# Tidal forces in the Simpson-Visser black-bounce and wormhole spacetimes

Dhruv Arora<EMAIL_ADDRESS>PDPIAS, Charusat University, Anand-388421
(Gujarat), India  Parth Bambhaniya<EMAIL_ADDRESS>PDPIAS, Charusat
University, Anand-388421 (Gujarat), India  International Center for Cosmology,
Charusat University, Anand-388421 (Gujarat), India  Dipanjan Dey
<EMAIL_ADDRESS>Department of Mathematics and Statistics, Dalhousie
University, Halifax, Nova Scotia, Canada B3H 3J5  Pankaj S. Joshi
<EMAIL_ADDRESS>International Centre for Space and Cosmology, School of
Arts and Sciences, Ahmedabad University, Ahmedabad-380009 (Gujarat), India
International Center for Cosmology, Charusat University, Anand-388421
(Gujarat), India

###### Abstract

The concept of regular black holes has gained attention in recent years,
especially in the context of quantum gravity theories. In these theories, the
existence of singularities is paradoxical as they represent a breakdown of the
laws of physics. Motivated by the recent developments in this area, we study
the tidal force effects in one such family of regular geometries described by
the Simpson-Visser metric. We find the radial and angular force profiles for a
radially in-falling particle in this spacetime and calculate the variation of
the geodesic separation vector with the radial coordinate using two different
initial conditions. These results are then compared with that of Schwarzschild
black hole spacetime. We show that for a regular black hole, both radial and
angular tidal forces show a peak outside the horizon and then fall to
ultimately switch their behavior from stretching to compression and vice-
versa. Also, they are finite at $r=0$ unlike the Schwarzschild spacetime. It
is also seen that the angular deviation profile shows an oscillating behavior
for a particular initial condition. Our analysis can be used to distinguish
between regular black hole, one-way and two-way wormholes and a singular black
hole spacetimes.

Keywords

Regular black holes, Tidal force, Wormholes

##  I Introduction

Various observations have revealed the general relativistic effects of
Einstein’s gravity, including the great discoveries of gravitational waves
LIGOScientific:2016aoc , the first ever black hole shadow images of the M87
and Milky Way’s galactic centre EventHorizonTelescope:2022xnr ;
EventHorizonTelescope:2022tzy ; EventHorizonTelescope:2022ago ;
EventHorizonTelescope:2022vjs ; EventHorizonTelescope:2022wok ;
EventHorizonTelescope:2022urf ; EventHorizonTelescope:2022xqj ;
EventHorizonTelescope:2022gsd , stellar motions of S-stars around the Milky
Way galactic centre Do:2019txf ; GRAVITY:2018ofz ; GRAVITY:2020gka , and so
on. These discoveries have sparked a lot of interest in researching the the
causal structure of spacetime and its dynamics around the galactic centre.

Despite the fact that black holes were predicted by GR more than a century ago
as a solution to Einstein’s equations, the physical properties of black holes
remain very interesting to the scientific community. Many solutions to
Einstein’s equations followed, generalising Schwarzschild solutions: Kerr,
Reissner-Nordstrom and Kerr-Newman black holes.

However having these achievements, there are still some conceptual issues with
black holes in general relativity. The existence of physical singularity is
the main problem associated with black holes. These issues a still unresolved
as of today. Many physicists have speculated that a quantum gravity theory
could solve this problem. However, even after the EHT findings, the
phenomenology of black holes remains a hot topic. Regular black holes,
wormholes, naked singularities, grava-stars, and other black hole mimickers
are also becoming more common in the literature Janis:1968zz ; Joshi:2011zm ;
Morris:1988tu ; Hawking:1988ae ; Schunck:2003kk ; Simpson:2018tsi ;
Moti:2021vck .

