarzschild
singularity in reference dfzeng2017  and then quantised with the method of
shell-decomposition in references dfzeng2018a ; dfzeng2018b  to interpret the
origin of Bekenstein-Hawking entropy and resolve the black hole information
missing puzzle caused by hawking radiation. While in reference dfzeng2020 ,
exact solutions for the simple $AdS_{2+1}$-Schwarzshild are provided and the
corresponding Bekenstein-Hawking entropy formula is analytically proved by the
integer number partition formula.

Figure 6: To a co-moving local observer which uses proper time $\tau$ to
describe physic evolutions, the Schwarzschild singularity caused by
gravitational collapse is trivially traversable as the horizon is. So its
world line in the ${\tau,\rho}$ plane A,B,C,D$\cdots$,A′,B′,C′,D′ is
periodic. The Schwarzschild singularity represents not any terminal of physic
evolution. But the definition domain of proper time $\tau$ covers that of the
outside fixed position observers’ time $t$ infinite times. During the
infinitely long $\tau$-time evolution, the shell structure a freely falling
co-moving observer see would not be exactly periodic due to the many-body
essence of the interactions among the different shells. This would cause
ergodicity of the shell structure’s evolution.

The existence of our solutions family (III.1)-(III.6) does not contradict with
the singularity theorem of Penrose and Hawking Penrose1965 ; Hawking1976 ;
geroch1979 , because all matters described in this solutions family indeed
fall onto the central point and cause a spatial-like singularity in finite
proper time as expected. To the co-moving observer, such a singularity is
nothing but an equal-$\tau$ hyper-surface. Just like the horizon can be
trivially traversed, the spatial-singularity which occupies only a future
point on the co-moving observer’s world line is also trivially traversable in
almost no $\tau$-time, after which all kinematic parameters of the system get
their $\tau$-time reversed value routinely. Tidal forces affects only the
finite sized objects. However, just as we pointed out in the introduction
section, such effects are time inverse symmetric. So even when the finite
sized objects are involved in a physic collapsing process, the status
information encoded in such objects will be retrieved as the singularity is
traversed. This means that we can define the microscopic state of the black
hole with the initial status of its consisting matters. For an exactly
spherical symmetric Schwarzschild black holes, the only possible initial
status information is its matter contents’ layering structure and each layer’s
radial collapsing speed.

To the outside fixed position observer or physical probes, the asymptotic
horizon provides an ideal excuse for them to say that the system has to be
considered as an ensemble of many parallel universe, each contains a
collapsing star with equal mass but different initial radial mass profile. To
the freely falling co-moving observers, this excuse cannot be used. However,
the fact that the domain of proper time $\tau$’s definition covers that of the
coordinate time $t$ many times brings us another interpretation, ergodicity.
That is, during the infinitely long $\tau$-time evolution, the point
representing the microscopic state of the black hole will walk through each
point of the phase space that represents all possible microscopic states of
the collapsing star. See FIG.6 for references. To display the oscillations
inside the horizon more conveniently, we extended the rule of conventional
Penrose Carter diagram so that the east world and west world of the spacetime
are simultaneously displayed in the $\\{\tau,\rho\\}$ plane. We introduced
this diagram method firstly in reference dfzeng2021 . We will show in the
following that the microscopic state definition based on the ergodicity of co-
moving observers and that based on the uncertainty principle of the outside
fixed position observers are equivalent to each other.

In both FIG.3 and FIG.6, the horizon represented by red straight lines and the
singularity represented by red zigzags are physical realities only to the
investigators which can see the Future of infinitely Far away Future (FiFF),
we will call the viewpoint of such observers as global of God-type. While the
viewpoint adopted by the true physic probes will be called local. Both the
outside fixed position observer and the freely falling co-moving observers are
local. To them the horizon and singularity are only mathematical realities
instead of physical ones, because they have not any physical ways to detect
such realities’ existence.

##  IV Quantisation

As the microscopic state definition function, both the asymptotic
$M[\infty,r]$ in (II.3) and the co-moving mass profile $M[\varrho]$ in
(III.1)-(III.6) at the classic level are arbitrary thus un-countable. However,
at the quantum level, things are rather different. We have two methods to
quantise the system canonically. The first is to use the hamiltonian
formulation of general relativity to build a Wheeler-Dewitt like functional
Schrodinger equation which takes the instantaneous mass function $m(t,r)$ as
independent variable and the radial mass-energy density profile $\rho[t,r]$ as
eigenvalues.

$\big{[}\frac{i^{2}\hbar^{2}\delta^{2}}{\delta m(r)^{2}}+\frac{r^{2}32\pi
G_{\scriptscriptstyle\\!N}\rho}{m^{\prime}h^{2}}+0\big{]}\Psi[m(r)]=0$ (IV.1)

where $h\equiv 1-\frac{2m(r)}{r}$ and $m(r)$ is the value of $m(t,r)$ at any
given time $t$. The $3+1$ decomposition or foliation of the spacetime itself
defines the relation between $m(t,r)$ and the instantaneous energy density
$\rho(t,r)$.

$h^{-\frac{1}{2}}r^{2}\big{[}-\frac{2m^{\prime}}{r^{2}}+8\pi
G_{\scriptscriptstyle\\!N}\rho\big{(}h^{-2}\frac{\dot{m}^{2}}{m^{\prime
2}}+0\big{)}\big{]}=0$ (IV.2)

In the classic hamiltonian constraint $m$ should be understood as the function
of $t$ and $r$ simultaneously, while in its quantum version $m$ is understood
as the function of $r$ only. This quantisation method is a generalisation of
the minimal super space quantum cosmology. We firstly proposed this
generalisation in reference dfzeng2017  but with an error happens in the
functional Schrodinger equation corrected in dfzeng2021 . In their counter
parts of references dfzeng2017 ; dfzeng2021 , the zero term positions of
(IV.1) and (IV.2) are occupied by terms of $\rho^{2}$ and $1$ respectively.
This difference is due to the time coordinate adopted in this work to define
the hamiltonian constraint is the outside fixed position observers time $t$
instead of the proper time $\tau$ of the inside co-moving observers.

To directly discretise the independent variable $m(r)$ and resolve the
functional Schrodinger equation (IV.1) is very difficult, see reference
dfzeng2017  for our earlier attempts. Beginning from reference dfzeng2018a
and through out references dfzeng2018b ; dfzeng2020 ; dfzeng2021 ; dfzeng2022
, we proposed a new discretisation stratagem. In this new stratagem, we
decompose the whole collapsing star into many concentric shells $\\{m_{i}\\}$
and quantise each of them canonically. The shell decomposition here is non-
apriori. All schemes allowed by the following two conditions would be
considered,

$\displaystyle m_{1}+m_{2}+\cdots+m_{k}=M_{\mathrm{tot}}$ (IV.3)
$\displaystyle\frac{GM_{i}m_{i}}{(2-2\gamma_{i}^{2})^{\frac{1}{2}}\hbar}=n_{i}=1,2,3,\cdots$
(IV.4)

The former follows from the total mass sum rule, while the latter follows from
the square integrability of $m_{i}$’s wave function;
$M_{i}\equiv\sum\limits_{i^{\prime}\leqslant i}m_{i^{\prime}}$; $\gamma_{i}$
or $n_{i}$ is the radial quantum number of shell $m_{i}$’s wave function and
classically they are determined by the shell’s initial radius. Quantum
mechanically the mass function of the whole collapsing star can be written as

$M[r]=\sum_{i}m_{i}|\psi_{i}(r)|^{2}$ (IV.5)

since $\psi_{i}(r)$ here is the eigenfunction of the hamiltonian whose
conjugating time is that of the outside fixed position observers, this quantum
mass function corresponds to the asymptotic function $M[\infty,r]$ of
eq(II.3). This can also be taken as the initial mass function on the co-moving
grid since the initial time epoch on the co-moving grid is arbitrarily
choosable.

Obviously, the mass function’s quantum definition (IV.5) tells us that it is
the combination of the shell decomposition and initial radius assigning scheme
that defines the microscopic state of the collapsing star, neither one of them
alone is enough to do that work. So our number counting of the microscopic
state of the system should be made on

$\\#\mathrm{{\bf\\!p}artition}\\{m_{i}\\}\otimes\mathrm{{\bf\\!e}xitation}\\{n_{i}\\}$
(IV.6)

the partition here is order-concerned, e.g. $\\{m_{i}\\}=\\{1,0.7,0.3\\}$ and
$\\{m_{i}^{\prime}\\}=\\{1,0.3,0.7\\}$ are two different partitions of
$M_{\mathrm{tot}}=2$. Two attentions must be payed with the number counting.
The first is, the global maximal value position of each shell’s wave function
modular $x^{\psi_{i}}_{\mathrm{max}}$ must happen outside
$r^{i}_{h}=2G\sum\limits_{i^{\prime}=0}^{i}m_{i^{\prime}}$ but not too far
away from it. $r^{i}_{h}$ is the classic horizon defined by the masses the
shell $m_{i}$ wraps. “not too far” means that the global maximal position of
the outmost shell’s wave function should take the lowest value which lies
outside the classic horizon $r_{h}=2GM_{\mathrm{tot}}$. The second is, any
radial excitation scheme should not make the global maximal value of shell
$m_{i}$’s wave function $|\psi_{i}^{2}|$ happens outside that of the shell
$m_{i+1}$, for all $i$. The position of these maximal values corresponds to
the classic asymptotic radius of the shells. This implies that the the radial
excitation number $\\{n_{i}\\}$ cannot be set arbitrarily. It can be
numerically shown that given shell mass decomposition $\\{m_{i}\\}$, the
number of possible $\\{n_{i}\\}$-values’ assigning schemes which satisfy these
two requirements grows at most polynomially with the total mass
$M_{\mathrm{tot}}$ increases. This polynomial non-uniqueness is the origin of
logarithmic corrections to the area law formula in our microscopic state
definition for black hole.

From the viewpoint of the outside fixed position observers, the classic motion
of each shell $m_{i}$ is freely falling in an effective Schwarzschild geometry
determined by the total mass of shells inside $m_{i}$ (including $m_{i}$
itself)

$\displaystyle ds^{2}=-h_{i}dt^{2}+h_{i}^{-1}dr^{2}+r^{2}d\Omega^{2}$ (IV.7)
$\displaystyle
h_{i}=1-\frac{2GM_{i}}{r},~{}M_{i}=\sum_{i^{\prime}=1}^{i}m_{i^{\prime}}$
(IV.8)

Using $x$ and $x^{\mu}(\lambda)\equiv\\{t(\lambda),x(\lambda),0,0\\}$ to
denote the radius of shell $m_{i}$ and the world line of a representative
point’s moving on it, the standard geodesic equation and four velocity
normalisation of the representative point tell us

$\displaystyle\ddot{t}+\Gamma^{(i)t}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0\Rightarrow
h_{i}\dot{t}=\gamma_{i}=\mathrm{const}$ (IV.9) $\displaystyle-
h_{i}\dot{t}^{2}+h^{-1}_{i}\dot{x}^{2}=-1$ (IV.10)

where $\Gamma^{(i)\sigma}_{\mu\nu}$ and $\gamma_{i}$ are the usual Christoffel
symbol corresponding to the effective metric (IV.7) and an integration
constant of the geodesic equations respectively. $\gamma_{i}=0$ corresponds
that the shell is released from $r_{ini}=2GM_{i}$ and $\dot{r}_{ini}=0$;
$\gamma_{i}=1$ corresponds that the shell is released from $r_{ini}=\infty$
and $\dot{r}_{ini}=0$. The former implies that the surface
$r=r_{\mathrm{ini}}$ happens to be a horizon. The latter implies the masses
inside $r=r_{\mathrm{ini}}$ is far from forming horizon. From the two
equations of (IV.9) and (IV.10), we can easily derive out

$\gamma_{i}^{2}-\dot{x}^{2}-h_{i}=0$ (IV.11)

Multiplying both sides of this equation with the shell mass $m_{i}$ and
manipulate the resultant equation routinely, we will get

$m_{i}\dot{x}^{2}-\frac{GM_{i}m_{i}}{x}-m_{i}(\gamma_{i}^{2}-1)=0$ (IV.12)

This is nothing but the hamiltonian constraint of the shell’s motion.

