# A Possible Quantum Effect of Gravitation

Jarmo Mäkelä111Vaasa University of Applied Sciences, Wolffintie 30, 65200
Vaasa, Finland, email<EMAIL_ADDRESS>
###### Abstract

Beginning from the standard Arnowitt-Deser-Misner (ADM) formulation of general
relativity we construct a tentative model of quantum gravity from the point of
view of an observer with constant proper acceleration, just outside of a
horizon of spacetime. In addition of producing the standard results of black-
hole thermodynamics, our model makes an entirely new prediction that there is
a certain upper bound for the energies of massive particles. For protons, for
instance, this upper bound is around $1.1\times 10^{21}eV$. The result is
interesting, because this energy is roughly of the same order of magnitude as
are the highest energies ever measured for protons in cosmic rays.

##  1 Introduction

The most important aim of any theory of physics is to make predictions, which
could be proved either right or wrong by means of experiments and
observations. So far it has always been thought that all quantum effects of
gravity lie in the Planck regime, where the typical energies are of the order
of $10^{28}eV$, and therefore well beyond any possibilities for observations
and experiments. In this paper we speculate on a possiblity that there is at
least one effect of quantum gravity, where the typical energies are well below
the Planck energy, and measurable by means of the current observational
techniques.

To be more precise, we consider an idea that quantum mechanics, when applied
to Einstein’s general theory of relativity, could imply that there is a
certain upper bound for the velocities of massive particles, which is a bit
less than the speed of light. As a consequence, any massive particle with mass
$m$ has a certain maximum energy. We suggest that to make the predictions of
our model of quantum gravity consistent with the well-known results of black-
hole thermodynamics, this maximum energy must be:

$E_{max}=mc^{2}\cosh\left[\frac{2\pi^{2}}{\ln(2)}\right].$ (1.1)

For protons, for example, this maximum energy is around $1.1\times 10^{21}eV$,
and therefore seven orders of magnitude below the Planck energy. The result is
interesting, because the highest energies ever measured for protons in cosmic
rays are roughly of the same order of magnitude, bur still less than the upper
bound set by Eq. (1.1): Now and then energies of order $10^{19}eV$ have been
measured for protons, [1] and the highest energy ever measured has been
$(3.2\pm 0.9)\times 10^{20}eV$. [2] Thus the prediction given by Eq. (1.1)
could indeed be proved wrong by means of technologies available today. Should
one ever observe a proton with energy higher than $1.1\times 10^{21}eV$, the
model of quantum gravity leading to Eq. (1.1) would have been proved
erroneous.

The derivation of Eq. (1.1) in this paper is based on a Hamiltonian approach
to quantum gravity. We construct the Hamiltonian operator for the
gravitational field from the point of view of an observer just outside of a
horizon of spacetime. The key assumption of our model is that the proper
acceleleration $a$ of the observer is constant. In other words, no matter what
may happen in spacetime, the proper acceleration $a$ of the observer is always
the same. Our model of quantum gravity involves a free, dimensionless
parameter, which must be fixed such that the model reproduces the familiar
results of black-hole thermodynamics. It turns out that this parameter
determines an upper bound for the velocities of massive particles with respect
to the observer. It must be emphasized that even though our results may seem
startling, they have nevertheless been obtained by means of a straightforward
application of the standard rules of quantum mechanics to Einstein’s general
relativity.

Unless otherwise stated, we shall always use the natural units, where
$\hbar=G=c=k_{B}=1$.

##  2 Hamiltonian

The first successfull Hamiltonian formulation of Einstein’s general relativity
was constructed by Arnowitt, Deser and Misner in the year 1962. [3] The
starting point of this formulation is to write the line element of spacetime
as:

$ds^{2}=-N^{2}\,dt^{2}+q_{ab}(dx^{a}+N^{a}\,dt)(dx^{b}+N^{b}\,dt).$ (2.1)

In this line element, $q_{ab}$ $(a,b=1,2,3)$ is the metric tensor induced on
the spacelike hypersurface of spacetime, where the time coordinate
$t=constant$. $N$ is the lapse function, and $N^{a}$ is the shift vector. As
it is well known, the action in this so-called ADM formulation may be cast
into a Hamiltonian form as:

$S_{ADM}=\int(p^{ab}\dot{q}_{ab}-N_{a}\mathcal{H}^{a}-N\mathcal{H})\,d^{4}x,$
(2.2)

where we have integrated over the whole spacetime. The dot means the time
derivative, $p^{ab}$ is the momentum conjugate to $q_{ab}$, and we have
denoted:

$\displaystyle\mathcal{H}^{a}$ $\displaystyle:=$
$\displaystyle-2p^{as}_{\,\,\,|s},$ (2.3a) $\displaystyle\mathcal{H}$
$\displaystyle:=$
$\displaystyle\frac{\sqrt{q}}{16\pi}(K_{ab}K^{ab}-K^{2}-\mathcal{R}),$ (2.3b)

where ”$|$” means the covariant derivative on the hypersurface. $K_{ab}$ is
the exterior curvature tensor on the hypersurface, $K$ is its trace, and
$\mathcal{R}$ is the Riemann curvature scalar on the hypersurface.

Varying the ADM action $S_{ADM}$ with respect to the lapse $N$ gives the
Hamiltonian constraint

$\mathcal{H}=0,$ (2.4)

and if we vary the ADM action with respect to the shift vector $N^{a}$, we get
the three diffeomorphism constraints

$\mathcal{H}^{a}=0.$ (2.5)

The Hamiltonian constraint states the invariance of $S_{ADM}$ in the
reparametrization of the time coordinate $t$, and it is equivalent with
Einstein’s field equation $G_{00}=0$ in vacuum. The three differomorphism
constraints, in turn, state the invariance of $S_{ADM}$ with respect to the
choice of the spatial coordinate $x^{a}$ on the hypersurface, and they are
equivalent with the equations $G_{a0}=0$ $(a=1,2,3)$.

When the spacelike hypersurface, where the time coordinate $t=constant$ is
compact, we may obtain the equations $G_{ab}=0$, which are the real, dynamical
equations of general relativity in the ADM formulation, quite simply, by means
of the variation of $q_{ab}$ and its conjugate momentum $p^{ab}$. However, if
the spacelike hypersurface either is not compact, or the spatial region under
consideration possesses a boundary, we meet with a curious problem: The
curvature scalar $\mathcal{R}$ on the hypersurface has the property:

$\sqrt{q}\mathcal{R}=\sqrt{q}q^{mn}(\Gamma_{as}^{s}\Gamma_{mn}^{a}-\Gamma_{an}^{s}\Gamma^{a}_{ms})-[q^{mn}\sqrt{q}(\delta^{a}_{m}\Gamma_{ns}^{s}-\Gamma^{a}_{mn})]_{,a}.$
(2.6)

Since the Christoffel symbol $\Gamma_{mn}^{a}$ on the hypersurface involves
the first spatial derivatives of $q_{ab}$, the second term on the right hand
side of Eq. (2.6) involves its second derivatives. Hence it follows that when
varying $S_{ADM}$, we must fix not only the metric tensor $q_{ab}$, but also
its derivatives on the boundary. To overcome this problem we must compensate
the second term on the right hand side of Eq. (2.6) by means of an appropriate
boundary term. When the lapse $N\equiv 1$ and the shift $N^{a}\equiv 0$ on the
boundary $B$ of the hypersurface, which means that the time coordinate of an
observer at rest with respect to the spatial coordinates on the boundary
agrees with his proper time, an appropriate boundary term is the time integral
of the function:

$-H_{B}:=\frac{1}{16\pi}\oint_{B}[q^{mn}\sqrt{q}(\delta_{m}^{a}\Gamma_{ns}^{s}-\Gamma_{mn}^{a})]\,dS_{a},$
(2.7)

where we have integrated over the closed two-boundary $B$ of the hypersurface.
Hence the action takes the form:

$S=\int(p^{ab}\dot{q}_{ab}-N_{a}\mathcal{H}^{a}-N\mathcal{H})\,d^{3}x\,dt-\int
H_{B}\,dt,$ (2.8)

and we may write the Hamiltonian of the system as:

$H=\int(N_{a}\mathcal{H}^{a}+N\mathcal{H})\,d^{3}x+H_{B},$ (2.9)

where we have integrated over that part of the hypersurface, which is bounded
by the closed two-surface $B$.