All of the previously discovered black hole solutions have a curvature
singularity, where geometrical and physical quantities diverge. J. M. Bardeen
proposed a regular black hole solution and noted the existence of solutions
free of curvature singularities Rodrigues:2022qdp . This Bardeen black hole is
an exact solution of Einstein’s equations coupled to nonlinear
electrodynamics. One reason for looking into such compact objects is that they
could be black hole alternatives. Furthermore, as is well known, many of the
possible observational signatures of black holes, such as precession of
timelike bound orbits, gravitational lensing, shadow properties, and energy
extractions can be mimicked by black hole mimickers Liu:2021yev ;
Joshi:2013dva ; Bambhaniya:2021ugr ; Tahelyani:2022uxw ; Rahaman:2021kge ;
Harko:2008vy ; Harko:2009xf ; Kovacs:2010xm ; Guo:2020tgv ; Chowdhury:2011aa ;
Shaikh:2019jfr ; Shaikh:2018oul ; Virbhadra:2007kw ; Gyulchev:2008ff ;
Sahu:2012er ; Martinez ; tsirulev ; Joshi:2019rdo ; Bambhaniya:2019pbr ;
Dey:2019fpv ; Bam2020 ; Bambhaniya:2022xbz ; Vagnozzi:2019apd ; Chen:2022nbb ;
atamurotov_2015 ; abdujabbarov_2015b ; Vagnozzi:2022moj ; Saurabh:2022jjv ;
Patel:2022jbk ; Patel:2023efv ; Dey:2020bgo ; Dey:2020haf .

From a historical standpoint, Einstein and Rosen proposed the existence of
”bridges” through space-time using general relativity theory Einstein:1935tc .
These bridges connect two points in space-time, allowing Einstein-Rosen
bridges, also known as wormholes, to be created. Such wormholes have been
shown to be non traversable. The concept of traversable wormholes was later
explored in the seminal work of Morris and Throne Morris:1988cz . As a result,
we can use astrophysical data to test black holes and wormholes. The authors
of Ref. Jusufi:2021lei  investigated the possibility of testing wormhole
geometries in our galaxy using an in-falling and radiating gas accretion model
and the motion of the S2 star. Refs. Dai:2019mse ; Bambi:2021qfo  investigated
the possibility of testing wormholes by observing the galactic centre using
the in-falling gas model. References Simonetti:2020ivl  contain additional
interesting works on the formation and stability of wormholes.

Simpson and Visser proposed Simpson:2018tsi  a very simple theoretically
appealing spherically symmetric and static spacetime family derived from
Schwarzschild geometry that allows for a unique description of regular black
holes and wormholes via smooth interpolation between these two possibilities
via a length-scale parameter that drives the regularisation of the central
singularity. Its rotational form was also recently demonstrated Mazza:2021rgq
. Ref. Bronnikov:2021uta  shows that this spacetime can be obtained as a
solution to Einstein field equations sourced by a combination of a minimally
coupled phantom scalar field with a non-zero potential and a magnetic field in
the framework on non linear electrodynamics. It is now well understood that a
test body in free fall towards the centre of another massive body is stretched
in the radial direction while compressed in the angular direction. Gravity’s
tidal effect, which is caused by a variation in gravity’s strength between two
adjacent points, causes the stretching and compression. Tidal force phenomena
are extremely common in our universe and have been the subject of popular
scientific investigation for much of the twentieth century.

The authors of Crispino:2016pnv  investigated the effects of tidal force on a
freely falling particle in a Reissner-Nordstrom (RN) black hole and discovered
that the radial and angular components of the tidal force change signs at a
boundary known as the event horizon (a null hypersurface), which is not the
case in a Schwarzschild black hole. The authors of Gad:2010ion  compared the
tidal force effects in a stringy charged black hole to those in Schwarzschild
and RN black holes. The radial and angular components of the tidal force in
the Kiselev black hole were studied by Shahzad and Jawad Shahzad:2017vwi .
Motivated by the above developments, in this paper we study the tidal force
effects and geodesic deviation in the Simpson-Visser spacetime. We then
compare our results with the Schwarzschild case to look for and study any
significant changes that can shed some light on the nature of black hole
mimickers.

This paper’s overview is as follows: In Section (II), we discussed the
features of this spacetime, i.e. a synopsis of previous work on this metric
and its properties. We present the calculations for tidal force effects for a
radially in-falling particle in this spacetime in section (III) and obtain
radial and angular force profiles using tetrad formalism. Radial and angular
force profiles and their behavior is discussed in section (IV). We compared
the numerical solution of the geodesic deviation equation for the Simpson-
Visser metric with the Schwarzschild black hole metric in section (V).
Conclusions are given in section (VI). Throughout the paper, we have used
gravitational constant ($G$) and velocity of light ($c$) as unity.