Now replace the classic momentum $m\dot{x}$ into its operator form
$i\hbar\partial_{x}$ and introduce a wave function $\psi_{i}(x)$ to denote the
probability amplitude the shell $m_{i}$ be measured at radius being $x$
status, we will get the quantum version of the Hamiltonian constraint (IV.12)

$[-\frac{\hbar^{2}}{2m_{i}}\partial_{x}^{2}-\frac{GM_{i}m_{i}}{x}-m_{i}(\gamma_{i}^{2}-1)]\psi_{i}(x)=0$
(IV.13)

This quantises the motion of shell $m_{i}$ canonically. All other shells can
be quantised similarly. Directly multiplying their wave-functions together, we
will get the wave functional of the whole collapsing star as follows

$\displaystyle\Psi[M(r)]=\psi_{0}\otimes\psi_{1}\otimes\psi_{2}\cdots,\sum_{i}m_{i}=M_{\mathrm{tot}}$
(IV.14)

Equation (IV.13) is almost the standard eigenstate Schrodinger equation with a
coulomb like potential except the normalisation condition

$\int_{0}^{\infty}|\psi_{i}|^{2}dx=1$ (IV.15)

which involves no angular space factor $4\pi x^{2}$ because $\psi_{i}(x)$ is
the probability amplitude of the spherical shell as a whole to lie on the
state of radius taking $x$ value instead of the probability amplitude a point
particle to be measured at position $(x,\theta,\phi)$. The eigenfunction and
eigenvalue of (IV.13) can be written down as

$\displaystyle\psi_{i}=N_{i}e^{-\bar{x}}\bar{x}L_{n_{i}-1}^{1}(2\bar{x}),\bar{x}\equiv
m_{i}x(2{\rule[1.5pt]{3.0pt}{0.6pt}}2\gamma_{i}^{2})^{\frac{1}{2}}/\hbar$
(IV.16) $\displaystyle
m_{i}(\gamma_{i}^{2}-1){\equiv}E_{i}=-\frac{(GM_{i}m_{i})^{2}m_{i}}{2n_{i}^{2}\hbar^{2}}$
(IV.17)

where $n_{i}=1,2,3\cdots$ take positive integer values and
$L_{n_{i}-1}^{1}(2x)$ is the associated Lagurre polynomial; $N_{i}$ is the
normalisation factor following from (IV.15). Equation (IV.17) explains the
origin of our microscopic state number counting constraint (IV.4).

Although the radial quantisation condition (IV.17) or (IV.4) of each component
shell provides countability for the microscopic state of the system, the
number of state counted basing on equations (IV.3)-(IV.4) is still infinite.
This is because both the two conditions do not impose discreteness on the
masses $\\{m_{i}\\}$ of the component shells, but only on the quotients
$\\{\frac{GM_{i}m_{i}}{(2-2\gamma_{i}^{2})^{1/2}\hbar}\\}$ and $\gamma_{i}$
are not required to be zero exactly. As a regularisation for this countable
infinite, we introduce the following concept of partitions’ distinguishability

$\displaystyle
two~{}partitions~{}of~{}equal~{}number~{}of~{}shells~{}\\{m_{i}\\}\&$ (IV.18)
$\displaystyle\\{m^{\prime}_{i}\\}~{}are~{}indistinguishable~{}on~{}precision~{}\varepsilon~{}if$
$\displaystyle{\scriptstyle\Delta}m_{i}\equiv|m_{i}-m^{\prime}_{i}|\leqslant\min\\{\frac{\varepsilon}{GM_{i}},\frac{\varepsilon}{GM^{\prime}_{i}}\\}~{}for~{}all~{}i;$
$\displaystyle{two~{}partitions~{}of~{}unequal~{}shell~{}numbers}$
$\displaystyle{are~{}always~{}distinguishable}$

By this definition, we can quantitatively state that to precision
$\varepsilon=1$, the finest way of decomposing the collapsing star into
concentric shells is that defined by the condition that for all $i$,
$\frac{GM_{i}m_{i}}{(2-2\gamma_{i}^{2})^{1/2}\hbar}=1$ and $\gamma_{i}=0$. We
will call this decomposition scheme as fundamental. By this fundamental
scheme, the number of shells the collapsing star can be partitioned into is
$k\approx\frac{GM_{\mathrm{tot}}^{2}}{2\sqrt{2}}$. This is because

$\displaystyle\hskip 28.45274ptGM_{i}(M_{i}-M_{i-1})=\sqrt{2}\Rightarrow$
(IV.19) $\displaystyle
M_{i}\xrightarrow[enough]{i~{}large}\big{(}\frac{2\sqrt{2}i}{G}\big{)}^{\frac{1}{2}},i^{\mathrm{max}}_{\mathrm{fix}M_{\mathrm{tot}}}\approx\frac{GM^{2}_{\mathrm{tot}}}{2\sqrt{2}}\equiv
k$ (IV.20)

For the fundamental shell decomposition above, to assure that the global
maximal value of each shell $m_{i}$’s wave function modular square happens
outside the classic horizon $r^{i}_{h}\equiv 2GM_{i}$, the lowest level radial
quantum number assigning scheme is to set $n_{i}=2$ for all $i$. This is
because in this case, the global maximal value of $|\psi^{2}_{n_{i}}(x)|$
happens on
$x^{\psi_{n_{i}}}_{\mathrm{max}}=\frac{3+\sqrt{5}}{GM_{i}m_{i}^{2}}=2.618GM_{i}$.
If we set any of the $n_{i}$s to $3$, then the corresponding
$x^{\psi_{n_{i}}}_{\mathrm{max}}=6.525GM_{i}$ will go outside its nearest
outer neighbour. If we set all of the $n_{i}$s to $3$, then the corresponding
wave functional of the whole system would have a too fat non-zero tail outside
the classic horizon $r_{h}=2GM_{\mathrm{tot}}$. The object described by the
corresponding wave functional would be distributed in a spherical spatial
region of radius $r=6.525GM_{\mathrm{tot}}$ instead of being localised inside
the classic horizon. So we will not take such highly excited quantum state as
the microscopic state of black holes defined by the classic horizon
$r_{h}=2GM_{\mathrm{tot}}$. We will call microscopic state defined by the
fundamental shell decomposition and pure $2$ radial quantum number as the
fundamental state of a classic black hole. For latter usages, we introduce the
following diagram to represent this fundamental state

$\displaystyle
M_{k~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{156.49014pt}{11.38109pt}\,\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{k},n_{k}{=}2$
$\displaystyle
M_{k^{\\!-}}{\color[rgb]{0,1,1}\rule[-4.0pt]{142.26378pt}{11.38109pt}\,\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{k^{\\!-}},n_{k^{\\!-}}{=}2$
$\displaystyle
M_{i~{}}\cdots\cdots\rule{56.9055pt}{0.0pt}{\color[rgb]{0,1,1}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{i}\,,n_{i}{=}2$
(IV.21) $\displaystyle
M_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{28.45274pt}{11.38109pt}\,\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{1},n_{1}{=}2$
$\displaystyle
M_{0~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{14.22636pt}{11.38109pt}\,\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{0},n_{0}{=}2$

where $k\\!^{-}{\equiv}k{-}1$ and $GM_{i}m_{i}=\sqrt{2}$ for all $i$. By this
representation, the most general radial excitation scheme can be written as

$\displaystyle
M_{k~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{150.79959pt}{11.38109pt}{\color[rgb]{1,0,0}\xrightarrow{~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{k},{\color[rgb]{1,0,0}n_{k}{\geqslant}2}$
$\displaystyle
M_{k^{\\!-}}{\color[rgb]{0,1,1}\rule[-4.0pt]{136.57323pt}{11.38109pt}{\color[rgb]{1,0,0}\xrightarrow{~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{k^{\\!-}},{\color[rgb]{1,0,0}n_{k^{\\!-}}{\geqslant}2}$
$\displaystyle
M_{i~{}}\cdots\cdots\rule{51.21495pt}{0.0pt}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}~{}}}{\color[rgb]{0,1,1}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{i}\,,{\color[rgb]{1,0,0}n_{i}{\geqslant}2}$
(IV.22) $\displaystyle
M_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{28.45274pt}{11.38109pt}\,{\color[rgb]{1,0,0}\xrightarrow{~{}~{}~{}~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{2},{\color[rgb]{1,0,0}n_{1}{\geqslant}2}$
$\displaystyle
M_{0~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{14.22636pt}{11.38109pt}\,{\color[rgb]{1,0,0}\xrightarrow{~{}~{}~{}~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{1},{\color[rgb]{1,0,0}n_{0}{\geqslant}2}$

However, by the explicit form of each shell’s wave function (IV.16), it can be
easily verified that one cannot find any radial quantum numbers’ combination
$\\{n_{i}\\}$ which is different from that of the fundamental state (IV.21)
and for all $i$, the position of $|\psi_{i}^{2}|$’s maximal value does not go
outside that of the shell $m_{i+1}$’s $|\psi^{2}_{i+1}|$. This means that the
fundamental state of the system is unique.

By choosing two or more inner shells and merging them into some new single
shell, we have ways to excite the inner structure of the fundamental state
without change the total ADM-mass/energy of the system. The most simple choice
is selecting two neighbouring shells and merging them into a single one. For
example, merging the outmost two shells

$\displaystyle
M_{k^{\\!-}}{\color[rgb]{0,1,1}\rule[-4.0pt]{133.72795pt}{11.38109pt}\,\rule[-4.0pt]{28.45274pt}{11.38109pt}}m_{k^{\\!-}}~{},n_{k^{\\!-}}{\approx}4$
$\displaystyle
M_{i~{}}\cdots\cdots\rule{56.9055pt}{0.0pt}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}{\color[rgb]{0,1,1}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{i}\,,{\color[rgb]{1,0,0}n_{i}{\geqslant}2}$
(IV.23) $\displaystyle
M_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{28.45274pt}{11.38109pt}\,{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{1},{\color[rgb]{1,0,0}n_{1}{\geqslant}2}$
$\displaystyle
M_{0~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{14.22636pt}{11.38109pt}\,{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{0},{\color[rgb]{1,0,0}n_{0}{\geqslant}2}$

or more generally

$\displaystyle
M_{k^{\\!-}}{\color[rgb]{0,1,1}\rule[-4.0pt]{147.95433pt}{11.38109pt}\,\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{k^{\\!-}},n_{k^{\\!-}}{=}2$
$\displaystyle
M_{i~{}}\cdots\cdots\rule{42.67912pt}{0.0pt}{\color[rgb]{0,1,1}\rule[-4.0pt]{28.45274pt}{11.38109pt}}m_{i}~{}~{}~{},~{}n_{i}{\approx}4$
(IV.24) $\displaystyle
M_{2~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{28.45274pt}{11.38109pt}\,{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{2},{\color[rgb]{1,0,0}n_{2}{\geqslant}2}$
$\displaystyle
M_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{14.22636pt}{11.38109pt}\,{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{1},{\color[rgb]{1,0,0}n_{1}{\geqslant}2}$

$n_{k^{\\!-}}{\equiv}\frac{GM_{k\\!^{-}\\!}(m_{k\\!^{-}\\!}{+}m_{k})}{(2-2\gamma_{k\\!^{-}\\!}^{2})^{1/2}\hbar}{\approx}4$
or
$n_{i}{\equiv}\frac{GM_{i}(m_{i}{+}m_{i\\!^{+}\\!})}{(2-2\gamma_{i}^{2})^{1/2}\hbar}{\approx}4$
means that as the two shells are merged, $n_{k\\!^{-}\\!}$ and $n_{i}$ can be
set to $4$ if we tune the value of $\gamma_{k\\!^{-}}$ or $\gamma_{i}$ to some
value which is larger than their value in the fundamental state. This is
rational because when we merge two shells, the resultant shell must be put on
a state whose radial quantum number is roughly the summation of the merger-
before members’ quantum number to assure the global maximal value of the
resultant shell’s wave function happens outside the horizon
$r^{i^{\prime}}_{h}=2GM_{i^{\prime}}$. Before the two shells’ recombination,
all $n_{i}$ equals to $2$. The radial quantum number of shells inside the
recombined shell can now be set to values greater than 2 because as the latter
takes now a larger radial quantum number, the formers have the possibility to
be excited to higher radial quantum number level and still keep their global
maximal value inside the that of the latter. This means that a given
recombined shell decomposition scheme could have more than one radial
excitation ways.

