# Gravity Induced Spontaneous Radiation

Ding-fang Zeng<EMAIL_ADDRESS>Beijing University of Technology, Chaoyang,
Bejing 100124, P.R. China,
Niels Bohr Institute, Blegdamsvej 17, Copenhagen, 2300, Denmark

###### Abstract

We suggest that behind the black hole information paradox is a new and
universal radiation mechanism, Gravity Induced Spontaneous Radiation, or GISR
hereafter. This mechanism happens to all kinds of compositional objects and it
requires only their microscopic structure as the basis. It’s always
accompanied with such inner structures’ variation and allows for explicitly
hermitian hamiltonian description. For black holes, by Wigner-Wiesskopf
approximation we show that such a radiation has a thermal spectrum exactly the
same as hawking radiation; while through numeric integration, we show that the
variation of the radiation particles’ entropy exhibits all features of Page
curve as expected. We also provide exact and analytic solutions to the
Einstein equation describing microscopic structures required by the GISR of
black holes and show that, after quantisation the degeneracy of wave functions
corresponding with those solutions are consistent with the area law formula of
Bekenstein-Hawking entropy.

Among his many works, the discovery of black hole radiation and the related
information missing puzzle may be S. Hawking’s most profound contribution to
modern theoretical physics hawking1975cmp . Historically, calculations of the
gray-body factor of black-brane radiation provide the most hard-core evidence
for Anti-de Sitter/Conformal Field Theory correspondence AdScftReview99 .
Recently, explorations of the information missing puzzle bring us new ideas
such as ER=EPREREPR  and Island formulae et al ReplicaWormholeReview2006 .
This work will, additionally, propose that underlying this radiation and the
information missing puzzle is a universal mechanism rarely investigated in the
literature and encodes rich information on black hole microscopic structure,
singularity theorem and quantum gravity. To understand this statement better,
we need going back to S. Mathur’s small correction theorem fuzzball2009mathur
for a while.

Mathur’s theorem says that as long as Hawking radiation happens through pair
production and escaping mechanism around the no-hair horizon, then no matter
how ingeniously small corrections are added to the evolution of a black hole,
the entropy of its hawking particles will increase monotonically until the
black hole evaporates away. Basing on this theorem, it’s very natural to
conclude that some type of new radiation mechanism is the only way to solve
the information missing puzzle dfzeng2018a ; dfzeng2018b ; stojkovic2020unp .
We will call this mechanism GISR, as dubbed in the abstract of this work, see
FIG.1 for references. We conjecture that GISR is a universal radiation
mechanism that happens to all compositional objects instead of black holes
exclusively, see the appendix for reasons for this mechanism’s universality.
However, in this work black holes will be taken as proxies for all such
objects and we will talk about mainly three aspects of this mechanism. That
is, its explicitly hermitian hamiltonian description, its resolution to the
information missing puzzle and its implication for black hole inner structures
and quantum gravitation theory.

Just as many other no-go theorems forbidding some working directions, e.g.
Coleman-Mandula theorem’s forbidding boson-fermion symmetries in classic non-
graded Lie algebra WeinbergSUSYtxtbook , Mathur’s theorem also contains
implicit assumptions, especially, the factorisation doing which writes the
intermediate state of an evaporating black hole system as direct product of
the black hole’s and that of the hawking particles. This is a very important
assumption adopted implicitly in many arguments for information missing
puzzle, such as AMPS(S) firewall argument fireworksAMPS2012 ;
fireworksAMPS2013 . Just as we pointed out in refs.dfzeng2021  and will
emphasise in the current work, when this assumption is get rid of and the
intermediate state is written as entanglement superposition of B&R (black hole
and radiation) combinations of different mass ratio, the missing information
will be retrieved very naturally. Refs.Papadodimas2013 ; Stephen1302 ;
Stephen1308 ; Stephen2022 ; NVW2013a ; NVW2013b  point out that Mathur’s
theorem contains another kind of factorisation which writes
$U\sum_{d}\psi_{d}(i)\rightarrow\sum_{d^{\prime}}U\psi_{d^{\prime}}(i)$, where
$U$ means evolution and $d$ distinguishes macroscopic states of the radiating
black hole due to the back reaction from hawking particles. These works argue
that unitarity of the full quantum state’s evolution does not imply that of
the single quantum state basis. Getting rid of this assumption has the
potential to retrieve information from hawking particles.

Many technique details referred to in this work can be found in our long paper
dfzeng2021 , new points of this work is, i) for the first time we realise and
express explicitly that, GISR may be a universal mechanism which happens to
all kinds of compositional objects. Black holes are just a special example;
ii) for the first time, we realise and write out properly the concrete form of
couplings between the radiating black hole and radiated particles through
GISR. Our previous writing

$H_{\mathrm{int}}=\frac{\mathrm{SimilarityFactor}[\psi_{\mathrm{ini}},\psi_{\mathrm{fnl}}]}{\sqrt{G}}*b^{\dagger}_{uv}a_{u-v}$
(1)

has the fault of all black holes radiating at a uniform rate. In this work, we
change the couplings into

$H_{\mathrm{int}}=\frac{\mathrm{SimilarityFactor}[\psi_{\mathrm{ini}},\psi_{\mathrm{fnl}}]}{G*\mathrm{Max}[M_{\mathrm{ini}},M_{\mathrm{fnl}}])}*b^{\dagger}_{uv}a_{u-v}$
(2)

which will allow the black holes to radiate at a rate ${\propto}M^{-2}$, thus
coincide better with the black hole thermodynamics and Stefan-Boltzmann law,
$d^{2}M/dt*dA=\sigma*T^{4}$. This coupling has intuitive explanation that GISR
happens like a classic diffusion. The bigger the radiating body is, the slower
it diffuses; iii) only in the current work, we realise and express definitely
what was done erroneously in the conventional argument for information missing
in Hawking radiation is kind of Schrödinger’s cat misunderstanding.

