# Barrow entropy and spacetime foam

Yu.L. Bolotin  National Science Center Kharkov Institute of Physics and
Technology, 1 Akademicheskaya str., Kharkov, 61108, Ukraine  V.V.Yanovsky
<EMAIL_ADDRESS>V. N. Karazin Kharkiv National University, 4 Svobody
Sq., Kharkiv, 61022, Ukraine  Institute for Single Crystals, NAS Ukraine, 60
Nauky Ave., Kharkov, 31001, Ukraine

###### Abstract

Quantum gravitational effects, on the one hand, lead to a limitation in the
accuracy of measuring spatial and time intervals, and, on the other hand, they
generate a discrete of spacetime structure (quantum foam). The common source
of both measurement limitations and discreteness of space-time are quantum
fluctuations, so their characteristics must be related to each other. We study
such a relationship using Barrow entropy as a source of fractal space-time
structure. The minimum inaccuracy in measuring space-time intervals is
expressed through the Barrow entropy parameter. The connection between the
level of fractality and the speed of information processing is considered.

Barrow entropy, fractal structure, spacetime, horizon, quantum fluctuations

##  I Introduction

Planck was the first to propose combining the fundamental constants $c,\hbar$,
which are independent of dynamics, with a constant $G$ that describes the
gravitational interaction in the nonrelativistic limit. This made it possible
to construct a fundamental unit of length $l_{p}=\sqrt{\frac{\hbar
G}{c^{3}}}$.

There are two ways to include the fundamental length in the structure of
space-time and, accordingly, two types of physical models [1, 2].

The currently dominant approach treats space-time as continuous (a pseudo-
Riemannian manifold), but does not allow physical measurements of lengths
below some scale, much smaller than any scale currently available. In models
of this type, the Planck length plays role of the fundamental length. An
alternative approach is to consider space-time to be inherently fundamentally
discrete. The implementation of such an approach, even the classical level, is
a difficult task, requiring a complete restructuring of the mathematical
apparatus [3, 4, 5].

As an intermediate stage of the transition from continuous to discrete space-
time, we can consider the transformation of smooth surfaces into fractal ones.
Even on the smallest scales currently observable, spacetime appears smooth and
structureless. However, the opinion that continuous space-time is just an
idealization is gradually winning out. One of the main reasons forcing us to
abandon the concept of continuum is the existence of quantum fluctuations. It
is natural to assume that quantum effects on Planck scales transform
continuous space-time into a complex (turbulent) structure, called space-time
foam (quantum foam) [6].

This restructuring of space-time structure is especially important in the
vicinity of the black hole event horizon or cosmological horizons. The surface
area of the horizon, which obviously changes during such a restructuring,
determines the dynamics and thermodynamics of the object limited by the
horizon.

##  II Measurement limitations

Taking into account quantum gravitational effects requires a transition from
the Heisenberg uncertainty principle to the generalized uncertainty principle
(GUP) [7, 8, 9, 10].

According to GUP, uncertainty in position $\Delta x$ and momentum in $\Delta
p$ one dimension are related by

$\Delta x\geq\frac{\hbar}{\Delta p}+l_{P}^{2}\frac{\Delta p}{\hbar}$ (1)

The dramatic consequence of taking gravity into account in the measurement
process is the emergence of such a fundamental concept as the minimum length,
$\Delta x_{min}\approx l_{P}$. The physics of this parameter is extremely
simple. At small photon momenta (large wavelengths the localization of the
measured object will be bad. For large photon momenta, its gravitational
interaction with the measured object will again negatively affect the process
of measurements. Between these two extremes, we can choose the photon
momentum, which optimizes the measurement process.

The existence of a minimum length and, as a consequence, the discrete
structure of space is, on the one hand, an almost unlimited source of new
effects that are absent in the spatial continuum, and, on the other hand, it
allows us to discover previously unknown connections between already studied
phenomena. In addition, any theory with a minimum length inevitably
contradicts a number of traditional formulations of both quantum mechanics and
general relativity.

Taking into account quantum gravitational effects leads to a fundamental
limitation on the accuracy of measuring $\delta l$ length $l$, [11]

$\delta l\geq\left(ll_{P}^{2}\right)^{1/3}$ (2)

We emphasize that restriction (2) is universal and does not depend on the
method of measuring length. Relations (2) can be written in terms of
uncertainty $\delta t=\frac{\delta l}{c}$ arising when measuring the time
interval $t=\frac{l}{c}$

$\delta t\geq\left(tt_{P}^{2}\right)^{1/3}$ (3)

##  III The holographic principle and dynamic lattice

The dependence of the accuracy of length measurement on the characteristic
size of the measured object (2) is equivalent to the holographic principle
[12, 13].