##  II Simpson-Visser SPACETIME

The Simpson-Visser metric represents a spherically symmetric and static class
of black hole mimickers with a minimal surface in place of the central
singularity. It is defined by the metric Simpson:2018tsi :

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+(r^{2}+l^{2})(d\theta^{2}+\sin^{2}\theta
d\phi^{2})$ (1)

where

$\displaystyle f(r)=1-\frac{2M}{\sqrt{r^{2}+l^{2}}}$ (2)

In the above equation, $M\geq 0$ is the ADM mass and $l>0$ is the
regularization parameter of the central singularity, and possibly reflects
quantum gravity effects. This spacetime is carefully designed to be the
minimalist modification of the Schwarzschild spacetime. By varying the
parameter $l$, the metric can represent the following cases:

  1. 1.

The ordinary Schwarzschild spacetime ($l=0$).

  2. 2.

A regular black hole geometry with one-way spacelike throat ($l<2M$).

  3. 3.

A one-way wormhole geometry with external null throat ($l=2M$).

  4. 4.

A traversable wormhole geometry with two-way timelike throat ($l>2M$).

The geometry in the region where it represents a regular black hole is unusual
in that it describes a bounce into a future incarnation of the universe rather
than a bounce back into our own Barcelo:2014cla ; BenAchour:2020gon . This
metric does not correspond to a traditional regular black hole geometries such
as the Bardeen, Bergmann-Roman, Frolov, or Hayward Hayward:2005gi ;
Frolov:2017rjz ; Cano:2018aod ; Bardeen:2018frm . Instead, it is either a
regular black hole (bouncing into a future incarnation of the universe) or a
traversable wormhole, depending on the value of the parameter $l$. Latest
studies done by authors in Zhou:2022yio  argue on the geodesic incompleteness
of some popular regular black holes such as analytically extended Hayward
black hole and the extension of the Culetu-Simpson-Visser’s non-analytic
smooth black hole. Similar analysis can also be extended to the metric given
in Eq. (1) to test its geodesic completeness. The coordinate location of the
horizons in this spacetime when using Schwarzschild coordinates
($t$,$r$,$\theta$,$\phi$) is given by imposing $g^{rr}=0$ which gives us:

$\displaystyle r_{\pm}=\sqrt{(2M)^{2}-l^{2}}.$ (3)

Also, by studying the radial null curves in this geometry i.e. setting
$ds^{2}=0$, $d\theta=d\phi=0$ we get:

$\displaystyle\frac{dr}{dt}=\pm\bigg{(}1-\frac{2M}{\sqrt{r^{2}+l^{2}}}\bigg{)}.$
(4)

This equation also defines the coordinate speed of light in this metric. For
all values of $l<2M$, we get a pair of horizons at symmetrically placed $r$
coordinate given by Eq. (3). If $l=2M$, then as $r\rightarrow 0$, we have
$\frac{dr}{dt}\rightarrow 0$. This means there is a horizon at $r=0$ in the
case of the one-way wormhole with an external null throat. If $l>2M$, then
$\frac{dr}{dt}\neq 0$ for any value of $r$. This suggests that the geometry is
a two-way traversable wormhole.

We can use scalar curvature invariants to detect the event horizon for the
above-mentioned Simpson-Visser spacetime. Describing the presence of a horizon
using scalar curvature invariants for a given spacetime is very useful since
this description is coordinate-independent Abdel ; Coley:2017woz ; mcnut . We
can write down the following dimensionless scalar invariant Abdel :

$\displaystyle\mathcal{Q}_{2}=\frac{1}{27}\frac{I_{5}~{}I_{6}-I_{7}^{2}}{\left(I_{1}^{2}+I_{2}^{2}\right)^{\frac{5}{2}}}\,\,,$
(5)