Figure 1: In conventional understanding, hawking particles arise from vacuum
fluctuation and escape from the no-hair horizon. Collecting and measuring
their energy provide us energy-time curves universal to all microscopic black
holes. In GISR, radiations arise from their coupling with the microscopic
state of black holes embodied in their Bekenstein-Hawking entropy. Collecting
and measuring their energy will provide us a microscopic state dependent
energy-time curve. In this work, microscopic state of the black hole exhibits
as many modes of over cross oscillation of the black hole matter contents
measured in proper time.

##  I Explicitly Hermitian Hamiltonian Description

Different from the usual atomic radiation which happens through electric
dipole couplings between the atom and photons, GISR happens through
gravitational monopole couplings between the black hole and to-be radiated
particles,

$H=H_{\mathrm{BH}}+H_{\mathrm{vac}}+H_{\mathrm{int}}$ (3)
$=\begin{pmatrix}w^{i}\\\ &w_{{\rule[1.5pt]{3.0pt}{0.6pt}}}^{j}\\\ &&\ddots\\\
&&&{\scriptstyle\it 0}^{\scriptscriptstyle\it
1}\end{pmatrix}\\!+\\!\sum_{q}\hbar\omega_{q}a^{\dagger}_{q}a_{q}{+}\\!\\!\sum_{u\;\\!\\!^{n}{\rule[1.5pt]{3.0pt}{0.6pt}}v^{\ell}}^{\hbar\omega_{q}=}g_{u\;\\!\\!^{n}v^{\ell}}b^{\dagger}_{u\;\\!\\!^{n}v^{\ell}}a_{q}$
(4)
$g_{u\;\\!\\!^{n}v^{\ell}}{\propto}{-}\frac{\hbar}{G\\{M_{u},M_{v}\\}^{\mathrm{max}}}\mathrm{Siml}\\{\Psi[M_{u\;\\!\\!^{n}}\\!(r)],\Psi[M_{v\;\\!\\!^{\ell}}\\!(r)]\\}$
(5)

In this hamiltonian, $w$,
$w_{\rule[1.5pt]{3.0pt}{0.6pt}}{=}w{{\rule[1.5pt]{3.0pt}{0.6pt}}}{\scriptstyle
1}$ et al denote the degeneracy or eigenvalue of black hole mass/energy
regarding contexts; $i$, $j$ et al distinguish microscopic states of equal
mass black holes, $i=1,2\cdots,w$, $w=\exp[\frac{4\pi
r_{\\!h}^{2}}{4G\\!_{N}}]$,
$j=1,2\cdots,w{{\rule[1.5pt]{3.0pt}{0.6pt}}}{\scriptstyle
1},w{{\rule[1.5pt]{3.0pt}{0.6pt}}}1=\exp[\frac{4\pi r_{\\!h}^{\prime
2}}{4G\\!_{N}}]$; the symbol ${\scriptstyle\it 0}^{\scriptscriptstyle\it 1}$
represents quantum state of the system when the black hole totally evaporate
away; $a_{q}^{\dagger}$ & $a_{q}$ are operators describing the vacuum
fluctuation, $b^{\dagger}_{u\;\\!\\!^{n}v^{\ell}}$ & $a_{q}$ take responses
for the black hole mass/energy level’s lowering or increasing and the vacuum
fluctuation mode $\omega^{n}_{q}$’s realisation or inverse. The analogue logic
behind the coupling $g_{uv}$ and the electric dipole coupling is
$\frac{\hbar}{GM}\mathrm{Siml}\\{\cdots\\}\sim{}\vec{d}{\cdot}\vec{A}$. The
similarity factor plays similar roles of electric-dipoles while
$\frac{\hbar}{GM}$ measures the energy reward if the black hole radiates a
particle away or absorbs one in. More massive the black hole is, more
negligible it will be rewarded from such a process. See refdfzeng2021  for the
definition of similarity factors in our concrete model of black hole inner
structures. In some sense, GISR is a diffusion-like phenomena, so we
conjecture it a universal effects. For the same reason, the energy profit
becomes zero when the black hole becomes infinitely massive or its horizon
covers the the whole universe. Well as is motivated physically, we must note
here that the concrete form of (5) is an intuitive ansatz. Currently we still
do not know if there is any first principle from which one can derive it out
exactly. The idea of reference Stephen2022  provides a possible working
direction.

Focusing on spherically symmetric radiations only, so the hawking particles’
spatial-momentum can be ignored and their quantum state will be characterised
by their energy exclusively111As was firstly noted by D.Page and emphasised in
Stephen1302 ; Stephen1308 , during most hawking radiation events, the center
of mass position and velocity of the BH will change unavoidably due to
radiation back reaction. Considering such effect helps us little in
understanding physics involved in the information missing puzzle. However,
such effect will change indeed some details of our work, e.g. it will make
Rabi-like oscillations on our $r_{h}(t)$ curve and $s(t)$ curve in FIG.2
almost not happen., the basis of Hilbert space for an evaporating black hole
and the corresponding radiations can be written as

$\displaystyle\\{w^{i}\otimes{\it\phi},w^{j}_{{\rule[1.5pt]{3.0pt}{0.6pt}}}\otimes\omega_{1}^{i},w^{k}_{{{\rule[1.5pt]{3.0pt}{0.6pt}}\,{\rule[1.5pt]{3.0pt}{0.6pt}}}}\otimes\omega_{1}^{j}\omega_{1}^{i},w^{j}_{{{\rule[1.5pt]{3.0pt}{0.6pt}}\,{\rule[1.5pt]{3.0pt}{0.6pt}}}}\otimes\omega_{2}^{i},\cdots$
$\displaystyle,u\;\\!\\!^{n}\otimes
q^{k}{p}^{j}\cdots{o}^{i}(u{+}q{+}p\cdots{+}o=w),\cdots\;$ (6)
$\displaystyle{\scriptstyle\it 0}^{\scriptscriptstyle\it
1}\otimes\omega_{1}^{z}\cdots\omega_{1}^{j}\omega_{1}^{i},{\scriptstyle
0}^{\scriptscriptstyle
1}\otimes\omega_{1}^{y}\cdots\omega_{2}^{i},\cdots,{\scriptstyle
0}^{\scriptscriptstyle 1}\otimes\omega_{w}^{i}\\}$