When the Hamiltonian constraint $\mathcal{H}=0$ and the three diffeomorhism
constraints $\mathcal{H}^{a}=0$ are satisfied, the Hamiltonian of the
gravitational field inside of the closed two-surface $B$ reduces to the form:

$H=H_{B}.$ (2.10)

Solving the Hamiltonian and the diffeomorphism constraints we may separate the
gauge degrees of freedom of the gravitational field from its real, physical
degrees of freedom. In general, the metric tensor $q_{ab}$ has six independent
components at each spacetime point, and because there are four constraints at
each point, the gravitational field has just $6-4=2$ real, physical degrees of
freedom at each point of spacetime. In principle, we may attempt to solve the
gauge degrees of freedom from the constraints in terms of the physical degrees
of freedom, and then substitute in $H_{B}$. Denoting those physical degrees of
freedom by $\alpha^{s}$, we may therefore write the Hamiltonian $H_{B}$ in Eq.
(2.10) as:

$H_{B}=H_{B}(\alpha^{s}).$ (2.11)

The Hamiltonian reduction described above has really been performed in
spherically symmetric, asymptotically flat vacuum spacetimes by Kuchar. [4]
For asymptotically flat vacuum spacetimes $H_{B}$ reduces to the ADM energy of
spacetime. The only spherically symmetric, asymptotically flat vacuum solution
to the Hamiltonian and the diffeormorphism constraints is the Schwarzschild
solution, which describes spacetime involving the Schwarzschild black hole.
The ADM energy in such spacetime agrees with the Schwarzschild mass $M$ of the
hole. A similar Hamiltonian reduction has also been carried out in spherically
symmetric spacetime interacting with spherically symmetric electromagnetic
field. [5, 6] In general, for spacetime involving a black hole the physical
degrees of freedom may be chosen to be the mass, electric charge and angular
momentum of the hole. In what follows, we shall take the function
$H_{B}(\alpha^{s})$ as the real, physical Hamiltonian of the gravitational
field inside of a closed, spacelike two-surface embedded into a spacelike
hypersurface, where the time coordinate $t=constant$, and see, where this will
take us. 222For the proof that the boundary term really gives, under certain
conditions, the physical Hamiltonian of the gravitational field, see Ref. [9].

We shall now obtain an expression for the Hamiltonian $H_{B}(\alpha^{s})$
defined in Eq. (2.7), when the closed two-surface $B$ lies just outside of a
horizon of spacetime. As the starting point we shall use the properties of the
Rindler spacetime. As it is well known, Rindler spacetime is just a two-
dimesional, flat Minkowski spacetime equipped with Rindler coordinates $t$ and
$x$. When written in terms of these coordinates, the line element takes the
form: [7]

$ds^{2}=-x^{2}\,dt^{2}+dx^{2}.$ (2.12)

The Rindler coordinates $t$ and $x$ are related to the flat Minkowski
coordinates $X$ and $T$ such that:

$\displaystyle t$ $\displaystyle:=$
$\displaystyle\tanh^{-1}\left(\frac{T}{X}\right),$ (2.13a) $\displaystyle x$
$\displaystyle:=$ $\displaystyle\sqrt{X^{2}-T^{2}}.$ (2.13b)

Eq. (2.13a) implies that we may understand the Rindler time coordinate $t$ as
the boost angle of an observer in the Minkowski spacetime. The spatial
coordinate $x$, in turn, determines the proper acceleration of the observer:
An observer with constant $x$ has constant proper acceleration

$a=\frac{1}{x}.$ (2.14)

So we find that at the Rindler horizon, where $x=0$, and therefore $T=\pm X$,
the proper acceleration $a$ tends to infinity.

Almost any spacetime equipped with a horizon may be approximated, just outside
of the horizon, by means of an appropriate four-dimensional generalization of
the Rindler spacetime. The four-dimensional generalization used in this paper
is the one with the line element [8]

$ds^{2}=-y\,dt^{2}+\frac{(y^{\prime})^{2}}{4y}\,d\lambda^{2}+h_{jk}\,d\chi^{j}\,d\chi^{k}.$
(2.15)

When obtaining this line element we have replaced the Rindler coordinate $x$
by a new coordinate

$y:=x^{2},$ (2.16)

which, in turn, is assumed to be a function of a spatial coordinate $\lambda$,
and parameters $\alpha^{s}$ $(s=1,2,\dots,M)$ determining the geometrical
properties of spacetime. In other words,

$y=y(\lambda,\alpha^{1},\alpha^{2},\dots,\alpha^{M}).$ (2.17)

For the Reissner-Nordström spacetime, for instance, which is an asymptotically
flat, electrovacuum spacetime involving the Reissner-Nordström black hole, the
parameters may be taken to be the mass and the electric charge of the hole.
The coordinates $\chi^{j}$ $(j=1,2)$, in turn, are the coordinates on the
spacelike two-surface, where $\lambda=constant$. $h_{jk}$ $(j,k=1,2)$ is the
metric tensor induced on this two-surface, and we have assumed that $h_{jk}$
is a function of the spatial coordinates $\lambda$ and $\chi^{j}$, but not of
the parameters $\alpha^{s}$. In Eq. (2.15) we have denoted:

$y^{\prime}:=\frac{\partial y}{\partial\lambda}.$ (2.18)

In general, the proper acceleration vector field of an observer is

$a^{\mu}=u^{\sigma}u^{\mu}_{;\sigma},$ (2.19)

where $u^{\mu}$ is the future pointing unit tangent vector field of the world
line of the observer. For an observer at rest with respect to the coordinates
$\lambda$ and $\chi^{j}$ the only non-zero component of $u^{\mu}$ is
$u^{t}=1/\sqrt{y}$, and therefore the only non-zero component of $a^{\mu}$ for
such observer is:

$a^{\lambda}=\frac{2}{y^{\prime}}.$ (2.20)

Hence the norm of $a^{\mu}$ is:

$a=\sqrt{a_{\lambda}a^{\lambda}}=\frac{1}{\sqrt{y}}.$ (2.21)

We shall assume that the proper acceleration $a$ of our observer is always a
constant, no matter what may happen in spacetime. According to Eq. (2.21)
constant $a$ means constant $y$, and hence it follows that when the physical
quantities $\alpha^{s}$ take on infinitesimal changes $d\alpha^{s}$, the
coordinate $\lambda$ must take on a change $d\lambda$ such that