where $I_{5}=k_{\mu}k^{\mu}$ where $k_{\mu}=-\nabla_{\mu}I_{1}$,
$I_{6}=n_{\mu}n^{\mu}$ where $n_{\mu}=-\nabla_{\mu}I_{2}$, and
$I_{7}=k_{\mu}n^{\mu}$. Here, $I_{1}=C_{abcd}C^{abcd}$ and
$I_{2}=C_{abcd}^{*}C^{abcd}$ where $C_{abcd},~{}C_{abcd}^{*}$ are the Weyl
curvature tensor and its dual tensor, respectively. The dimensionless scalar
invariant $\mathcal{Q}_{2}$ becomes zero at $r=0$ and at
$r=\sqrt{(2M)^{2}-l^{2}}~{}$ i.e., at the event horizon when $l<2M$. This is
true for any values of angular co-ordinates, however, at
$\theta=\frac{\pi}{2}$ plane, $\mathcal{Q}_{2}$ is identically zero $\forall
r$. When $l>2M$, $\mathcal{Q}_{2}$ becomes zero at $r=0$ only which suggests
the absence of the event horizon. Therefore, in Simpson-Visser spacetime,
$\mathcal{Q}_{2}=0,\forall r>0,\forall\theta\neq\frac{\pi}{2}$ implies the
presence of event horizon (i.e., a null hypersurface $\Sigma=\partial
J^{-}(\mathcal{I}^{+})\cap\mathcal{M}$, where $J^{-}(\mathcal{I}^{+})$ is the
causal past of the future null infinity $\mathcal{I}^{+}$) which is true in
any coordinate system.

Another very crucial property of any spacetime that needs to be verified is
whether its ADM mass is finite since it is related to the asymptotic flatness
of that spacetime. An asymptotically Minkowskian spacetime would also be
asymptotically flat if the ADM mass of that spacetime is finite. Spacetimes
which are asymptotically Minkowskian but don’t have finite ADM mass are known
as quasi-flat spacetime Sud . The ADM mass for a spacetime manifold
($\mathcal{M},g_{\alpha\beta}$) can be written as:

$M_{ADM}=-\frac{1}{8\pi}\lim_{S_{t}\to\infty}\oint_{S_{t}}(K-K_{0})\sqrt{q}\quad
d^{2}y$ (6)

where $S_{t}$ is the bounded spacelike two-surface, $K$ is the extrinsic
curvature of the two-surface $S_{t}$ embedded in the three-dimensional
spacelike hypersurface $\Sigma_{t}$, $K_{o}$ is the extrinsic curvature of the
two-surface $S_{t}$ embedded in the flat space, and the infinitesimal area of
the two-surface is given by
$\sqrt{q}~{}d^{2}y=\left(r^{2}+l^{2}\right)sin\theta~{}d\theta~{}d\phi$, where
$q$ is the determinant of the induced metric on the two-surface. The extrinsic
curvature of $S_{t}$ for the Simpson-Visser space-time is:

$\displaystyle
K=\frac{2r}{l^{2}+r^{2}}\bigg{(}1-\frac{2M}{\sqrt{r^{2}+l^{2}}}\bigg{)}^{\frac{1}{2}}\,\,,$
(7)

and $K_{0}=\frac{2}{r}$. Therefore, the ADM mass

$M_{ADM}=\lim_{r\to\infty}\left[r\left(1+\frac{l^{2}}{r^{2}}\right)-r\sqrt{1-\frac{2M}{\sqrt{r^{2}+l^{2}}}}\right]=M,$

which is finite. One can also verify that the Simpson-Visser spacetime is
asymptotically Minkowskian, since
$\lim_{r\to\infty}g_{tt}=-1,~{}\lim_{r\to\infty}g_{rr}=1$, and
$\lim_{r\to\infty}g_{\theta\theta}=r^{2},~{}\lim_{r\to\infty}g_{\phi\phi}=r^{2}\sin^{2}\theta$.
Therefore, this space-time is asymptotically flat.

Figure 1: Penrose diagram for $l>2M$ corresponding to a two-way wormhole
Simpson:2018tsi .

Figure 2: Penrose diagram for $l=2M$ corresponding to a one-way wormhole
Simpson:2018tsi .

##  III TIDAL FORCE EFFECTS

In this section, we will delve into the complexities of the equations
involving tidal forces in our spacetime. To investigate the equation for the
distance between two infinitesimally close and free falling particles, we use
the following equation for the spacelike components of the geodesic deviation
vector $\eta^{\mu}$

$\displaystyle\frac{D^{2}\eta^{\mu}}{D\tau^{2}}=R^{\mu}_{\alpha\beta\gamma}v^{\alpha}v^{\beta}\eta^{\gamma},$
(8)

where $R^{\mu}_{\alpha\beta\gamma}$ is the Riemann curvature tensor and
$v^{\mu}$ is the unit tangent vector to the geodesic. It is known that a test
body moving in space solely under the influence of gravity follows a geodesic
Falcon-Gomez:2022wze . Now, due to the difference in curvature at every point
on the geodesic and the fact that each point on the test body tends to follows
a unique geodesic, the body experiences a difference in acceleration, leading
to a stretching and squeezing effect known as tidal forces. This is the reason
that the Riemann curvature tensor is used to study the tidal interactions due
to gravity.