The simplest formulation of the holographic principle contains two statements.
Firstly, all information contained in a certain region of space can be
recorded (presented) on the border of this region, (a holographic screen).
Secondly, the theory at the boundary of the studied region of space should
contain no more than one degree of freedom per Planck area. In other words,
the total number of degrees of freedom N obeys the inequality

$N\leq\frac{A}{l_{P}^{2}}$ (4)

where $A$ is the area of the horizon.

Figure 1: The division of volume into unit cells.

The holographic principle as a strict statement is valid only for black holes.
In other cases, it is just a hypothesis. Therefore, it is of interest to
answer the question: is the holographic principle valid in a theory that
combines quantum mechanics and gravity? On the one hand, such a theory
generates a minimum length and provides a natural transition to a
3-dimensional spatial lattice. But, on the other hand, for the validity of the
holographic principle it is required that all volumetric information can be
recorded on a 2-dimensional surface lattice. Let’s make sure that the
inclusion of gravitational effects provides effective 2-dimensionality [14,
15]. To do this, consider some 3-dimensional volume with a characteristic
linear size $l$. In a theory with a minimum length $l_{min}\approx l_{P}$, it
is natural to assume that the volume is divided into cells of size
$l_{P}^{3}$. However, such a conclusion is erroneous. Due to a fundamental
limitation $\delta l\geq(ll_{P}^{2})^{1/3}$, the minimum edge of a cell into
which a cube with an edge $l$ can be divided is (see Fig. 1)

Thus, spacetime foam leads to the formation of a dynamic lattice with a period
depending on the size of the lattice. Therefore, the maximum number of cells
in such a lattice

$N=\left[\frac{l}{\left(ll_{P}^{2}\right)^{1/3}}\right]^{1/3}=\frac{l^{2}}{l_{P}^{2}}$
(5)

If we associate one degree of freedom of the system with each cell, then the
resulting relation reproduces the holographic principle. The connection
between the “quantization” of the horizon area and the holographic principle
is directly determined by the evolution of horizon. Let a mass $\delta m$ be
thrown over the horizon. It is clear that this mass should be enough to
increase the area of the horizon by the Planck unit area. Then

$l_{P}^{2}\leq\left(\frac{2G}{c^{2}}\right)^{2}\left[(m+\delta
m)^{2}-m^{2}\right]$ (6)

It is convenient to replace the Planck length with the Planck mass. Then
inequality (6) will take the form

$\delta m^{2}+2m\delta m-m_{P}^{2}\geq 0$ (7)

This, in turn, means that the radius of the black hole is discrete

$r_{g,n+1}=\sqrt{r_{g,n}^{2}+r_{g,P}^{2}}$ (8)

where $r_{g,P}=\frac{2Gm_{P}}{c^{2}}$ is the quantum of change in the radius
of the black hole. Considering this relation is a recurrence equation with
respect to the variable $n$, you can look for its solution

$r_{g,n}=\sqrt{const+nr_{g,P}^{2}}$

where $const$ is an arbitrary constant that determines the minimum radius of
the black hole. If $const=0$ or $n\gg 1$ then

$r_{g,n}\propto\sqrt{n}$ (9)

This result actually reproduces the holographic principle in the form (5).

##  IV Barrow entropy and spacetime foam

Barrow [16] proposed a phenomenological model of the formation of the fractal
structure of the horizon for a stationary black hole. Burrow’s reasoning was
as follows. The surface area of a Schwarzschild black hole $A_{g}=4\pi
r_{g}^{2}$ determines its the entropy
$S=A_{g}\frac{c^{3}}{4G}\approx\frac{A_{g}}{A_{Pl}}$. The entropy is
proportional to the number of Planck squares on the surface of the horizon.
Consequently, quantum gravitational effects leading to the fractal structure
of the horizon can significantly increase its entropy compared to the standard
Bekenstein entropy. Of course, the increase in the surface area of a black
hole during its evolution is consistent with the second law of thermodynamics.