$dy=\frac{\partial y}{\partial\lambda}\,d\lambda+\frac{\partial
y}{\partial\alpha^{s}}\,d\alpha^{s}=0.$ (2.22)

Let us now obtain an expression for $H(\alpha^{s})$ just outside of the
horizon, beginning from Eq. (2.15). Using Eq. (2.7) we find:

$\begin{split}H_{B}&=-\frac{1}{16\pi}\oint_{B}\left(\frac{2\sqrt{y}}{y^{\prime}}\Gamma_{\lambda
j}^{j}-\frac{y^{\prime}}{2\sqrt{y}}h^{jk}\Gamma^{\lambda}_{jk}\right)\sqrt{h}\,d^{2}\chi\\\
&=-\frac{1}{8\pi}\oint_{B}\frac{\sqrt{y}}{y^{\prime}}h^{jk}{h^{\prime}}_{jk}\sqrt{h}\,d^{2}\chi,\end{split}$
(2.23)

where we have integrated the coordinates $\chi^{1}$ and $\chi^{2}$ over the
two-surface $B$. It is interesting to observe, what happens to $H_{B}$, when
the quantities $\alpha^{s}$ take on infinitesimal changes $d\alpha^{s}$. The
resulting change in the Hamiltonian $H_{B}$ is:

$\begin{split}dH_{B}&=\frac{\partial
H_{B}}{\partial\alpha^{s}}\,d\alpha^{s}\\\
&=-\frac{1}{8\pi}\oint_{B}\left[\frac{1}{2\sqrt{y}y^{\prime}}\frac{\partial
y}{\partial\alpha^{s}}-\frac{\sqrt{y}}{(y^{\prime})^{2}}\frac{\partial
y^{\prime}}{\partial\alpha^{s}}\right]h^{jk}{h^{\prime}}_{jk}\sqrt{h}\,d^{2}\chi\,d\alpha^{s}.\end{split}$
(2.24)

Because we assume that the proper acceleration $a=constant$ on the two-surface
$B$, Eq. (2.22) implies:

$\frac{\partial y}{\partial\alpha^{s}}\,d\alpha^{s}=-y^{\prime}\,d\lambda.$
(2.25)

When we are just outside of the horizon, where $y=0$, and $y^{\prime}\neq 0$,
we may ignore the second term indiside of the brackets in Eq. (2.24). Hence we
may write, in effect,

$dH_{B}=\frac{a}{8\pi}\,dA,$ (2.26)

where

$dA=\frac{dA}{d\lambda}\,d\lambda=\frac{1}{2}\oint_{B}h^{jk}{h^{\prime}}_{jk}\sqrt{h}\,d\lambda$
(2.27)

is the change taken by the area

$A=\oint_{B}\sqrt{h}\,d^{2}\chi$ (2.28)

of the two-surface $B$, when the coordinate $\lambda$ takes on the change
$d\lambda$. Since the proper acceleration $a$ of the observer is kept as a
constant, we may thus write the gravitational Hamiltonian from the point of
view of such observer as:

$H=\frac{a}{8\pi}A,$ (2.29)

where $A$ is the area of the horizon.

##  3 Classical Dynamics

We shall now proceed to the study of the classical Hamiltonian dynamics of
spacetime implied by Eq. (2.29). As it is well known, the so-called densitized
triads

$\tilde{E}^{m}_{I}:=\sqrt{q}E^{m}_{I}$ (3.1)

may be used as the coordinates of the configuration space in the Hamiltonian
formulation of general relativity. In that case the quantities

$-\frac{1}{16\pi}K_{m}^{I},$

where

$K^{I}_{m}:=E^{Is}K_{sm},$ (3.2)

are the corresponding coordinates of the momentum space. In Eqs. (3.1) and
(3.2) the vector fields $E^{m}_{I}$ are the triads on the spacelike
hypersurface of spacetime, where the time coordinate $t$ is a constant. They
are related to the metric tensor $q_{mn}$ induced on that hypersurface as:

$E_{I}^{m}E_{J}^{n}q_{mn}=\delta_{IJ}$ (3.3)

for all $I,J=1,2,3$. Indeed, there is a canonical transformation from the
phase space coordinates $(q_{mn},p^{mn})$ of general relativity to the
coordinates $(\tilde{E}^{m}_{I},-\frac{1}{16\pi}K^{I}_{m})$, provided that the
resulting formulation of general relativity is assumed to remain invariant in
the local SO(3) rotations of the triads. [10] Conversely, we may use the
densitized triads $\tilde{E}_{I}^{m}$ as the coordinates of the momentum
space, and the quantities $\frac{1}{16\pi}K^{I}_{m}$ as the corresponding
coordinates of the configuration space.

In the line element (2.15) we have:

$\displaystyle q_{\lambda\lambda}$ $\displaystyle=$
$\displaystyle\frac{(y^{\prime})^{2}}{4y},$ (3.4a) $\displaystyle q_{jk}$
$\displaystyle=$ $\displaystyle h_{jk},$ (3.4b)

for all $j,k=1,2$. We may thus choose the triads such that

$\displaystyle E^{\lambda}_{3}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{y}}{y^{\prime}},$ (3.5a) $\displaystyle
E^{j}_{I}E^{k}_{J}h_{jk}$ $\displaystyle=$ $\displaystyle\delta_{IJ},$ (3.5b)

for all $I,J=1,2$. So we find:

$\tilde{E}^{\lambda}_{3}=\sqrt{h},$ (3.6)

and we may write the Hamiltonian $H$ in Eq. (2.29) as:

$H=\frac{a}{8\pi}\oint_{B}\tilde{E}^{\lambda}_{3}\,d^{2}\chi.$ (3.7)

It is interesting that because the proper acceleration $a$ is assumed to be a
constant, our Hamiltonian of the gravitational field may be expressed as a
function of just one independent variable at each point of the spacelike two-
surface $B$. Actually, this is something one might have expected: The
Hamiltonian $H$ in Eq. (2.29) depends on the area $A$ of the two-surface $B$
only which, in turn, is determined by the metric tensor $h_{jk}$ induced on
$B$. The metric tensor $h_{jk}$ has 3 independent components, but the
requirement of the invariance of the gravitational action on the choice of the
two coordinates $\chi^{1}$ and $\chi^{2}$ on $B$ brings along 2 constraints
between these components. As a result, we are left with just $3-2=1$
independent, physical degree of freedom.

We now define the variables $p(\chi)$ associated with the points
$\chi=(\chi^{1},\chi^{2})$ of the spacelike two-surface $B$ as:

$p(\chi):=\frac{1}{8\pi}\tilde{E}_{3}^{\lambda}(\chi).$ (3.8)

By means of these variables we may write the area of $B$ as:

$A=8\pi\oint_{B}p(\chi)\,d^{2}\chi,$ (3.9)

and the Hamiltonian of the gravitational field takes the form:

$H=a\oint_{B}p(\chi)\,d^{2}\chi.$ (3.10)

We shall now use the variables $p(\chi)$ as the coordinates of the momentum
space. The corresponding coordinates of the configuration space we shall
denote by $\phi(\chi)$. The quantity $\phi(\chi)$ associated with the point
$\chi$ of $B$ obeys the Hamiltonian equation of motion:

$\dot{\phi}(\chi)=\frac{\delta H}{\delta p(\chi)}=a,$ (3.11)

where the dot denotes the derivative with respect to the proper time $\tau$ of
the observer. The general solution to Eq. (3.11) is:

$\phi(\chi)=a\tau+C(\chi),$ (3.12)

where $C(\chi)$ is a constant function of the proper time $\tau$, and depends
only on the point $\chi$ of $B$. Since the Hamiltonian $H$ in Eq. (3.10) does
not involve $\phi(\chi)$ at all, the Hamiltonian equation of motion written
for the momentum variable $p(\chi)$ reads as:

$\dot{p}(\chi)=-\frac{\delta H}{\delta\phi(\chi)}=0,$ (3.13)

and therefore the variables $p(\chi)$ are constants of motion of the system.