The most general shell decomposition and radial excitation scheme can be
generated from the fundamental state by shell recombination and radial quantum
numbers’ rearranging. The resultant microscopic state of the system will have
equal total mass as but different inner profiles from the fundamental one.
They can be represented as

$\displaystyle
M_{\ell~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{142.26378pt}{11.38109pt}\,\rule[-4.0pt]{28.45274pt}{11.38109pt}}m_{\ell}~{}~{},~{}{\color[rgb]{1,0,0}n_{\ell}}$
$\displaystyle
M_{i~{}}\cdots\cdots\rule{34.1433pt}{0.0pt}{\color[rgb]{1,0,0}\xrightarrow{~{}}}{\color[rgb]{0,1,1}\rule[-4.0pt]{34.1433pt}{11.38109pt}}m_{i}~{}~{}~{},~{}{\color[rgb]{1,0,0}n_{i}{\geqslant}2}$
(IV.25) $\displaystyle
M_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{28.45274pt}{11.38109pt}\,{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{1},{\color[rgb]{1,0,0}n_{1}{\geqslant}2}$
$\displaystyle
M_{0~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{14.22636pt}{11.38109pt}\,{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{0},{\color[rgb]{1,0,0}n_{0}{\geqslant}2}$

The number of shell recombinations from the fundamental state is the number of
ordered partition of integer $k=\frac{GM_{\mathrm{tot}}^{2}}{2\sqrt{2}}$,
which is $2^{k-1}$ exactly, including the fundamental state itself. So the
total number of microscopic state of the system can be written as

$W=\\#\mathrm{{\bf\\!p}artition}\\{m_{i}\\}\otimes\mathrm{{\bf\\!e}xitation}\\{n_{i}\\}=2^{k-1}\cdot
X(k)$ (IV.26)

where $X(k)$ is a correction factor which arises from the non-uniqueness of
the radial quantum number assigning scheme in the specified shell
decomposition scheme. In the case of neglecting the requirement that
$|\psi^{2}_{i}|$’s global maximal value happens inside that of
$|\psi^{2}_{i+1}|$, references dfzeng2021  proves that $X(k)$, which is named
$F(k)$ there, grows exponentially as $k$ increases. However, when this
requirement is considered, it can be verified that $F(k)$’s growing trend is
at most polynomial way. See the next section for a concrete working example.

So by the distinguishability definition of (IV.18) and the result (IV.26), we
can now claim that to precision $\varepsilon$, the number of all
distinguishable ways of decomposing a collapsing star with asymptotic horizon
into concentric shells $\\{m_{i}\\}$ and setting each shell’s radial quantum
number $\\{n_{i}\\}$ is

$W=\exp\\{c(\varepsilon)[\frac{A}{4G}+\ln
X(A)]\\},A=4\pi(2GM_{\mathrm{tot}})^{2}$ (IV.27)

where $M_{\mathrm{tot}}$ is only required to be mildly larger than
$G^{-\frac{1}{2}}{\equiv}M_{\mathrm{pl}}$. Since the decomposition precision
parameter $\varepsilon$ linearly affects the number of shells of the
fundamental decomposition, $c(\varepsilon)$ is a linear function of
$\varepsilon$. This means that we can easily get the logarithmically corrected
Bekenstein-Hawking formula with $c(\varepsilon)=1$ by simply setting
$\varepsilon\approx\frac{1}{8\sqrt{2}\pi}$. To get the explicit formula of the
logarithmic correction $X(A)$, we need quantitatively work out the relation
between the number of radial quantum number assigning scheme $\\{n_{i}\\}$ and
the collapsing star’s total mass $M_{\mathrm{tot}}$ for specified shell
decomposition $\\{m_{i}\\}$ under the constraint that for all $i$, the global
maximal value of $|\psi^{2}_{i}|$ happens not outside that of
$|\psi^{2}_{i+1}|$, and the global maximal value of all $|\psi^{2}_{i}|$
happens outside the classic horizon of $r^{i}_{h}=2GM_{i}$, with
$n_{i-outmost}$ takes the minimal allowed value. This may be a challenging
work but operable in principle.

Equation (IV.17) or (IV.4) implies that the outmost shell has a minimal
allowed mass

$\displaystyle\frac{GM_{\mathrm{tot}}m_{\mathrm{outmost}}}{(2-2\gamma_{\mathrm{outmost}}^{2})^{1/2}}=n_{\mathrm{outmost}}^{\mathrm{min}}=2\Rightarrow$
(IV.28) $\displaystyle
m_{\mathrm{outmost}}=\frac{2(2-2\gamma^{2}_{\mathrm{outmost}})^{\frac{1}{2}}}{GM_{\mathrm{tot}}}$
(IV.29)

Since the shell with $\gamma_{\mathrm{outmost}}{=}0$ has the maximal
probability be found on the horizon and that with
$\gamma_{\mathrm{outmost}}=1$ has the maximal probability be found on the
infinity $r$ region. Exact horizon boundary condition implies that
$\gamma_{\mathrm{outmost}}{=}0$. This means that

$m^{\mathrm{min}}_{\mathrm{outmost}}=\frac{2\sqrt{2}}{GM_{\mathrm{tot}}}$
(IV.30)

This is also the typical mass of most of the outer layered shells. This
explains why we have chances to get the area law entropy by counting the bulk
motion degrees of freedom of the black hole matter contents. Because the
number of such shells reads

$N\sim\frac{M_{\mathrm{tot}}}{(GM_{\mathrm{tot}})^{-1}}\propto GM^{2}$ (IV.31)

Note that for all macroscopic black holes, the value of
$m^{\mathrm{min}}_{\mathrm{outmost}}$ is far less than the standard model
particles which consists of the collapsing star. So what the Bekenstein-
Hawking entropy reflects is not the motion degrees of freedom of these
particles, but the collection motion modes of them which we call as composite
shells. This fact is pointed out and emphasised in our series of works
dfzeng2017 ; dfzeng2018a ; dfzeng2018b ; dfzeng2020 ; dfzeng2021 ; dfzeng2022
.

Equation (IV.30) also tells us that $m^{\mathrm{min}}_{\mathrm{outmost}}$ has
the same order as the hawking temperature of a mass $M_{\mathrm{tot}}$ black
hole. This means that this shell can be easily radiated away as hawking
radiations. FIG.7 displays the wave function modular square of the fundamental
microscopic state of a $3M_{\mathrm{pl}}$ black hole caused by a dust ball’s
gravitational collapse. From the figure we easily see the out most shell has
very large probability be found outside the horizon. Of course this outmost
shell’s appearance outside the horizon itself is not hawking radiation. The
true hawking radiation arises from the microscopic state’s change of the
system and can be described as the spontaneous radiation of usual atoms with
explicitly hermitian hamiltonian, which is named gravitation induced
spontaneous radiation in dfzeng2022 , see references dfzeng2018a ; dfzeng2018b
; dfzeng2021  for more earlier and detailed investigations.

The physical picture and calculations up to here provide a quantum mechanics
based general relativity interpretation for the Bekenstein-Hawking entropy and
area law formula. This interpretation contrasts with the common belief that
Bekenstein-Hawking entropy has pure geometric origin, i.e. the fundamental
degrees of freedom of black holes is carried by their horizon, each planck
unit of area element carries one bit of information. However, our calculation
shows that such fundamental degrees of freedom may completely be carried by
the bulk motion modes of the matter contents. As long as we find the right
carrier the area law formula flows logically and naturally. Since in a
collapsing star consists of dust only, no other forces except gravitation is
involved, the matter’s motion and fluctuation is an exact and faithful
reflection of the geometric feature of the spacetime underground. As results,
the quantisation of these matters’ motion is also the quantisation of
spacetime itself. In a sense what we provide in this work is a quantum
gravitation theory indeed. Since it focuses on the origin of Bekenstein
Hawking entropy exclusively, we consider it a mini-version of some underlying
full quantum gravitation theories.

##  V A working example for the quantisation

Let us consider the inner structure’s quantisation of an
$M_{\mathrm{tot}}=3M_{\mathrm{pl}}$ black hole as a concrete example for the
microscopic state definition ideas above. For this black hole, to precision
$\varepsilon=1$, its matter contents can be divided into 4 distinguishable
concentric shells at most, under the condition that $n_{i}{=}2$,
$\gamma_{i}{=}0$ for all $i$. The fundamental state of the whole system is
defined by the following shell mass assigning and radial quantum level
number’s setting scheme,

$\displaystyle\\{m_{i}\\}=\\{1.25118,0.718127,0.559288,0.471405\\}$ (V.1)
$\displaystyle\\{M_{i}\\}=\\{1.25118,1.96931,2.5286,3\\}$ (V.2)
$\displaystyle\\{\frac{GM_{i}m_{i}}{\sqrt{2}}\\}=\\{1,1,1,1\\}$ (V.3)
$\displaystyle\\{(1-\gamma_{i}^{2})^{\frac{1}{2}}\\}=\\{0.5,0.5,0.5,0.5\\}$
(V.4) $\displaystyle\\{n_{i}\\}=\\{2,2,2,2\\}$ (V.5)
$\displaystyle\\{\psi_{i}\\}\propto\\{e^{-\bar{x}}\bar{x}\mathrm{LaguerreL}_{0}^{1}(2\bar{x})\\}$
(V.6)
$\displaystyle\\{x_{\mathrm{max}}^{\psi_{i}}\\}=\\{2.67,5.16,6.62,7.85\\}$
(V.7)

Note that in the wave function of (V.6),
$\bar{x}{\equiv}m_{i}(2{-}2\gamma_{i}^{2})^{\frac{1}{2}}x$. By the
representation method of (IV.21), this ground state can be represented as

$\displaystyle
M_{3~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{91.04881pt}{11.38109pt}\,\rule[-4.0pt]{14.22636pt}{11.38109pt}}m_{3},n_{3}{=}2$
$\displaystyle
M_{2~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{68.2866pt}{11.38109pt}\,\rule[-4.0pt]{19.91692pt}{11.38109pt}}m_{2},n_{2}{=}2$
$\displaystyle
M_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{36.98857pt}{11.38109pt}\,\rule[-4.0pt]{28.45274pt}{11.38109pt}}m_{1},n_{1}{=}2$
$\displaystyle
M_{0~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{36.98857pt}{11.38109pt}}m_{0},n_{0}{=}2$

Figure 7: The modular square of the ground state wave function of an
$M=3M_{\mathrm{pl}}$ black hole.

FIG.7 displays the modular square of the wave function of each sub-shell. In
the figure, three things are worthy of noticing. The first is, the positions
of the global maximal value of each sub-shell’s wave function are ordered from
inside to the outside. The second is, the global maximal value of all shell
$m_{i}$’s wave function happens outside $r^{i}_{h}$. This is nothing but the
requirement that classically each shell lies outside the horizon
$r^{i}_{h}=2G\sum_{i^{\prime}=0}^{i}m_{i^{\prime}}$ determined by the matter
mass they wrap. The third is, the probability of each shells’ being found
outside the global horizon defined by $r_{h}=2GM_{\mathrm{tot}}$ is rather
remarkable. Keeping the condition
$x^{\psi_{i}}_{\mathrm{max}}<x^{\psi_{i+1}}_{\mathrm{max}},\forall i$ and
exciting all shells $m_{i}$ to their higher level quantum state uniformly by
setting $n_{i}=3$ or more larger is possible, but that would make the
probability of these shell’s being found outside the global horizon becomes
too larger than the ground state characterised by $n_{i}=2,\forall i$.

Now if we let some but not all of the inner shells such as $m_{0}$ to be
excited, we will find that even only let $n_{0}=3$, the global maximal value
of $\psi_{0}(x)$ would happen outside that of $\psi_{1}(x)$. So under the
requirement that all other shell’s quantisation conditions are kept invariant,
i.e. $\frac{GM_{i}m_{i}}{\sqrt{2}}=2$($i{\neq}0\&1$), the only possible way of
exciting $m_{0}$ is letting it be bounded with $m_{1}$ as a single shell.
Similarly, if we want excite other one or more shells independently, the best
way is also to let them be bounded with their closest outer neighbour into a
single shell. For the $M=3M_{\mathrm{pl}}$ black hole whose fundamental state
is defined by the four $n_{i}=2$($i{=}0,1,2,3$) shells, we have only
$7=2^{4-1}-1$ possible ways to recombine those shells. Let us consider them
one by one.

Recombination A,
$\\{m_{0}+m_{1}{\equiv}m^{\prime}_{0},m_{2}{\equiv}m^{\prime}_{1},m_{3}{\equiv}m_{2}^{\prime}\\}$,
remember in the wave function (V.14),
$\bar{x}{\equiv}m_{i}(2{-}2\gamma_{i}^{2})^{\frac{1}{2}}x$. This is the same
as (V.6) and similarly in the other recombination schemes.