On this basis, the quantum state of a black hole and its radiations at
arbitrary middle epoch can be written as

$|\psi(t)\rangle=\sum_{u=w}^{0}\sum_{n=1}^{u}\sum_{{\scriptscriptstyle\sum}o^{i}}^{w{{\rule[1.5pt]{3.0pt}{0.6pt}}}u}e^{-iut-i\omega
t}c_{u\;\\!\\!^{n}}^{\vec{\omega}}(t)|u\;\\!\\!^{n}\otimes\vec{\omega}\rangle$
(7)

where ${\vec{\omega}}\equiv\\{o^{i},\cdots p^{j}q^{k}\\}$ is an abbreviation
of the radiation particles’ quantum state, with the index $i,j$ and $k$ et al
inherited from the parent body $w^{i}$, $w^{j}_{{\rule[1.5pt]{3.0pt}{0.6pt}}}$
and $v^{k}$ et al correspondingly and the total energy given by
$\omega{\equiv}o+{\cdots}p+q=w-u$. Evolutions of this wave function are
determined by the standard Schrödinger equation as follows

$i{\partial}_{t}c^{\vec{\omega}}_{u\;\\!\\!^{n}}(t){=}\sum_{v\neq
u}^{v+{{}^{\prime}\\!}\omega=w}\sum_{\ell=1}^{v}g_{u\;\\!\\!^{n}v^{\ell}}c^{{{}^{\prime}\\!}{\vec{\omega}}}_{v^{\ell}}(t)$
(8)

where $\hbar$ has been set to $1$ and ${{{}^{\prime}}{\vec{\omega}}}$ differs
from $\vec{\omega}$ only by the last emitted or absorbed particles. Without
loss of generality, we will set

$c_{w^{1}}^{\scriptscriptstyle\it\phi}(0)=1,c_{w^{i\neq
1}}^{\scriptscriptstyle\it\phi}(0)=0,c_{u\;\\!\\!^{n}}^{\vec{\omega}\scriptscriptstyle\neq\phi}(0)=0$
(9)

That is, we let our black hole lie on eigenstate $w^{1}$ at initial time
$t=0$.

For the first one or few particles’ radiation, we can set all
$c_{v\;\\!\\!^{\ell}}^{{{}^{\prime}\\!}\vec{\omega}\neq\phi,o{\scriptscriptstyle
1}}{=}0$ and focus on the evolution of $c^{\phi}_{w^{1}}(t)$ and
$c_{u\;\\!\\!^{n}}^{o{\scriptscriptstyle 1}}(t)$ exclusively. In this case, by
the standard Wigner-Wiesskopf approximation dfzeng2021 ; ScullyQOtextbook , we
can easily prove that

$c^{\phi}_{w^{1}}(t){=}e^{-\Gamma t},c_{u\;\\!\\!^{n}}^{o{\scriptscriptstyle
1}}(t){=}\frac{ig_{u\;\\!\\!^{n}\\!w\;\\!\\!^{1}}}{\Gamma}(e^{{\rule[1.5pt]{3.0pt}{0.6pt}}\Gamma
t}{\rule[1.5pt]{3.0pt}{0.6pt}}1),c_{v\;\\!\\!^{\ell}}^{{{}^{\prime}\\!}\vec{\omega}\neq\phi,o{\scriptscriptstyle
1}}{=}0$ (10)
$\sum_{n}\frac{g^{2}_{u\;\\!\\!^{n}\\!w\;\\!\\!^{1}}}{\Gamma^{2}}\approx\frac{u[{=}e^{4\pi
G_{\scriptscriptstyle N}(M-o{\scriptscriptstyle
1})^{2}}]}{w_{\rule[1.5pt]{3.0pt}{0.6pt}}({=}e^{4\pi G_{\scriptscriptstyle
N}M^{2}}{\rule[1.5pt]{3.0pt}{0.6pt}}1){+}\cdots 2{+}1}\approx\frac{e^{-8\pi
GMo{\scriptscriptstyle 1}}}{e^{4\pi G_{\scriptscriptstyle N}M^{2}}}$ (11)

where
$\Gamma{\equiv}\\!\sum_{u,n}^{u{\\!\neq\\!}w}|g_{w\;\\!\\!^{1}\\!u\;\\!\\!^{n}}\\!|^{2}\delta(w{-}u{-}o{\scriptstyle
1})$ and the first step in (11) uses the fact that
$g_{u\;\\!\\!^{n}\\!w\;\\!\\!^{1}}$ is approximately equal for most radiation
channels. Setting $8\pi G_{\scriptscriptstyle
N}M\equiv(k_{\scriptscriptstyle\\!B\\!}T)^{-1}$, the energy of the radiated
particle reads

$\langle
E\rangle_{t\rightarrow}^{\infty}{=}\\!\\!\\!\\!\\!\sum_{o{\scriptscriptstyle
1},n}^{u+o{\scriptscriptstyle 1}=w}\\!\\!\\!{o\scriptstyle
1}|c_{u\;\\!\\!^{n}}^{o{\scriptscriptstyle
1}}|^{2}{=}\\!\sum_{k}\\!\frac{k\omega
e^{-\frac{k\omega}{k_{\scriptscriptstyle\\!B\\!}T}}}{e^{{{\rule[1.5pt]{3.0pt}{0.6pt}}}\frac{k\omega}{k_{\scriptscriptstyle\\!B\\!}T}}{+}\cdots
1}{=}\frac{\omega}{e^{\frac{k\omega}{k_{\scriptscriptstyle\\!B\\!}T}}{\pm}1}$
(12)

The second step assumes that the radiated particle is quantised so that
$o{\scriptstyle 1}=k\omega$ with $k=0,1$ for fermions and $k=0,1,2\cdots$ for
bosons. Eq(12) implies that the power spectrum of GISR of black holes is
completely the same as Hawking radiation. By our conjecture GISR happens to
all kinds of compositional objects, whose effective temperature can be read
out from the derivations in (11)

