# A Proof of Merca’s Conjectures on Sums of Odd Divisor Functions

Kaya Lakein and Anne Larsen  Department of Mathematics, Stanford University,
Stanford, CA 94305<EMAIL_ADDRESS>Department of Mathematics, Harvard
University, Cambridge, MA 02138<EMAIL_ADDRESS>
###### Abstract.

In a recent paper, Merca posed three conjectures on congruences for specific
convolutions of a sum of odd divisor functions with a generating function for
generalized $m$-gonal numbers. Extending Merca’s work, we complete the proof
of these conjectures.

Euler’s partition function $p(n)$ is defined by the number of partitions of
any nonnegative integer $n$, and its generating function is given by

$\displaystyle\sum_{n=0}^{\infty}p(n)q^{n}=\prod_{k=1}^{\infty}\frac{1}{1-q^{k}},\quad\absolutevalue{q}<1.$

The properties of the function $p(n)$, such as its asymptotic behavior and its
parity, have been an object of study for a long time. For instance, Ballantine
and Merca [1] recently made a conjecture on when

$\sum_{ak+1\text{ is a square}}p(n-k)$

is odd, which was proved by Hong and Zhang [2].

The function $p(n)$ is linked to the divisor function

$\sigma(n):=\sum_{d\mid n}d,$

whose generating function is given by

$\displaystyle\sum_{n=1}^{\infty}\sigma(n)q^{n}=\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{n}}.$

In particular, $p(n)$ and $\sigma(n)$ satisfy the following convolution
identities, which differ only in the values of $p(0)$ and $\sigma(0)$:

$\displaystyle\sum_{k=-\infty}^{\infty}(-1)^{k}p(n-P_{5}(k))$
$\displaystyle=\delta_{0,n},\text{ with }p(0)=1,$
$\displaystyle\sum_{k=-\infty}^{\infty}(-1)^{k}\sigma(n-P_{5}(k))$
$\displaystyle=0,\text{ with }\sigma(0)\text{ replaced by }n,$

where $\delta_{ij}$ is the Kronecker delta, and $P_{m}(k)$ is the $k$th
generalized $m$-gonal number

(1) $\displaystyle
P_{m}(k):=\left(\frac{m}{2}-1\right)k^{2}-\left(\frac{m}{2}-2\right)k.$

Motivated by these identities as well as the fact that the divisor functions
$\sigma(n)$ and

$\sigma_{\mathrm{odd}}(n):=\sum_{\begin{subarray}{c}d\mid n\\\ d\text{
odd}\end{subarray}}d,$

where $\sigma_{\mathrm{odd}}(n):=0$ for $n\leq 0$, have the same parity, Merca
recently studied the relationship between $\sigma_{\mathrm{odd}}(n)$ and the
generalized $m$-gonal numbers. More specifically, he investigated for which
positive integers $m$ the following congruences hold for all
$n\in\mathbb{Z}^{+}$:

(2) $\displaystyle\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(n-P_{m}(k))$
$\displaystyle\equiv\begin{dcases}n\allowbreak\mkern 8.0mu({\operator@font
mod}\mkern 6.0mu2)&\text{if }n=P_{m}(j),\,j\in\mathbb{Z},\\\
0\allowbreak\mkern 8.0mu({\operator@font mod}\mkern
6.0mu2)&\text{otherwise},\\\ \end{dcases}$ (3)
$\displaystyle\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(n-P_{5}(k))$
$\displaystyle\equiv\begin{dcases}n\allowbreak\mkern 8.0mu({\operator@font
mod}\mkern 6.0mum)&\text{if }n=P_{5}(j),\,j\in\mathbb{Z},\\\
0\allowbreak\mkern 8.0mu({\operator@font mod}\mkern
6.0mum)&\text{otherwise},\end{dcases}$ (4)
$\displaystyle\sum_{k=-\infty}^{\infty}(-1)^{P_{3}(-k)}\sigma_{\mathrm{odd}}(n-P_{5}(k))$
$\displaystyle\equiv\begin{dcases}(-1)^{P_{3}(-j)}\cdot n\allowbreak\mkern
8.0mu({\operator@font mod}\mkern 6.0mum)&\text{if
}n=P_{5}(j),\,j\in\mathbb{Z},\\\ 0\allowbreak\mkern 8.0mu({\operator@font
mod}\mkern 6.0mum)&\text{otherwise}.\end{dcases}$

In particular, Merca posed the following conjectures:

######  Conjecture.

The following are true:

  1. (i)

The congruence (2) holds for all $n\in\mathbb{Z}^{+}$ if and only if
$m\in\\{5,6\\}$.

  2. (ii)

The congruence (3) holds for all $n\in\mathbb{Z}^{+}$ if and only if
$m\in\\{2,3,6\\}$.

  3. (iii)

The congruence (4) holds for all $n\in\mathbb{Z}^{+}$ if and only if
$m\in\\{2,4\\}$.

Merca showed the if condition for each of these conjectures. Using his work,
we obtain the following theorem:

######  Theorem 1.

Merca’s conjectures are true.

###### Proof.

We begin by proving (ii). Merca showed that (3) holds if $m\in\\{2,3,6\\}$ [3,
Theorem 3], hence it suffices to show that if $m\notin\\{2,3,6\\}$, then there
exists some $n\in\mathbb{Z}^{+}$ such that $n\neq P_{5}(j)$ for all
$j\in\mathbb{Z}$ and

(5) $\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(n-P_{5}(k))\not\equiv
0\allowbreak\mkern 8.0mu({\operator@font mod}\mkern 6.0mum).$

Since $\sigma_{\mathrm{odd}}(n-P_{5}(k))=0$ whenever $n-P_{5}(k)\leq 0$, the
sum in (5) is in fact finite, and we easily compute
$\sum_{k}\sigma_{\mathrm{odd}}(3-P_{5}(k))=6$, where $3\neq P_{5}(j)$ for
$j\in\mathbb{Z}$. Thus, (5) holds unless $6\equiv 0\allowbreak\mkern
8.0mu({\operator@font mod}\mkern 6.0mum)$. But this is the case only if
$m\in\\{2,3,6\\}$.

Next, we prove (iii). Again, Merca proves that (4) holds if $m\in\\{2,4\\}$
[3, Theorem 4], hence it suffices to show that if $m\notin\\{2,4\\}$, then
there exists some $n\in\mathbb{Z}^{+}$ such that $n\neq P_{5}(j)$ for all
$j\in\mathbb{Z}$ and

(6)
$\sum_{k=-\infty}^{\infty}(-1)^{P_{3}(-k)}\sigma_{\mathrm{odd}}(n-P_{5}(k))\not\equiv
0\allowbreak\mkern 8.0mu({\operator@font mod}\mkern 6.0mum).$

We compute $\sum_{k}(-1)^{P_{3}(-k)}\sigma_{\mathrm{odd}}(3-P_{5}(k))=4$,
where $3\neq P_{5}(j)$ for $j\in\mathbb{Z}$, and so (6) holds unless $4\equiv
0\allowbreak\mkern 8.0mu({\operator@font mod}\mkern 6.0mum)$. But this is the
case only if $m\in\\{2,4\\}$.

Finally, we prove (i). Since $\sigma_{\mathrm{odd}}(n)$ is odd if and only if
$n$ is a square or a twice square (see [3, p. 3]), we have that

(7)
$\sum_{n=1}^{\infty}\sigma_{\mathrm{odd}}(n)q^{n}\equiv\sum_{n=1}^{\infty}q^{n^{2}}+\sum_{n=1}^{\infty}q^{2n^{2}}\allowbreak\mkern
8.0mu({\operator@font mod}\mkern 6.0mu2).$

The $n$th coefficient of

$\Big{(}\sum_{\ell=1}^{\infty}\sigma_{\mathrm{odd}}(\ell)q^{\ell}\Big{)}\Big{(}\sum_{k=-\infty}^{\infty}q^{P_{m}(k)}\Big{)}=\sum_{\ell=1}^{\infty}\Big{(}\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(\ell)q^{\ell+P_{m}(k)}\Big{)}=\sum_{n=1}^{\infty}\Big{(}\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(n-P_{m}(k))\Big{)}q^{n}$

is given by $\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(n-P_{m}(k))$. On
the other hand, the $n$th coefficient of

$\displaystyle\Big{(}\sum_{\ell=1}^{\infty}q^{\ell^{2}}+\sum_{\ell=1}^{\infty}q^{2\ell^{2}}\Big{)}\Big{(}\sum_{k=-\infty}^{\infty}q^{P_{m}(k)}\Big{)}=\sum_{\begin{subarray}{c}\ell\geq
1\\\
k\in\mathbb{Z}\end{subarray}}\left(q^{\ell^{2}+P_{m}(k)}+q^{2\ell^{2}+P_{m}(k)}\right)$

is given by $a_{m}(n)+b_{m}(n)$, where

$\displaystyle a_{m}(n)=\absolutevalue{A_{m}(n)}$
$\displaystyle:=\\#\\{(\ell,k)\in\mathbb{Z}^{+}\times\mathbb{Z}:\,\ell^{2}+P_{m}(k)=n\\},$
$\displaystyle b_{m}(n)=\absolutevalue{B_{m}(n)}$
$\displaystyle:=\\#\\{(\ell,k)\in\mathbb{Z}^{+}\times\mathbb{Z}:\,2\ell^{2}+P_{m}(k)=n\\}.$

Thus, due to (7) we must have

$\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(n-P_{m}(k))\equiv
a_{m}(n)+b_{m}(n)\allowbreak\mkern 8.0mu({\operator@font mod}\mkern 6.0mu2).$

Suppose first that $m\geq 7$. Then we claim that $P_{m}(0)=0,P_{m}(1)=1$, and
$P_{m}(k)>3$ for all $k\notin\\{0,1\\}$. From (1), it is clear that
$P_{m}(0)=0$ and $P_{m}(1)=1$. To see that $P_{m}(k)>3$ for all
$k\notin\\{0,1\\}$, note that since the leading term of $P_{m}(x)$ is positive
and its vertex is at $0<\frac{m-4}{2m-4}<1$, we have $P_{m}(k)\geq
P_{m}(2)=m\geq 7$ for $k\geq 2$ and $P_{m}(k)\geq P_{m}(-1)=m-3\geq 4$ for
$k\leq-1$.

Now, let $n=3$. Then the above shows that $n$ is not a generalized $m$-gonal
number for $m\geq 7$, and so for (2) to hold, we must have
$\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(3-P_{m}(k))\equiv
0\allowbreak\mkern 8.0mu({\operator@font mod}\mkern 6.0mu2)$. If $(\ell,k)\in
A_{m}(3)$, then $\ell^{2}=3-P_{m}(k)$, so that in particular $\ell^{2}\leq 3$,
which forces $\ell=1$. But then we must have $P_{m}(k)=2$, which we have seen
to be impossible. Hence, $A_{m}(3)$ is empty, and $a_{m}(3)\equiv
0\allowbreak\mkern 8.0mu({\operator@font mod}\mkern 6.0mu2)$. On the other
hand, if $(\ell,k)\in B_{m}(3)$, we must again have $\ell=1$. It follows that
$P_{m}(k)=1$, which is the case if and only if $k=1$. Hence
$B_{m}(3)=\\{(1,1)\\}$, and $b_{m}(3)\equiv 1\allowbreak\mkern
8.0mu({\operator@font mod}\mkern 6.0mu2)$. We conclude that
$\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(3-P_{m}(k))\equiv
a_{m}(3)+b_{m}(3)\equiv 1\not\equiv 0\allowbreak\mkern 8.0mu({\operator@font
mod}\mkern 6.0mu2)$.

Merca showed that (2) holds for $m\in\\{5,6\\}$, and for $m\in\\{1,2\\}$, the
sum in (2) diverges, hence it remains to consider $m\in\\{3,4\\}$. Suppose
first that $m=3$, and note that $P_{3}(k)=\frac{1}{2}(k^{2}+k)$. We have
$3=P_{3}(-3)=P_{3}(2)$, so for (2) to hold, we must have
$\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(3-k)\equiv 3\equiv
1\allowbreak\mkern 8.0mu({\operator@font mod}\mkern 6.0mu2)$. If $(\ell,k)\in
A_{3}(3)$, then $\ell=1$ and $P_{3}(k)=2$, which is impossible. Hence
$A_{3}(3)$ is empty. If $(\ell,k)\in B_{3}(3)$, then $\ell=1$ and
$P_{3}(k)=1$, which is the case if and only if $k\in\\{-2,1\\}$. Thus
$B_{3}(3)=\\{(1,-2),(1,1)\\}$, and
$\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(3-k)\equiv
a_{3}(3)+b_{3}(3)\equiv 0\not\equiv 1\allowbreak\mkern 8.0mu({\operator@font
mod}\mkern 6.0mu2)$.