$\displaystyle\\{m^{\prime}_{i}\\}=\\{1.96931,0.559288,0.471405\\}$ (V.9)
$\displaystyle\\{M^{\prime}_{i}\\}=\\{1.96931,2.5286,3.\\}$ (V.10)
$\displaystyle\\{\frac{GM^{\prime}_{i}m^{\prime}_{i}}{\sqrt{2}}\\}=\\{2.74228,1,1\\}$
(V.11)
$\displaystyle\\{(1-{\gamma^{\prime}}_{i}^{2})^{\frac{1}{2}}\\}=\\{\Gamma_{0},0.5,0.5\\},0\leqslant\Gamma_{0}\leqslant
1$ (V.12) $\displaystyle\\{n^{\prime}_{i}\\}=\\{2.74228/\Gamma_{0},2,2\\}$
(V.13)
$\displaystyle\\{\psi^{\prime}_{i}\\}\propto\\{e^{-\bar{x}}\bar{x}\mathrm{LaguerreL}_{n_{i}-1}^{1}(2\bar{x})\\}$
(V.14)
$\displaystyle\\{x_{\mathrm{max}}^{\psi_{i}^{\prime}}\\}=\\{3.94{\leqslant}x^{\psi^{\prime}_{0}}_{\mathrm{max}}{\leqslant}6.62,6.62,7.85\\}$
(V.15)

To satisfy the constraint
$2GM^{\prime}_{0}{=}3.94{\leqslant}x^{\psi^{\prime}_{0}}_{\mathrm{max}}{\leqslant}6.62$,
we find that $n^{\prime}_{0}=5$ are allowed, other values are not. So this
shell recombination scheme contributes only one microscopic state to the
system. By the representation method of (IV.21), this state can be represented
as

$\displaystyle
M^{\prime}_{2~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{91.04881pt}{11.38109pt}\,\rule[-4.0pt]{14.22636pt}{11.38109pt}}m^{\prime}_{2},n_{2}{=}2$
(V.16) $\displaystyle
M^{\prime}_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{68.2866pt}{11.38109pt}\,\rule[-4.0pt]{19.91692pt}{11.38109pt}}m^{\prime}_{1},n_{1}{=}2$
$\displaystyle
M^{\prime}_{0~{}}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}{\color[rgb]{0,1,1}\rule[-4.0pt]{54.06023pt}{11.38109pt}}m^{\prime}_{0},{\color[rgb]{1,0,0}n_{0}{=}5}$

Recombination B,
$\\{m_{0}{\equiv}m^{\prime}_{0},m_{1}{+}m_{2}{\equiv}m^{\prime}_{1},m_{3}{\equiv}m_{2}^{\prime}\\}$,
still remember that $\bar{x}{\equiv}m_{i}(2{-}2\gamma_{i}^{2})^{\frac{1}{2}}x$

$\displaystyle\\{m^{\prime}_{i}\\}=\\{1.25118,1.27742,0.471405\\}$ (V.17)
$\displaystyle\\{M^{\prime}_{i}\\}=\\{1.25118,2.5286,3\\}$ (V.18)
$\displaystyle\\{\frac{GM^{\prime}_{i}m^{\prime}_{i}}{\sqrt{2}}\\}=\\{1.10694,2.284,1\\}$
(V.19)
$\displaystyle\\{(1-{\gamma^{\prime}}_{i}^{2})^{\frac{1}{2}}\\}=\\{0.55347,\Gamma_{1},0.5\\},0\leqslant\Gamma_{1}\leqslant
1$ (V.20) $\displaystyle\\{n^{\prime}_{i}\\}=\\{2,2.28406/\Gamma_{1},2\\}$
(V.21)
$\displaystyle\\{\psi^{\prime}_{i}\\}\propto\\{e^{-\bar{x}}\bar{x}\mathrm{LaguerreL}_{n_{i}{-}1}^{1}(2\bar{x})\\}$
(V.22)
$\displaystyle\\{x_{\mathrm{max}}^{\psi_{i}^{\prime}}\\}=\\{2.67,5.06{\leqslant}x^{\psi^{\prime}_{1}}_{\mathrm{max}}{\leqslant}7.85,7.85\\}$
(V.23)

To satisfy the condition
$2GM^{\prime}_{2}{=}5.06<x^{\psi^{\prime}_{1}}_{\mathrm{max}}<7.85$, we find
that $n^{\prime}_{1}=4$ are allowed, other values are not. This means that the
shell recombination scheme B also contributes only one microscopic state to
the system. The diagram representation of this state can be plotted as follows

$\displaystyle
M^{\prime}_{2~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{91.04881pt}{11.38109pt}\,\rule[-4.0pt]{14.22636pt}{11.38109pt}}m^{\prime}_{2},n_{2}{=}2$
(V.24) $\displaystyle
M^{\prime}_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{36.98857pt}{11.38109pt}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{39.83385pt}{11.38109pt}}m^{\prime}_{1},{\color[rgb]{1,0,0}n_{1}{=}4}$
$\displaystyle
M^{\prime}_{0~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{36.98857pt}{11.38109pt}}m^{\prime}_{0},n_{0}{=}2$

Recombination C,
$\\{m_{0}{\equiv}m^{\prime}_{0},m_{1}{\equiv}m^{\prime}_{1},m_{2}{+}m_{3}{\equiv}m_{2}^{\prime}\\}$

$\displaystyle\\{m^{\prime}_{i}\\}=\\{1.25118,0.718127,1.03069\\}$ (V.25)
$\displaystyle\\{M^{\prime}_{i}\\}=\\{1.25118,1.96931,3\\}$ (V.26)
$\displaystyle\\{\frac{GM^{\prime}_{i}m^{\prime}_{i}}{\sqrt{2}}\\}=\\{1.10694,1,2.18643\\}$
(V.27)
$\displaystyle\\{(1-{\gamma^{\prime}}_{i}^{2})^{\frac{1}{2}}\\}=\\{0.55347,0.5,\Gamma_{2}\\},0\leqslant\Gamma_{2}\leqslant
1$ (V.28) $\displaystyle\\{n^{\prime}_{i}\\}=\\{2,2,2.18643/\Gamma_{2}\\}$
(V.29)
$\displaystyle\\{\psi^{\prime}_{i}\\}\propto\\{e^{-\bar{x}}\bar{x}\mathrm{LaguerreL}_{n_{i}-1}^{1}(2\bar{x})\\}$
(V.30)
$\displaystyle\\{x_{\mathrm{max}}^{\psi_{i}^{\prime}}\\}=\\{2.67,5.16,6{\leqslant}x^{\psi^{\prime}_{2}}_{\mathrm{max}}\\}$
(V.31)

To satisfy the condition
$2GM^{\prime}_{2}{=}6{\leqslant}x^{\psi^{\prime}_{2}}_{\mathrm{max}}$, we find
that $n^{\prime}_{2}=4$ is the lowest requirement. Other greater than $4$
values for $n^{\prime}_{2}$ could also satisfy this constraint. But the
corresponding wave function will have a very fat non-zero tail outside the
global horizon thus contradict with definition of black hole in general
relativity. We will exclude such state as the microscopic state of the black
holes characterised by the classic horizon $r_{h}=2GM_{\mathrm{tot}}$. This
means that the shell recombination scheme C also contributes only one
microscopic state to the system. The diagram representation of this state can
be plotted as follows

$\displaystyle
M^{\prime}_{2~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{71.13188pt}{11.38109pt}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{25.60747pt}{11.38109pt}}m^{\prime}_{2},{\color[rgb]{1,0,0}n_{2}{=}4}$
(V.32) $\displaystyle
M^{\prime}_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{36.98857pt}{11.38109pt}\,\rule[-4.0pt]{31.29802pt}{11.38109pt}}m^{\prime}_{1},n_{1}{=}2$
$\displaystyle
M^{\prime}_{0~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{36.98857pt}{11.38109pt}}m^{\prime}_{0},n_{0}{=}2$

Recombination D,
$\\{m_{0}{+}m_{1}{+}m_{2}{\equiv}m^{\prime}_{0},m_{3}{\equiv}m^{\prime}_{1}\\}$

$\displaystyle\\{m^{\prime}_{i}\\}=\\{2.5286,0.4714\\}$ (V.33)
$\displaystyle\\{M^{\prime}_{i}\\}=\\{2.5286,3\\}$ (V.34)
$\displaystyle\\{\frac{GM^{\prime}_{i}m^{\prime}_{i}}{\sqrt{2}}\\}=\\{4.5211,1\\}$
(V.35)
$\displaystyle\\{(1-{\gamma^{\prime}}_{i}^{2})^{\frac{1}{2}}\\}=\\{\Gamma_{0},0.5\\},0\leqslant\Gamma_{0}\leqslant
1$ (V.36) $\displaystyle\\{n^{\prime}_{i}\\}=\\{4.5211/\Gamma_{0},2\\}$ (V.37)
$\displaystyle\\{\psi^{\prime}_{i}\\}\propto\\{e^{-\bar{x}}\bar{x}\mathrm{LaguerreL}_{n_{i}-1}^{1}(2\bar{x})\\}$
(V.38)
$\displaystyle\\{x_{\mathrm{max}}^{\psi_{i}^{\prime}}\\}=\\{5.06{\leqslant}x^{\psi^{\prime}_{0}}_{\mathrm{max}}{\leqslant}7.85,7.85\\}$
(V.39)

To satisfy the condition
$2GM^{\prime}_{0}{=}5.06{\leqslant}x^{\psi^{\prime}_{0}}_{\mathrm{max}}{\leqslant}7.85$,
we find that $n^{\prime}_{0}=7,8$ are both allowed. Other values are not. So
this recombination contributes 2 microscopic states to the system. The diagram
representation of these two states can be plotted as follows

$\displaystyle
M^{\prime}_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{71.13188pt}{11.38109pt}\,\rule[-4.0pt]{14.22636pt}{11.38109pt}}m^{\prime}_{1},n_{1}{=}2$
(V.40) $\displaystyle
M^{\prime}_{0~{}}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}~{}}}{\color[rgb]{0,1,1}\rule[-4.0pt]{54.06023pt}{11.38109pt}\,}m^{\prime}_{0},{\color[rgb]{1,0,0}n_{0}{=}7,8}$

Recombination E,
$\\{m_{0}{+}m_{1}{\equiv}m^{\prime}_{0},m_{2}{+}m_{3}{\equiv}m^{\prime}_{1}\\}$

$\displaystyle\\{m^{\prime}_{i}\\}=\\{1.96931,1.03069\\}$ (V.41)
$\displaystyle\\{M^{\prime}_{i}\\}=\\{1.96931,3\\}$ (V.42)
$\displaystyle\\{\frac{GM^{\prime}_{i}m^{\prime}_{i}}{\sqrt{2}}\\}=\\{2.742,2.186\\}$
(V.43)
$\displaystyle\\{(1-{\gamma^{\prime}}_{i}^{2})^{\frac{1}{2}}\\}=\\{\Gamma_{0},\Gamma_{1}\\},0\leqslant\Gamma_{0},\Gamma_{1}\leqslant
1$ (V.44)
$\displaystyle\\{n^{\prime}_{i}\\}=\\{2.742/\Gamma_{0},2.186/\Gamma_{1}\\}$
(V.45)
$\displaystyle\\{\psi^{\prime}_{i}\\}\propto\\{e^{-\bar{x}}\bar{x}\mathrm{LaguerreL}_{n_{i}-1}^{1}(2\bar{x})\\}$
(V.46)
$\displaystyle\\{x_{\mathrm{max}}^{\psi_{i}^{\prime}}\\}=\\{3.94{\leqslant}x^{\psi^{\prime}_{0}}_{\mathrm{max}},6{\leqslant}x^{\psi^{\prime}_{1}}_{\mathrm{max}}\\}$
(V.47)

To satisfy the constraints
$2GM^{\prime}_{0}{=}3.94{\leqslant}x^{\psi^{\prime}_{0}}_{\mathrm{max}}$,
$2GM^{\prime}_{1}{=}$ $6{\leqslant}x^{\psi^{\prime}_{1}}_{\mathrm{max}}$ and
$x^{\psi^{\prime}_{0}}_{\mathrm{max}}<x^{\psi^{\prime}_{1}}_{\mathrm{max}}$,
we find that $n^{\prime}_{0}=5$, $n^{\prime}_{1}=4$ are allowed. Other values
for $n^{\prime}_{0}$ and $n^{\prime}_{1}$ which satisfy this three constraints
are also possible, but all of them have the problem of giving the
corresponding wave function a too fat nonzero tail outside the horizon thus
contradict with the definition of black holes characterised by their classic
horizon. So such highly excited state will not be considered as the
microscopic state of the class black holes. The diagram representation of this
state can be plotted as follows

$\displaystyle
M^{\prime}_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{56.9055pt}{11.38109pt}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{22.76219pt}{11.38109pt}}m^{\prime}_{1},{\color[rgb]{1,0,0}n_{1}{=}4}$
(V.48) $\displaystyle
M^{\prime}_{0~{}}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}~{}}}{\color[rgb]{0,1,1}\rule[-4.0pt]{39.83385pt}{11.38109pt}\,}m^{\prime}_{0},{\color[rgb]{1,0,0}n_{0}{=}6}$

Recombination F,
$\\{m_{0}{\equiv}m^{\prime}_{0},m_{1}{+}m_{2}{+}m_{3}{\equiv}m^{\prime}_{1}\\}$