$k_{\mathrm{B}}T_{\mathrm{GISR}}=W\frac{dM}{dW},k_{\scriptscriptstyle\\!B}\log
W=S.$ (13)

For black holes, this implies $k_{\mathrm{B}}T_{\mathrm{GISR}}=\frac{1}{8\pi
GM}=k_{\mathrm{B}}T_{\mathrm{hwk}}$ very naturally. While for non black hole
objects, to get this temperature, we need to know the radiating bodies mass
v.s. entropy as a prerequisite. It is worth emphasising that the thermal
feature of GISR of black holes is a universal short-time behaviour of the
radiating master’s exponential degeneracy of microscopic states instead of the
Wigner-Wiesskopf approximation we adopted. The exactly exponential degeneracy
of the black holes’ microscopic state is due to their big mass value and
owning of horizon instead of any artificial assumption. The latter fact allows
their microscopic state wave function be written as the direct product of many
concentric shells’, all are constrained to peaks inside the horizon due to
conditions eq(33) latter. Different from black holes, eigenstates of most
other compact objects degenerate only approximately, usually with
exponentially small mass/energy split. However, this split does not form
obstacles for us to describe their GISR with a uniform hamiltonian of
eqs(3)-(5).

For the long term behaviour of the black hole and its GISR, we only need to
integrate equations (8)-(9) numerically. We provide in FIG.2 results of this
integration explicitly. From the figure, we easily see that the variation of
the radiation particles’ entropy has indeed first increasing then decreasing
feature, just as Page curve manifests for unitarily evolving black holes. New
features in GISR are, the trends have a late time non-monotonic behaviour.
This is because in its hamiltonian description (4), the $u>v$ radiation and
$u<v$ absorption terms are equally allowed. This means that the earlier
radiated particles have non-zero probability to come back and cause
$c^{\vec{\omega}}_{u^{n}}(t)$’s Rabi-like oscillation. However, as $w$
increases, this late time oscillation will become negligible relative to the
first cycle of increase-decrease variation. At the same time, the size of the
matrix form Schrödinger equation (8) grows exponentially.

Figure 2: Evolutions of the horizon size and radiation products’ entropy of 6
degenerating black hole eigenstate following from exact numeric integrations
of eq(8) with initial condition like (9). $A_{\mathrm{ini}}$ is the horizon
area of initial black holes.

##  II Information Missing Puzzle

In GISR and its full quantum mechanical description, the causes of potential
information missing effects are almost transparent. Firstly, the Wigner-
Wiesskopf approximation, also known as Markovian approximation, includes
forgetting history effects. Forgetting history implies information missing
naturally. Technically dfzeng2021 , this happens in obtaining (10) when
shifting $c^{\phi}_{w^{1}}$’s history out of the integration symbol,

$\displaystyle c^{o{\scriptscriptstyle
1}}_{u\;\\!\\!^{n}}(t)\approx-i\int_{0}^{t}g_{u\;\\!\\!^{n}\\!w\;\\!\\!^{1}}^{u{+}\omega=w}c^{\phi}_{w^{1}}(t^{\prime})dt^{\prime}\rule{14.22636pt}{0.0pt}$
(14) $\displaystyle\approx-
ig_{u\;\\!\\!^{n}\\!w\;\\!\\!^{1}}c^{\phi}_{w^{1}}(t)\int_{0}^{t\rightarrow\infty}e^{-i[\omega-(w-u)]t^{\prime}}dt^{\prime}$
(15)

This doing introduces non-hermitian to the process and is the direct cause of
thermal spectrum. Secondly or more importantly, in Hawking type arguments, a
black hole’s radiation evolution is considered a sequence of events like

(16)

While in the true quantum world, at any given time, the black hole size is not
specifiable and the system can only be considered superposition of BR(black
hole & radiation)- combinationss of different mass-ratio,

(17)

where PC-diagram with different length of zigzag part denotes different size
black holes and their corresponding radiation products. Considerations (16)
ignore entanglement superposition among BR-combinations of different mass-
ratio, thus introducing entropies to the system artificially.

Mathematically, the process (17) can be written as

$\displaystyle|\psi_{0}\rangle\rightarrow|\psi_{0}\rangle{+}c_{b}^{t_{1}}|b\;\\!\\!^{b}\\!r\;\\!\\!^{b}\rangle{+}c_{m}^{t_{1}}|b\;\\!\\!^{m}\\!r\;\\!\\!^{m}\rangle{+}c_{s}^{t_{1}}|b\;\\!\\!^{s}\\!r\;\\!\\!^{s}\rangle{+}|\psi_{1}\rangle$
(18)
$\displaystyle\rightarrow|\psi_{0}\rangle{+}c_{b}^{t_{2}}|b\;\\!\\!^{b}\\!r\;\\!\\!^{b}\rangle{+}c_{m}^{t_{2}}|b\;\\!\\!^{m}\\!r\;\\!\\!^{m}\rangle{+}c_{s}^{t_{2}}|b\;\\!\\!^{s}\\!r\;\\!\\!^{s}\rangle{+}|\psi_{1}\rangle\rightarrow\cdots$

where $|\psi_{0}\rangle$, $|\psi_{1}\rangle$ are the radiation-before,
evaporation-after black hole state respectively,
$|{b\;\\!\\!^{b}r\;\\!\\!^{b}}\rangle$,
$|{b\;\\!\\!^{m}r\;\\!\\!^{m}}\rangle$, $|{b\;\\!\\!^{s}r\;\\!\\!^{s}}\rangle$
are three typical but non-exhaustive intermediate state of big, median, small
black holes with their radiation products. Tracing out the microscopic state
of the black hole, we can write

$\displaystyle|\psi^{t}_{r}\rangle\sim\sqrt{\rho^{0}}\oplus
c^{t}_{b}\sqrt{\rho^{b}}\oplus c^{t}_{m}\sqrt{\rho^{m}}\oplus
c^{t}_{s}\sqrt{\rho^{s}}\oplus\sqrt{\rho^{1}}$ (19)
$\displaystyle\rho^{b}=\mathrm{tr}_{b^{b}}|b^{b}r^{b}\rangle\langle{b^{b}r^{b}}|,~{}\rho^{m}=\mathrm{tr}_{b^{m}}|b^{m}r^{m}\rangle\langle{b^{m}r^{m}}|,~{}\cdots$