Finally, suppose $m=4$, and note that $P_{4}(k)=k^{2}$. Since $4=P_{4}(2)$,
for (2) to hold we must have
$\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(4-k)\equiv 4\equiv
0\allowbreak\mkern 8.0mu({\operator@font mod}\mkern 6.0mu2)$. If $(\ell,k)\in
A_{4}(4)$, then either $\ell=1$ and $P_{4}(k)=3$, which is impossible, or
$\ell=2$ and $P_{4}(k)=0$, which is the case if and only if $k=0$. Thus,
$A_{4}(4)=\\{(2,0)\\}$. On the other hand, if $(\ell,k)\in B_{4}(4)$, then
$\ell=1$ and $P_{3}(k)=2$, which is impossible. Thus $B_{4}(4)$ is empty, and
$\sum_{k=-\infty}^{\infty}\sigma_{\mathrm{odd}}(4-k)\equiv
a_{4}(4)+b_{4}(4)\equiv 1\not\equiv 0\allowbreak\mkern 8.0mu({\operator@font
mod}\mkern 6.0mu2)$. ∎

## Acknowledgements

We would like to thank Ken Ono for suggesting this project, and for several
helpful conversations. We thank William Craig and Badri Pandey, as well as the
referee, for their comments on the exposition in this note. Finally, we are
grateful for the generous support of the National Science Foundation (DMS
2002265 and DMS 205118), the National Security Agency (H98230-21-1-0059), the
Thomas Jefferson Fund at the University of Virginia, and the Templeton World
Charity Foundation.

## References

  * [1] Cristina M. Ballantine and Mercia Merca, _Parity of Sums of Partition Numbers and Squares in Arithmetic Progressions_ , Ramanujan J. 44 (2017), 617-630.
  * [2] Letong Hong and Shengtong Zhang, _Proof of the Ballentine-Merca Conjecture and Theta Function Identities Modulo 2_ , https://arxiv.org/abs/2101.09846, 2021.
  * [3] Mircea Merca, _Congruence Identities Involving Sums of Odd Divisors Function_ , Proc. Rom. Acad. Ser. A 22, no. 2 (2021), 119-125.


# On the Karlsson-Nussbaum conjecture for resolvents of nonexpansive mappings

Aleksandra Huczek  Department of Mathematics, Pedagogical University of
Krakow, PL-30-084 Cracow, Poland<EMAIL_ADDRESS>and  Andrzej
Wiśnicki  Department of Mathematics, Pedagogical University of Krakow,
PL-30-084 Cracow, Poland<EMAIL_ADDRESS>
###### Abstract.

Let $D\subset\mathbb{R}^{n}$ be a bounded convex domain and $F:D\rightarrow D$
a $1$-Lipschitz mapping with respect to the Hilbert metric $d$ on $D$
satisfying condition $d(sx+(1-s)y,sz+(1-s)w)\leq\max\\{d(x,z),d(y,w)\\}$. We
show that if $F$ does not have fixed points, then the convex hull of the
accumulation points (in the norm topology) of the family
$\\{R_{\lambda}\\}_{\lambda>0}$ of resolvents of $F$ is a subset of $\partial
D.$ As a consequence, we show a Wolff-Denjoy type theorem for resolvents of
nonexpansive mappings acting on an ellipsoid $D$.

###### Key words and phrases:

Karlsson–Nussbaum conjecture, Wolff–Denjoy theorem, Geodesic space, Hilbert’s
projective metric, Resolvent, Nonexpansive mapping.

###### 2020 Mathematics Subject Classification:

Primary 53C60; Secondary 37C25, 47H09, 51M10

##  1\. Introduction

The study of dynamics of nonlinear mappings started by considering iterates of
holomorphic mappings on one-dimensional bounded domains. In this field, one of
the first theorem is the classical Wolff–Denjoy theorem which describes
dynamics of iteration of holomorphic self-mappings on the unit disc of the
complex plane. It asserts that if $f:\Delta\rightarrow\Delta$ is a holomorphic
map of the unit disc $\Delta\subset\mathbb{C}$ without a fixed point, then
there is a point $\xi\in\partial\Delta$ such that the iterates $f^{n}$
converge locally uniformly to $\xi$ on $\Delta$. Generalizations of this
theorem in different directions have been obtained by numerous authors (see
[1, 6, 9, 14, 15] and references therein). One such generalization was
formulated by Beardon who noticed that the Wolff–Denjoy theorem can be
considered in a purely geometric way depending only on the hyperbolic
properties of a metric and gave its proof using geometric methods (see [4]).
In [5], Beardon extended his approach for strictly convex bounded domains with
the Hilbert metric. Considering the notion of the omega limit set
$\omega_{f}(x)$ as the set of accumulation points of the sequence $f^{n}(x)$
and the notion of the attractor $\Omega_{f}=\bigcup_{x\in D}\omega_{f}(x)$, we
can formulate a generalization of the Wolff-Denjoy theorem known as the
Karlsson-Nussbaum conjecture, which was formulated independently by Karlsson
and Nussbaum (see [10, 14]). This conjecture states that if $D$ is a bounded
convex domain in a finite-dimensional real vector space and $f:D\rightarrow D$
is a fixed point free nonexpansive mapping acting on the Hilbert metric space
$(D,d_{H})$, then there exists a convex set $\Omega\subseteq\partial D$ such
that for each $x\in D$, all accumulation points $\omega_{f}(x)$ of the orbit
$O(x,f)$ lie in $\Omega$.

The aim of this note is to show a variant of the Karlsson-Nussbaum conjecture
for resolvents of nonexpansive ($1$-Lipschitz) mappings. For this purpose we
construct in Section 3 the family of resolvents of a nonexpansive mapping and
prove its main properties: nonexpansivity and the resolvent identity. In the
literature, the resolvents usually occur in the context of Banach spaces or
geodesic spaces that are Busemann convex, see e.g., [3, 17]. Then their
construction is based on the Banach contraction principle. Since a Hilbert
metric space $(D,d_{H})$ is in general not Busemann convex, our construction
of resolvents is a little more complicated and exploits the argument related
to Edelstein’s theorem [8].

In Section 4 we formulate and prove the main theorem of this work. We show
that if $D\subset\mathbb{R}^{n}$ is a bounded convex domain and
$F:D\rightarrow D$ is a fixed point free nonexpansive mapping with respect to
the Hilbert metric $d_{H}$ on $D$ satisfying condition

(D) $d_{H}(sx+(1-s)y,sz+(1-s)w)\leq\max\\{d_{H}(x,z),d_{H}(y,w)\\},$

then the convex hull of the accumulation points of the family
$\\{R_{\lambda}\\}_{\lambda>0}$ of resolvents of $F$ is a subset of $\partial
D$. Since a Hilbert metric space $(D,d_{H})$ is Busemann convex if and only if
$D$ is ellipsoid, we obtain as a corollary a Wolff-Denjoy type theorem for
resolvents of nonexpansive mappings acting on an ellipsoid $D$.

##  2\. Preliminaries

Let $V$ be a finite dimensional real vector space, $D\subset V$ a convex
bounded domain and $(D,d)$ a metric space. A curve $\sigma:[a,b]\rightarrow D$
is said to be geodesic if
$d(\sigma(t_{1}),\sigma(t_{2}))=\left|t_{1}-t_{2}\right|$ for all
$t_{1},t_{2}\in[a,b]$. We will use the same name for the image
$\sigma([a,b])\subset D$ of $\sigma$, denoted by $[\sigma(a),\sigma(b)].$ We
say that $D$ is a geodesic space if every two points of $D$ can be joined by a
geodesic. A map $F:D\rightarrow D$ is called contractive if
$d(F(x),F(y))<d(x,y)$ for any distinct points $x,y\in D$. A map
$F:D\rightarrow D$ is called nonexpansive if for any $x,y\in D,$
$d(F(x),F(y))\leq d(x,y)$.

We recall the definition of the Hilbert metric space. If $x,y\in D$, consider
the straight line passing through $x$ and $y$ that intersects the boundary of
$D$ in precisely two points $a$ and $b$. Assuming that $x$ is between $a$ and
$y$, and $y$ is between $x$ and $b$, we define the cross-ratio metric

$d_{H}(x,y)=\log\bigg{(}\frac{||y-a||\,||x-b||}{||x-a||\,||y-b||}\bigg{)},\qquad
x\neq y.$

Furthermore, we put $d_{H}(x,y)=0$ if $x=y$.

Following Beardon [5] we consider the subsequent lemmas.

######  Lemma 2.1.

Let $D_{1},D_{2}\subset V,\,D_{1}\subset D_{2}$ be bounded convex domains and
$(D_{1},d_{1}),(D_{2},d_{2})$ be Hilbert metric spaces, then $d_{2}\leq
d_{1}$. Furthermore, for distinct points $x,y\in D_{1},d_{1}(x,y)=d_{2}(x,y)$
iff the segment $L_{xy}\cap D_{1}$ coincides with $L_{xy}\cap D_{2}$.

######  Lemma 2.2.

Suppose that $(D,d_{H})$ is a Hilbert metric space, $x_{0}\in D$ and
$l\in[0,1)$. Then the mapping $g(x)=x_{0}+l(x-x_{0})$ is contractive.

###### Proof.

Fix $x_{0}\in D$ and $l\in[0,1)$. Let $x,y\in D$ and consider the straight
line passing through $x$ and $y$ that intersects $\partial D$ in two points
$x^{\prime}$ and $y^{\prime}$ such that $x$ is between $x^{\prime}$ and $y$,
and $y$ is between $x$ and $y^{\prime}$. Take two points
$z^{\prime}=(1-l)x_{0}+lx^{\prime}\in\partial
g(D),w^{\prime}=(1-l)x_{0}+ly^{\prime}\in\partial g(D)$, and note that the
points $z^{\prime},g(x),g(y),w^{\prime}$ are collinear such that $g(x)$ is
between $z^{\prime}$ and $g(y)$, and $g(y)$ is between $g(x)$ and
$w^{\prime}$. Since $g(D)$ lies in a compact subset of $D$, it follows from
Lemma 2.1 that $d_{H}(g(x),g(y))<d_{2}(g(x),g(y))$, where $d_{2}$ denotes the
Hilbert metric in $g(D)$. By definition of the Hilbert metric space we have

$d_{H}(x,y)=\log\bigg{(}\frac{||x^{\prime}-y||\,||x-y^{\prime}||}{||x^{\prime}-x||\,||y-y^{\prime}||}\bigg{)}=\log\bigg{(}\frac{||z^{\prime}-g(y)||\,||w^{\prime}-g(x)||}{||z^{\prime}-g(x)||\,||w^{\prime}-g(y)||}\bigg{)}=d_{2}(g(x),g(y)).$

Therefore we get $d_{H}(g(x),g(y))<d_{H}(x,y)$. ∎

Note that if $D\subset V$ is a bounded convex domain, then the Hilbert metric
$d_{H}$ is locally equivalent to the euclidean norm in $V$. Furthermore, for
any $w\in D$, if $\\{x_{n}\\}$ is a sequence in $D$ converging to
$\xi\in\partial D=\overline{D}\setminus D$, then

$d_{H}(x_{n},w)\rightarrow\infty$

(see [5, 9]). The above property is equivalent to properness of $D$, that is,
every closed and bounded subset of $(D,d_{H})$ is compact. It is not difficult
to show that for $x,y,z\in D$ and $s\in[0,1],$

(C) $d_{H}(sx+(1-s)y,z)\leq\max\\{d_{H}(x,z),d_{H}(y,z)\\}.$

In what follows, we will assume a more restrictive condition: for all
$x,y,z,w\in D$ and $s\in[0,1],$

(D) $d_{H}(sx+(1-s)y,sz+(1-s)w)\leq\max\\{d_{H}(x,z),d_{H}(y,w)\\}.$

##  3\. Resolvents of nonexpansive mappings

In this section we describe the construction of a resolvent of a nonexpansive
mapping acting on a Hilbert metric space. Let $D\subset V$ be a convex bounded
domain and $F:D\rightarrow D$ a nonexpansive mapping with respect to the
Hilbert metric $d$ on $D$. Recall that the topology of $(D,d)$ coincides with
the Euclidean topology and $(D,d)$ is proper metric space, that is, every
closed ball $\bar{B}(x_{0},r),\,x_{0}\in D,$ is compact. We fix $x\in D$,
$\lambda>0$, and define a mapping

$G_{x,\lambda}(y)=\frac{1}{1+\lambda}x+\frac{\lambda}{1+\lambda}F(y),\qquad
y\in D.$

It follows from Lemma 2.2 that $G_{x,\lambda}$ is contractive.

We show that $G_{x,\lambda}(D)$ is bounded. For this purpose, we select $w\in
G_{x,\lambda}(D)$. Note that there exists $y\in D$ such that
$w=\frac{1}{1+\lambda}x+\frac{\lambda}{1+\lambda}F(y)$. We show that
$B(w,\frac{1}{1+\lambda}d)\subset D$, where $d=\inf_{v\in\partial D}||v-x||$.
Choose any $w^{\prime}\in B(w,\frac{1}{1+\lambda}d)$. Then there exists $z\in
D$ such that $w^{\prime}=\frac{1}{1+\lambda}z+\frac{\lambda}{1+\lambda}F(y)$.
Note that

$||w-w^{\prime}||=\bigg{|}\bigg{|}\frac{1}{1+\lambda}x+\frac{\lambda}{1+\lambda}F(y)-\frac{1}{1+\lambda}z+\frac{\lambda}{1+\lambda}F(y)\bigg{|}\bigg{|}=\frac{1}{1+\lambda}||x-z||.$

If $||w-w^{\prime}||<\frac{1}{1+\lambda}$, then
$||x-z||=(1+\lambda)||w-w^{\prime}||<d$. It implies that $z\in D$ and hence
$w^{\prime}\in D$. It follows that for all $w\in G_{x,\lambda}(D),$

(3.1) $\inf_{v\in\partial D}||v-w||\geq\frac{1}{1+\lambda}d.$

Take a sequence $\\{w_{n}\\}\subset G_{x,\lambda}(D)$. Since $\overline{D}$ is
compact in the Euclidean topology, there exists a subsequence
$\\{w_{n_{k}}\\}$ and $x_{0}\in\overline{D}$ such that
$||w_{n_{k}}-x_{0}||\rightarrow 0$, if $k\rightarrow\infty$. It follows from
(3.1) that $x_{0}\in D$, and hence $d(w_{n_{k}},x_{0})\rightarrow 0$ since the
topology of $(D,d)$ coincides with the Euclidean topology. Therefore,
$G_{x,\lambda}(D)$ is bounded in $(D,d)$ and by properness of $D$ we have that
$\overline{G_{x,\lambda}(D)}$ is compact in $(D,d)$.