$\displaystyle\\{m^{\prime}_{i}\\}=\\{1.25118,1.74882\\}$ (V.49)
$\displaystyle\\{M^{\prime}_{i}\\}=\\{1.25118,3\\}$ (V.50)
$\displaystyle\\{\frac{GM^{\prime}_{i}m^{\prime}_{i}}{\sqrt{2}}\\}=\\{1.10694,3.70981\\}$
(V.51)
$\displaystyle\\{(1-{\gamma^{\prime}}_{i}^{2})^{\frac{1}{2}}\\}=\\{0.55347,\Gamma_{1}\\},0\leqslant\Gamma_{1}\leqslant
1$ (V.52) $\displaystyle\\{n^{\prime}_{i}\\}=\\{2,3.70981/\Gamma_{1}\\}$
(V.53)
$\displaystyle\\{\psi^{\prime}_{i}\\}\propto\\{e^{-\bar{x}}\bar{x}\mathrm{LaguerreL}_{n_{i}-1}^{1}(2\bar{x})\\}$
(V.54)
$\displaystyle\\{x_{\mathrm{max}}^{\psi_{i}^{\prime}}\\}=\\{2.67,6{\leqslant}x^{\psi^{\prime}_{1}}_{\mathrm{max}}\\}$
(V.55)

To satisfy the condition
$2GM^{\prime}_{1}{=}6{\leqslant}x^{\psi^{\prime}_{1}}_{\mathrm{max}}$, we find
that $n^{\prime}_{1}=6$ is the lowest level required. Other greater than $6$
levels for $n^{\prime}_{1}$ which satisfy this condition are also possible,
but has the problem of giving the corresponding wave function a too fat tail
outside the classic horizon. So they will still not be taken as the
microscopic state of class black holes. The diagram representation of this
state can be plotted as follows

$\displaystyle
M^{\prime}_{1~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{34.1433pt}{11.38109pt}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}\rule[-4.0pt]{45.5244pt}{11.38109pt}}m^{\prime}_{1},{\color[rgb]{1,0,0}n_{1}{=}6}$
(V.56) $\displaystyle
M^{\prime}_{0~{}}{\color[rgb]{0,1,1}\rule[-4.0pt]{34.1433pt}{11.38109pt}\,}m^{\prime}_{0},n_{0}{=}2$

Recombination G, $\\{m_{0}{+}m_{1}{+}m_{2}{+}m_{3}{\equiv}m^{\prime}_{0}\\}$

$\displaystyle\\{m^{\prime}_{i}\\}=\\{3\\},\\{M^{\prime}_{i}\\}=\\{3\\}$
(V.57)
$\displaystyle\\{\frac{GM^{\prime}_{i}m^{\prime}_{i}}{\sqrt{2}}\\}=\\{6.36396\\}$
(V.58)
$\displaystyle\\{(1-{\gamma^{\prime}}_{i}^{2})^{\frac{1}{2}}\\}=\\{\Gamma_{0}\\},0\leqslant\Gamma_{1}\leqslant
1$ (V.59) $\displaystyle\\{n^{\prime}_{i}\\}=\\{6.36396/\Gamma_{0}\\}$ (V.60)
$\displaystyle\\{\psi^{\prime}_{i}\\}\propto\\{e^{-\bar{x}}\bar{x}\mathrm{LaguerreL}_{n_{i}-1}^{1}(2\bar{x})\\}$
(V.61)
$\displaystyle\\{x_{\mathrm{max}}^{\psi_{i}^{\prime}}\\}=\\{6{\leqslant}x^{\psi^{\prime}_{0}}_{\mathrm{max}}\\}$
(V.62)

To satisfy the condition
$2GM^{\prime}_{0}{=}6{\leqslant}x^{\psi^{\prime}_{1}}_{\mathrm{max}}$, we find
that $n^{\prime}_{0}=10$ is the lowest level required. Other greater than $10$
levels for $n^{\prime}_{0}$ satisfying this condition are also possible, but
has the problem of giving the corresponding wave function a too fat nonzero
tail outside the classic horizon. So still can not be taken as microscopic
states of the classic black holes. The diagram representation of this state
can be plotted as follows

$M^{\prime}_{0~{}}{\color[rgb]{1,0,0}\xrightarrow{~{}~{}}}{\color[rgb]{0,1,1}\rule[-4.0pt]{56.9055pt}{11.38109pt}}m^{\prime}_{0},{\color[rgb]{1,0,0}n_{0}{=}10}$
(V.63)

From the calculations above, we can see that to precision $\varepsilon=1$, an
$M=3M_{\mathrm{pl}}$ black hole can be divided into 4 distinguishable
concentric shells at most under the condition that $n_{i}{=}2$ and
$\gamma_{i}{=}0$ for all $i$. Letting all this four shells lie on the quantum
state of $n_{i}=2$ yields the fundamental state of the system. In this
fundamental state, the global maximal value of each shells’ wave function
modular square happens outside the classic horizon determined by the masses
the shell wraps $r^{i}_{h}{=}2GM_{i}$, $M_{i}{=}\sum_{i^{\prime}=0}^{i}m_{i}$.
To excite each shell independently and preserve the ordering of the excited
shells is equivalent to recombine shells from the fundamental state and tuning
the resultant shell’s radial quantum number. Including the fundamental state
itself, we have $2^{4-1}=8$ ways to to such a recombination. Except the
recombination scheme D, the number of ways to excite all shells in all the
recombinations is unique. Generalising to the large mass case, this is our
conclusion proved analytically in the previous section, the matter contents of
a spherical symmetric black hole of mass $M$(mildly larger than
$M_{\mathrm{pl}}$) with asymptotic horizon $r=2GM$ can be divided into
$k=\frac{GM^{2}}{2\sqrt{2}}$ distinguishable concentric shells to precision
$\varepsilon=1$. These fundamental shells can be recombined and excited at
$2^{k-1}X(k)$ number of ways. $X(k)$ is at most a polynomial function of $k$.

##  VI Black hole complementarity principle

By our proof and calculations above, one can easily understand that in the co-
moving observer’s description, the gravitational collapse passes through the
horizon and Schwarzschild singularity fluently in finite and seeable proper
time. The resulting black hole owns an quasi-periodically oscillatory matter
core and multiply possible mass profiles which is ergodically accessed by the
evolution of system. While in the outside fixed position observers’ time
definition domain, a collapsing star can infinitely approach but will never
contract into the horizon defined by its total mass; the system forms only an
asymptotically implementable horizon and asymptotically linear mass function,
the microscopic state of the system is characterised by its mass function’s
deviation from the exact linear profile $M[r]=\frac{r}{2G}$. The gap between
this two observers’ description is filled up as long as we note that, the
horizon and singularity encountered by the co-moving observer happen only in
the Future of infinitely Far away Future (FiFF) by the outside fixed position
observers’ physical time definition. They are realities only in the parallel
universe or ensemble spacetime of these outside fixed position observer or
detectors.

This fact naturally brings us to the idea of black hole complementarity
principle complementarity1990thooft ; complementarity1993verlinde ;
complementarity1993schoutens ; complementarity1993 ; complementarity1994 ,
which conjectures that both the co-moving observer and the outside fixed
position observer’s description of a collapsing star’s evolution are equally
true. What we want to complement here is that, when the ergodicity feature of
the co-moving observer and its observables’ interaction dynamics and the
uncertainty principle implied parallel universe or ensemble spacetime
interpretation for the outside fixed position observers’ measurements are
considered, the two types of observer’s definition for the black hole
microscopic state will not only be equally true, but also be equally complete.
In another word, the two observers’ description are equivalent. By the
extended Penrose-Carter diagram, we display this equivalence in FIG.8.

Figure 8: Equivalence between two types of observers microscopic state
definition for the black holes. Relative to FIG.3, the right hand part of this
figure adopts the extended rule of conventional Penrose Carter diagram so that
the east world and west world of the spacetime are simultaneously displayed in
the $\\{\tau,\rho\\}$ plane.

In the quantisation stratagem of references dfzeng2018a ; dfzeng2018b ;
dfzeng2020 ; dfzeng2021  we assumed $\gamma_{i}^{2}<0$ for each single shell
when do the microscopic state number counting for black holes caused by
gravitational collapse. Technically, this stratagem is nothing but the
requirement that classically all shells consisting of the collapsing star
oscillate inside the horizon defined by $r<r^{i}_{h}\equiv 2GM_{i}$ so that
for these shells both $\dot{t}$ and $\dot{x}$ take imaginary values but the
four velocity normalisation condition $h\dot{t}^{2}-h^{-1}\dot{r}^{2}=1$ holds
just as the shell is moving outside the horizon $r^{i}_{h}$. While in our
proofs of this work, section IV, this stratatem is given up and the same area
law is obtained successfully. Physically this means that we do not need any
shell $m_{i}$ to move inside its horizon $r^{i}_{h}=2GM_{i}$, we only need to
consider their motion of falling towards there. The fact that the two proofs
lead to the same area law entropy formula implies that, there is an
equivalence between the two pictures of (i) the motion of matters consisting
of a black hole is their oscillation over-crossing the central point and (ii)
the motion of matters consisting of a black hole is its composite shells’
falling towards their own’s horizon. In the latter picture, the quantum
fluctuation implied by the uncertainty principle has the power of causing
various shells to escape away from their asymptotical horizon thus making
their inside horizon space looks like something just be dug out from the whole
spacetime and anti-podal point identified tHooft2021 ; tHooft2019 ; tHooft1996
; tHooftISSP2016 ; tHooft1601 ; tHooft1605 ; tHooft1612 ; tHooft1804 ;
tHooft1809 ; tHooftFP2018 ; tHooftISSP2018 .

Although our proof or illustration of the equivalence relation between the two
observers’ definition of the black hole microscopic states is based on the
physical black holes with non-singular inner mass distribution. We emphasise
here that, even for the mathematical black holes defined as the exact
solutions to the vacuum Einstein equation thus characterised by the apriori
horizon and singularity. Similar complementarity principle must also be
possible be understood from the two types observers’ microscopic state
definition. Because as long as such microscopic state are physical, they must
be observable in principle. For such black holes, the idea of black hole
complementarity principle is challenged by the firewall paradoxes in recent
years fireworksAMPS2012 ; fireworksAMPS2013

By definition, the black hole in the fire wall paradox is nothing but a
horizon wrapped singularity and between the horizon and the singularity is a
vacuum region with wrongly signatured space-time metric. By the fire wall
paradox’s reasoning, the hawking particle arises from the particle pair
production and partial escaping from the vacuum fluctuation around the
horizon. The escapers have nothing to do the microscopic state of the black
hole. At late times any given escaper is entangled with both its partner
falling into the horizon and the early escaping successful colleagues as a
whole. This two side entanglement would break the monodromy of the quantum
entanglement thus calls for protections from mechanisms such as firewalls.
Obviously, for black holes caused by gravitational collapse, exact horizon and
singularity never form to the outside observer or detectors. To them the black
holes are inner-structured and characterised by special radial mass profiles.
Referring FIG.9, late time hawking particles can be recorded only when the in-
going member of the early produced pairs get annihilated with matter particles
consisting of the collapsing star because otherwise we will have vacuum decay
inevitably. Effectively, this makes each hawking particle looks like arising
from the microscopic state change of the black hole instead of completely
random fluctuation of the vacuums around it. References dfzeng2018a ;
dfzeng2018b ; dfzeng2021  provide an explicitly hermitian hamiltonian
description for this radiation mechanism which is named gravity induced
spontaneous radiation in dfzeng2022 .

Figure 9: The vacuum fluctuation and partial escaping mechanism for hawking
radiation is a policy of deficit financing. The cost of the outside Gieger
counter’s recording hawking particles is the vacuum definition standard’s
lowering, an analog of the currency releasing standard’s lowering in
conventional financial systems. An event of particle pair’s annihilation would
uplift this vacuum standard, analog of the currency releasing standard’s up-
tuning. The potential curve $V[\phi]$’s dash to solid style change denotes the
vacuum definition standard’s change. For a black hole caused by gravitational
collapse, the horizon is only an asymptotically realisable but not a truly
implemented physic surface so matter particles of the system can be found
outside of it. The time separation ${\Delta}t$ between the Gieger counter’s
recording particle and the ingoing particle’s annihilation with matter
particles consisting of the collapsing star is very short so that
${\Delta}t{\cdot}{\Delta}E\approx\frac{\hbar}{2}$ is holden very well.  Figure
10: For black holes defined as exact solutions to the vacuum Einstein
equation, the horizon is absolute and apriori. The in-going particles arising
from vacuum pair production needs infinite $t$-time to go across the horizon
and annihilate with matters inside to uplift the standard of vacuum
definition. During this period, if the outside fixed position Gieger counter
records hawking particles continuously, the vacuum definition standard would
be observed to be lowered remarkably. This would cause vacuum decay. Particles
would be observable everywhere, just like currency inflations in the
conventional financial system. So the deficit financing policy interpretion
for hawking radiation’s production in these black holes does not work at all.
That is, the outside fixed position observer could not see the black hole’s
evaporation as the result of vacuum fluctuation caused particle production and
partial escaping.