So the entropy of the radiation product can be calculated routinely
$s=\mathrm{tr}_{rr^{\prime}}\langle\psi^{t}_{r}|\psi^{t}_{r^{\prime}}\rangle\ln\langle\psi^{t}_{r}|\psi^{t}_{r^{\prime}}\rangle$,
with result

$s(t)=|c^{t}_{b}|^{2}s_{b\;\\!\\!^{b}r\;\\!\\!^{b}}+|c^{t}_{m}|^{2}s_{b\;\\!\\!^{m}r\;\\!\\!^{m}}+|c^{t}_{s}|^{2}s_{b\;\\!\\!^{s}r\;\\!\\!^{s}}\Rightarrow\raisebox{-5.69054pt}{\includegraphics[totalheight=22.76219pt]{fig0MathPageCurve.png}}$
(20)

where $s_{b\;\\!\\!^{b}r\;\\!\\!^{b}}$, $s_{b\;\\!\\!^{m}r\;\\!\\!^{m}}$,
$s_{b\;\\!\\!^{s}r\;\\!\\!^{s}}$ denote entanglement entropies of the
intermediate state black hole and their radiation products. At early times,
$c_{b}^{t_{1}}\gg c^{t_{1}}_{m}\&c^{t_{1}}_{s}$, so $s(t)$ is dominated by
$s_{b\;\\!\\!^{b}r\;\\!\\!^{b}}$ and increases with time. As time passes by,
$s(t)$ will be dominated by $s_{b\;\\!\\!^{m}r\;\\!\\!^{m}}$ and reach maximum
on Page epoch, then decrease due to dominations of
$s_{b\;\\!\\!^{s}r\;\\!\\!^{s}}$, and then Rabbi oscillate, as FIG.2 displays.

From the viewpoint of GISR, the so called island formula or replica wormhole
method ReplicaWormholeReview2006  are nothing but equivalent accounting for
interferences between BR-combinations of different mass-ratio

(21)

This description yields Page-curve for Hawking radiation in 1+1 dimensional JT
gravity, but it is not fundamental either on quantum mechanic or general
relativity level mathur2021 ; giddings2022 ; stojkovic2020prd . It cannot tell
us where the “missing” information go. That is, when contributions from the
replica wormhole channel is included, what should we measure to recover
information stored in the initial black hole? In GISR’s resolution, the answer
to such questions are clear. The horizon size v.s. time relation of a black
hole is determined by the initial state of the black hole through series of
similarity factors appearing the hamiltonian of GISR and is measurable to
retrieve initial state of the black hole directly, see FIG.2 for concrete
examples.

Essentially, what one mistakes in Hawking type argument for information
missing is a kind of Schrödinger’s cat misunderstanding. Binding one cat with
each hawking particle will make this fact more easily understood. Since the
hawking particles’ emission is quantum mechanical, the intermediate state of
an evaporating black hole can only be considered superposition of many cat-
groups with different number of died ones.

 (22)

But viewing the intermediate state as many different mass semi-classic black
holes and their corresponding hawking particles, one implicitly measures the
state of the cat-group

 (23)

Each of this measurement gives up knowledges on the quantum state of the
hawking particles, especially the time interval between two successive cats’
death, which is determined by similarity factors connecting the serial of
initial-final state wave function of the black hole before and after each
hawking particle’s emission. Neglecting this information will turn the
beginning pure state to ending mixed one or beginning mixed state to ending
more highly-mixed ones and produces entropy

${\Delta}S=(\frac{A_{i}}{4G}-\frac{A_{i+1}}{4G})-0$ (24)

where $0$ denotes the entropy of initial pure state, $A_{i}$ and $A_{i+1}$ are
the horizon area of the black hole on $i$ and $i{+}1$th measurement. Their
difference quantise entropy of hawking particles emitted between this two
measurements. The positivity of ${\Delta}S$ will cause the entropy of hawking
particles increase monotonically.

##  III Inside black holes

The idea of GISR dates back to 1970s Bekenstein1974a ; Mukhanov1986 ;
Bekenstein1997b , during which Mukhanov and Bekenstein speculated atomic
physics like interpretation for hawking radiation. However, due to their
simple quantisation of black hole masses basing on adiabatic invariance of the
horizon area, MB derived out discrete line shape spectrum for the radiation,
which contradicts Hawking’s thermal and continuous spectrum obviously.
However, just as was shown in dfzeng2018a ; dfzeng2018b ; stojkovic2020unp ;
dfzeng2021 ; dfzeng2017 ; dfzeng2020 , the microscopic state of black holes
can be quantised in such a way that their mass spectrum is continuous. So
contradictions plaguing MB are avoidable. To see this directly, we start with
the following exact black hole solutions family to the Einstein equation with
nontrivial inside horizon structure

$\displaystyle
ds^{2}_{\mathrm{in}}{=}{-}d\tau^{2}{+}\frac{\big{[}1{-}\big{(}\frac{2GM}{\varrho^{3}}\big{)}\\!^{\frac{1}{2}}\\!\frac{M^{\prime}\\!\varrho}{2M}\tau\big{]}^{2}\\!d\varrho^{2}}{a[\tau,\varrho]}{+}a[\tau,\varrho]^{2}\varrho^{2}d\Omega^{2}_{2}$
(25) $\displaystyle
a[\tau,\varrho]\\!=\\!\big{[}1\\!-\\!\frac{3}{2}\big{(}\frac{2GM[\varrho]}{\varrho^{3}}\big{)}\\!^{\frac{1}{2}}\tau\big{]}^{\\!\frac{2}{3}},~{}M[\varrho\\!\geqslant\\!\varrho_{\mathrm{max}}]{=}M_{\mathrm{tot}}$
(26) $\displaystyle
a[\tau\\!\in\\!|_{\frac{p\;\\!\\!^{\varrho}}{4}}^{\frac{p\;\\!\\!^{\varrho}}{2}},\varrho]{=}{-}a[\frac{p\;\\!\\!^{\varrho}}{2}{-}\tau,\varrho],a[\tau|_{\frac{p\;\\!\\!^{\varrho}}{2}}^{\frac{p\;\\!\\!^{\varrho}}{1}},\varrho]{=}{-}a[p^{\varrho}{-}\tau,\varrho]$
(27) $\displaystyle
a[\tau,\varrho]=a[\tau{+}p^{\varrho},\varrho],p^{\varrho}\equiv\frac{8}{3}\big{(}\frac{\varrho^{3}}{2GM[\varrho]}\big{)}\\!^{\frac{1}{2}}\rule{42.67912pt}{0.0pt}$
(28)