Note that $D\supset G_{x,\lambda}(D)\supset G_{x,\lambda}^{2}(D)\supset...$,
which means that the orbits of $G_{x,\lambda}$ are bounded. Fix $y\in D$.
Since $\overline{G_{x,\lambda}(D)}$ is compact, there exists a subsequence
$\\{G_{x,\lambda}^{n_{k}}(y)\\}$ of $\\{G_{x,\lambda}^{n}(y)\\}$ converging to
some $z\in D$. Let

$d_{n}=d(G_{x,\lambda}^{n}(y),G_{x,\lambda}^{n+1}(y)).$

Since $G_{x,\lambda}$ is contractive, the sequence $\\{d_{n}\\}$ is decreasing
and hence it converges to some $\zeta$, as $n\rightarrow\infty$. Hence

$\zeta\leftarrow
d_{n_{k}}=d(G_{x,\lambda}^{n_{k}}(y),G_{x,\lambda}^{n_{k}+1}(y))\rightarrow
d(G_{x,\lambda}(z),z),$

and

$\zeta\leftarrow
d_{n_{k}+1}=d(G_{x,\lambda}^{n_{k}+1}(y),G_{x,\lambda}^{n_{k}+2}(y))\rightarrow
d(G_{x,\lambda}^{2}(z),G_{x,\lambda}(z)).$

We get

$d(G_{x,\lambda}^{2}(z),G_{x,\lambda}(z))=d(G_{x,\lambda}(z),z)=\zeta.$

Since the map $G_{x,\lambda}$ is contractive, $G_{x,\lambda}(z)=z.$ Moreover,
$z$ is the unique fixed point of $G_{x,\lambda}$. Indeed, otherwise if
$z_{1},z_{2}\in D$, $z_{1}\neq z_{2}$ are fixed points of $G_{x,\lambda}$,
then

$d(z_{1},z_{2})=d(G_{x,\lambda}(z_{1}),G_{x,\lambda}(z_{2}))<d(z_{1},z_{2}),$

and we obtain a contradiction. Define $z=R_{\lambda}(x).$ We refer to the
mapping $R_{\lambda}:D\rightarrow D$ as the resolvent of $F.$ We have

$z=G_{x,\lambda}(z)=\frac{1}{1+\lambda}x+\frac{\lambda}{1+\lambda}F(z),\qquad
x\in D,\;\lambda>0,$

and hence

(3.2)
$R_{\lambda}(x)=\frac{1}{1+\lambda}x+\frac{\lambda}{1+\lambda}F(R_{\lambda}(x)),\qquad
x\in D,\,\lambda>0.$

Furthermore, any converging subsequence $G_{x,\lambda}^{m_{k}}(y)$ has the
limit $z$ (the unique fixed point), as $k\rightarrow\infty.$ This gives the
formula:

(3.3) $\lim_{n\rightarrow\infty}G_{x,\lambda}^{n}(y)=R_{\lambda}(x),\qquad
y\in D.$

It turns out that if $(D,d)$ is sufficiently regular, then the resolvent of a
nonexpansive mapping is also nonexpansive.

######  Lemma 3.1.

Let $(D,d)$ be a Hilbert metric space satisfying condition (D),
$F:D\rightarrow D$ a nonexpansive mapping, and $\lambda>0$. Then the resolvent
$R_{\lambda}:D\rightarrow D$ is nonexpansive.

###### Proof.

Fix $z_{0},z_{1},z_{2}\in D$. First we show that
$d(G_{z_{1},\lambda}^{n}(z_{0}),G_{z_{2},\lambda}^{n}(z_{0}))\leq
d(z_{1},z_{2})$ for each $n$. We proceed by induction. For $n=1$, it follows
from condition (C) that

$\displaystyle d(G_{z_{1},\lambda}(z_{0}),G_{z_{2},\lambda}(z_{0}))$
$\displaystyle=$ $\displaystyle
d\bigg{(}\frac{1}{1+\lambda}z_{1}+\frac{\lambda}{1+\lambda}F(z_{0}),\frac{1}{1+\lambda}z_{2}+\frac{\lambda}{1+\lambda}F(z_{0})\bigg{)}$
$\displaystyle\leq$
$\displaystyle\max\\{d(z_{1},z_{2}),d(F(z_{0}),F(z_{0}))\\}=d(z_{1},z_{2}).$

Fix $n\in\mathbb{N}$ and suppose that
$d(G_{z_{1},\lambda}^{n}(z_{0}),G_{z_{2},\lambda}^{n}(z_{0}))\leq
d(z_{1},z_{2})$. Then it follows from (D) that

$\displaystyle
d(G_{z_{1},\lambda}^{n+1}(z_{0}),G_{z_{2},\lambda}^{n+1}(z_{0}))$
$\displaystyle=$ $\displaystyle
d\bigg{(}\frac{1}{1+\lambda}z_{1}+\frac{\lambda}{1+\lambda}F(G_{z_{1},\lambda}^{n}(z_{0})),\frac{1}{1+\lambda}z_{2}+\frac{\lambda}{1+\lambda}F(G_{z_{2},\lambda}^{n}(z_{0}))\bigg{)}$
$\displaystyle\leq$
$\displaystyle\max\\{d(z_{1},z_{2}),d(G_{z_{1},\lambda}^{n}(z_{0}),G_{z_{2},\lambda}^{n}(z_{0}))\\}=d(z_{1},z_{2}).$

Now the formula (3.3) yields

$\displaystyle d(R_{\lambda}(z_{1}),R_{\lambda}(z_{2}))$ $\displaystyle=$
$\displaystyle\lim_{n\rightarrow\infty}d\bigg{(}G_{z_{1},\lambda}^{n}(z_{0}),G_{z_{2},\lambda}^{n}(z_{0})\bigg{)}$
$\displaystyle\leq$ $\displaystyle d(z_{1},z_{2}),$

which shows that $R_{\lambda}$ is a nonexpansive mapping. ∎

We will also use the following property called the resolvent identity.

######  Proposition 3.2.

Suppose that $F:D\rightarrow D$ is a nonexpansive mapping. Then its resolvent
$R_{\lambda}$ satisfies

$R_{\lambda}(x)=R_{\mu}\bigg{(}\frac{\lambda-\mu}{\lambda}R_{\lambda}(x)+\frac{\mu}{\lambda}x\bigg{)},\qquad
x\in D,$

for all $\lambda>\mu>0$.

###### Proof.

Fix $x\in D$ and $\lambda,\mu>0$ such that $\lambda>\mu$. Define

(3.4) $y:=\frac{\lambda-\mu}{\lambda}R_{\lambda}(x)+\frac{\mu}{\lambda}x.$

It follows from (3.2) that there exists the unique point

(3.5) $z:=R_{\mu}(y)=\frac{1}{1+\mu}y+\frac{\mu}{1+\mu}F(R_{\mu}(y)).$

On the other hand, we have

(3.6)
$\tilde{z}:=R_{\lambda}(x)=\frac{1}{1+\lambda}x+\frac{\lambda}{1+\lambda}F(R_{\lambda}(x)).$

From the above and (3.4) we get $\lambda
y-\mu\tilde{z}(1+\lambda)=(\lambda-\mu)\tilde{z}-\lambda\mu F(\tilde{z})$,
which implies

$\tilde{z}=\frac{1}{1+\mu}y+\frac{\mu}{1+\mu}F(\tilde{z}).$

Therefore, from the uniqueness of the construction of the point $z$ and by
(3.5) and (3.6), we have

$R_{\mu}(y)=z=\tilde{z}=R_{\lambda}(x).$

∎

For any $x\in D$, $F:D\rightarrow D$, the set of accumulation points (in the
norm topology) of the sequence $\\{F^{n}(x)\\}$ is called the omega limit set
of $x$ and is denoted by $\omega_{F}(x)$. In a similar way, if
$R_{\lambda}:D\rightarrow D,\lambda>0,$ is a family of resolvents of $F$, we
define

$\omega_{\\{R_{\lambda}\\}_{\lambda>0}}(x)=\\{y\in\overline{D}:\left\|R_{\lambda_{n}}(x)-y\right\|\rightarrow
0\text{ for some increasing sequence
}\\{\lambda_{n}\\},\lambda_{n}\rightarrow\infty\\},$

and the attractor of $\\{R_{\lambda}\\}_{\lambda>0},$

$\Omega_{{}_{\\{R_{\lambda}\\}_{\lambda>0}}}=\bigcup_{x\in
D}\omega_{{}_{\\{R_{\lambda}\\}_{\lambda>0}}}(x).$

######  Lemma 3.3.

Suppose that $F:D\rightarrow D$ is a nonexpansive mapping without fixed points
and $R_{\lambda}:D\rightarrow D$, $\lambda>0$ is a family of resolvents of
$F$. Then $\Omega_{{}_{\\{R_{\lambda}\\}_{\lambda>0}}}\subset\partial D.$

###### Proof.

On the contrary, we suppose that there exists $y\in D$ such that
$\left\|R_{\lambda_{n}}(x)-y\right\|\rightarrow 0$ for some $x\in D$ and an
increasing sequence $\\{\lambda_{n}\\},\lambda_{n}\rightarrow\infty.$ Then

(3.7)
$||R_{\lambda_{n}}(x)-F(R_{\lambda_{n}}(x))||=\frac{1}{1+\lambda_{n}}||x-F(R_{\lambda_{n}}(x))||\rightarrow
0,$

as $n\rightarrow\infty.$ Since the topology of $(D,d)$ coincides with the norm
topology, $F:D\rightarrow D$ is norm-continuous, and hence

$||F(y)-y||\leq||F(y)-F(R_{\lambda_{n}}(x))||+||F(R_{\lambda_{n}}(x))-R_{\lambda_{n}}(x)||+||R_{\lambda_{n}}(x)-y||\rightarrow
0,$

as $n\rightarrow\infty$. Thus $F(y)=y$, and we obtain a contradiction. ∎

##  4\. Main theorem

We begin by recalling one of the fundamental properties of a Hilbert metric
space that allows Karlsson and Noskov to extend Beardon’s Wolff–Denjoy theorem
for bounded strictly convex domains (see [11, Theorem 5.5],[13, Proposition
8.3.3]).

######  Lemma 4.1.

Let $D\subseteq V$ be an open bounded convex set and $d$ a Hilbert metric on
$D$. If $\\{x_{n}\\}$ and $\\{y_{n}\\}$ are convergent sequences in $D$ with
limits $x$ and $y$ in $\partial D$, respectively, and the segment
$[x,y]\nsubseteq\partial D$, then for each $z\in D$ we have

$\lim_{n\rightarrow\infty}[d(x_{n},y_{n})-\max\\{d(x_{n},z),d(y_{n},z)\\}]=\infty.$

We also need the following standard argument that can be found for example in
[7, Lemma 5.4].

######  Lemma 4.2.

Let $(D,d)$ be a separable metric space and let $a_{n}:D\rightarrow\mathbb{R}$
be a nonexspansive mapping for each $n\in\mathbb{N}$. If for every $x\in D$,
the sequence $\\{a_{n}(x)\\}$ is bounded, then there exists a subsequence
$\\{a_{n_{j}}\\}$ of $\\{a_{n}\\}$ such that
$\lim_{j\rightarrow\infty}a_{n_{j}}(x)$ exists for every $x\in D$.

Fix $x_{0}\in D$ and consider a sequence $\\{x_{n}\in D:n\in\mathbb{N}\\}$
contained in $D$. Define $a_{n}(x)=d(x,x_{n})-d(x_{n},x_{0})$ for any
$n\in\mathbb{N}$. Note that

$|a_{n}(y)-a_{n}(x)|\leq d(x,y),$

i.e., $a_{n}$ is nonexpansive and the sequence $\\{a_{n}(y)\\}$ is bounded (by
$d(y,x_{0})$) for every $y\in D$. It follows from Lemma 4.2 that there exists
a subsequence $\\{x_{n_{j}}\\}$ of $\\{x_{n}\\}$ such that
$\lim_{j\rightarrow\infty}a_{n_{j}}(x)$ exists for any $x\in D$, i.e.,

(4.1) $\lim_{j\rightarrow\infty}d(x,x_{n_{j}})-d(x_{n_{j}},x_{0})$

exists for every $x\in D$.

Now we are in a position to prove a variant of the Karlsson–Nussbaum
conjecture for resolvents of nonexpansive mappings.