For the black holes defined as exact solutions to the vacuum Einstein equation
so carry absolute horizon and singularity apriori, hawking particles cannot be
produced continuously as the results of particle pair production and partial
escaping from the vacuum fluctuation around the horizon. This is because the
hawking radiation recorded by the outside fixed position observers comes from
the vacuum definition standard’s lowering. The particle pairs annihilation
take the responsibility of uplifting this back thus retrieving such standard.
To those observers, each member of the particle pairs falling towards the
horizon needs infinite time to get across the horizon and reach the central
point which accumulates all matter contents of the black hole to annihilate
with particles there so that the vacuum definition standard can be uplifted
back, see FIG.10 for intuitions. If an outside fixed position observer records
hawking particles continuously in his/her detector, then his/her vacuum
definition standard would be lowered continuously without compensation. As
results, he/she would see infinite number of particles everywhere as the
results of vacuum decaying. So, for black holes defined as exact solutions to
the vacuum Einstein equation, the situation imagined in the firewall paradoxes
where the black hole radiated half of its mass will not appear at all. In fact
the fraction of this black hole’s mass change due to hawking radiation is
strictly constrained by the uncertainty principle
${\Delta}M{\cdot}{\Delta}t\approx\frac{\hbar}{2}$.

So the complementarity principle is right enough to be taken as a basis for
understanding the black hole inner structures and their microscopic state
definition. What the firewall paradox challenges is not the complementarity
principle, but the underlying mechanism of hawking particle’s generation
mechanism. The right mechanisms such as gravity induced spontaneous radiation
must contain some ingredients which couples the hawking particle’s
observational feature such as interaction amplitude with the microscopic state
change of the radiating black holes, so that the information encoded in the
initial state of the black hole can be transferred into observational features
of the hawking particles. See references dfzeng2018a ; dfzeng2018b ;
dfzeng2021 ; dfzeng2022  for more concrete and detailed implementation of this
idea.

##  VII The physical reality’s observer dependence

Why is the singularity theorem chosen by so many people as the basis for
understanding the inner structure and microscopic state of black holes in
general relativity? Why is it not the complementarity principle
interpretation? Pushing aside the establishing time reason, the other one
important reason for this status may be people’s mistaking of the global
viewpoint as local viewpoints and negligence of the physical reality’s
observer dependence, under the name of the physic law’s general coordinate
invariance.

It is well known that in a black hole formed through gravitational collapse,
the outside fixed position observers will not see the formation of horizon and
singularity in their any finite future. These observers can be any outside
test particle or probes whose motion is affected by the gravitational field of
the to be black hole collapsing star, such as the other partner in a binary
merger event and the light rays skimming over the surface of a single black
hole to be imaged. During the whole lifetime of these test particle’s
existence, their motion is affected by the gravitational field provided by the
matter contents of the collapsing star being measured. So they see no black
holes defined by the horizon and singularity at all, they see just a to be
black hole collapsing star.

Due to the co-moving observers existence and the singularity theorem’s being
proven, many people believe that the fate of all heavy enough collapsing stars
in the nature is the horizon wrapped singularity definitely. The logic behind
this belief is, the physics laws are invariant under the general coordinate
transformation. In that the singularity and horizon form in the finite future
of the collapsing matter themselves’ time definition, their effects on the
motion of the outside fixed position observers must also happen in the form of
horizon and singularity naturally. The outside fixed position observers cannot
see such objects appearance only because they adopt a bad coordinate system or
time definition.

However, this popular belief makes two mistakes rather hiddely, (i) taking the
global viewpoint of the horizon’s existence as the viewpoint of local
observers and (ii) taking the physics law’s general coordinate invariance as
the physical reality’s observer independence. The physics law here refers to
relations like $G_{\mu\nu}=8\pi G_{N}T_{\mu\nu}$ which bridge observables
following from at least two types of physical measurement. The physical
reality refers to any physic quantity or physical features which is measured
through only one type of physical measurement. These are two totally different
concepts, the invariance of the physics laws under the general coordinate
transformation does not imply the physical reality’s observer independence in
any sense. In fact, the physical reality’s observer independence is not a fact
at all. Let us elaborate this point through three examples.

$\begin{array}[]{l}ds^{2}{=}{-}hdt^{2}{+}h^{-1}dr^{2}{+}\cdots$,$h=1{-}\frac{2GM}{r}\\\
k^{\mu}=(\frac{\omega_{0}h_{0}^{\\!1\\!/\\!2}}{h},\omega_{0}h_{0}^{\\!1\\!/\\!2},0,0)\end{array}$
---
$u^{\mu}_{\mathrm{fx}}{=}(h^{{-}{1\\!/\\!2}}_{\mathrm{fx}},0,0,0)$ | $u^{\mu}_{\mathrm{fr}}{=}(\frac{\sqrt{h_{0}}}{h},\sqrt{h_{0}{-}h},0,0)$
$\omega_{\mathrm{fx}}{=}{-}u_{\mathrm{fx}}{\cdot}k=\frac{\omega_{0}h_{0}^{\\!1\\!/\\!2}}{h_{\mathrm{fx}}^{\\!1\\!/\\!2}}$ | $\omega_{\mathrm{fr}}{=}{-}u_{\mathrm{fr}}{\cdot}k{=}\frac{\omega_{0}[h_{0}{-}(h_{0}^{2}{-}h_{0}h)^{\\!1\\!/\\!2}]}{h}$
$\begin{array}[]{l}ds^{2}{=}{-}hdv^{2}{+}2dvdr{+}\cdots$,$h=1{-}\frac{2GM}{r}\\\
\bar{k}^{\mu}=(\frac{2\omega_{0}h_{0}^{\\!1\\!/\\!2}}{h},\omega_{0}h_{0}^{\\!1\\!/\\!2},0,0)\end{array}$
$\bar{u}^{\mu}_{\mathrm{fx}}{=}(h^{{-}{1\\!/\\!2}}_{\mathrm{fx}},0,0,0)$ | $\bar{u}^{\mu}_{\mathrm{fr}}{=}(\frac{\sqrt{h_{0}}{+}\sqrt{h_{0}{-}h}}{h},\sqrt{h_{0}{-}h},0,0)$
$\bar{\omega}_{\mathrm{fx}}{=}{-}\bar{u}_{\mathrm{fx}}{\cdot}\bar{k}=\frac{\omega_{0}h_{0}^{\\!1\\!/\\!2}}{h_{\mathrm{fx}}^{\\!1\\!/\\!2}}$ | $\begin{array}[]{l}\bar{\omega}_{\mathrm{fr}}{=}{-}\bar{u}_{\mathrm{fr}}{\cdot}\bar{k}{=}\frac{\omega_{0}[h_{0}{-}(h_{0}^{2}{-}h_{0}h)^{\\!1\\!/\\!2}]}{h}\end{array}$
Table 1: Two types of observer’s measurement of a light signal emitted from
the surface of a freely collapsing star whose outside geometry is exactly
Schwarzschild type. The upper is in the standard Boyer-Linderquist coordinate,
the lower is in the Eddington-Finckelstein coordinate. As a physical reality,
the light signal’s frequency is invariant under general coordinate
transformation but observer dependent. $h_{0}$ is the value of $h$ at the
light signal’s starting point. $h_{\mathrm{fx}}$ is the value of $h$ at the
point the fixed position observer sits on. $k^{\mu}$ is the wave number of the
light signal, whose components are determined by
$\dot{k}^{0}+\Gamma^{0}_{\mu\nu}k^{\mu}k^{\nu}=0$ and $k{\cdot}k=0$.
$u^{\mu}_{\mathrm{fx}}$ is the four velocity of the fixed position observer
far away, $u^{\mu}_{\mathrm{fr}}$ is the four velocity of the freely falling
observer outside the star. The components of $u^{\mu}$ are determined by
$\dot{u}^{0}+\Gamma^{0}_{\mu\nu}u^{\mu}u^{\nu}=0$ and $u{\cdot}u=-1$

The first is a concrete measurement example. Consider two observer’s measuring
of a same light signal’s frequency emitted from an atom fixed on a freely
collapsing star whose outside geometry is written in two different coordinate
system. The first observer is a fixed position one, the second is a freely
falling one. TABLE 1 lists these two observer’s definition and measurement
mathematics. From the Table, we easily see that, the frequencies measured by
the two observers $\omega_{\mathrm{fx}}$ and $\omega_{\mathrm{fr}}$ are
obviously unequal, although they are invariant before and after the coordinate
is transformed from the Boyer-Lindrquist to the Edington-Finkelstein type,
$\omega_{\mathrm{fx}}=\bar{\omega}_{\mathrm{fx}}$,
$\omega_{\mathrm{fr}}=\bar{\omega}_{\mathrm{fr}}$. In this example, the
frequency of the light signal, and even the redshift’s dependence on the
radial coordinate $r$ is what we called physical realities. These realities
are obviously observer dependent, because their value follows from the inner
product of the observer’s four velocity and the tensor representing the
observable itself.

$\begin{array}[]{l}ds^{2}{=}{-}hdt^{2}{+}h^{-1}dr^{2}{+}\cdots,h{=}1{-}\frac{2GM}{r}$,$2GM{\equiv}r_{s}\\\
T^{\mu\nu}{=}\left[\begin{array}[]{cc}\rho\dot{t}\dot{t}&\rho\dot{t}\dot{x}\\\
\rho\dot{t}\dot{x}&\rho\dot{x}^{2}\end{array}\right],T^{\theta\theta}{=}0,T^{\phi\phi}{=}0\\\
h\dot{t}{\equiv}\gamma,\dot{x}^{2}{=}\gamma^{2}{-}h,\rho{=}\frac{m\delta(r{-}x)}{4\pi
x^{2}},m{\ll}M\end{array}$
---
$u^{\mu}_{\mathrm{fx}}=(h_{\mathrm{fx}}^{-{1/2}},0,0,0)$ | $u^{\mu}_{\mathrm{co}}{=}(\dot{t},\dot{x},0,0)$
$\begin{array}[]{l}p_{\mathrm{fx}}=T^{\mu\nu}\frac{u^{\mathrm{fx}}_{\mu}u^{\mathrm{fx}}_{\nu}{+}g^{\mathrm{fx}}_{\mu\nu}}{3}\\\ {=}\frac{\rho}{3}(h_{\mathrm{fx}}^{-1}\dot{t}^{2}{-}h_{\mathrm{fx}}\dot{t}^{2}{+}h^{-1}_{\mathrm{fx}}\dot{x}^{2})\\\ =\frac{\rho}{3}\frac{(h_{\mathrm{fx}}^{-1}{-}h_{\mathrm{fx}})\gamma^{2}}{h(x)^{2}}{+}\frac{\gamma^{2}{-}h}{h_{\mathrm{fx}}}\\\ \xrightarrow{x\rightarrow r_{s}}\infty\end{array}$ | $\begin{array}[]{l}p_{\mathrm{co}}{=}T^{\mu\nu}\frac{u^{\mathrm{co}}_{\mu}u^{\mathrm{co}}_{\nu}{+}g^{\mathrm{co}}_{\mu\nu}}{3}\\\ {=}{\rho}u^{\mu}_{\mathrm{co}}u^{\nu}_{\mathrm{co}}\frac{u_{\mu}^{\mathrm{co}}u_{\nu}^{\mathrm{co}}+g^{\mathrm{co}}_{\mu\nu}}{3}\\\ {\equiv}0\\\ {}\end{array}$
$\begin{array}[]{l}d\bar{s}^{2}{=}{-}d\tau^{2}{+}\frac{r_{s}^{2/3}d\varrho^{2}}{[\frac{3}{2}(\varrho-\tau)]^{2/3}}{+}\cdots,r{=}\big{[}\frac{3}{2}(\varrho{-}\tau)\big{]}^{\frac{2}{3}}(2GM)^{\frac{1}{3}}\\\
\begin{array}[]{l}\bar{T}^{\mu\nu}{=}\Bigg{[}\\!\begin{array}[]{cc}\rho(\dot{t}{+}\dot{x})^{2}\\!&\\!\frac{\rho(\dot{t}{+}\dot{x})}{h}\frac{r_{s}\dot{t}{+}r\dot{x}}{\sqrt{r_{s}r}}\\\
\mathrm{symm}\\!\\!&\\!\\!\\!\frac{\rho}{h^{2}}\frac{r}{r_{s}}(\frac{r_{s}}{r}\dot{t}{+}\dot{r})^{2}\end{array}\\!\Bigg{]},\bar{T}^{\theta\theta}{=}\bar{T}^{\phi\phi}{=}0\end{array}\end{array}$
$\begin{array}[]{l}\bar{u}^{\mu}_{\mathrm{fx}}=(h^{-1/2}_{\mathrm{fx}},\frac{(r_{s}/r_{\mathrm{fx}})^{1/2}}{h^{3/2}_{\mathrm{fx}}},0,0)\\\ \bar{p}_{\mathrm{fx}}{=}\bar{T}^{\mu\nu}\frac{\bar{u}^{\mathrm{fx}}_{\mu}\bar{u}^{\mathrm{fx}}_{\nu}{+}\bar{g}^{\mathrm{fx}}_{\mu\nu}}{3}\\\ =\frac{\rho}{3}\frac{(h_{\mathrm{fx}}^{-1}{-}h_{\mathrm{fx}})\gamma^{2}}{h(x)^{2}}{+}\frac{\gamma^{2}{-}h}{h_{\mathrm{fx}}}\\\ \xrightarrow{x\rightarrow r_{s}}\infty\end{array}$ | $\begin{array}[]{l}{\bar{u}}^{\mu}_{\mathrm{co}}{=}(\dot{t}{+}\dot{x},\frac{r_{s}\dot{t}{+}r\dot{x}}{h\sqrt{r_{s}r}},0,0)\\\ \bar{p}_{\mathrm{co}}{=}\bar{T}^{\mu\nu}\frac{\bar{u}^{\mathrm{co}}_{\mu}\bar{u}^{\mathrm{co}}_{\nu}{+}\bar{u}^{\mathrm{co}}_{\mu\nu}}{3}\\\ {\equiv}0\\\ {}\end{array}$
Table 2: Two types of observers’ measurement of the pressure of a dust type
mass $m$ test spherical shell freely falling towards the horizon of a pre-
existing Schwarszchild black hole of mass $M$. The upper is in the Boyer-
Lindquist coordinate, the downer is in the Lemaitre coordinate. As a physical
reality, the test mass shell’s pressure is invariant under the general
coordinate transformation, but observer dependent. $T^{\mu\nu}$ is the energy
momentum tensor of the test mass shell; $u^{\mu}_{\mathrm{fx}}$ and
$u^{\mu}_{\mathrm{fr}}$ are the four velocity of the fixed position observer
and the freely falling observer respectively. $\bar{T}^{\mu\nu}$,
$\bar{u}^{\mu}_{\mathrm{fx}}$ and $\bar{u}^{\mu}_{\mathrm{fr}}$ are their
counter parts in the Lemaitre coordinate system.