Now, what is $a\tau$? Eqs. (2.12) and (2.14) imply that during the lapse $dt$
of the Rindler time $t$ the elapsed proper time of the observer with constant
proper acceleration $a$ is:

$d\tau=\frac{1}{a}\,dt,$ (3.14)

and therefore:

$dt=a\,d\tau.$ (3.15)

So we have, possibly up to an additive constant:

$a\tau=t,$ (3.16)

and Eq. (3.12) takes the form:

$\phi(\chi)=t+C(\chi).$ (3.17)

So we may identify the quantity $\phi(\chi)$, possibly up to an additive
constant, as the boost angle of our observer at the point $\chi$.

##  4 Quantization

When going over from the classical to the quantum-mechanical treatment of our
model we replace the classical variables $\phi(\chi)$ and $p(\chi)$ by the
corresponding operators $\hat{\phi}(\chi)$ and $\hat{p}(\chi)$ obeying the
canonical commutation relations:

$\displaystyle[\hat{\phi}(\chi),\hat{\phi}(\chi^{\prime})]$ $\displaystyle=$
$\displaystyle[\hat{p}(\chi),\hat{p}(\chi^{\prime})]=0,$ (4.1a)
$\displaystyle[\hat{\phi}(\chi),\hat{p}(\chi^{\prime})]$ $\displaystyle=$
$\displaystyle i\delta(\chi-\chi^{\prime}),$ (4.1b)

at all points $\chi,\chi^{\prime}\in B$. In Eq. (4.1b)
$\delta(\chi-\chi^{\prime})$ is the delta function on the two-surface $B$. Eq.
(3.10) implies that the Hamiltonian operator of the gravitational field takes
the form:

$\hat{H}=a\oint_{B}\hat{p}(\chi)\,d^{2}\chi,$ (4.2)

and hence the time-independent Schrödinger equation

$\hat{H}|\psi\rangle=E|\psi\rangle$ (4.3)

written for the energy eigenstates $|\psi\rangle$ becomes to:

$a\oint_{B}\hat{p}(\chi)|\psi\rangle\,d^{2}\chi=E|\psi\rangle.$ (4.4)

Since we may identify the energy $E$ with the classical Hamiltonian $H$ in Eq.
(3.10), Eq. (4.4) reduces to:

$\oint_{B}\hat{p}(\chi)|\psi\rangle\,d^{2}\chi=\oint_{B}p(\chi)|\psi\rangle\,d^{2}\chi.$
(4.5)

A solution to this equation is the one, where the state $|\psi\rangle$ is a
common eigenstate of all of the operators $\hat{p}(\chi)$, i. e.

$\hat{p}(\chi)|\psi\rangle=p(\chi)|\psi\rangle$ (4.6)

for all $\chi\in B$. In this equation $\hat{p}(\chi)$ is the operator
associated with the point $\chi$ and $p(\chi)$ the momentum variable at that
point. We shall assume that the state $|\psi\rangle$ is normed to unity, i. e.

$\langle\psi|\psi\rangle=1.$ (4.7)

In this paper we shall consider a model of quantum gravity, where the energy
eigenstates $|\psi\rangle$ satisfy both of the Eqs. (4.6) and (4.7), and see,
where this will take us.

In the position representation the quantum state $|\psi\rangle$ is described
by a functional $\psi[\phi(\chi)]$ of the function $\phi(\chi)$. It may
regarded as the wave function of the gravitational field. Between the states
$|\psi_{1}\rangle$ and $|\psi_{2}\rangle$ represented by the wave functions
$\psi_{1}[\phi(x)]$ and $\psi_{2}[\phi(\chi)]$, respectively, we define the
inner product

$\langle\psi_{1}|\psi_{2}\rangle:=\int\psi^{*}_{1}[\phi(\chi)]\psi_{2}[\phi(\chi)]\,\mathcal{D}[\phi(\chi)],$
(4.8)

where $\mathcal{D}[\phi(\chi)]$ is an approriate integration measure in the
space of the functions $\phi(\chi)$. In the position repesentation we write
the operators $\hat{p}(\chi)$ as functional differential operators:

$\hat{p}(\chi)=-i\frac{\delta}{\delta\phi(\chi)},$ (4.9)

and hence Eq. (4.6) takes the form:

$-i\frac{\delta\psi[\phi(\chi)]}{\delta\phi(\chi)}=p(\chi)\psi[\phi(\chi)].$
(4.10)

The general solution to this equation is:

$\psi[\phi(\chi)]=\mathcal{N}\exp\left[i\oint_{B}p(\chi)\phi(\chi)\,d^{2}\chi\right],$
(4.11)

where $\mathcal{N}$ is an arbitrary complex number. Since the eigenstates
associated with the different eigenvalues must be orthogonal, it follows that
if we denote by $\psi[\phi(\chi)]$ the wave function associated with function
$p(\chi)$, and by $\psi^{\prime}[\phi(\chi)]$ the wave function associated
with function $p^{\prime}(\chi)$, then

$\int\psi^{*}[\phi(\chi)]\psi^{\prime}[\phi(\chi)]\,\mathcal{D}[\phi(\chi)]=1,$
(4.12)

when

$p(\chi)\equiv p^{\prime}(\chi),$ (4.13)

and

$\int\psi^{*}[\phi(\chi)]\psi^{\prime}[\phi(\chi)]\,\mathcal{D}[\phi(\chi)]=0,$
(4.14)

otherwise.

$p(\chi)$ and $\phi(\chi)$ are functions of the coordinates
$(\chi^{1},\chi^{2})$ of the points of a certain planar region $D$. We divide
the planar region $D$ in $N$ non-intersecting subsets $D_{j}$
$(j=1,2,\dots,N)$ of $D$. In other words, $D_{j}\cap D_{k}=\emptyset$ for all
$j\neq k$, and

$D=D_{1}\cup D_{2}\cup\cdots\cup D_{N}.$ (4.15)

The area of the region $D$ is:

$\mathcal{A}=\int_{D}d^{2}\chi.$ (4.16)

We shall assume that all of the subsets $D_{j}$ have the same area

$\Delta\chi:=\frac{\mathcal{A}}{N},$ (4.17)

and that the largest diameter of the subsets tends to zero, when $N$ tends to
infinity. From each subset $D_{j}$ we pick up a point $\chi_{j}$. With these
assumptions we may write the integral in the exponential in Eq. (4.11) as a
Riemann integral:

$\oint_{B}p(\chi)\phi(\chi)\,d^{2}\chi=\lim_{N\rightarrow\infty}\left(\sum_{j=1}^{N}p_{j}\phi_{j}\right)\,\Delta\chi,$
(4.18)

where we have denoted:

$\displaystyle\phi_{j}$ $\displaystyle:=$ $\displaystyle\phi(\chi_{j}),$
(4.19a) $\displaystyle p_{j}$ $\displaystyle:=$ $\displaystyle p(\chi_{j}),$
(4.19b)

for all $j=1,2,\dots,N$. Hence we may write the wave functional
$\psi[\phi(\chi)]$ as:

$\psi[\phi(\chi)]=\mathcal{N}\lim_{N\rightarrow\infty}\prod_{j=1}^{N}[\exp(ip_{j}\phi_{j}\,\Delta\chi)].$
(4.20)

Eq. (4.20) enables us to define the integral on the left hand side of Eq.
(4.12):

$\int\psi^{*}[\phi(\chi)]\psi^{\prime}[\phi(\chi)]\,\mathcal{D}[\phi(\chi)]:=\mathcal{N}^{*}\mathcal{N}^{\prime}\lim_{N\rightarrow\infty}\prod_{j=1}^{N}\left\\{\int_{-\frac{L}{2}}^{\frac{L}{2}}\exp[i(p^{\prime}_{j}-p_{j})\phi_{j}\,\Delta\chi]\,d\phi_{j}\right\\}.$
(4.21)

In this equation we have denoted $p^{\prime}_{j}:=p^{\prime}(\chi_{j})$ for
all $j=1,2,\dots,N$. $L$ is a positive real and $\mathcal{N}^{\prime}$ an
arbitrary complex number. We could, of course, make $L$ to tend to infinity,
which would mean that we integrate every variable $\phi_{j}$ from the negative
to the positive infinity. Such an integration, however, would give infinite
results, making impossible to satisfy the fundamental postulate (4.7) of our
model. Because of that we shall assume from this point on that $L$ is a fixed,
positive real, and see, where this will take us. In this sense $L$ is a free
parameter of our model, and it must be fixed such that the predictions of the
model are consistent with everything else we know about quantum gravity.

If we want the functions $\psi[\phi(\chi)]$ and $\psi^{\prime}[\phi(\chi)]$ in
Eq. (4.21) to satisfy the requirements (4.12) and (4.14), we must have:

$p_{j}\,\Delta\chi=n_{j}\frac{2\pi}{L},$ (4.22)

where $n_{j}=0,\pm 1,\pm 2,\dots$ for all $j=1,2,\dots,N$. Eq. (3.9) implies
that we may write the area of the spacelike two-surface $B$ as:

$A=8\pi\lim_{N\rightarrow\infty}\left(\sum_{j=1}^{N}p_{j}\,\Delta\chi\right),$
(4.23)

and so we may understand the quantity

$A_{j}:=8\pi p_{j}\,\Delta\chi$ (4.24)

as the area of the piece, or constituent, of the two-surface $B$ associated
with the subset $D_{j}$ of the planar region $D$. According to Eq. (4.22) the
area eigenvalues of those constituents are of the form:

$A_{j}=n_{j}\frac{16\pi^{2}}{L}.$ (4.25)

The concept of negative area obviously does not make sense, and therefore we
shall abandon, from this point on, the negative values of the quantum numbers
$n_{j}$, and keep only the non-negative ones. In other words, we shall assume
that $n_{j}=0,1,2,\dots$ for all $j=1,2,\dots,N$.

Eq. (4.25) implies that the area of an individual constituent of the surface
$B$ is has a discrete spectrum with an equal spacing. Unless there is an
infinite number of constituents with zero area on the surface $B$, we are
therefore compelled to infer that the surface $B$ consists of a finite number
of constituents, each of them having an area spectrum given by Eq. (4.25).
This number we shall denote by $N$. The total area of the surface is the sum
of the areas of its constituents. In other words, we have:

$A=A_{1}+A_{2}+\cdots+A_{N},$ (4.26)

which means, through Eq. (4.24), that the area eigenvalues of the surface are
of the form:

$A=\frac{16\pi^{2}}{L}(n_{1}+n_{2}+\cdots+n_{N}).$ (4.27)

Hence we have shown that the fundamental postulates (4.6) and (4.7) of our
model imply that the surface $B$, and thereby the horizon, has a discrete
structure: The surface $B$ consists of a finite number of separate
constituents, all of them having a discrete, equally spaced area spectrum.
Consequently, the wave functional $\psi[\phi(\chi)]$ of spacetime in Eq.
(4.11) becomes replaced by a function $\psi(\phi_{1},\phi_{2},\dots,\phi_{N})$
of the $N$ boost angles $\phi_{j}$ $(j=1,2,\dots,N)$ at the constituents of
the surface such that:

$\psi(\phi_{1},\phi_{2},\dots,\phi_{N})=\frac{1}{\sqrt{L^{N}}}\exp\left[\frac{i}{8\pi}(A_{1}\phi_{1}+A_{2}\phi_{2}+\cdots+A_{N}\phi_{N})\right].$
(4.28)

To satisfy the requirement (4.12) we have chosen the constant $\mathcal{N}$ to
be $1/\sqrt{L^{N}}$.

We have now completed the quantization of our model in the position
representation, where the wave function is expressed as a functional
$\psi[\phi(\chi)]$ of the function $\phi(\chi)$. In the momentum
representation the wave function is expressed as a functional
$\tilde{\psi}[p(\chi)]$ of the function $p(\chi)$. When operating on
$\tilde{\psi}[p(\chi)]$ by the operator $\hat{p}(\chi)$, we just multiply
$\tilde{\psi}[p(\chi)]$ by $p(\chi)$. in other words,

$\hat{p}(\chi)\tilde{\psi}[p(x)]=p(\chi)\tilde{\psi}[p(\chi)],$ (4.29)

no matter what is $\tilde{\psi}[p(\chi)]$. For instance, if we denote by
$\tilde{\psi}_{p^{\prime}}[p(\chi)]$ the eigenfunction associated with
$p^{\prime}(\chi)$, we must have:

$\hat{p}(\chi)\tilde{\psi}_{p^{\prime}}[p(\chi)]=p(\chi)\tilde{\psi}_{p^{\prime}}[p(\chi)].$
(4.30)

However, since $\tilde{\psi}_{p^{\prime}}[p(\chi)]$ is an eigenfunction
associated with $p^{\prime}(\chi)$, we must also have:

$\hat{p}(\chi)\tilde{\psi}_{p^{\prime}}[p(\chi)]=p^{\prime}(\chi)\tilde{\psi}_{p^{\prime}}[p(\chi)],$
(4.31)

and therefore the eigenfunction $\tilde{\psi}_{p^{\prime}}[p(\chi)]$ must have
the property:

$p(\chi)\tilde{\psi}_{p^{\prime}}[p(\chi)]=p^{\prime}(\chi)\tilde{\psi}_{p^{\prime}}[p(\chi)],$
(4.32)

no matter what is $p(\chi)$.