######  Theorem 4.3.

Let $D\subset V$ be a bounded convex domain. Suppose that $(D,d)$ is a Hilbert
metric space satisfying condition (D) and $R_{\lambda}:D\rightarrow
D,\lambda>0,$ is a family of resolvents of a nonexpansive mapping
$F:D\rightarrow D$ without fixed points. Then
$\operatorname{co}\Omega_{\\{R_{\lambda}\\}_{\lambda>0}}\subseteq\partial D$.

###### Proof.

Suppose on the contrary that there exist $z_{1},\ldots,z_{m}\in
D,\zeta^{1}\in\omega_{\\{R_{\lambda}\\}_{\lambda>0}}(z_{1}),\ldots,\zeta^{m}\in\omega_{\\{R_{\lambda}\\}_{\lambda>0}}(z_{m})$
and $0<\alpha_{1},\ldots,\alpha_{m}<1$ with $\sum_{i=1}^{m}\alpha_{i}=1$ such
that $\sum_{i=1}^{m}\alpha_{i}\zeta^{i}\in D$. Since $F$ does not have fixed
points, it follows from Lemma 3.3 that the omega limit sets
$\omega_{\\{R_{\lambda}\\}_{\lambda>0}}(z_{i})\subseteq\partial D$,
$i=1,\ldots,m,$ and, following [13, Theorem 8.3.11], we can assume that $m\geq
2$ is minimal with the property that $\sum_{i=1}^{m}\alpha_{i}\zeta^{i}\in D$.
It follows that $R_{\lambda_{j}^{i}}(z_{i})\rightarrow\zeta^{i}\in\partial D$
for some increasing sequences
$\\{\lambda_{j}^{i}\\}_{j},\lambda_{j}^{i}\rightarrow\infty$, as
$j\rightarrow\infty,i=1,\ldots,m.$ We put $\zeta=\zeta^{1}$ and
$\eta=\sum_{i=2}^{m}\mu_{i}\zeta^{i}$, where
$\mu_{i}=\frac{\alpha_{i}}{1-\alpha_{1}}$ for $i\in[2,m]$. Let
$\eta^{j}=\sum_{i=2}^{m}\mu_{i}R_{\lambda_{j}^{i}}(z_{i})$ for all $j\geq 1$.
Since $m$ is minimal, we have $\zeta,\eta\in\partial D$ and
$\alpha_{1}\zeta+(1-\alpha_{1})\eta\in D$. Since $D$ is convex, we get
$\alpha\zeta+(1-\alpha)\eta\in D$ for all $\alpha\in(0,1)$. By passing to a
subsequence we can assume from (4.1) that for every $x\in D$ there exists the
limit

(4.2)
$g(x)=\lim_{j\rightarrow\infty}d(x,R_{\lambda_{j}^{1}}(z_{1}))-d(R_{\lambda_{j}^{1}}(z_{1}),z_{1}).$

Since

$\bigg{|}\bigg{|}y-\frac{\lambda_{j}^{1}-\mu}{\lambda_{j}^{1}}y-\frac{\mu}{\lambda_{j}^{1}}z_{1}\bigg{|}\bigg{|}=\frac{\mu}{\lambda_{j}^{1}}||y-z_{1}||\rightarrow
0,\ \ \ \mbox{as}\;\lambda_{j}^{1}\rightarrow\infty,$

and topologies of $(D,d)$ and $(\overline{D},||\cdot||)$ coincide on $D$, we
have

(4.3)
$d\bigg{(}y,\frac{\lambda_{j}^{1}-\mu}{\lambda_{j}^{1}}y+\frac{\mu}{\lambda_{j}^{1}}z_{1}\bigg{)}\rightarrow
0,$

if $\lambda_{j}^{1}\rightarrow\infty.$ According to Lemma 3.1, Proposition
3.2, (4.3) and condition (C), we get

$\displaystyle g(R_{\mu}(y))$ $\displaystyle=$
$\displaystyle\lim_{j\rightarrow\infty}d(R_{\mu}(y),R_{\lambda_{j}^{1}}(z_{1}))-d(R_{\lambda_{j}^{1}}(z_{1}),z_{1})$
$\displaystyle=$
$\displaystyle\lim_{j\rightarrow\infty}d\bigg{(}R_{\mu}(y),R_{\mu}\bigg{(}\frac{\lambda_{j}^{1}-\mu}{\lambda_{j}^{1}}R_{\lambda_{j}^{1}}(z_{1})+\frac{\mu}{\lambda_{j}^{1}}z_{1}\bigg{)}\bigg{)}-d(R_{\lambda_{j}^{1}}(z_{1}),z_{1})$
$\displaystyle\leq$
$\displaystyle\limsup_{j\rightarrow\infty}d\bigg{(}y,\frac{\lambda_{j}^{1}-\mu}{\lambda_{j}^{1}}R_{\lambda_{j}^{1}}(z_{1})+\frac{\mu}{\lambda_{j}^{1}}z_{1}\bigg{)}-d(R_{\lambda_{j}^{1}}(z_{1}),z_{1})$
$\displaystyle=$
$\displaystyle\lim_{j\rightarrow\infty}d\bigg{(}\frac{\lambda_{j}^{1}-\mu}{\lambda_{j}^{1}}y+\frac{\mu}{\lambda_{j}^{1}}z_{1},\frac{\lambda_{j}^{1}-\mu}{\lambda_{j}^{1}}R_{\lambda_{j}^{1}}(z_{1})+\frac{\mu}{\lambda_{j}^{1}}z_{1}\bigg{)}-d(R_{\lambda_{j}^{1}}(z_{1}),z_{1})$
$\displaystyle\leq$
$\displaystyle\lim_{j\rightarrow\infty}d(y,R_{\lambda_{j}^{1}}(z_{1}))-d(R_{\lambda_{j}^{1}}(z_{1}),z_{1})$
$\displaystyle=$ $\displaystyle g(y).$

From the above we have $g(R_{\mu}(y))\leq g(y)\leq d(y,z_{1})$ for every $y\in
D$ and $\mu>0$. It follows from (C) that for any $k\in\mathbb{N}$,

$g(\eta^{k})=g\bigg{(}\sum_{i=2}^{m}\,\mu_{i}R_{\lambda_{k}^{i}}(z_{i})\bigg{)}\leq\max_{i=2,\ldots,m}g(z_{i})\leq\max_{i=2,\ldots,m}d(z_{i},z_{1})=M.$

Consequently, by diagonal method, there exists a subsequence
$\lambda_{j_{1}}^{1}\leq\lambda_{j_{2}}^{1}\leq\ldots\leq\lambda_{j_{k}}^{1}\leq\ldots$
of $\\{\lambda_{j}^{1}\\}$ such that

(4.4)
$\limsup_{k\rightarrow\infty}d(\eta^{k},R_{\lambda_{j_{k}}^{1}}(z_{1}))-d(R_{\lambda_{j_{k}}^{1}}(z_{1}),z_{1})\leq
M+1.$

Since $R_{\lambda_{j}^{i}}(z_{i})\rightarrow\zeta^{i}$, as
$j\rightarrow\infty$ for any $i=1,\ldots,m$, we have

$||\eta^{j}-\eta||=\bigg{|}\bigg{|}\sum_{i=2}^{m}\mu_{i}R_{\lambda_{j}^{i}}(z_{i})-\sum_{i=2}^{m}\mu_{i}\zeta^{i}\bigg{|}\bigg{|}\leq\sum_{i=2}^{m}\mu_{i}||R_{\lambda_{j}^{i}}(z_{i})-\zeta^{i}||\rightarrow
0,\quad j\rightarrow\infty,$

which implies that $||\eta^{j}-\eta||\rightarrow 0,j\rightarrow\infty$.
Moreover, since $[\zeta,\eta]\nsubseteq\partial D$ it follows from Lemma 4.1
that

$\liminf_{k\rightarrow\infty}d(\eta^{k},R_{\lambda_{j_{k}}^{1}}(z_{1}))-d(R_{\lambda_{j_{k}}^{1}}(z_{1}),z_{1})=\infty.$

However, the above formula contradicts (4.4). ∎

We can use Theorem 4.3 to show a Wolff-Denjoy type theorem for resolvents of
nonexpansive mappings. Let $(D,d)$ be a geodesic metric space and
$[x,y],[x^{\prime},y^{\prime}]$ two arbitrary geodesic segments in $D$. For
every $\alpha\in[0,1]$, consider the point $z=\alpha x+(1-\alpha)y$ on segment
$[x,y]$ such that $d(\alpha x+(1-\alpha)y,y)=\alpha d(x,y)$ and in the same
way, the point $z^{\prime}=\alpha x^{\prime}+(1-\alpha)y^{\prime}$ on segment
$[x^{\prime},y^{\prime}]$ such that $d(\alpha
x^{\prime}+(1-\alpha)y^{\prime},y^{\prime})=\alpha d(x^{\prime},y^{\prime})$.
Recall that a geodesic space $(D,d)$ is called Busemann convex if

$d(z,z^{\prime})\leq(1-\alpha)d(x,x^{\prime})+\alpha d(y,y^{\prime})$

for every $x,y,x^{\prime},y^{\prime}\in D$ and $\alpha\in[0,1]$.

Combining Corollary 3.3 and Proposition 3.4 in [2], we obtain the following
proposition (see also [12], [16, p. 191]).

######  Proposition 4.4.

Let $D\subset V$ be a bounded convex domain. A Hilbert metric space $(D,d)$ is
Busemann convex if and only if $D$ is an ellipsoid.

Since in Hilbert’s metric spaces every straight-line segment is a geodesic, it
follows from Proposition 4.4 that $(D,d)$ satisfies condition (D), whenever
$D$ is an ellipsoid. This leads to the following Wolff-Denjoy type theorem for
resolvents of nonexpansive mappings.

######  Corollary 4.5.

Suppose that $D\subset V$ is an ellipsoid and $R_{\lambda}:D\rightarrow
D,\lambda>0,$ is the resolvent of a nonexpansive mapping $F:D\rightarrow D$
(with respect to Hilbert’s metric) without fixed points. Then there exists
$\xi\in\partial D$ such that $\\{R_{\lambda}\\}_{\lambda>0}$ converge
uniformly on bounded sets of $D$ to $\xi$.

###### Proof.

It follows from Theorem 4.3 that
$\operatorname{co}\Omega_{\\{R_{\lambda}\\}_{\lambda>0}}\subseteq\partial D.$
Since $D$ is strictly convex, $\Omega_{\\{R_{\lambda}\\}_{\lambda>0}}$
consists of a single element $\xi\in\partial D.$ The proof of uniform
convergence on bounded sets is standard (see, e.g., [5]): suppose, on the
contrary, that there exist an open neighbourhood $U\subset\overline{D}$ of
$\xi$, a bounded set $K\subset D$ and a sequence $\\{y_{\lambda_{n}}\\}\subset
K$ $(\lambda_{n}\rightarrow\infty)$ such that
$R_{\lambda_{n}}(y_{\lambda_{n}})\notin U$ for each $n$. Then

$d(R_{\lambda_{n}}(y_{\lambda_{n}}),R_{\lambda_{n}}(y))\leq
d(y_{\lambda_{n}},y)\leq\operatorname{diam}K$

for any $y\in K$ and, since $R_{\lambda_{n}}(y)\rightarrow\xi$, we deduce from
Lemma 4.1 that
$R_{\lambda_{n}}(y_{\lambda_{n}})\rightarrow\xi\in\overline{D}\setminus U$, a
contradiction. ∎

Acknowledgements The first author was partially supported by National Science
Center (Poland) Preludium Grant No. UMO-2021/41/N/ST1/02968.

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# Consequences of a minimal length in a pseudo-complex extension of General
Relativity

Leila Maghlaoui  Physics Deptartment, University of Mentouri, Constantine 1,
Constantine P.O. box, 325 Ain El Bey Way, 25017 Constnantine, Algeria
and
Mathematical and Subatomic Physics Laboratory (LPMPS), University of Mentouri,
Constantine 1, Algeria  Peter O. Hess  Instituto de Ciencias Nucleares,
Universidad Nacional Autónoma de México, Ciudad Universitaria, Circuito
Exterior S/N, A.P. 70-543, 04510 México D.F. Mexico.
and
Frankfurt Institute for Advanced Studies, J. W. von Goethe University, Hessen,
Germany,
<EMAIL_ADDRESS>
(Day Month Year; Day Month Year)

###### Abstract

The effects of a minimal length are investigated within an algebraically
extended theory of General Relativity (GR). Former attempts, to include a
minimal length in GR are first resumed, with a conformal factor of the metric
as a consequence. Effective potentials for various black hole masses (as
ratios to the minimal length) are deduced. It is found that the existence of a
minimal length has, for a small mass black hole, important effects on the
effective potential near the event horizon, creating barriers which inhibit
that particles can pass the event horizon. Further, a new limit for the
minimal mass of a black hole is derived.

###### keywords:

Pseudo-complex General Relativity, maximal acceleration, Schwarzschild
geometry

PACS numbers:

##  1 Introduction

General Relativity (GR) is a well established theory for describing numerous
observations in the solar system [1] and in cosmology, it is one of the best
tested theories.