The second is still a concrete measurement example. But this time let us
consider two observers measurement of a same test shell’s pressure when it is
freely falling towards the horizon of a pre-existing Schwarzschild black hole
characterised by the exact horizon and central singularity. The first observer
is a fixed position observer which is sitting far away from the horizon of the
black hole. The second is a freely falling observer co-moving with the test
shell. We display these two observers’ four velocity and measurements of the
test mass shell’s pressure in TABLE 2. In this example, the pressures of the
test mass shell is our physical reality. From the table, we can easily see
that this physical reality is also observer dependent
$p_{\mathrm{fx}}\neq{}p_{\mathrm{co}}$, although coordinate invariant,
$p_{\mathrm{fx}}=\bar{p}_{\mathrm{fx}}$ and
$p_{\mathrm{co}}=\bar{p}_{\mathrm{co}}$. $p_{\mathrm{fx}}$ has the property of
approaching to infinite as the shell falls infinitely close to the horizon. So
to the far away fixed position observers, the asymptotic mathematical
horizon’s existence makes the black hole look like an incompressible but
finitely sized objects. $p_{\mathrm{co}}$ has the property of being zero
constantly, which means that to the co-moving observer, the horizon exhibit
not any observational effect at al.

Figure 11: The Face to Face Communication Difficulty (F2FCD). In an spatial
flat and homogeneous Friedman-Robertson universe, Alice and Bob are a couple
who date frequently. Behind each of them is their ex-boy/girl friend Robert
and Walker respectively, two spherical giants of radius $H_{0}^{-1}$ and
average mass density $\frac{3H_{0}^{2}}{8\pi G_{N}}$, $H_{0}$ denotes just the
Hubble constant of the current universe. Although divorced for almost
$1.37\times 10^{10}$ years already, the gravitation between them and their own
ex-friend still makes Alice and Bob recede from each other everyday. From
Alice and Bob, this is nothing but the usual freely falling. Charle, their
common friend or a global observer, also thinks this is nothing but the
conventional big bang. The question is, why is it possible Alice and Bob get
known to each other and begin to date? Because each of them lives inside the
horizon caused by her or his ex-friend but outside that of the other’s.

The third is a conceptual example. Let us consider the thought experiment
cartooned in FIG.11 which contains a paradox we called F2FCD, see the figure
and captions carefully. This paradox magnifies the error of the common belief
that the horizon and singularity are observer independent physical realities
to all black holes. Charle’s confusion arises from two factors. (I) believing
that in Alice’s freely falling reference frame, Robert has a horizon and
similarly in Bob and Walker’s case. This is an error of taking the global
viewpoint of Charle himself as the viewpoint of local observers such as Alice
and Bob. (II) believing that the horizon-carrying feature of Robert to Alice
is also the case to Bob and vice versa. This is an error of taking the physic
laws’ general coordinate invariance as the physical reality’s observer
independence. In fact, as an outside observer, in any finite time of his own,
Bob can not see or detect Robert’s horizon at all. So he can get known with
Alice without any difficulty. This F2FCD paradox is firstly introduced in
reference dfzeng2018b  to necessitate the horizon’s fuzziness to resolve the
information missing puzzle. But its value may be more right here to help
understanding the physical reality’s observer dependence.

As long as we know that the horizon-carrying is not a physical reality of
black holes independent of observers, we can easily understand that the
microscopic state picture of black holes is observer dependent. By our
exposition in section II,III and IV, to the outside fixed position observer,
the matter contents of physical black holes live inside a spatial region which
is infinitely close to that wrapped by the mathematic surface
$r_{h}=2GM_{\mathrm{tot}}$. But they live there just like some in-compressible
and static fluid. Because inside that region, matters inside each sphere of
smaller radius has the same feature. This is nothing but the concept of the
old frozen star frozenStar1971 ; Buchdahl1959 ; chand1964a ; chand1964b ;
bondi1964 , see references BrusteinFrozen2301 ; Brustein2304 ; Brustein2310
for recent investigations. Except that the in-compressibility now is
understood as a result of the infinite gravitational time delaying instead of
any physical exclusive forces. To the co-moving observer, these matters just
freely falling towards the central point, over cross each other there and then
oscillate periodically. But these inside horizon motion and oscillation happen
in the future of infinitely far away future by the outside fixed position
observers’ physic time definition. They are physical realities only in the
sense of parallel universe or statistic ensembles. Just as we showed above and
in our earlier works dfzeng2017 ; dfzeng2018a ; dfzeng2018b ; dfzeng2020 ;
dfzeng2021 ; dfzeng2022 , quantisation of these matters’ fluctuation on the
frozen star background or oscillations across the central point both lead to
the right area law featured Bekenstein Hawking entropy, thus form equivalent
and complementarity definitions for the microscopic state of black holes.

##  VIII Banana shape deformation

It is very important to emphasise that the observers referred to in this work
are not limited to human beings but are proxies of any probe or detectors,
especially those living outside the matter occupation region of the being
probed objects. They can be the imaging photons skimming over a single black
hole or the other partner involved in a BHB merger event. These test objects’
motion is determined by the gravitational field produced by the matters
consisting of the being tested black holes which never fully collapse and
cause singularity in any finite future in the domain of the physic time
definition of the outside fixed position observers. These being tested objects
carry approximate horizon which is formed so long that during the short
duration of the imaging process or binary merger process the imaging photon’s
perturbation or the inspiral partner’s tidal force has no chances to cause
disintegration. Because longer formation time means smaller matter occupation
region radius thus more remarkable time delaying from the matter occupation
region’s boundary to the test particle’s position, so that more time consuming
for their matter contents to be affected by the imaging photon or merger
partner’s gravitation, and vice versa. In the BHB meger events, this provides
us chances to treat dynamics of the binary system with the adiabatic
approximation. By this approximation, we look the motion of the binary system
as a combination of each participant’s mass central’s motion and their
different parts’ relative motion. The latter arises from the inhomogeneous
radiation back reaction forces and will cause banana shape deformation of
FIG.12.

Figure 12: When two black holes caused by gravitational collapse merge, both
participants are outside observers of the other. So none of them can see the
other’s horizon as physical realities, instead each of them will receive light
signal and thus gravitations from the matter contents of the other. The
inhomogeneous back reaction of gravitational wave radiation would cause non-
uniform orbit decay thus leading to banana shape-deformation in their matter
contents’ spatial distribution. The resulting black holes’ quadrupole can be
approximated by that of the two arced strings.  Figure 13: When two black
holes defined as exact solutions to the vacuum Einstein equation merge, both
participants have absolute and apriori horizon. None of them can see the
other’s inside horizon structure. The back reaction force of the gravitational
wave radiation can only affect the orbit of each other as a whole. None of
them will experience inner matters distribution’s spatial change. Only their
horizons will experience droplet shape deformation and merge into a single one
finally. Considering the inhomogeneity of the time delaying effect, the true
horizon shape deformation tidalDeformationNumeric  would be two comma like way
of ‘’.

The key reason for the banana shape deformation is the inhomogeneity of the
gravitational wave radiation back reaction. Referring to FIG.12, let us
consider the newtonian motion of a representative point such as $\vec{x}_{i}$
on the participant B and its counter partner on participant A as an
illustration for this phenomena. By the standard quadrupole formula for the
back reaction force arising from the system’s inspiral motion and
gravitational wave radiation,

$\displaystyle
F_{i}^{\mathrm{reac}}=\frac{G}{c^{5}}m_{i}\big{\\{}-\frac{2}{5}x^{i}\frac{d^{5}Q_{i}}{dt^{5}}+\mathcal{O}[\frac{1}{c^{2}}]\big{\\}}$
(VIII.1) $\displaystyle\frac{d^{5}Q_{i}}{dt^{5}}=\omega^{5}m_{i}x_{i}^{2}$
(VIII.2)

we can write down the dynamic equation controlling its inspiral orbit’s decay
and get solutions

$\displaystyle\frac{d}{dt}\big{(}-\frac{Gm_{i}^{2}}{2x_{i}}\big{)}=F_{i}^{\mathrm{reac}}\omega
x_{i}\Rightarrow$ (VIII.3) $\displaystyle
x_{i}=x_{i}^{0}(1-t/t_{0})^{\frac{1}{4}},t_{0}=\frac{(x^{0}_{i})^{4}c^{5}}{20G^{3}m_{i}^{3}}$
(VIII.4)

The syncrhonicity of all different composite points’ inspiral motion
$\frac{Gm_{i}^{2}}{4x_{i}^{2}}=m_{i}\omega^{2}x_{i}$ implies that
$\frac{Gm_{i}}{4x_{i}^{3}}=\omega^{2}$ is an approximate constant for
different point $i$. So as the inspiral motion progresses on those points with
larger initial $x_{i}$ will reduce their orbit radius more remarkably

${\Delta}x=x_{i}^{0}-x_{i}^{0}(1-t/t_{0})^{1/4},t_{0}=\frac{c^{5}}{1280\omega^{6}(x_{i}^{0})^{5}}$
(VIII.5)

In the case of all points inspiral at the same angular frequency, this will
cause the compression of the system along the $\vec{x}_{i}$ direction, the net
effect is a banana shape deformation. The true case in general relativity
would be much more complicated than the newtonian illustrations here and can
only be simulated numerically. But the qualitative physic picture will not
change.

Obviously, when the two participating black holes experience banana shape
deformation as in FIG.12, the radiation activity of the binary system will
decrease inevitably, because in this case the radiation relevant quadrupole of
the system decreases manifestly. The extremal case is when the two ends of the
banana shape deformed black holes contact each other. In such cases the
gravitational wave radiation will stop completely as the result of rotating
symmetry’s retrieving. For physic black holes originating from gravitational
collapse, non-singular inner mass profile will make this banana shape
deformation inevitable and will become very important as the two participants
inspiral to the merger and ring down stage. From equation (VIII.5), we can see
that this effect is of $\mathcal{O}[G^{0}]$, so it is non-perturbative and not
accounted for in the higher multipole moments appearing as the response of the
participating black holes to the tidal forces caused by their partners and
written into the tidal love numbers perturbationally. The tidal love numbers
are well known to be zero for non-spinning black holes defined as exact
solutions to the vacuum Einstein equation tidalLoveNumber2009Poisson ;
tidalLoveNumber2009Dmaour ; tidalLoveNumber2015 ; tidalLoveNumber2021 . See
FIG.13 for intuitive pictures.