This describes a dust cluster’s Over-Cross-Oscillation inside the horizon
instead of aggregating on the central point and forming eternal singularities
there. The energy-momentum tensor seeds the metric (25)-(28) has forms
$T_{\mu\nu}\\!=\\!\mathrm{diagonal}\\{\rho,0,0,0\\}$ with
$\rho\\!=\\!\frac{M^{\prime}[\varrho]/8\pi\varrho^{2}}{a^{\\!\frac{3}{2}}\\!\\!+\\!\\!\frac{3GM^{\prime}\tau^{2}}{4\varrho^{2}}\\!-\\!\big{(}\frac{GM}{\varrho^{3}}\big{)}\\!^{\frac{1}{2}}\frac{M^{\prime}\varrho\tau}{2M}}$.
Although dust is chosen here as proxies for matters consisting our black
holes, when gravitation dominates all other interaction, this is a precise
enough approximation for general matter sources. Reasons for their OCO can be
attributed to the forward scattering amplitude domination instead of any
unknown quantum gravitation effects.

It is very important to note that OCO is a finite periodic oscillation only
when measured with proper time defined on the black hole matter contents
themselves. So in the above formulas, $\tau$, $\varrho$ and $\Omega_{2}$ are
proper time and co-moving coordinate of the matter contents’ volume element,
$a[\tau,\varrho]$ the scale factor, $M[\varrho]$ the co-moving radial mass
profile of the cluster at some arbitrary initial time whose concrete form is
arbitrary on classic levels. To those outside horizon and coordinate time used
observers, the black hole matter contents’ oscillation happens in the Future
of Infinite Future (FINF), see FIG.3, upper left part for illustration. In
reference dfzeng2021 , we resort uncertainty principle to allow this
oscillation happen in finite coordinate time. This is essentially a parallel
universe or ensemble spacetime interpretation for uncertainty principle. So to
those observers, the oscillation of $a(\tau)$ and multiple possibility of
$M[\varrho]$ is meaningful only in the sense of ensemble spacetime, or
parallel universe implied by uncertainty principle.

Figure 3: In the outside observers’ measurement, the formation of horizon and
singularity due to gravitational collapse happens in the FINF, upper left. But
in the parallel universe or ensemble spacetime implied by uncertainty
principles, such events happen in their finite future. After quantisation, the
matter contents will exhibit $\exp(A/4G_{N})$ ways of over cross oscillation
inside the horizon determined by their total mass, upper right. Each happens
in a parallel universe. Lower part is our extended PC-diagram in which the
east and west semi-sphere of a full $R^{3}$ appear on the diagram
independently, and the white hole region $\scriptstyle\mathrm{IV}$ is shifted
from the past to future and becomes $\scriptstyle\widetilde{\mathrm{IV}}$. The
parallel universe arises through general coordinate transformation like
$r\rightarrow r^{in\pi}$. On this PC-diagram, a representation volume point of
the outmost matter shell moves along traces like A-B-C-D-E-F-G-H-A$\cdots$, C
is the antipodal of B, E is identified with D, and et al.

Outside the matter occupation region our metric joins to the usual
Schwarzschild black hole in Lemaïtre coordinate smoothly

$ds^{2}_{\mathrm{out}}=-d\tau^{2}{+}\frac{d\varrho^{2}}{(1{-}\frac{3\tau}{2r_{h}})^{\frac{2}{3}}}{+}(1{-}\frac{3\tau}{2r_{h}})^{\frac{4}{3}}r_{h}^{2}d\Omega_{2}^{2}$
(29)

with
$r_{h}=\frac{3}{8}p^{\varrho_{\mathrm{max}}}_{\mathrm{eriod}}=2GM_{\mathrm{tot}}$.
This means that our inside horizon metric (25)-(28) satisfies requirements of
the no-hair theorem. By shifting the white hole region to the future of black
hole region and plotting the east&west semi-sphere separately, we revise the
usual Penrose Carter diagram and plot our black hole with OCO matter cores and
parallel universe extension in FIG. 3, lower part. Our revision of Penrose-
Carter diagram with east and west semi-sphere displayed independently on the
map contain ideas similar with ’t Hooft’s antipodal identification tHooft2021
. From the figure, we easily see that our solutions respect the singularity
theorem very well, i.e. all matters consisting of or falling into the black
hole will reach the singularity in finite proper time Penrose1965 ;
Hawking1976 ; geroch1979 . But such singularity is a periodic phenomena only
instead an eternal one and end of physics.

Quantisation of the OCO structure (25)-(28) can be done canonically. Looking
the whole matter core as a direct sum of many concentric shells, each moves
freely under gravitations due to itself and more inside partners.