Nevertheless, it is not decided if GR is complete to describe phenomena in
very strong gravitational fields, as in the vicinity of a black hole. For
example, quantization effects are expected to play a role. There are further
problems, as the loss of information and the appearance of an event horizon
which disconnects an outer observer to the interior of a black hole (though,
many do not think that this is a problem).

Therefore, in very strong gravitational fields one can speculate on how
possibly to extend GR. Einstein himself intended to modify his theory such
that it contains Electrodynamics [2, 3]. In order to do so, he introduced a
complex metric. Another approach was followed by M. Born [4, 5], who
complained about the dominant role of coordinates in GR, contrary to Quantum
Mechanics where the momentum is treated on an equal footing. M. Born tried to
solve the problem by introducing the concept of complementarity, modifying the
length element such that it is defined in the coordinate-momentum space.

Later, Caianiello and his coworkers retook the ideas of M. Born and wrote a
series of papers about a theory called Quantum Geometry [6, 7, 8, 9, 10, 11,
12] (for the formulation in differential geometry see [13]). In [14] a pseudo-
complex extension of GR (pcGR) was proposed, with many applications [15], and
in [16] the most recent review is given including several predictions. M.
Born’s theory corresponds to a particular limit in pcGR [16]. It was shown
that deviations to GR, for macroscopic black holes, only appear near the event
horizon (not having yet being resolved due to resolution problems). In all
applications of pcGR the minimal length was neglected, because its effects
where considered to be negligible.

In earlier contributions [17] (and references therein) the effects of a
minimal length on the structure near the event horizon were investigated
within GR. The questions, we would like to answer, are: What happens when GR
is extended algebraically to pcGR, which includes the effects of a dark energy
in the vicinity of a black hole? In what range of the black hole mass the
effects of a minimal length are noticeable? In [17] the effects of the maximal
acceleration, as a parameter, were studied and not the role of a minimal
length and its relation to the mass of the black hole (though, they are
related). As we will see the minimal length affects small mass black holes,
which may have been created during the Big Bang.

The inclusion of a minimal length in the extended theory is quite involved and
we have to recur to approximations, as restricting to the Schwarzschild
solution only.

The paper is structured as follows: In Section 2 the theory proposed by E. R.
Caianiello is resumed. In Section 3 the effects of the minimal length in pcGR
in the Schwarzschild case is discussed and finally in Section 4 the results
are interpreted and conclusions are drawn.

##  2 The theory proposed by E. R. Caianiello

This model [6, 7, 8, 9, 10, 11, 12, 17] interprets the quantization via a
curvature of the relativistic eight-dimensional space-time tangent bundle
$TM=M_{4}+TM_{4}$, with $M_{4}$ as the usual flat space-time manifold with the
metric $\eta_{\mu\nu}$ (see also [13]). This description satisfies the Born
reciprocity principle [4, 5] and it incorporates the notion that the proper
accelerations of massive particles along their worldliness with upper limit
$A_{m}$, referred to as maximal acceleration (MA). Indeed the usual Minkowski
line element $d^{2}s=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$ is replaced by the line
element in the eight-dimensional space-time tangent bundle TM, with
$\eta_{\mu\nu}$ substituted by $g_{\mu\nu}$,

$d^{2}w=g_{AB}dX^{A}dX^{B},\text{ \ }A,B=0,....7,\text{\ }$ (1)

where

$\displaystyle g_{AB}$ $\displaystyle=g_{\mu\nu}\otimes g_{\mu\nu},$
$\displaystyle X^{A}$
$\displaystyle=\left(x^{\mu},\frac{c^{2}}{A_{m}}\frac{dx^{\mu}}{ds}\right),\text{
\ }\mu=0,..3~{}~{}~{},$ (2)

$x^{\mu}=(ct,\overrightarrow{x})$ is the usual space-time four-vector and
$\overset{\cdot}{x}^{\mu}=u_{\mu}=\frac{dx^{\mu}}{ds}$ the four-velocity, $c$
is the speed of light in the vacuum. The velocity components are strictly
speaking only valid in flat space [16] and it is a consequence of the
dispersion relation. As noted above, E.R. Caianiello substituted the
$\eta_{\mu\nu}$ by a general metric $g_{\mu\nu}$, where the relation is
approximate.

The line element in the ordinary four-dimensional space-time $\mathbf{M}_{4}$,
in which a particles moves when the constraints of a maximal acceleration is
present, is rewritten such that the length element has the usual form
augmented by a conformal factor. In fact, we can calculate the effective four
dimensional metric $\widetilde{g}_{\mu\nu}$ on the hypersurface locally
embedded in $\mathbf{TM}_{4}$. In this case, the line element becomes

$\displaystyle d^{2}w$
$\displaystyle=(1+\frac{g_{\mu\nu}a^{\mu}a^{\nu}}{A_{m}^{2}})d^{2}s$ (3)
$\displaystyle d^{2}w$ $\displaystyle=\widetilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}$
(4) $\displaystyle\text{with }d^{2}s$
$\displaystyle=g_{\mu\nu}dx^{\mu}dx^{\nu}\text{ and
}\widetilde{g}_{\mu\nu}=(1+\frac{g_{\mu\nu}a^{\mu}a^{\nu}}{A_{m}^{2}})g_{\mu\nu}=\sigma^{2}(r)g_{\mu\nu}~{}~{}~{},$
(5)

with the four-acceleration
$\overset{\cdot\cdot}{x}^{\mu}=a^{\mu}=c^{2}\frac{d^{2}x^{\mu}}{d^{2}s}$
($ds=cd\tau$, with $\tau$ as the eigentime).

It is quite involved to obtain the exact form for the conformal factor
$\sigma^{2}(r)$ and one recurs to an iterative procedure: In the first step,
the equations of motion with $\sigma^{2}(r)=1$ are deduced in the usual
manner, solving the geodesic equations. These give the components of the
4-acceleration $a^{\mu}_{(0)}$. Substituting these components into
$\sigma^{2}(r)$ results in

$\sigma_{(1)}^{2}(r)=1+\frac{g_{\mu\nu}a_{(0)}^{\mu}a_{(0)}^{\nu}}{A_{m}^{2}}$
(6)

The index $(i)$ denotes the order of the iteration. We can iterate this
procedure by resolving the equations of motion with the $\sigma^{2}_{(1)}$,
obtaining the next iteration $a^{\mu}_{(1)}$ for the 4-acceleration. As a
result we obtain the next iterative expression for
$\sigma^{2}=\sigma^{2}_{(2)}$, namely

$\sigma_{(2)}^{2}(r)=1+\frac{g_{\mu\nu}a_{(1)}^{\mu}a_{(1)}^{\nu}}{A_{m}^{2}}~{}~{}~{}.$
(7)

This iteration procedure can be continued, however, in general it is stop
after the second iteration and study the effects on the solutions, the form of
the effective potential, etc.

After a second iteration, the modified length element acquires the form

$\displaystyle d^{2}w$
$\displaystyle=\left(1+\frac{g_{\mu\nu}a_{(1)}^{\mu}a_{(1)}^{\nu}}{A_{m}^{2}}\right)d^{2}s$
(8) $\displaystyle\text{ }\widetilde{g}_{\mu\nu}$
$\displaystyle=\left(1+\frac{g_{\mu\nu}a_{(1)}^{\mu}a_{(1)}^{\nu}}{A_{m}^{2}}\right)g_{\mu\nu}$
(9)

Limiting values for the accelerations were also derived by several authors on
different grounds and applied to many branches of physics such as string
theory, cosmology, quantum field theory, black hole physics etc [18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 13]. In [17] this maximal acceleration is
obtained as $A_{m}=\frac{2mc^{3}}{\hbar}$, where the causal limit was taken
into account. Another group starts from an eight dimensional space by
introducing pseudo-complex variables and projecting this space to a four
dimensional physical space [30, 31, 32, 33].

Up to here, the notion of a maximal acceleration is restricted to GR,
formulated with the coordinates $x^{\mu}$. The central question of our
contribution is: What happens when not only a maximal acceleration is
introduced, but also the coordinates are algebraically extended [34] to
pseudo-complex coordinates

$X^{\mu}=x^{\mu}+Iy^{\mu}~{}~{}~{},$ (10)

with $I^{2}=1$ [34].

In [14] just such an extension of the general relativity was proposed, called
the pseudo-complex General Relativity (pcGR), with the most recent review
given in [16]. The pseudo-complex extension is proven to be the only
consistent algebraic extension of GR [34], not having neither ghost nor
tachyon solutions. In pseudo-complex General Relativity the space-time
coordinates are of the form $X_{\mu}=x_{\mu}+I\frac{l}{c}u_{\mu}$, with its
pseudo-real part $x^{\mu}$ and its pseudo-imaginary component, chosen to have
the form $y^{\mu}=\frac{l}{c}u_{\mu}$ in analogy to [18]. The component
$\frac{l}{c}u_{\mu}$ is an approximation, strictly speaking valid only in flat
space, and can be associated to the components of the tangent vector (four
velocity vector) at a given space-time point. The factor $l$ has the unit of a
length, which is introduced due to dimensional reasons. The consequence of
that is the appearance of a minimal length scale $l$ within the theory which
implies a maximal acceleration. As mentioned above. we assume that the minimal
length scale is of order of the Planck length ($l_{P}=\sqrt{\frac{\hbar
k}{c^{3}}=}1.616199\times 10^{-35}{\rm m}$, $k$ is the gravitational constant
and $\hbar$ is the reduced Planck constant).

The line element in pcGR is given by

$d^{2}w=g_{\mu\nu}(X,P)DX^{\mu}DX^{\nu}~{}~{}~{},$ (11)

where $X^{\mu}$ and $P_{\nu}$ are pseudo-complex.

Further assumptions to the metric are applied, namely that
$g_{\mu\nu}(X)=g_{\mu\nu}(x)$, i.e. that it does not depend on the four-
velocity, and with the constraint that the length element itself is pseudo-
real, which is the case when the dispersion relation is satisfied [16]. With
this, the length element in pcGR acquires the form

$d^{2}w=g_{\mu\nu}(x)\left[dx^{\mu}dx^{\nu}+\left(\frac{l}{c}\right)^{2}du^{\mu}du^{\nu}\right]$
(12)

In this extension the geometry is understood as a consequence of curvature in
eight-dimensional phase space, in which the coordinates of the velocity-space
manifold are the components of the four-velocity with a dimensional factor $l$
of the order of the Planck length $l_{P}$. In (12) the length element
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$ is extracted and we use $ds=cd\tau$, with
$d\tau$ as the eigentime. We obtain

$\displaystyle d^{2}w$
$\displaystyle=d^{2}s\left(1-\left(\frac{l^{2}}{c^{4}}\right)\left|g_{\mu\nu}a^{\mu}a^{\nu}\right|\right)$
(13) $\displaystyle d^{2}w$
$\displaystyle=\widetilde{g}_{\mu\nu}dx^{\mu}dx^{\nu}$
$\displaystyle\widetilde{g}_{\mu\nu}$
$\displaystyle=\left(1-\left(\frac{l^{2}}{c^{4}}\right)\left|g_{\mu\nu}a^{\mu}a^{\nu}\right|\right)g_{\mu\nu}~{}~{}~{},$
(14)

with $\widetilde{g}_{\mu\nu}$ representing the modified metric and $a^{\mu}$
the acceleration component $\frac{du^{\mu}}{d\tau}$, with $\tau$ as the
eigentime.

Therefore, one can rewrite the length element as

$\displaystyle d^{2}w$ $\displaystyle=$ $\displaystyle\sigma^{2}(r)d^{2}s$
with $\displaystyle\sigma^{2}(r)$ $\displaystyle=$
$\displaystyle\left(1-\left(\frac{l^{2}}{c^{4}}\right)\left|g_{\mu\nu}a^{\mu}a^{\nu}\right|\right)~{}~{}~{}.$
(15)

It is clear that the maximal acceleration is included automatically in pseudo-
complex general relativity ($A_{m}=\frac{c^{2}}{l}$). The new metric
$\widetilde{g}_{\mu\nu}$ depends on the coordinates $x_{\mu}$ and on the
acceleration field $a_{\mu}$. In this case, the generalized proper-time
interval (13) becomes

$d^{2}w=\left(1-\frac{l^{2}}{c^{4}}\left|a\right|^{2}\right)d^{2}s~{}~{}~{},$
(16)

which restricts the acceleration, as before, to a finite interval with a
maximal acceleration $\frac{c^{2}}{l}$ = $\frac{2mc^{3}}{\hbar}$ [17, 6, 7, 8,
9, 10, 11, 12]. In [35] it was applied to a pc-field theory with the important
feature that it is by construction regularized, due to the appearance of a
minimal length scale $l$. Due to the fact that this minimal length scale is a
parameter, it is not affected by a Lorentz transformation and, thus, all
symmetries are preserved, which is a huge simplification compared to theories
which require a violation of the Lorentz symmetry. In [16, 36] this fact was
used in the hope that it prevents the formation of a black hole. Indeed the
formation of a black hole is avoided in this theory, but not due to the
minimal length scale but due to the appearance of dark energy which appears
naturally within this theory. The same effects appear when it was applied to
the Robertson-Walker universe [15, 36]. In all cases, the variational
principle introduced contributions, which can be interpreted as dark energy,
acting repulsively such that the formation of an event horizon and a
singularity at the center is avoided. This is a most important result, as any
proper theory should not contain singularities.