Since both of the merger participants are outside observers of their inspiral
partner, neither of them would see the other’s horizon. Both of them would see
the other as in-compressible frozen star. So their compression along
$\vec{x}_{i}$ direction implies stretching along the $\vec{v}_{\mathrm{insp}}$
direction inevitably. In a sense, the space inside a spherical symmetric
Schwarzshild horizon is the minimal volume we can compress matters into. This
can be seen from the following fact that a spherically symmetrical collapsings
star with linear radial mass profile $M(r)\approx\frac{r}{2G}$ implement the
lowest possible self gravitating potential energy without causing the physical
horizon’s formation

$\displaystyle\mathcal{E}_{sph}=-\\!\int_{0}^{a}\frac{GM(r)\rho_{\mathrm{sph}}(r)4\pi
r^{2}dr}{r}[M(r){=}\frac{r}{2G}]$ (VIII.6)
$\displaystyle=-\frac{GM^{2}_{\mathrm{tot}}}{a},a=2GM_{\mathrm{tot}},$
(VIII.7)

Note that $M(r)\approx\frac{r}{2G}$ implies
$\rho_{\mathrm{sph}}(r)\approx\frac{1}{8\pi Gr^{2}}$ necessarily. Compressing
this star and tuning its mass value so that the resulting ellipsoid has equal
Newton potential strattonEMTbook  as the spherical symmetric star
$-\frac{GM}{a}=-\frac{1}{2}$, the self gravitation energy of the resultant
object will become

$\displaystyle\mathcal{E}_{ell}=-\\!\int_{0}^{1}\\!\frac{GM(\lambda)F(k,\varphi)\rho_{\mathrm{ell}}(\lambda)d\mathbf{v}}{\lambda\sqrt{a^{2}-c^{2}}}=-\frac{GM^{2}F(k,\varphi)}{\sqrt{a^{2}-c^{2}}}$
(VIII.8) $\displaystyle
k\equiv\frac{\sqrt{a^{2}{-}b^{2}}}{\sqrt{a^{2}{-}c^{2}}},\varphi\equiv\arcsin\sqrt{1{-}\frac{c^{2}}{a^{2}}}$
(VIII.9)

where $F(k,\varphi)$ is the elliptic integral of the first kind and
$d\mathbf{v}=abc\lambda^{2}d\lambda\sin\theta d\theta d\phi$. In this
calculation, the star is considered as many elliptic shells of equal axis
length ratio {$a\lambda,b\lambda,c\lambda$} and radial mass function
$M(\lambda)=M\cdot\lambda$ or $\rho_{\mathrm{ell}}(\lambda)=\frac{M}{4\pi
abc\lambda^{2}}$. By our convention, the star is compressed along $y$ and $z$
direction but kept invariant along the $x$-direction. Letting the surface
potential of the resulting ellipsoid take the same value as that of the
Schwarzschild black hole
$\frac{GMF(k,\varphi)}{\sqrt{a^{2}-c^{2}}}=\frac{GM}{a}=\frac{1}{2}$ with the
same $x$-directional size, we will find that
$M_{\mathrm{ell}}<M_{\mathrm{sph}}$ and
$\mathcal{E}_{\mathrm{ell}}/\mathcal{E}_{\mathrm{sph}}<1$. See FIG.14 for
concrete numerics. Conversely, it’s very natural to understand that if we keep
the total mass invariant when compressing the star along one direction, then
the star must be stretched along the other so that the horizon formation
condition $\frac{GMF(k,\varphi)}{\sqrt{a^{2}-c^{2}}}=\frac{1}{2}$ will not be
realised as any outside test body cannot detect it in the finite future of
their domain of time definition.

Figure 14: The ratio of self gravitational potential energy of ellipsoid and
banana shape deformed star over that of the Schwarzshild black hole with equal
surface potential $-GM/a=-\frac{1}{2}$ and $x$-directional size. The
$y$,$z$-directional size are set to $b=c=a\eta$. The right is for banana shape
deformation; $\psi$ is the bending angle opened by the arc of the banana
relative to the radius of inspiral motion.

Considering the banana shape deformation, we will get similar conclusion. In
the case the ellipsoid star are parameterised as,

$\displaystyle
x{=}a\lambda\sin\theta\cos\phi,y{=}b\lambda\sin\theta\sin\phi,z{=}c\lambda\cos\theta$
(VIII.10) $\displaystyle\lambda\in(0,1),\theta\in(0,\pi),\phi\in(0,2\pi)$
(VIII.11)

the banana shape deformation can be written as

$\displaystyle
x=(\frac{\ell}{2}{+}a\lambda\sin\theta\cos\phi)\cdot\cos\big{(}\frac{b\lambda\sin\theta\sin\phi}{\frac{\ell}{2}+a\lambda\sin\theta\cos\phi}\big{)}$
(VIII.12) $\displaystyle
y=(\frac{\ell}{2}{+}a\lambda\sin\theta\cos\phi)\cdot\sin\big{(}\frac{b\lambda\sin\theta\sin\phi}{\frac{\ell}{2}+a\lambda\sin\theta\cos\phi}\big{)}$
(VIII.13) $\displaystyle z=c\lambda\cos\theta$ (VIII.14)

Similar with the ellipsoid shape deformation case, still looking the banana as
many concentric banana-peel whose own volume has the form $4\pi
a_{\mathrm{bnn}}b_{\mathrm{bnn}}c_{\mathrm{bnn}}\lambda^{2}d\lambda$ and wrap
mass $M\cdot\lambda$ inside, then the self-gravitational potential energy of
the system can be calculated approximately as

$\displaystyle\mathcal{E}_{\mathrm{bnn}}=-\int_{0}^{1}\frac{Gm(\lambda)\rho_{\mathrm{bnn}}(\lambda)d\mathbf{v}_{\mathrm{bnn}}}{\lambda\sqrt{a^{2}-c^{2}}}F_{\mathrm{bnn}}(k,\varphi)$
(VIII.15)
$\displaystyle\approx-\frac{GM^{2}}{\sqrt{a^{2}_{\mathrm{bnn}}-c^{2}_{\mathrm{bnn}}}}F(k_{\mathrm{bnn}},\varphi_{\mathrm{bnn}})$
(VIII.16) $\displaystyle
k_{\mathrm{bnn}}\equiv\frac{\sqrt{a^{2}_{\mathrm{bnn}}{-}b^{2}_{\mathrm{bnn}}}}{\sqrt{a^{2}_{\mathrm{bnn}}{-}c^{2}_{\mathrm{bnn}}}},\varphi\equiv\arcsin\sqrt{1{-}\frac{c^{2}_{\mathrm{bnn}}}{a^{2}_{\mathrm{bnn}}}}$
(VIII.17) $\displaystyle
a_{\mathrm{bnn}}=\frac{a\sin{\psi/2}}{\psi/2},b_{\mathrm{bnn}}=b,c_{\mathrm{bnn}}=c$
(VIII.18)

where $\psi$ is the bending angle of the banana, referring to FIG 14 for
physical pictures; subscripts $\scriptscriptstyle\mathrm{bnn}$ means banana
shape deformation. In the case of $\psi\ll\pi$, equation (VIII.16) provides
rather good approximation to the underlying exact result. For large $\psi$,
equation (VIII.16) is not a good approximate, but still catches the
qualitatively feature of banana shape deformation costing extra energy
relative to the ellipsoid deformation.

Figure 15: The radiation active quadrupole of a banana shape deformed BHB
system is suppressed by a factor of $\zeta=\frac{\sin\alpha}{\alpha}$.
$4GM_{i}$ and $4GM_{i}z_{i}$ are the deformation before and after
diameter/length of the black holes respectively.

Referring to FIG.15, when we consider a banana shape deformed BHB system’s
rotation and gravitational wave radiation, the time dependent quadrupole will
become

$Q_{xx}=\sum_{i=}^{1,2}\\!\\!\int_{-\frac{\alpha}{2}+0/\pi+\omega
t}^{\frac{\alpha}{2}+0/\pi+\omega t}\\!\\!\frac{M_{i}}{\alpha
a_{i}}(a_{i}\cos\theta)^{2}a_{i}d\theta$ (VIII.19)
$=\big{(}M_{1}a_{1}^{2}{+}M_{2}a_{2}^{2}\big{)}\frac{\sin\\!\alpha\cos\\!2{\omega}t{+}\alpha}{2\alpha}\sim\mu
a^{2}\frac{\sin\\!\alpha\cos 2{\omega}t}{2\alpha}$
$Q_{yy}{=}\sum_{i=}^{1,2}\\!\\!\int\\!\\!\frac{M_{i}}{\alpha
a_{i}}(a_{i}\sin\theta)^{2}a_{i}d\theta{\sim}{-}\mu
a^{2}\frac{\sin\\!\alpha\cos 2{\omega}t}{2\alpha}$

where $\mu\equiv\frac{M_{1}M_{2}}{M}$ and $a_{1,2}{=}\frac{M_{2,1}a}{M}$,
$M{=}M_{1}{+}M_{2}$. To assure the shape deformed black holes open equal
angles $\alpha_{1}\equiv\frac{4GM_{1}z_{1}}{a_{1}}=$
$\frac{4GM_{2}z_{2}}{a_{2}}\equiv\alpha_{2}=\alpha$ with respect to their
inspiral central, the elongation factor $z_{i}$ is required to satisfy
$\frac{z_{1}M_{1}}{M_{2}}=\frac{z_{2}M_{2}}{M_{1}}\equiv z$. As results, the
radiation active quadrupole of the system can be written as

$Q^{\mathrm{radiation}}_{\mathrm{active}}=\zeta\cdot\mu
a^{2},\zeta=\frac{\sin\\!\alpha}{\alpha}=\frac{\sin\\!4GMz/a}{4GMz/a}$
(VIII.20)

Just as we pointed out previously, during the inspiral and merger process the
banana shape deformed black hole would be elongated under the inhomogeneous
radiation back reaction force, so $z$ is a growing function of the time
coordinate $t$. In this work, we will not try to determine the function form
of $z(t)$ dynamically but choose to parameterise it as various power functions
of $a(t)$

$z[a(t)]=0,1,a(t)^{-1},a(t)^{-2},\cdots,$ (VIII.21)

As time passes by, $a(t)$ would decrease due to the gravitational wave
radiation caused orbit decay. So $a(t)^{-n}(0{<}n)$s are increasing functions
of time. $z=0$ corresponds to the case the black hole has point like singular
structure. $z=1$ corresponds to the case the black hole has extended inner
structure but such structures experience only banana shape bending but no
elongations as the system inspiral and emit gravitational wave radiations.
Black holes with $z{\propto}a(t)^{-1}$ experience more banana shape elongation
during the early inspiral stage than those with $z{\propto}a(t)^{-2}$ so are
more soft than the latter. It is necessary to notice that although the black
holes with non-singular inner structure experience banana shape deformation,
the protection provided by their asymptotic horizon would prevent them from
tidal disintegration during most time of their binary merger process, from the
early inspiral to the very late ringdown stage. However, to show this inner
structure truly exists and happens there, we need to develop new theoretical
tools to calculate their observational signal analytically.

##  IX An exact one body method

The full process of a BHB system’s evolution can be divided into three stages,
i.e. inspiral, merger and ringdown. The post netwonian approximation and EOB
method can be used to describe evolutions of the system during the first and
early part of the second stage. Numeric relativity can be used to model the
system’s evolution during the whole process but is usually applied only for
the second stage due to computational resource limits. The practical signal
wave form for observations are compositions of the three method’s output, with
the joining condition carefully tuned by hand.

Even one day, our computational resources get rich enough to calculate the
wave form of the merger process with numeric method exclusively, the popular
belief that Schwarzschild black hole has singular inner mass distribution is
still not a conclusion extractable from such calculations. The reason is, Kerr
black holes have extended inner structure characterised by their singular
ring. If such a structure follows from the merger events of two Schwarzschild
black holes, then the inner mass distribution of the Schwarzschild black hole
must not be point-like singular. Instead it must be extended and will
experience stretching and bending during the binary merger process. Signals of
such structures would be hidden inside the boundary data NR2002 ; NR2004 ;
NR2007 ; NR2018  of numeric relativity. To understand the existence of such an
inner structure picture observationally, what we should do is not to escalate
the precision of the current numeric relativity’s simulation by finding more
precise initial data basing on higher order post newtonian expansions or
something else. What we should do is to develop some more self consistent and
full stage applicable model description or analytical wave form generating
techniques which directly translate the inner structure picture of black holes
to the gravitational wave forms measurable from the real BHBs’ merger events.

The basic idea of conventional EOB method eob1998 ; eob2000 ; eob2009  is to
map the general relativity two bodies’ motion onto those of a total mass
particle’s shifting and a reduced mass particle’s inspiral. Due to the non-
linearity of general relativity, the gravitational field controlling the
reduced mass particle’s inspiral is not a Schwarzschild metric sourced by the
total mass particle, but an effective and corrected but still static and
Schwarzschild-like one. This effective metric must be constructed by requiring
that the reduced mass particle’s motion in it given by the world line action
coincide with the two particle’s relative motion in the original general
relativity two-body problem under the post newtonian expansion. For the
inspiral and early merger stage, this method can yield gravitational wave
forms coincide with those following from numeric relativities very precisely.