Citation: Hatefi, E.; Hatefi, A.
Estimation of Critical Collapse
Solutions to Black Holes with
Nonlinear Statistical Models.
Mathematics 2022, 10, 4537.
https://doi.org/10.3390/
math10234537
Academic Editor: Stoytcho Stoyanov
Yazadjiev
Received: 25 October 2022
Accepted: 28 November 2022
Published: 30 November 2022
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Article
Estimation of Critical Collapse Solutions to Black Holes with
Nonlinear Statistical Models
Ehsan Hatefi
1,2,
* and Armin Hatefi
3
1
GRAM Research Group, Department of Signal Theory and Communications, University of Alcala,
28805 Alcala de Henares, Spain
2
Scuola Normale Superiore and I.N.F.N, Piazza dei Cavalieri 7, 56126 Pisa, Italy
3
Department of Mathematics and Statistics, Memorial University of Newfoundland,
St. John’s, NL A1C 5S7, Canada
* Correspondence: ehsanhatefi@gmail.com
Abstract:
The self-similar gravitational collapse solutions to the Einstein-axion–dilaton system have
already been discovered. Those solutions become invariants after combining the spacetime dilation
with the transformations of internal SL(2, R). We apply nonlinear statistical models to estimate the
functions that appear in the physics of Black Holes of the axion–dilaton system in four dimensions.
These statistical models include parametric polynomial regression, nonparametric kernel regression
and semi-parametric local polynomial regression models. Through various numerical studies, we
reached accurate numerical and closed-form continuously differentiable estimates for the functions
appearing in the metric and equations of motion.
Keywords: mathematical physics; black holes; statistical analysis
MSC: 46N55
1. Introduction
As the end state of gravitational collapse, black holes are defined by their mass, angular
momentum and their charge. M. Choptuik [
1
] explored the so-called critical phenomena
in gravitational collapse, as well as Choptuik scaling. He made a breakthrough in the
subject of numerical relativity. Indeed, Choptuik scaling [
1
,
2
] is a property that occurs in
various systems that experience gravitational collapse. He discovered that there might be
a fourth universal quantity that establishes the critical collapse. Choptuik followed the
study of the spherically symmetric collapse of scalar fields and explored a critical behaviour
that demonstrates the discrete spacetime self-similarity. By taking the amplitude of the
scalar field
p
, he derived a critical value
p
crit
where a black hole forms as
p
exceeds
p
crit
.
Furthermore, as
p
goes beyond the threshold, the mass of the black hole
M
bh
illustrates the
scaling law
M
bh
(p) ∝ (p − p
crit
)
γ
, (1)
where the Choptuik exponent was found to be
γ '
0.37 [
1
] in four dimensions and for a
real scalar field. Various numerical computations with different matter content have also
been discovered [3–7].
Motivated by string theory, the axion–dilaton system can also experience the same
gravitational collapse process. The study of the Choptuik phenomenon in the axion–dilaton
system was initiated in [
8
–
10
]. The AdS/CFT correspondence [
11
–
13
] is viewed as the first
motivation to investigate critical collapse solutions, especially for the axion–dilaton system.
The AdS/CFT correspondence correlates the critical exponent and the imaginary part of
quasi normal modes, as well as the dual conformal field theory [
14
]. The second motivation
relies on the holographic description of black hole formation [
15
], particularly in the physics
of black holes and their implications [
16
–
18
]. From the IIB string theory point of view, we
Mathematics 2022, 10, 4537. https://doi.org/10.3390/math10234537 https://www.mdpi.com/journal/mathematics