$\displaystyle\Bigg{\\{}\begin{array}[]{l}h_{i}\dot{t}^{2}-h^{-1}_{i}\dot{r}^{2}=1,~{}h_{i}=1-\frac{2GM_{i}}{r}\\\
\ddot{t}+\Gamma^{(i)t}_{tr}\dot{t}\dot{r}=0\Rightarrow
h_{i}\dot{t}=\gamma_{i}=\mathrm{const}\end{array}$ (32)
$\displaystyle\Rightarrow\dot{r}^{2}-\gamma^{2}_{i}+h_{i}=0,~{}\gamma_{i}^{2}\leqslant
0\rule{36.98857pt}{0.0pt}$ (33)

where $h_{i}$ is the function appearing in the effective geometry felt by the
$i$-th shell, $ds^{2}=-h_{i}dt^{2}+h_{i}^{-1}dr^{2}+r^{2}d\Omega^{2}$,
$\Gamma_{i}$ is the corresponding Christoffel symbol. For each shell $i$, we
quantise equation (33) by looking it as an operator version hamiltonian
constraint and introduce a wave function $\psi_{i}(r)$ to denote probability
amplitude the shell be found of $r$-size, so that

$\big{[}{-}\frac{\hbar^{2}}{2m_{i}}\partial_{r}^{2}{-}\frac{GM_{i}m_{i}}{r}{-}\frac{\gamma^{2}_{i}\\!-\\!1}{2}m_{i}\big{]}\psi_{i}(r){=}0,~{}0{\leqslant}r{<}\infty$
(34)

where $M_{i}$ is the mass of the $i$-th shell and its inner partners, while
$m_{i}$ that of the $i$-th shell only. Square integrability of $\psi_{i}$
requires that

$\displaystyle\psi_{i}=N_{i}e^{-x}xL_{n_{i}-1}^{1}(2x),x\equiv
mr(1{\rule[1.5pt]{3.0pt}{0.6pt}}\gamma_{i}^{2})^{\frac{1}{2}}/\hbar$ (35)
$\displaystyle
n_{i}=\frac{GM_{i}m_{i}}{\hbar(1{\rule[1.5pt]{3.0pt}{0.6pt}}\gamma_{i}^{2})^{\frac{1}{2}}}=0,1,2,\cdots,\gamma_{i}^{2}\leqslant
0\rule{28.45274pt}{0.0pt}$ (36)

where $L_{n_{i}-1}^{1}(2x)$ is the associated Lagurre polynomial and $N_{i}$
its normalisation in standard mathematics. We then direct-product all
$\psi_{i}$s to get the wave functional of the whole matter core as follows

$\Psi[M(r)]=\psi_{0}\otimes\psi_{1}\otimes\psi_{2}\cdots,\sum_{i}m_{i}=M_{\mathrm{tot}}$
(37)

We will call objects described by this wave functional as OCO fuzzy ball,
which is a pictorially similar but mathematically more precise(in the sense we
can write out its wave function explicitly) fuzzy ball than string theory
fuzzy balls. The quantisation condition (36) and the sum rule in (37) do not
require discreteness of the black hole mass spectrum. However, they indeed
form complete constraints on how the matter core of a black hole can be
considered big number of concentric shells and from what initial radial
position each shell is released to freely OCO inside the horizon, i.e.
degeneracies of the wave function(37). We provide in FIG.3, upper right part
evidences that, the number of such degeneracies is consistent with the area
law formula of Bekenstein-Hawking entropy,
$Log[\\#degen]\approx\frac{M_{\mathrm{tot}}^{2}}{2}\propto{Area}$, see
dfzeng2018a ; dfzeng2018b ; dfzeng2020 ; dfzeng2017  for more calculation
details and dfzeng2021  for a schematic proof road.

Obviously by replacing the conventional Schwarzschild singularity with their
matter contents’ OCO motion, we assign black hole an atomic like inner
structure thus allowing their atomic like radiation practicable. Just as we
pointed out previously, this replacement does not break the singularity
theorem. Instead, the classic OCO motion extends connotations of the
singularity theorem; while the quantum OCO not only resolves the Schwarzschild
singularity but also yields proper area law formula for Bekenstein-Hawking
entropy. In this sense, we claim that OCO or OCO fuzzball a discovered
structure instead of introduced picture for black hole microscopics. We expect
it to appear also in a full quantum gravitational description of black holes.
It allows us to understand black hole physics the same way as we do with other
objects, instead of customly building a new set as was done in the replica
wormhole or quantum extremal surface ReplicaWormholeReview2006 .

##  IV Conclusion

We proposed in this work GISR as a new and universal mechanism for hawking
radiation and a simple resolution to the black hole information missing
puzzle. We provide an explicitly hermitian hamiltonian description for this
mechanism and the microscopic structure of black holes supporting it which we
call OCO fuzzy balls. Our hamiltonian description indicates that hawking
particles carry information away from the black hole through changing its
inner structures hence its evaporation progression. Viewing an evaporating
black hole as series of time-dependent semi-classic objects, the conventional
arguments for information missing ignore quantum interferences between BR-
combinations of different mass-ratio thus introducing entropies to the system
artificially and cause information missing puzzle. OCO fuzzy ball has a
similar physical picture to string theory fuzzy balls but exactly spelled-out
wave function and countable degeneracies consistent with the Bekenstein-
Hawking entropy formula. OCO is a finite period oscillation only when measured
with proper time associated with the volume element of the black hole matter
contents. When measured with the outside horizon observer’s coordinate time,
this oscillation happens in the FINF, or parallel universe or ensemble
spacetime required by the statistic interpretation of uncertainty principle.
Both GISR and OCO fuzzy ball resort no physics beyond the standard general
relativity and quantum mechanics. This means that, potentials of the canonic
quantum mechanic approach to the theory of quantum gravity is worth of digging
up further while new ideas such as replica wormholes on the basic feature of
quantum gravitation need be examined more carefully.

As discussion, we first note that GISR is a universal radiation which happens
to all kinds of compositional objects. So it is very interesting to explore
features or find evidences for such a radiation in, e.g. the direct S-wave to
S-wave transition of usual atoms. Secondly, tidal effects of the OCO
structured black holes are also observable from gravitational waves following
from such objects’ binary body merger cardoso2016 ; cardoso2017 . Thirdly, the
OCO fuzzy ball interpretation of BH entropy is generalisable to higher
dimensions and will give definite predictions dfzeng2018b ; dfzeng2021  for
some real-number partition questions defined by constraints like (36) and
(37). In one word, to accept that behind the information missing puzzle
happened GISR is a big deal of small capital. Otherwise we need to answer, how
to distinguish particles found around the black hole are due to Hawking
Radiation or GISR? If we refute GISR from the beginning, anther question
arises, why gravitation couplings between black holes and those to-be radiated
particles do not cause radiation while electromagnetic couplings between the
charged particles and photons do? See the following appendix for more
reasonings on the rationality of GISR’s universality.