Finally, one has to note that the second term in the conformal factor
$\sigma^{2}$ is only a covariant quantity in flat space, where the dispersion
relation $\eta_{\mu\nu}dx^{\mu}dy^{\mu}=0$, with
$y^{\mu}=\left(\frac{l}{c}\right)u^{\mu}$ ($u^{\mu}$ is the 4-velocity) is
valid [17]. In a curved space, one has to solve a complicated equation, which
is the result of the constraint that the pseudo-complex length element is
real, i.e., the pseudo-imaginary term has to vanish. For more details and
first attempts to solve it, please consult [16]. This is a price to pay when
the iteration procedure is used.

##  3 The Pseudo-complex Schwarzschild geometry

In pcGR it is assumed that a central mass generates, due to quantum effects, a
distribution of dark energy in its vicinity, which is based on theoretical
grounds using semi-classical Quantum Mechanics [37]. The distribution is
parametrized as $B_{n}/r^{n}$, with two phenomenological parameters, $B_{n}$
and $n$. The $B_{n}=bm^{n}$ describes the coupling of the central mass to the
dark energy and $n$ its fall-off as a function of the radial distance. This is
the simplest ansatz and one easily can add further complicated dependencies in
$r$. In [14] $n$ was set to 2, which results in a metric already excluded by
solar system experiments [1]. In [14, 38, 39] the $n$ was equal to $3$ and
several observable predictions were made. Using the first observation of
gravitational waves [40], in [41, 42] it is shown that $n=3$ is also excluded,
thus $n=4$ is assumed now. In this case the pseudo-complex Schwarzschild
solution is

$d^{2}s=\left(1-\frac{2m}{r}+\frac{\Omega}{2r}\right)d^{2}t-\frac{d^{2}r}{\left(1-\frac{2m}{r}+\frac{\Omega}{2r}\right)}-r^{2}(d^{2}\theta+\sin^{2}\theta
d^{2}\phi)$ (17) $\Omega(r)=\frac{B_{4}}{3}r^{-3}$ (18)

The results for $n=3$ are quite similar, changing only details.

In [43] the $B_{4}$ parameter was varied within the pc-Kerr solution from zero
(GR) to a maximal value, from which on no event horizon exists anymore. The
transition from the existence of an event horizon to its disappearance could
be related to a phase transition. However, again the effects of a minimal
length were neglected. Here, we also will consider the whole range of $B_{4}$,
i.e, investigating the transition from GR to pcGR, but now with the inclusion
of a minimal length.

With the approximation applied, the line element with a minimal length $l$ is
given by the relation (16), which is similar in form as given by E. R.
Caianiello.

In the following sub-section the factor $\sigma^{2}(l)$ will be investigated
further.

###  3.1 The factor $\sigma^{2}(r)$

In order to calculate the corrections to the pc-Schwarzschild metric,
experienced by a particle and its motion along a geodesic, one must determine
the factor $\sigma^{2}(r)$ of the order of $l^{2}$. Before that, we will
impose some conditions on the factor $\sigma^{2}(r).$ By using the embedding
procedure mentioned above of the order $l^{2}$, the line element acquires the
form (16), which has to be positive definite, i.e.,

$d^{2}w>0\Longleftrightarrow\left|a^{2}\right|<\frac{c^{4}}{l^{2}}~{}~{}~{},$
(19)

This implies that the acceleration is limited from above by $A_{m}=c^{2}/l$,
called the maximal acceleration. With this condition, the factor
$\sigma^{2}(r)$ is limited by 0 from below, when the acceleration is maximal,
and by 1 from above, when the acceleration is zero:

$0\leq\sigma^{2}(r)\leq 1$ (20)

As a consequence, the pcGR implies a maximal acceleration, as in GR [6, 7, 8,
9, 10, 11, 12].

In order to deduce the explicit expression of $\sigma^{2}(r)$ we recur again
to an iteration procedure (restricting now to the first iteration), 5black
which provides an approximate solution for this factor: In the first
iteration, to which we will restrict, the metric with $\sigma^{2}=1$ is taken
and the corresponding geodesic equations are derived and solved. To obtain
these geodesic equations, we start from the variational integral principle

$\delta\int\left[\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)\left(\overset{\cdot}{t}_{0}\right)^{2}-\frac{1}{\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)}\left(\overset{\cdot}{r}\right)^{2}-r^{2}(\left(\overset{\cdot}{\theta}\right)^{2}+\sin^{2}\theta\left(\overset{\cdot}{\phi}\right)^{2})\right]ds^{2}~{}=~{}0~{}~{}~{},$
(21)

where the dot refers to the derivative with respect to the variable s. The
variation leads to equations of motion

$\displaystyle\frac{d}{ds}(r^{2}\overset{\cdot}{\theta}^{2})$
$\displaystyle=r^{2}\sin\theta\cos\theta\overset{\cdot}{\phi}^{2}$ (22)
$\displaystyle\frac{d}{ds}(r^{2}\sin^{2}\theta\overset{\cdot}{\phi})$
$\displaystyle=0$ (23)
$\displaystyle\frac{d}{ds}\left[\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)\overset{\cdot}{t}\right]$
$\displaystyle=0~{}~{}~{},$ (24)

from where the accelerations $a_{(0)}^{\mu}$ in this first iteration are
obtained.

In this completely relativistic metric of the pc-Schwarzschild, all orbits
will be still in the orbital plane defined by

$\theta=\frac{\pi}{2};\text{ }\overset{\cdot}{\theta}=0~{}~{}~{}.$ (25)

With this, the equations of motion for the energy and the angular momentum,
respectively, are

$\displaystyle\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)\overset{\cdot}{t}$
$\displaystyle=E$ (26) $\displaystyle r^{2}\overset{\cdot}{\phi}$
$\displaystyle=L~{}~{}~{}.$ (27)

In addition, we can use $\frac{ds^{2}}{ds^{2}}=1$, which leads to a third
equation, namely

$1=\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)c^{2}\overset{\cdot}{t}^{2}-\frac{1}{\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)}\overset{\cdot}{r}^{2}-r^{2}\overset{\cdot}{\phi}^{2}~{}~{}~{}.$
(28)

Substituting Eqs. (26) and (27) into (28), we obtain

$\overset{\cdot}{r}^{2}=E^{2}-\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)\left(\frac{L^{2}}{r^{2}}+1\right)~{}~{}~{}.$
(29)

Substituting the last results into $\sigma^{2}$ =
$\left(1-\frac{l^{2}}{c^{4}}\left|a^{2}\right|\right)$ leads to
$\sigma_{(1)}^{2}$ in terms of $l$, $\overset{\cdot\cdot}{t}$,
$\overset{\cdot\cdot}{r},$ and $\overset{\cdot\cdot}{\phi}$

$\displaystyle\sigma_{(1)}^{2}(r)$ $\displaystyle=$
$\displaystyle\left\\{1-\frac{l^{2}}{c^{4}}\left|-c^{2}\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)\overset{\cdot\cdot}{t}^{2}+\frac{1}{\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)}\overset{\cdot\cdot}{r}^{2}+r^{2}\overset{\cdot\cdot}{\phi}^{2}\right|\right\\}$
(30)

with

$\displaystyle\overset{\cdot\cdot}{t}$
$\displaystyle=\frac{-E}{c^{2}\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)^{2}}\left(\frac{2m}{r^{2}}-\frac{2B_{4}}{3r^{5}}\right)\overset{\cdot}{r}$
$\displaystyle\overset{\cdot\cdot}{r}$
$\displaystyle=\left(-\frac{m}{r^{2}}+\frac{L^{2}}{r^{3}}-\frac{3mL^{2}}{r^{4}}+\frac{B_{4}}{3r^{5}}+\frac{2B_{4}L^{2}}{3r^{7}}\right)$
$\displaystyle\overset{\cdot\cdot}{\phi}$
$\displaystyle=-\frac{2L}{r^{3}}\overset{\cdot}{r}$ $\displaystyle\text{
}\overset{\cdot}{r}^{2}$
$\displaystyle=\left[\frac{E^{2}}{c^{2}}-\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)\left(\frac{L^{2}}{r^{2}}+1\right)\right]$
(31)

The $m$ is the mass of the source in units of length, $L$ the angular momentum
and $E$ the energy, with $E$ and $L$ being constant of motions, in units of
the particle mass $M$. In what follows, we will present some plots of the
factor $\sigma^{2}(r)$, in order to show how $\sigma^{2}$ varies as a function
of the a-dimensional variables

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{r}{m}~{},~{}\lambda~{}=~{}\frac{L}{m}~{},~{}\epsilon~{}=~{}\frac{l}{m}~{},~{}\alpha~{}=~{}\frac{B}{m^{4}}~{}~{}~{}.$
(32)

The dependence of $\sigma^{2}$ on different parameter values $\epsilon$ ($l$)
and $B$ ($\alpha$) is shown in Figs. 1 and 2. As can be seen, there are two
effects: i) The deviations to $\sigma^{2}=1$ become noticeable the more
$\alpha$ approaches from above $\frac{81}{8}$, where still an event horizon
exists, for larger $\alpha$ the deviations are smoothed out. When $\alpha$ is
smaller than $\frac{81}{8}$ the deviations increase. In all cases,
$\sigma^{2}$ is strongly lowered below $\frac{r}{m}=2$. For large
$\frac{r}{m}$ the $\sigma^{2}$ approaches 1. Note, that $\epsilon$ =
$\frac{l}{m}$ = 0.001 implies a small mass only thousand time larger than the
minimal length. For a large mass, for example of the size of a regular star,
only small or no effects are seen.

Figure 1: Dashed line: The factor $\sigma^{2}(\rho)$ for
$\lambda=0,\alpha=81/8,\epsilon=0.1$ and $E=1$. Solid line: The factor
$\sigma^{2}(\rho)$ for $\lambda=0,\alpha=81/8,\epsilon=0.001$ and $E=1$.
Figure 2: Dashed line: the factor $\sigma^{2}(\rho)$ for $\lambda=0$,
$\alpha=8,\epsilon=0.1$ , $E=1$. Solid line: the factor $\sigma^{2}(\rho)$ for
$\lambda=0,\alpha=81/8,\epsilon=0.1$ and $E=1$ and Dotted line: the factor
$\sigma^{2}(\rho)$ for $\lambda=0,\alpha=12,\epsilon=0.1$ and $E=1$.

To resume, $\sigma^{2}(\rho=\frac{r}{m})\longrightarrow 1$ as $\rho$ =
$\frac{r}{m}$ = $\longrightarrow\infty$, as it should be because at large $r$
the result of standard GR has to be reproduced and $\sigma^{2}$ has to acquire
the value 1. Taking $\alpha=(81/8)$ and $\alpha=8$, the factor
$\sigma^{2}(\rho)$ shows a divergence near $\rho=3/2$ and near
$\rho_{+}=\frac{(1+\sqrt{5})}{2},$ $\rho_{-}=1$ respectively (see Fig. 1 and
2). In Fig. 1 two lines (dashed and solid) are depicted. The values $\lambda$,
$\alpha$ and $E$ are kept constant, while the dashed line is for
$\epsilon=0.1$ (black hole mass ten times $l$) and the solid line for
$\epsilon=0.001$ (a thousand times $l$). For $\sigma^{2}$ the behavior is
regular and finite for $\alpha=12$, above the critical value, even when the
mass is only ten times $l$, indicating the (see Fig 2) vanishing of event
horizons.

A vanishing $d^{2}w$ also indicates that the effective distance at this point
is zero.

###  3.2 Event Horizons

In order to give an interpretation for the divergences of the factor
$\sigma^{2}(r)$, where it passes zero and tending to $-\infty$ (note, that a
negative value of $\sigma^{2}$ is not allowed, i.e., it corresponds to a non-
physical region), it is useful to determine the position of the event
horizon(s) (there may be several) for different values of $B_{4}=\alpha
m^{4}$. For example, in the GR-Schwarzschild solution, the surface $r=2m$ is
an event horizon. It is also a surface of infinite red-shift. As will be see
below, these surfaces are different in the pc-Schwarzschild solution [43]. We
begin by investigating the possible positions of the event horizons more
explicitly.

A null surface satisfies the equation

$f(x^{0},x^{1},x^{2},x^{3})=0~{}~{}~{},$ (33)

with $x^{0}=ct$, $x^{1}=r$, $x^{2}=\theta$ and $x^{3}=\phi$.

In the case of the Schwarzschild solution in GR the function $f$ for the event
horizon is given by

$f=r-2m=x^{1}-2m$

Let us turn to pc-Schwarzschild solution, for which $g_{\mu\nu}$ is given by

$g_{\mu\nu}=\left(\begin{array}[]{cccc}\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)&0&0&0\\\
0&-\frac{1}{\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)}&0&0\\\
0&0&-r^{2}&0\\\ 0&0&0&-r^{2}\sin^{2}\theta\end{array}\right)$ (34)

The $B_{4}$ can, in principle, acquires any value and we will consider the
whole range, from 0 to above $\frac{81}{8}m^{4}$.

According to the possible values of $B_{4}=\alpha m^{4}$, we can distinguished
three cases:

  * •

1\. When $0\leq\alpha<\frac{81}{8},$ there are two event horizons at
$r_{\pm,}$, for example for $\alpha=8$ we get $r_{+}=1.75211m$ and
$r_{-}=1.17221m$, which agrees with [43].