The only solution to this equation is the ”delta functional”
$\delta[p(\chi)-p^{\prime}(\chi)]$, which has the property:

$\delta[p(\chi)]=0,$ (4.33)

whenever the function $p(\chi)$ is not identically zero. It also has the
properties:

$\displaystyle\int\delta[p(\chi)]\,\mathcal{D}[p(\chi)]$ $\displaystyle=$
$\displaystyle 1,$ (4.34a)
$\displaystyle\int\delta[p(\chi)]F[p(\chi)]\,\mathcal{D}[p(\chi)]$
$\displaystyle=$ $\displaystyle F[0]$ (4.34b)

for an arbitrary functional $F[p(\chi)]$ of the function $p(\chi)$.
$\mathcal{D}[p(\chi)]$ is an appropriate integration measure in the space of
the functions $p(\chi)$. However, as we saw in Eq. (4.22), the quantities
$p(\chi)$ may actually take just discrete values at the points of the surface
$B$. Hence our delta functional $\delta[p(\chi)-p^{\prime}(\chi)]$ is just an
ordinary Kronecker delta with the properties:

$\delta[p(\chi)-p^{\prime}(\chi)]=1,$ (4.35)

whenever $p(\chi)\equiv p^{\prime}(\chi)$, and

$\delta[p(\chi)-p^{\prime}(\chi)]=0,$ (4.36)

otherwise.

As the reader may have noticed, quantization of gravity in our model from the
point of view of an observer with constant proper acceleration just outside of
a horizon of spacetime is really very similar to the quantization of a system
of free particles in non-relativistic quantum mechanics: As in the system of
free particles, the coordinates of the momentum space were constants of
motion, whereas the corresponding coordinates of the configuration space were,
up to additive constants, linear functions of the time coordinate of the
observer. The constant $L$ introduced in this section may be understood as the
edge length of a box used in the box normalization of the wave function in the
position representation. However, whereas in the box normalization in non-
relativistic quantum mechanics the edge length $L$ is finally taken to
infinity at the end of the calculations, in our model $L$ is assumed to be a
fixed number.

##  5 Thermodynamical Properties of Spacetimes with Horizon

Our simple model about the quantum-mechanical properties of the horizons of
spacetime implied that the horizon of spacetime consists of separate
constituents, each of them contributing to the horizon an area, which is an
integer times a certain fundamental area. In the SI units the horizon area
takes the form:

$A=\alpha\ell_{Pl}^{2}(n_{1}+n_{2}+\cdots+n_{N}),$ (5.1)

where $N$ is the number of the constituents of the horizon, and

$\ell_{Pl}:=\sqrt{\frac{\hbar G}{c^{3}}}\approx 1.6\times 10^{-35}m$ (5.2)

is the Planck length. We have denoted:

$\alpha:=\frac{16\pi^{2}}{L}.$ (5.3)

The quantum numbers $n_{1},n_{2},\dots,n_{N}$ are non-negative integers, and
they determine the area eigenstates of the constituents of the horizon.
Constituent $j=1,2,\dots,N$ is in vacuum, if $n_{j}=0$; otherwise the
constituent in an excited state.

It is interesting that this very simple microscopic model of the horizons
implies, among other things, all of the well-known results, such as the
Hawking effect, for the thermodynamics of black holes. The only necessary
assumption needed for the derivation of these results is that the microscopic
states of the horizon are identified with the combinations of the excited
states of its constituents such that when the excited states of the
constituents, determined by the quantum numbers $n_{j}$, are permuted, the
microscopic state of the horizon will also change. That derivation has been
carried out in various ways and in various contexts in several papers.
$[11-16]$ For the benefit of the reader, we shall now outline its main points.

As we found in Section 2, beginning from the standard ADM formulation of
general relativity, the quantity

$H=\frac{a}{8\pi}A$ (5.4)

may be viewed as the classical Hamiltonian, and hence energy of the
gravitational field, inside of a closed spacelike two-surface with area $A$,
just outside of the horizon, from the point of view of an observer lying on
the surface with constant proper acceleration $a$. Since the two-surface is
assumed to lie just outside of the horizon, we may identify its area $A$ with
the area of the horizon.

The thermodynamical properties of any system may be obtained from its
partition function

$Z(\beta):=\sum_{n}e^{-\beta E_{n}},$ (5.5)

where $\beta$ is the temperature parameter, and we have summed over all energy
eigenstates $n$ with energy eigenvalues $E_{n}$ of the system. Using Eqs.
(5.1) and (5.4) and employing the assumptions stated above we find that the
partition function of the horizon takes, from the point of view of our
observer, the form: $[11,14,16]$

$\begin{split}Z(\beta)=&\sum_{n_{1}=1}^{\infty}\exp\left(-n_{1}\frac{\alpha\beta
a}{8\pi}\right)\\\
+&\left[\sum_{n_{1}=1}^{\infty}\exp\left(-n_{1}\frac{\alpha\beta
a}{8\pi}\right)\right]\left[\sum_{n_{2}=1}^{\infty}\exp\left(-n_{2}\frac{\alpha\beta
a}{8\pi}\right)\right]\\\ +&\cdots\\\
+&\left[\sum_{n_{1}=1}^{\infty}\exp\left(-n_{1}\frac{\alpha\beta
a}{8\pi}\right)\right]\cdots\left[\sum_{n_{N}=1}^{\infty}\exp\left(-n_{N}\frac{\alpha\beta
a}{8\pi}\right)\right].\end{split}$ (5.6)

In the first term on the right hand side of Eq. (5.6) just one of the
constituents of the horizon is in an excited state. In the second term two of
the constituents are in excited states. Finally, in the last term all $N$ of
the constituents of the horizon are in excited states.

The partition function written in Eq. (5.6) may be evaluated explicitly. A
straightforward calculation, based on the elementary properties of the
geometric series, gives the result: [16]

$Z(\beta)=\frac{1}{2-2^{\beta T_{C}}}\left[1-\left(\frac{1}{2^{\beta
T_{C}}-1}\right)^{N}\right],$ (5.7)

where we have defined the characteristic temperature

$T_{C}:=\frac{\alpha a}{8\pi\ln(2)}$ (5.8)

of the horizon. Eq. (5.7) is valid, whenever $\beta\neq\frac{1}{T_{C}}$. If
$\beta=\frac{1}{T_{C}}$, we have $Z(\beta)=N$. From this partition function
one may obtain, for instance, an expression for the energy of the
gravitational field per a constituent of the horizon:

$\begin{split}\bar{E}&:=-\frac{1}{N}\frac{\partial}{\partial\beta}\ln\left[Z(\beta)\right]\\\
&=\left[\frac{1}{N}\frac{2^{\beta T_{C}}}{2^{\beta T_{C}}-2}+\frac{2^{\beta
T_{C}}}{2^{\beta T_{C}}-1-(2^{\beta
T_{C}}-1)^{N+1}}\right]T_{C}\ln(2).\end{split}$ (5.9)

In all of our calculations, which so far have been carried explicitly, with no
approximations made, we have always assumed that the number $N$ of the
constituents of the horizon is very large. With this assumption we find that
the first term on the right hand side of Eq. (5.9) will always vanish, when
$\beta T_{C}\neq 1$. The second term in the brackets will also vanish in the
large $N$ limit, whenever $\beta T_{C}>1$. This means that the constituents of
the horizon are effectively in the vacuum, when the temperature of the horizon
is, from the point of view of our observer, less than the characteristic
temperature $T_{C}$. However, something very strange happens to $\bar{E}$,
when $\beta T_{C}<1$, which means that the temperature exceeds the
characteristic temperature $T_{C}$. When $\beta T_{C}<1$, the quantity
$(2^{\beta T_{C}}-1)^{N+1}$ vanishes in the large $N$ limit, and we may write
the average energy $\bar{E}$ in Eq. (5.9), in effect, as:

$\bar{E}=\frac{2^{\beta T_{C}}}{2^{\beta T_{C}}-1}T_{C}\ln(2).$ (5.10)

So we find that when $\beta T_{C}$ tends to 1 from the left hand side, which
means that the temperature $T$ of the horizon tends to $T_{C}$ from the right
hand side, we have:

$\bar{E}=2T_{C}\ln(2)=\frac{2\alpha a}{8\pi},$ (5.11)

where we have used Eq. (5.8). This means that at the characteristic
temperature $T_{C}$ the constituents of the horizon jump, in average, from the
vacuum to the second excited states. In other words, the horizon performs a
phase transition at the characteristic temperature $T_{C}$. Since the
consituents are effectively in the vacuum, when $T<T_{C}$, and jump to the
second excited states, when $T=T_{C}$, we may view the characteristic
temperature $T_{C}$, defined in Eq. (5.8), as the lowest possible temperature
of the horizon.