##  Appendix A About the Universality of GISR

Many people think that hawking radiation is a privilege of black hole like
objects with horizon. However, the vacuum definition and calculation of
quantum field theory in curved spacetime BirrelDaviesTextbook ; Carroll ;
Jacobson1996 ; Traschen  indicate that the prerequisite for hawking radiation
is a future or past horizon instead of both. Almost all compositional objects
with masses can have future horizon which can be used as the boundary of
vacuum definition and cause hawking radiation. It is just basing on this fact
that we conjecture GISR happens to all compositional objects with inner
structures. To understand the logics behind this conjecture well, let us go
through the relevant calculation and reasonings step by step.

i) A uniform accelerating observer in Minkowski spacetime can see thermal
particles from the vacuum state defined by static observers. This is the so
called Unruh effects.

ii) Hawking radiation can be looked as an Unruh effect measured by static
observers sitting on fixed radial positions in a Schwarzshild (just an
example) black hole outside, such observers are accelerating otherwise will
fall towards the black hole due to gravitations.

iii) The definition of particles or vacuum is related with the global feature
of spacetime the observer is living in and fields he/she is examining. Three
types of vacuums are known explicitly, Hartle-Hawking, Unruh and Boulware
ones. All are defined through regularity conditions in Hadamard sense. That
is, the two point correlation functions have asymptotics
$G(x_{1},x_{2})\rightarrow\frac{U(x_{1},x_{2})}{4\pi^{2}\sigma_{\epsilon}}+V(x_{1},x_{2})\ln\sigma_{\epsilon}+W(x_{1},x_{2})$
as
$|x_{1}{-}x_{2}|\equiv{}dx_{12}^{2}+2i\epsilon(t_{1}-t_{2})+\epsilon^{2}\rightarrow
0$, where $U$, $V$ and $W$ are regular functions.

Figure 4: Figures for Hartle-Hawking vacuum definition

iv) Hartle-Hawking vacuum is defined on eternal black hole background. It is
regular on the past and future horizon $H^{\pm}$ at $r=2GM$, also on the past
and future null infinity $\mathcal{J}^{\pm}$. The static observers in region I
can see thermal radiations emitting from the black hole and equal flux of
radiations from past null infinity. So the black hole is in thermal
equilibrium with its environment, thus the nomenclature of eternal. At the the
same time of radiate they also absorb.

Figure 5: Figures for Unruh vacuum definition

v) Unruh vacuum is singular on the $H^{-}$ but regular in $H^{+}$. For this
reason it is considered more appropriate for black holes forms through
gravitational collapse. In such black holes $H^{-}$ is just an imagined
surface instead of a happend one, see the dashed lines in Fig.5. The static
observers sitting outside the horizon can see thermal radiations from $H^{+}$
but not from $\mathcal{J}^{-}$. However, there is a key point worthy of
noticing here. That is, the equal $r$-surfaces inside the horizon are more
horizontal than vertical like those outside the horizon, see FIG.5 for
illustrations. For this reason, the inside horizon part of the matter
occupation region should be plotted as the right panel instead of the left one
in the figure.

vi) Boulware vacuum is such a state that static observers sitting outside the
horizon will not see particles escaping away or falling towards
$\mathcal{J}^{\pm}$, thus they will not see hawking radiation. However, such a
vacuum is singular on both $H^{+}$ and $H^{-}$. Its existence tells us that,
regularity conditions on the horizon play key roles for our definition of
vacuum state and thermal radiation flux.

vii) For reason vi), many people think that objects without horizon, such as
neutron stars will not emit hawking radiation. However, this thought neglects
an important fact that, in real black holes forms through gravitational
collapse, the horizon is such a surface appears only in the infinite future of
observers sitting

Figure 6: The formation of horizon through gravitational collapses needs
infinite long Schwarzschild time. Hawking radiation happens during the
collapsing process instead of the post horizon-formation era.

outside the collapsing star. See Fig.6 for references. Note that $H^{+}$
corresponds to $t=+\infty$ surface.

viii) General compositional objects with inner structures can always be
considered a black hole to be, we can use their future event horizon $H^{+}$
as the place to impose regular conditions for vacuum state definition and get
thermal radiation fluxes for observers sitting outside the matter occupation
region.

ix) A possible argument against the previous reasoning may be, for objects
without horizon, we have a time-like Killing vector throughout the spacetime.
So we can use it to define positive frequency modes everywhere and match the
Minkiowski modes at infinity. The resulting vacuum state would look like
exactly of Boulware type, free of fluxes both at $\mathcal{J}^{+}$ and
$\mathcal{J}^{-}$. The fact that Boulware vacuum is singular on the horizon
will not bring us any trouble, since in this case, we have no horizon at all.

x) Answers to the counterview ix) are as follows, although we have a time like
Killing vector through out the spacetime, its joint condition across the
interface of matter occupation region happens in such a way that positive-
frequency modes inside the matter region mixes positive and negative frequency
ones outside it and vice versa. References Stojkovic2007 ; Stojkovic2008a ;
Stojkovic2008b ; Stojkovic2009 ; Stojkovic2013 ; Stojkovic2014 ; Stojkovic2015
provides a big amount of calculations and arguments for this fact. According
to these works, hawking radiation happens during the collapsing process
instead of the post horizon formation era or the “black holes” radiate away
before they forms definitely.

Basing on the above analysis and reasoning, we think it may be rationale to
assume that all compositional objects with inner structures can emit GISR.
Nevertheless, we do not claim that GISR equals hawking radiation completely.

## Acknowledgements

This work is supported by NSFC grant no. 11875082. The last stage of this work
is finished at Niels Bohr Institute. We thank very much for warm hosts
provided by Vitor Cardoso and support from Villum Investigator program
supported by the VILLUM Foundation (grant no. VIL37766) and the DNRF Chair
program (grant no. DNRF162) by the Danish National Research Foundation.

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