  * •

2\. For $\alpha=\frac{81}{8}$ the two event horizons $r_{+}$ and $r_{-}$ merge
into a single surface at $r_{e}=\frac{3m}{2}$ as in [43]. This case is mainly
taken for the applications of pcGR [16].

  * •

3\. For $\alpha>\frac{81}{8}$ there are no event horizons.

For all cases, we investigate the effects within the pc-Schwarzschild
geometry. The null hypersurface is determined by
$\left(1-\frac{2m}{r}+\frac{B_{4}}{6r^{4}}\right)=0.$ It is clear that, the
factor $\sigma^{2}(r)$ diverges at certain points for $\alpha\leq\frac{81}{8}$
on the two existing event horizons. For example $\alpha=8$, the factor
$\sigma^{2}(r)$ has two divergences at $r_{+}=1.75211m$ and $r_{-}=1.17221m$,
respectively (see Fig. 2, dashed line). Below and above each sigunlarity, the
$\sigma^{2}(r)$ function has to go through zero. According to the position of
the divergences $r_{\pm}$ we call these positions, where $\sigma^{2}(r)$ has a
zero, $r_{\pm 1}$ and $r_{\pm 2}$, respectively.

In fact, in the case $B_{4}<\frac{81}{8}m^{4}$ (more specifically, the example
$\alpha=8$ is shown) the factor $\sigma^{2}(r)=0$ has five possible values for
$r$ (as explained above), one is near center ($r=0)$ $r_{0}$, two $r_{-1},$
$r_{-2}$ are below and above the $r_{-}$, thus $r_{-1}<r_{-}<r_{-2}$ and two
$r_{+1},$ $r_{+2}$ are below and above the $r_{+}$, i.e.
$r_{+1}<r_{+}<r_{+2}$. For $B_{4}$ = $\frac{81}{8}m^{4}$
($\alpha=\frac{81}{8}$) there are three solutions $r$ where $\sigma^{2}(r)=0$.
One is always near the center $r=0$ and two $r_{e1}$ $r_{e2}$ are near
$r_{e}$, where the index e refers to the event horizon. For
$B_{4}>\frac{81}{8}m^{4}$ (more specific $\alpha=12$), there is one value of
$\ r$ near the center in which $\sigma^{2}(r)=0.$ Therefore, the factor
$\sigma^{2}(r)$ is always negative between $(0,$ $r_{0}),$ $(r_{-1},r_{-2})$
and $(r_{+1},$ $r_{+2})$. As is known, the factor $\sigma^{2}(r)=0$ represents
a particle with a maximal acceleration $\frac{c^{2}}{l}$, thus, the zeros of
the $\sigma^{2}(r)$ reveals these positionsof maximal acceleration.

###  3.3 The effective Potential

In order to make a direct comparison with the motion in the pc-Schwarzschild
geometry possible, we adopt the same procedure as before. The new action is
given by

$S=\int\sigma^{2}(r)\left\\{\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)\overset{\cdot}{t}^{2}-\frac{1}{\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)}\overset{\cdot}{r}^{2}-r^{2}(\overset{\cdot}{\theta}^{2}+\sin^{2}\theta\overset{\cdot}{\phi}^{2})\right\\}dw$
(35)

The equation of motions become ($\theta=\frac{\pi}{2})$

$\displaystyle\sigma^{2}(r)\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)\overset{\cdot}{t}$
$\displaystyle=E$ (36) $\displaystyle\sigma^{2}(r)r^{2}\overset{\cdot}{\phi}$
$\displaystyle=L$ (37)

In addition we use $\frac{dw^{2}}{dw^{2}}=1$, i.e.,

$1=\sigma^{2}(r)\left\\{\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)\left(\frac{dt}{dw}\right)^{2}-\frac{1}{\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)}\left(\frac{dr}{dw}\right)^{2}-r^{2}\left(\frac{d\phi}{dw}\right)^{2}\right\\}$
(38)

Substituting the equations (29) and (30) into equation (31), we get

$\frac{1}{\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)}\left(\frac{dr}{dw}\right)^{2}=\frac{E^{2}}{\sigma^{4}(r)\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)}-\frac{L^{2}}{\sigma^{4}(r)r^{2}}-\frac{1}{\sigma^{2}(r)}~{}~{}~{}.$
(39)

Solving for $\left(\frac{dr}{dw}\right)^{2}$, we obtain

$\displaystyle\left(\frac{dr}{dw}\right)^{2}$
$\displaystyle=\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)\left[\frac{E^{2}}{\sigma^{4}(r)\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)}-\frac{L^{2}}{\sigma^{4}(r)r^{2}}-\frac{1}{\sigma^{2}(r)}\right]$
$\displaystyle=E^{2}-V_{eff}^{2}(_{r})~{}~{}~{}.$ (40)

Comparing both sides we obtain for the effective potential

$V_{eff}^{2}(_{r})=E^{2}-\frac{E^{2}}{\sigma^{4}(r)}+\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)\left(\frac{L^{2}}{\sigma^{4}(r)r^{2}}+\frac{1}{\sigma^{2}(r)}\right)~{}~{}~{}.$
(41)

In a more traditional approach, one writes

$\frac{1}{2}\left(\frac{dr}{dw}\right)^{2}+P_{eff}(r)=\omega~{}~{}~{},$ (42)

where

$P_{eff}(r)=\frac{1}{2}(V_{eff}^{2}(_{r})-1)\text{ and
}\omega=\frac{1}{2}(E^{2}-1)$ (43)

In order to plot the effective potential, it is preferable to use the
a-dimensional variables (32) and the effective potential becomes, with
$\rho=\frac{r}{m}$,

$V_{eff}^{2}(_{r})=E^{2}-\frac{E^{2}}{\sigma^{4}(\rho)}+\left(1-\frac{2}{\rho}+\frac{\alpha}{6\rho^{4}}\right)\left(\frac{\lambda^{2}}{\sigma^{4}(l,\rho)\rho^{2}}+\frac{1}{\sigma^{2}(\rho)}\right)$
(44)

Inspecting (44), one notes that the potential exhibits singularities as soon
as $\sigma^{2}(r)$ approaches zero. On the other hand, when
$\sigma^{2}(r)\rightarrow-\infty$ the two last term tends to zero and the
effective potential approaches $E^{2}$.

For a selected set of parameter values, the effective potential is plotted in
the Figs. 3 to 11. In all examples one notes the important feature that for
all $\alpha$ and small $\epsilon$ (large masses) the effective potential
behaves smoothly, showing a repulsive behavior toward small distances. It
shows a minimum between 1 and 1.5 which finally increases toward larger $r$,
i.e., it has the typical form of an attractive ”molecular” potential.

Figure 3: Solid line: the effective potential $V_{eff}^{2}(\rho)$ for
$\lambda=0$, $\alpha=8,\epsilon=10^{-35}$ and $E=1$. Dashed line: the
effective potential $V_{eff}^{2}(\rho)$ for $\lambda=0,\alpha=8,\epsilon=0.1$
and $E=1$. Figure 4: Solid line: the effective potential $V_{eff}^{2}(\rho)$
for $\lambda=0$, $\alpha=81/8,\epsilon=10^{-35}$ and $E=1$. Dashed line: the
effective potential $V_{eff}^{2}(\rho)$ for
$\lambda=0,\alpha=81/8,\epsilon=0.1$ and $E=1$. Figure 5: Solid line: the
effective potential $V_{eff}^{2}(\rho)$ for $\lambda=0$,
$\alpha=12,\epsilon=10^{-35}$ and $E=1$. Dashed line: the effective potential
$V_{eff}^{2}(\rho)$ for $\lambda=0,\alpha=12,\epsilon=0.1$ and $E=1$. Figure
6: Dashed line: the effective potential $V_{eff}^{2}(\rho)$ for $\lambda=0$,
$\alpha=81/8,\epsilon=0.1$, $E=1$. Solid line: the effective potential
$V_{eff}^{2}(\rho)$ for $\lambda=0,\alpha=81/8,\epsilon=0.1$ and $E=2$ and
Dotted line: the effective potential $V_{eff}^{2}(\rho)$ for
$\lambda=0,\alpha=81/8,\epsilon=0.1$ and $E=4$. Figure 7: Dashed line: the
effective potential $V_{eff}^{2}(\rho)$ for $\lambda=0$,
$\alpha=12,\epsilon=0.1$ and $E=1$. Solid line: the effective potential
$V_{eff}^{2}(\rho)$ for $\lambda=0,\alpha=12,\epsilon=0.01$ and $E=1$. Dotted
line: the effective potential $V_{eff}^{2}(\rho)$ for
$\lambda=0,\alpha=12,\epsilon=0.001$ and $E=1$.  Figure 8:  A zoom of the
effective potential near $\frac{r}{m}=1.5$, with the intention to note the
existence of a potential barrier still for rather large black hole masses. For
very large masses, e.g. $\epsilon=10^{-20}$. the barrier seems to have
vanished. But effects are present up to $\epsilon=10^{-18}$ (not shown here).
Dashed line: the effective potential $V_{eff}^{2}(\rho)$ for $\lambda=0$,
$\alpha=81/8,\epsilon=10^{-8}$ and $E=1$. Solid line: the effective potential
$V_{eff}^{2}(\rho)$ for $\lambda=0,\alpha=81/8,\epsilon=10^{-9}$ and $E=1$.
Dotted line: the effective potential $V_{eff}^{2}(\rho)$ for
$\lambda=0,\alpha=81/8,\epsilon=10^{-20}$ and $E=1$.  Figure 9: Dashed line:
the effective potential $V_{eff}^{2}(\rho)$ for $\lambda=90$,
$\alpha=12,\epsilon=0.1$ and $E=10$. Solid line: the effective potential
$V_{eff}^{2}(\rho)$ for $\lambda=90,\alpha=12,\epsilon=0.1$ and $E=15$. Dotted
line: the effective potential $V_{eff}^{2}(\rho)$ for
$\lambda=90,\alpha=12,\epsilon=0.1$ and $E=20$.  Figure 10: Dashed line: the
effective potential $V_{eff}^{2}(\rho)$ for $\lambda=90$,
$\alpha=(81/8,\epsilon=0.000001$ and $E=10$. Solid line: the effective
potential $V_{eff}^{2}(\rho)$ for $\lambda=100,\alpha=12,\epsilon=0.000001$
and $E=10$. For $\frac{r}{m}\rightarrow 0$ the effective potential approaches
$E^{2}=100$.  Figure 11: Dashed line: the effective potential
$V_{eff}^{2}(\rho)$ for $\lambda=90$, $\alpha=12,\epsilon=10^{-35}$ and
$E=10$. Solid line: the effective potential $V_{eff}^{2}(\rho)$ for
$\lambda=90,\alpha=8,\epsilon=10^{-35}$ and $E=10$. Dotted line: the effective
potential $V_{eff}^{2}(\rho)$ for $\lambda=90,\alpha=81/8,\epsilon=10^{-35}$
and $E=10$. The nearly vertical line at small $\frac{r}{m}$ is the potential
coming down from large values and approaching $E^{2}=100$ at $\frac{r}{m}=0$.
Due to the poor resolution, the bending of the curve near 0 is not resolved.

The following discussion is easier to understand, noting that the effective
potential in (44) has two important terms: the term $-E^{2}/\sigma^{4}$
becomes dominant when the $\sigma^{2}$ tends to zero. In this case, the
potential goes to $-\infty$. When $\sigma^{2}(r)$ tends to $-\infty$, the only
term which survives is the first one, namely $E^{2}$, which the effective
potential finally acquires.

In Figs. 3 and 4 we studied the effect of changing $\alpha$ from 8, below the
critical limit, to $\alpha=\frac{81}{8}$, the critical limit, and $\alpha=12$,
above the critical limit. In these figures the solid line represents a
macroscopic black hole and it shows a typical behavior of an attractive,
molecular type of potential. In reference to it, in Fig. 3 for $\alpha$ we
used the value of 8, in Fig. 4 the value $\frac{81}{8}$ and in Fig. 5 the
$\alpha$ is above the critical value. We see that contrary to [17] several
potential barriers appear. This is due to the behavior of the
$\sigma^{2}(r)$-function which exhibits several zeros and also the potential
exhibits various zeros where $\sigma^{2}(r)$ tends to $-\infty$. We also see
that when the $\alpha$ changes, also the number of singularities change.

Interesting is how the height of the barrier changes with increasing energy
$E$ of the particle, see Fig. 6. As in [17] the height of the barrier
increases with $E$, indicating that the particle is always inhibited to enter
the area below the first position of the barrier. Note, that near the
singularity of $\sigma^{2}(r)$ the effective potential barrier approaches
$E^{2}$ (see the discussion above), thus, the barrier increases with $E^{2}$.

In Fig. 7 we studied the effect of lowering $\epsilon$ (increasing the mass of
the black hole), from $\epsilon=0.1$ on, which is ten times the minimal value
possible. We see, that with increasing mass (smaller $\epsilon$) the barrier
structure with its singularities vanish. the effective potential for large
black hole masses is plotted in the vicinity of $\rho=\frac{3}{2}$, where the
event horizon still exists. We note a sharp barrier, which is the result of
introducing a minimal length. Thus, no particle can pass this point, while
without the minimal length a particle can fall in. Here we see an important
effect in taking into account the presence of a minimal length.