As we found in Eq. (5.8), the characteristic temperature $T_{C}$ depends on
the parameter $\alpha$. It is interesting to observe that if we put:

$\alpha=4\ln(2),$ (5.12)

then

$T_{C}=\frac{a}{2\pi},$ (5.13)

which agrees with the Unruh temperature [17]

$T_{U}:=\frac{a}{2\pi}$ (5.14)

measured by an observer with proper acceleration $a$. To get the temperature
of the horizon from the point of view of a distant observer we must replace
the proper acceleration $a$ by the surface gravity $\kappa$ at the horizon.
The surface gravity $\kappa$ tells the proper acceleration at the horizon as
seen by the distant observer. If the horizon under consideration happens to be
a black hole event horizon, we therefore find that the lowest possible
temperature of the black hole is, from the point of view of our observer:

$T_{\infty}=\frac{\kappa}{2\pi}.$ (5.15)

This is one of the most fundamental results of black-hole thermodynamics. [18]

Eq. (5.13) has important implications to black hole entropy. As it is well
known, between the temperature parameter $\beta$, entropy $S$ and energy $E$
of any system there is the relationship:

$\beta=\frac{\partial S}{\partial E}.$ (5.16)

Identifying the right hand side of Eq. (5.4) as the energy $E$ of a black hole
from the point of view of an observer with proper acceleration $a$, just
outside of its event horizon, and putting $\beta=\frac{1}{T_{C}}$, we find:

$\frac{2\pi}{a}=\frac{\partial S}{\partial
A}\left(\frac{dE}{dA}\right)^{-1}=\frac{\partial S}{\partial
A}\frac{8\pi}{a},$ (5.17)

which implies:

$S=\frac{1}{4}A.$ (5.18)

In other words, the black hole entropy is, in the natural units, exactly one-
quarter of its event horizon area, when its temperature agrees with $T_{C}$.
[19] The same result may be obtained by means of a systematic analysis,
beginning from the partition function $Z(\beta)$ obtained in Eq. (5.7). [11]

As we have seen, our extremely simple model predicts all of the classic
properties of black-hole thermodynamics: With the choice (5.12) for the
parameter $\alpha$ we find that the black hole has temperature
$\frac{\kappa}{2\pi}$, and its entropy is one-quarter of its event horizon
area. Actually, the model does even more. There is a well-known problem
concerning the Schwarzschild black hole in a heat bath: When the hole absorbs
radiation from its environment, its Schwarzschild mass $M$ increases. However,
Eq. (5.15) implies that the temperature of the hole is

$T_{\infty}=\frac{1}{8\pi M},$ (5.19)

which means that the temperature of the hole decreases, when its mass $M$
increases, and therefore the hole absorbs heat faster and faster. One may show
that the mass $M$ of the Schwarzschild black hole becomes infinite in a finite
time. This is known as the singularity problem in the black hole mass. [20]

Our model solves the singularity problem in a very nice way: If we look at Eq.
(5.9), we observe that the average energy $\bar{E}$ per a constituent depends
on the temperature parameter $\beta$, and thus on the temperature $T$. When
the temperature $T$ is increased above $T_{C}$, the constituents begin to jump
to higher excited states, and the mass $M$ of the hole increases. Eq. (5.10)
implies that the relationship between the temperature $T_{\infty}$ measured by
a distant observer and the mass $M$ of the hole is, in the large $N$ limit,
not given by Eq. (5.19), but by an equation: [16]

$T_{\infty}=-\frac{1}{8\pi M}\frac{\ln(2)}{\ln\left[1-\frac{N\ln(2)}{4\pi
M^{2}}\right]},$ (5.20)

which is an increasing function of $M$. Hence it follows that when the black
hole absorbs heat and its mass increases, the temperature of the hole does not
really decrease, but it increases. During the process, where the hole tends to
a thermal equilibrium with the heat bath, its temperature increases, and it
absorbs heat all the time slower and slower. A study reveals that the mass of
the hole tends to a certain finite value, which depends on the temperature of
the heat bath. [16] So, there is no singularity in the black hole mass.

##  6 Consequences

As we have seen, our simple model of quantum gravity predicts, starting from
the basic principles, the horizon area spectrum in Eq. (5.1). Hence it
predicts, among other things, the well-known results of black-hole
thermodynamics, and it also provides a solution to the problem of the mass
singularity of black holes. When expressed in terms of the areas $A_{j}$ of
the constituents of the horizon the wave function of the gravitational field
takes a simple and beautiful form expressed in Eq. (4.28).

There is, however, a somewhat annoying feature in our model of quantum
gravity: It involves a new, dimensionless parameter $L$, which we introduced
by hand as the edge length of an $N$-dimensional box used in the box
normalization of the wave function of the gravitational field. The edge length
$L$ is related to the parameter $\alpha$ through Eq. (5.3), and if we use Eq.
(5.12), we find:

$L=\frac{4\pi^{2}}{\ln(2)}\approx 57.0.$ (6.1)

Our parameter is not any obvious combination of the natural constants, and we
did not take $L$ to infinity at the end of the calculations, either, as we do,
when we box normalize the wave function of the system of free particles in
non-relativistic quantum mechanics. Instead, $L$ is a fixed number, whose
value is dictated by the need of our model to predict the standard results of
black-hole thermodynamics. In this sense the parameter $L$ plays in our model
of quantum gravity the same role as does the Barbero-Immirzi parameter
$\gamma$ in loop quantum gravity. [21]

Curiously, these disadvantages of our model may actually be turned to its
advantage, but they take us to an analysis of motion in quantized spacetime.
The boost angle $\phi$ determines the velocity $v$ of an observer through the
relation:

$v=\tanh(\phi).$ (6.2)

It is natural to use $L$ as a bound for the allowed values of $\phi$ such that

$-\frac{L}{2}\leq\phi\leq\frac{L}{2}.$ (6.3)

In this way the points $(\phi_{1},\phi_{2},\dots,\phi_{N})$ of the
configuration space are really confined to lie in an $N$-dimesional box with
edge length $L$. We chose the allowed values of $\phi$ as in Eq. (6.3),
because there is no reason to favor any direction, when the observer moves in
spacetime. We could, of course, have chosen the allowed values of $\phi$ to
lie, for instance, within the interval $[0,L]$, but in that case any
possibility of motion to the negative direction would have been excluded,
which is absurd. For the same reason we did not perform the half-line
quantization, where the quantities $\phi_{j}$ would have been permitted to
take positive values only, to our system.