Abstracting from the specific model of the black hole horizon, the area of the
fractal surface of the horizon as a function of the gravitational radius
$r_{g}$ behaves as $r_{g}^{2+\Delta}$, where $0\leq\Delta\leq 1$. The
transition $(\Delta=0)\to(\Delta=1)$ realizes a transformation from a smooth
2-dimensional surface to 3–dimensional volume due to quantum deformation of
the horizon surface. In other words, $\Delta=0$ it corresponds to the simplest
structure of the horizon, and $\Delta=1$ corresponds to the greatest
complexity of the horizon surface.

The fractality of the horizon surface generates the corresponding entropy
(Barrow entropy)

$S_{B}=\left(\frac{A}{4A_{P}}\right)^{1+\frac{\Delta}{2}}$ (10)

where $A$ is the surface area of the horizon, $A_{P}$ is the Planck area, and
$\Delta$ is a free parameter of the model $0\leq\Delta\leq 1$. In the
undeformed case $(\Delta=0)$, the Burrow entropy coincides with the Bekenstein
entropy, and the value $\Delta=1$ corresponding to the maximum deformation,
leading to an increase in the fractal dimension of the horizon surface by one

$S\propto\frac{A}{A_{Pl}}\to
S\propto\frac{A^{\frac{3}{2}}}{A_{Pl}^{\frac{3}{2}}}$ (11)

It seems natural to connect the degree of fractality of the horizon surface
(parameter $\Delta$) with the period of the holographic lattice $\delta l$,
determined by relation (2). Such a connection must exist, since both the
fractal structure and space-time discreteness are generated by quantum
gravitational effects.

In any effective quantum field theory defined in a spatial region with a
characteristic size $L$ and using ultraviolet cutoff $\Lambda$, entropy
$S\propto L^{3}\Lambda^{3}$. If we assume that the entropy of any object of
linear size $l$ in such a theory should be less than the entropy of a black
hole of the same size, then

$S\leq S_{BH}\approx\left(\frac{l}{l_{P}}\right)^{2}$ (12)

It seems natural to identify the inverse scale of UV cutoff $\Lambda^{-1}$
with a limiting limitation on the accuracy of length measurements . Then the
relation (12) take the form

$\frac{l^{3}}{\delta l^{3}}\leq\left(\frac{l}{l_{P}}\right)^{2}$ (13)

Let us perform the same procedure for the Barrow entropy

$\frac{l^{3}}{\delta l^{3}}\leq\left(\frac{l}{l_{P}}\right)^{2+\Delta}$ (14)

From here, we will find generalization of the fundamental inequality (2) for
the Burrow entropy

$\delta l\geq\left(l^{1-\Delta}l_{P}^{2+\Delta}\right)^{1/3}$ (15)

Similarly, relation (3) is transformed into

$\delta t\geq\left(t^{1-\Delta}t_{P}^{2+\Delta}\right)^{1/3}$ (16)

Let us consider limiting cases for these relations. In the case $\Delta=0$ we
reproduce fundamental inequalities (2), (3). Moreover, in the case $\Delta=1$,
the accuracy of measuring space-time intervals ceases to depend on the size of
the measured interval, $\delta l\delta t\geq l_{P}t_{P}$.

It is convenient to go to dimensionless variables

$L\equiv\frac{l}{l_{P}},\,\delta\bar{l}\equiv\frac{\delta l}{l_{P}}$ (17)

$T\equiv\frac{t}{t_{P}},\,\delta\bar{t}\equiv\frac{\delta t}{t_{P}}$ (18)

In these variables, inequalities (15), (16) take the form

$\delta\bar{l}\geq\left(L^{1-\Delta}\right)^{1/3}$ (19)
$\delta\bar{t}\geq\left(T^{1-\Delta}\right)^{1/3}$ (20)

The physical meaning of the obtained limit relations is simple. For $\Delta=0$
we are dealing with a smooth surface of the horizon and, as a consequence, we
reproduce the inequalities (2), (3) generated by the Bekenstein entropy. For
the case $\Delta=1$ (maximum horizon deformation) $\delta l\delta t$ is
transformed into the minimal size of self-similarity $l_{P}t_{P}$.

Burrow’s concept of entropy can be extended to cosmological horizons by
introducing holographic dark energy with density $\rho\sim H^{2-\Delta}$ [17,
18, 19, 20, 21]. In this case, the parameter $\Delta$ can be expressed through
the current values of cosmographic parameters $q_{0}j_{0}$ [22, 23]

$\Delta=2\left[\frac{(1-j_{0})}{(1+q_{0})(1-2q_{0})}+1\right]$ (21)

where $q$ and $j$ are deceleration and jerk parameters.