In Fig. 8 a zoom to near $\frac{r}{m}=1.5$ of the effective potential is
depicted, for the critical $\alpha$ value of $\frac{81}{8}$. Without a minimal
length, the event horizon is at $\frac{r}{m}=\frac{3}{2}$, where according to
our calculations still a repulsive barrier is noted up to at least
$\epsilon=10^{-18}$ ($10^{18}$ times the Planck length), which is already of
macroscopic size, compared to the scale of an elementary particle! This
barrier is not seen anymore for $\epsilon=10^{-20}$. That the effective
potential approaches zero at $r=\frac{3}{2}m$ for $\alpha=\frac{81}{8}$ and
has there a minimum. For $\alpha=8$, at the position of the event horizons
$r_{+}=1.75211m$ and $r_{-}=1.17221m$, the potential is zero but does not
acquire a minimum there. These observations can be understood from the
expression of the effective potential in Eq. 41: The factor
$\left(1-\frac{2m}{r}+\frac{B}{6r^{4}}\right)$ is zero at the $r$ positions
mentioned and the effective potential reduces to
$E^{2}-\frac{E^{2}}{\sigma^{2}}$. The $\sigma^{2}(r)$ acquires there the value
of 1, thus, the two terms cancel and the effective potential is zero. This is
a unique behavior at the values of $\alpha$ mentioned, which disappears when
the $\alpha$-value is larger than its limiting value, because then the factor
in from of the last term in (41) is not zero anymore.

In Fig. 9 the angular momentum is now different from zero. Increasing the
Energy of the particle, with $\alpha=12$, the main effect is an increase of
the barrier, i.e., similar as discussed before. Also, introducing a large
angular momentum smooths out the singular structure of the potential.

Finally, in Figs. 10 and 11 nearly macroscopic black holes (very small
$\epsilon$) are studied, with a large angular momentum. Increasing $\alpha$,
only lifts the potential to higher values, which is due to the dark energy
alone, no $l$-dependence is observed. According to (44) the effective
potential always tends to $E^{2}$ at $r=0$. This is because the
$\sigma^{2}(r)$-function goes to $-\infty$, which eliminates the last terms in
(44). Approaching $r=0$, the potential comes from very large values, which can
be seen as a nearly vertical line in the figures.

###  3.4 Minimal mass of an accelerating object $A_{m}=c^{2}/l$

In [44] its is argued that there exists a value for the minimal mass of a
black hole, equal to the Plank mass. The main argument is that causality has
to be maintained and that for smaller mass quantum effects set in, thus the
classical theory is not valid any more [13, 17]. In this sub-section we show
that due to the minimal length the limit of a minimal black hole mass
increases.

The new mass scale is obtained when $m$ is equal to the minimal length, which
we will put here as the Planck length ($m=l$), or $m=l=10^{-35}$meter.

The Planck mass is determined as

$\displaystyle M_{P}$ $\displaystyle=$ $\displaystyle\sqrt{\frac{\hbar
c}{k}}=\left(\frac{0.105457\times 10^{-33}\times 3\times 10^{8}}{6.67384\times
10^{-11}}\right)^{\frac{1}{2}}$ $\displaystyle=$ $\displaystyle 0.217726\times
10^{-7}\text{ kg}$

Setting $m=l$ implies $\epsilon=1$, which in turn leads to (using
$m=\frac{kM_{\rm object}}{c^{2}}$, where $M_{\rm object}$ is the mass in kg)

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\frac{l}{m}=\frac{lc^{2}}{kM_{\text{object}}}$ (45)

Setting now the limit $\epsilon=1$ and solving for $M_{\rm object}$, we obtain
for the mass

$\displaystyle M_{\text{object}}$ $\displaystyle=$
$\displaystyle\frac{lc^{2}}{k}=\frac{9\times 10^{16}\times 6.62606\times
10^{-34}}{6.67384\times 10^{-11}}=8.93557\times 10^{-7}\text{ kg}~{}~{}~{}.$
(46)

5black

The ratio of $M_{\rm object}$ to $M_{P}$ is

$\frac{M_{{\rm object}}}{M_{P}}=41~{}~{}~{}.$ (47)

As a consequence, the minimal mass of an object, corresponding to the minimal
length scale, is $41$ times larger than the Planck mass, a more stringent
limit than the one proposed by S. Hawking [44].

##  4 Discussion and Conclusions

In this contribution we investigated on the effects of a minimal length scale
parameter in an extended version of General Relativity, called pseudo-complex
General Relativity, which is equivalent to introducing a $r$-dependent mass
[45]. Thus, the pc-GR can be seen as a representative other models which try
to extend General Relativity. In past applications of pcGR the minimal length
was ignored, but here we showed that it generates important differences for
small black holes, which we will resume in what follows. It is important to
stress that the minimal length is treated as a parameter and as a consequence
the Lorentz symmetry is maintained. This represents a great advantage to
theories treating the minimal length as a physical length, violating the
Lorentz symmetry resulting in laborious, involved theories. One may ask if the
use of a minimal length as a parameter is physical or not. In spite of this
question, even when the theory presented is thought as an effective theory
with a minimal length, the consequences discussed in this manuscript can serve
as an orientation for more general theories, thus, be very useful.

A minimal length is related to a maximal acceleration. Its effect is cast into
a conformal factor $\sigma^{2}(r)$ of the metric. When the acceleration of a
particle’s tends to its maximal value, then the metric correction factor
$\sigma^{2}(r)\longrightarrow 0$. At large distances the classical potential
of GR is recovered, i.e., $V_{eff}^{2}(r)\longrightarrow 1$ as
$r\longrightarrow\infty$.

As one main result, we showed that the effects of the minimal length can
mainly be noted when the mass is of the order of the Planck mass and a few
orders larger than it. For macroscopic black holes also some effect can be
seen when $\alpha=\frac{81}{8}$, up to $\epsilon=10^{-18}$, corresponding to a
length of $10^{-15}$fm, which is of the order of the size of a hadron.

But there is also an effect in pcGR due to the coupling of the central mass to
the size of the dark energy in the vicinity of a black hole. Depending on the
value of $B_{4}=bm^{4}$, we can distinguish three cases, given by particular
ranges of the parameter $B_{4}$, where we only resume the results mainly for
zero angular momentum ($\lambda=0$), though, also some remarks on $\lambda>0$
are included:

  * •

case a: $B_{4}<81m^{4}/8$

The range of $\epsilon=(l/m)$ is divided into two regions, the first is
$\epsilon_{0}\leq\epsilon\leq 1$, where $\epsilon_{0}$ marks the value above
which no singularity in $\sigma^{2}$ appears. In the second case
($\epsilon<\epsilon_{0}$) the conformal factor $\sigma^{2}$ s-hows
singularities, as can be appreciated in Fig. 1 and 2, where the factor shows
three and five points where $\sigma^{2}$ passes to zero to infinite negative
values. The region where $\sigma^{2}$ is negative is unphysical and has to be
excluded. The number and position of these singularities vary with $\epsilon$.
The exact value of $\epsilon_{0}$ cannot be given but only estimated.

We have two cases to consider. For the first one $\epsilon_{0}\leq\epsilon\leq
1$, the effective potential tends to $E^{2}$ near $r_{\pm}$ and the center
$r=0$. Moreover, the effective potential diverges when $\sigma^{2}(\rho)=0$ in
the range $r_{0}<r_{-1}<r_{-2}<r_{+1}<r_{+2}$. (The $r_{m}$ values refer to
the position where $\sigma^{2}(r)$ passes through zero, see the main text.)
The divergences appear as a potential barrier near $r_{\pm}$ and at the center
($r=0$). The potential barrier also increases with larger values of $E$ and,
as a consequence, the incoming massive particle of energy $E$ would never
falls into the black hole, i.e., the accretion of mass to the mini-black hole
is stopped. Fig. 3 and subsequent figures show that the effective potentials
can not be distinguished from each other at infinity.

For $\lambda\neq 0,$ the effective potential still has the value of $E^{2}$
near the center $r=0$ and to the horizons $r_{+}$ and $r_{-}$. There is a
critical values of $E$, which determines the sign of effective potential; it
becomes positive when
$E^{2}<\frac{\lambda^{2}}{\rho_{c}^{2}}(1-\frac{2}{\rho_{c}}+\frac{\alpha}{6\rho_{c}^{4}})$
where $\rho_{c}$ is the value of $\rho$ when coming from $\rho=0$ the
effective potential becomes negative (see Figs. 9-11). At a higher value of
the critical energy, there are two singularities in which the factor
$\sigma^{2}(r)=0.$

The second case is for $\epsilon\ll 1$, which corresponds to the classical
limit, i.e., the mass, in length units, of the black hole is much bigger than
the minimal length. This means that $\sigma^{2}(r)\longrightarrow 1.$ As
consequence, the potential barrier in this case disappears. Due to this, the
effects of the minimal length can be neglected.

  * •

case b: $B_{4}=81m^{4}/8$

In this case, according the value of $\epsilon=(l/m)$, we can also distinguish
two ranges: In the first one, we have $\epsilon_{0}$ $\leq\epsilon\leq 1$
(here $\epsilon_{0}$ represents the value of $\epsilon$ for which it makes the
transition from the first to the second range). The effective potential has
three singularities, one near zero and two degeneraste ones at $\rho_{e}$ =
$\frac{r_{e}}{m}$ = $\frac{3}{2}$ (the index $e$ refers to the event horizon)
but the effective potential has the maximum value $E^{2}$ for
$r_{e}=\frac{3m}{2}$ and near the center $r=0.$ These singularities act as a
potential barrier (see, for example, Fig. 3). When $\alpha=\frac{81}{8}$ the
effective potential for a large mass black hole still shows a repulsive
barrier at $\rho=\frac{3}{2}$, which inhibits the falling in of particles,
contrary to the case when no minimal length is taken into account, where the
particle can pass the horizon. It is noticeable that at the position of the
event horizon ($\rho_{e}=\frac{3}{2}$) there is still a very large barrier for
relative large black hole masses, at least up to $10^{18}l$ and the
penetration of a particle is suppressed. For much larger black hole of
$\epsilon=10^{-20}$ masses this barrier seems to disappear. An
$\epsilon=10^{-18}$ corresponds to a length of $10^{-15}$cm, which implies the
possibility that effects of the minimal length could be seen there, if
$\alpha=\frac{81}{8}$ and these small effect can be measured.

For the second range, we consider $\epsilon_{0}$ $\geqslant\epsilon,$ here
$\epsilon=(l/m)$ is very small (i.e., the black hole is of macroscopic size),
this means $\sigma^{2}(r)\longrightarrow 1,$ in this case the effective
potential has no singularities. This implies that there is also no potential
barrier. It is clear that the effect of a minimal length is large when
$l\approx m$, i.e., for microscopic black holes.

Fig. 1 shows that $\sigma^{2}(r)$ diverges for the same values of $\rho_{e}$.
The $\sigma^{2}(\rho)$ always appears in the denominator of the effective
potential (6) which thus equals $E^{2}$. This means that the velocity of any
incoming particle becomes zero (see equation (5)). The same behavior is seen
in Fig. 3. For the radial motion, there are always divergencies (i.e.,
singularities) produced by the zero of the factor $\sigma^{2}(\rho)$.

  * •

case c: $B_{4}>81m^{4}/8$

For this case, in the range $\epsilon_{0}$ $\leq\epsilon\leq 1$, the effective
potential has also a barrier in the potential, see Fig. 3 for $\lambda=0$. In
addition, for $\alpha=12$ ($B>81m^{4}/8$) the effective potential has always a
maximum at $E^{2}$ near $r=3m/2$ and near $r=0$.

In addition, for $\lambda\neq 0$ the effective potential has a critical value
for the energy, in which the factor $\sigma^{2}(r)=0$, for which we can
determined the sign of the effective potential: It is positive for
$E^{2}<\frac{\lambda^{2}}{\rho_{c}^{2}}(1-\frac{2}{\rho_{c}}+\frac{\alpha}{6\rho_{c}^{4}})$.
For $\epsilon_{0}$ $\geqslant\epsilon$, ($\rho_{c}=\frac{r_{c}}{m}$ is the
critical value of $\rho$ where the effective potential becomes negative, when
$\rho_{c}$ is approached from below) the effective potential is regular and
finite, which means that the potential barrier disappears in this case, but
the event horizon, which appears there, is not the effect of minimal length
$l$.

In conclusion, in all cases the effect of minimal length $l$, or maximal
acceleration, are noticeable only for small black hole masses, as a potential
barrier at the horizons $0<r_{0}<r_{-}<r_{+}$ in which the effective potential
diverges or has singularities. A ”small” black hole can still be relatively
large, as $m=10^{18}l$ corresponds to a black hole mass of about $10^{8}$kg.
The height of the barriers increase with the particle energy. As a consequence
, the formation of a larger black hole is stopped. Also the positions of
maximal accelerations $r_{\pm 1}$ and $r_{\pm 2}$ were revealed. At these
points, the effective potential has a barrier and a particle is prevented to
pass this barrier.

## Acknowledgments

P.O.H. acknowledges financial supoport from PAPIIT-DGAPA (IN100421).

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