Eq. (6.3) sets, through Eqs. (6.1) and (6.2), the following bounds to the
velocity $v$ of the observer:

$-c\tanh\left[\frac{2\pi^{2}}{\ln(2)}\right]\leq v\leq
c\tanh\left[\frac{2\pi^{2}}{\ln(2)}\right].$ (6.4)

Actually, these bounds are pretty wide. Because

$\tanh(\phi)=\sqrt{1-\frac{1}{\cosh^{2}(\phi)}},$ (6.5)

for positive $\phi$, we find that the maximum speed of the observer is:

$\begin{split}v_{max}&=c\sqrt{1-\frac{1}{\cosh^{2}\left[\frac{2\pi^{2}}{\ln(2)}\right]}}\\\
&\approx\left\\{1-\frac{1}{2\cosh^{2}\left[\frac{2\pi^{2}}{\ln(2)}\right]}\right\\}c\\\
&\approx(1-3.7\times 10^{-25})c.\end{split}$ (6.6)

Hence we note that the observer may still attain velocities, which are very
close to the speed $c$ of light. If a material particle raced with a photon
through a galaxy with diameter $10^{5}$ light years with this speed, it would
lag less than 0.4 mm behind the photon at the end of the race.

In addition of giving the maximum speed of the observer, Eq. (6.3) also gives
the maximum speed of massive particles with respect to the observer. To see
that this is the case, we first note that its always possible to pick up, at
the given point of spacetime, an inertial frame of reference for the observer.
In this frame the observer has a certain proper acceleration $a$, and a
certain boost angle $\phi$. The proper acceleration $a$ of the observer is
always the same, no matter how we choose the frame, but the boost angle $\phi$
is not, and it depends on the choice of the frame, which may be done
arbitrarily. It is always possible to pick up for a massive particle an
inertial frame of reference, where the particle is at rest. We may take that
frame as the inertial frame of reference of the observer. According to our
results it is impossible to accelerate our observer to a speed exceeding
$v_{max}$ in this frame. Hence the speed of the particle with respect to the
observer will not exceed $v_{max}$, either.

Since the energy $E$ of the particle with mass $m$ is related to its boost
angle $\phi$ as:

$E=mc^{2}\cosh(\phi),$ (6.7)

we find that the maximum energy of the particle is:

$E_{max}=mc^{2}\cosh\left[\frac{2\pi^{2}}{\ln(2)}\right].$ (6.8)

For the electron, for instance, the maximum energy is

$E_{max}^{electron}\approx 6.0\times^{17}eV,$ (6.9)

whereas for the proton the maximum energy is

$E_{max}^{proton}\approx 1.1\times 10^{21}eV.$ (6.10)

Hence our model makes a concrete and precise observational prediction: It is
impossible for an electron to reach energies higher than $6.0\times
10^{17}eV$, and for a proton energies higher than $1.1\times 10^{21}eV$.
Should there ever be an observation of an electron with energy higher than
$6.0\times 10^{17}eV$, or of a proton with energy higher than $1.1\times
10^{21}eV$, our model of quantum gravity would have been proved wrong. It is
interesting that both of these energies are well below the Planck energy,
which is

$E_{Pl}:=\sqrt{\frac{\hbar c^{5}}{G}}\approx 1.2\times 10^{28}eV.$ (6.11)

Our prediction may therefore be viewed as an entirely new, unexpected effect
of quantum gravity.

The most highly energetic particles have been found in cosmic rays. The
particles with extremely high energies are mostly protons. Now and then
protons with energies of the order of $10^{19}eV$ or higher are observed. [1]
The highest energy ever measured for a particle in cosmic radiation has been
$(3.2\pm 0.9)\times 10^{20}eV$. [2] It is not known, what this the so-called
”Oh-My-God particle” was, but it is believed to have been a proton. So we find
that the very highest energies ever measured for the protons in cosmic rays
are roughly of the same order of magnitude as in the prediction made in Eq.
(6.10) for their highest possible energies. The result may be viewed as an
observational support for our model of quantum gravity.

No doubt, even the particles with energies close to the maximum energy
$E_{max}$ given by Eq. (6.8) will undergo a red shift, if they arrive from
long enough distances. Employing for the red shift $z$ the standard formula

$z=\frac{H_{0}D}{c},$ (6.12)

where $H_{0}$ is the Hubble constant, one finds that the red-shifted maximum
energy of a particle arriving from distance $D$ is

$E^{\prime}_{max}=\frac{E_{max}}{\frac{H_{0}D}{c}+1}.$ (6.13)

For instance, if $D$ equals, say, 3 billion light years, or $3\times
10^{25}m$, and we put $2.2\times 10^{-18}s^{-1}$ for $H_{0}$, one obtains for
the red-shifted maximum energy an approximation

$E^{\prime}_{max}\approx 0.8E_{max}.$ (6.14)

##  7 Concluding Remarks

In this paper we have constructed, starting from the first principles and the
ADM formulation of general relativity, a simple model of quantum gravity from
the point of view of an observer with constant proper acceleration $a$, just
outside of a horizon of spacetime. Curiously, quantization of gravity
according to our model was very similar to the quantization of a system of
free particles in non-relativistic quantum mechanics. In the process we were
compelled to introduce a sort of box normalization to the quantum states of
the gravitational field. The edge length $L$ of the box then appeared as a
parameter of our model, and when $L$ was chosen appropriately, the model re-
produced all of the well-known results of black-hole thermodynamics. Actually,
the model did even more, since it provided a solution to the so-called mass
singularity problem of the Schwarzschild black hole.

The most startling result of our model, however, was that it predicted a
certain upper limit to the speed of material particles, which is a very tiny
amount less than the speed of light, depending on the choice for the parameter
$L$ of our model. This result was an outcome of the requirement, posed as a
postulate of our model, that the energy eigenstates of the gravitational field
must be normalizable. We found that to satisfy this requirement the horizon
must consist of a finite number of separate pieces, or constituents, each of
them having a discrete area spectrum with an equal spacing. The wave function
of the gravitational field was expressed in terms of the boost angles
associated with these constituents, and the normalizability of the wave
function put certain restrictions to the boost angle, and hence the velocity,
of the observer. Since velocity is relative, the restrictions to the velocity
of the observer may be viewed as restrictions to the velocity of the material
particles with respect to the observer.

The restrictions for the velocities of the material particles bring along
restrictions to the possible energies of the particles. In other words, the
energy of a material particle has a certain absolute upper bound. For protons,
for example, this upper bound was found to be $1.1\times 10^{21}eV$. The
result is interesting, because the very highest energies ever measured for
protons in cosmic rays have been around $10^{19}eV$, some of them being even
$10^{20}eV$. This is roughly of the same order of magnitude as is the highest
possible energy of the proton according to our model, still being very much
less than the Planck energy, which is around $10^{28}eV$.

It must be emphasized once more again that our results have been obtained by
means of a straightforward application of the standard rules of quantum
mecanics to Einstein’s general relativity. No doubt, our tentative model of
quantum gravity still has several shortcomings, such as our decision to
exclude the negative area eigenvalues ”by hand”. Nevertheless, our model
certainly has an advantage that it really makes a concrete and precise
prediction, whose validity may be tested by means of currently available
methods.

Data Availability Statement

The relevant data needed for the understanding of this paper is all included
either in the references listed in below, or in the calculations and the
analysis worked out in the text.

## References

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  * [2] D. J. Bird et al. Phys. Rev. Lett. 71 (1993) 3401.
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