Figure 2: Measurement accuracy as a function of the maximum duration of the
time interval.

Note that in SCM $j=1$, therefore $\Delta=2$, that goes beyond the permissible
values $\Delta$. In SCM, dark energy is realized in the form of a cosmological
constant, and is naturally $\rho_{e}=CH^{2-\Delta}$ for $\Delta=2$ transformed
into a constant. In this case entropy (it cannot be called Burrow entropy, for
which $0\leq\Delta\leq 1$)
$S_{B}=\left(\frac{A}{A_{0}}\right)^{1+\frac{\Delta}{2}}$ [24, 25]. Note that
the Barrow entropy represents a special case of the 4-parameter construction
that can reproduce the range of entropies used for horizon thermodynamics [26,
27].

##  V Application examples

As Burrow noted [16], based on an analysis of the process of evaporation of
black holes, the deformation of the horizon by quantum fluctuations should
accelerate the evaporation process. We will consider this statement from a
more general point of view.In its meaning, a time interval measured with a
given accuracy represents the time of correct functioning of a clock or, in
other words, its lifetime. Therefore, using inequality (3) we come to a far
from obvious conclusion: the more accurate the clock, the less time it
operates with a given accuracy. Let’s test this concept using a black hole
with Bekenstein entropy ($\Delta=0$) as a clock. Then the accuracy $\delta t$
of such a device naturally choose $r_{g}/c$. Now, using the upper limit of
inequality (3), we determine the lifetime of such a device (i.e., black hole)

$T_{BH}=\left(\frac{r_{g}}{c}t_{P}^{-2/3}\right)^{3}=\frac{8m^{3}G^{2}}{\hbar
c^{4}}$ (22)

The resulting lifetime of a black hole in this unusual way coincides with the
evaporation time of a black hole, which Hawking calculated. The only
difference is in the numerical factor. Turning to Burrow entropy, we note that
an increase in the degree of fractality (an increase in the parameter
$\Delta$) leads to a decrease $\delta t$ ( increase in measurement accuracy).
This, in turn, due to inequality (20), will reduce the time interval available
for measurement, i.e. lifetime of the object (see Fig. 2).

Increasing the speed of information processing and the volume of its storage
is a key problem in both fundamental and applied physics. The prospect of
practical implementation of quantum computing opens up new possibilities. It
is obvious that the information characteristics of holographic models (storage
volumes and processing speed) depend on the geometric structure of the horizon
(holographic screen). As we saw above, quantum fluctuations, which lead to
discrete space, play an important role far beyond the Planck scale. So for the
observable Universe ($l\sim 10^{28}$cm) $\delta
l\sim\left(ll_{P}^{2}\right)^{1/3}\sim 10^{-13}$cm. It is an almost
“macroscopic” value of the order of the characteristic nuclear sizes.

The existing physical limitations on the concentration of energy lead to a
direct ban on the processes of excessive concentration of information. These
restrictions can be obtained from the fundamental inequality (3) for the
Bekenstein entropy and inequality (20) in the presence of a fractal structure
of the horizon. Let us apply inequality (20) to assess the efficiency of the
computer. Quantity $\nu=\delta t$ can be interpreted as the operating
frequency, and $I=\frac{t}{\delta t}=\frac{T}{\delta\bar{t}}$ represents the
maximum amount stages of information processing. Then inequality (20) is
transformed into

$I^{1-\Delta}\nu^{2+\Delta}\leq t_{P}^{-(2+\Delta)}$ (23)

For $\Delta=0$ inequality (23) reproduces the well-known [14] constraint.

##  VI Conclusion

Barrow entropy model can be thought of as a way of generating fractal
structure in the form of space-time foam. We have shown that the period of the
dynamic lattice simulating this quantum foam directly depends on the only
model parameter of Barrow entropy and have found this dependence. Limiting
versions of this dependence were analyzed, corresponding to the undeformed
horizon surface and the maximum deformed surface by quantum fluctuations. The
inequality is used to estimate the lifetime of an object operating with a
specified accuracy of changes in time intervals. In relation to a black hole,
our assessment confirms Barrow’s conclusion [16]: an increase in the degree of
fractality reduces the time of evaporation of a black hole. The influence of
quantum fluctuations on the information processing process is briefly
considered.

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