To calculate the tidal forces in the falling body’s frame, we make use of the
tetrad formalism Mitsou:2019nhj . Tetrads are geometric objects that form a
set of local coordinate bases, i.e. a locally defined set of four linearly
independent vector fields known as tetrads or vierbien. At each point on a
geodesic, there is a tetrad frame that forms a local inertial reference frame,
where the laws of special relativity apply. Since, Lorentz transformations can
connect an infinite number of orthonormal bases at a specific point, they
cannot give enough information about the connection on the metric. To resolve
this, we use the transformation from the orthonormal basis to the coordinate
basis:

$\displaystyle\overrightarrow{e}_{\mu}=\hat{e}^{\hat{\mu}}_{\mu}\overrightarrow{e}_{\hat{\mu}},$
(9)

where $\overrightarrow{e}_{\mu}$ represents the coordinate basis,
$\overrightarrow{e}_{\hat{\mu}}$ represents the orthonormal basis and
$\hat{e}^{\hat{\mu}}_{\mu}$ are the tetrad components. The metric tensor
components in the tetrad basis are given as Mitsou:2019nhj :

$\displaystyle
g_{\mu\nu}=\eta_{\hat{\mu}\hat{\nu}}\hat{e}^{\mu}_{\hat{\mu}}\hat{e}^{\hat{\nu}}_{\nu}.$
(10)

This is the key equation that enables us to use orthonormal basis in curved
spacetime. The tetrad components for freely falling frames can be obtained
from Eq. (4) and Eq. (8).

$\hat{e}^{\mu}_{\hat{0}}=\bigg{\\{}\frac{E}{f(r)},-\sqrt{E^{2}-f(r)},0,0\bigg{\\}},$
(11)
$\hat{e}^{\mu}_{\hat{1}}=\bigg{\\{}\frac{-\sqrt{E^{2}-f(r)}}{f(r)},E,0,0\bigg{\\}},$
(12) $\hat{e}^{\mu}_{\hat{2}}=\bigg{\\{}0,0,\frac{1}{r},0\bigg{\\}},$ (13)
$\hat{e}^{\mu}_{\hat{3}}=\bigg{\\{}0,0,0,\frac{1}{r\sin\theta}\bigg{\\}},$
(14)

where the tetrad components follow the following rule:

$\hat{e}^{\mu}_{\hat{\nu}}\hat{e}^{\hat{\rho}}_{\mu}=\delta^{\hat{\rho}}_{\hat{\nu}}.$
(15)

We notice that the component $\hat{e}^{\mu}_{\hat{0}}=v^{\mu}$ is the tangent
vector to the geodesic i.e. the four velocity of the particle. Also, the
geodesic deviation vector follows the transformation rule from global
coordinates to local orthonormal coordinates:

$\eta^{\mu}=\hat{e}^{\mu}_{\hat{\nu}}\eta^{\hat{\nu}}.$ (16)

Figure 3: In this Figure, $r_{-},r_{+},R^{rtf}_{0},R^{atf}_{0}$ plotted as a
function of $l$. The intersection point of $r_{+}$ and $R^{rtf}_{0}$ happens
at $\pm\frac{l}{\sqrt{2}}$, where the radial tidal force inverts its
direction. Plots are made at a value of M = 1.

The non vanishing independent components of Riemann tensor for spherically
symmetric spacetimes, including Simpson-Visser are given by:

$\displaystyle
R^{1}_{212}=\frac{-rf(r)}{2}-f(r)+\frac{r^{2}f(r)}{r^{2}+l^{2}},$ (17)
$\displaystyle R^{1}_{010}=\frac{f(r)f^{\prime\prime}(r)}{2},$ (18)
$R^{1}_{313}=\frac{-rf^{\prime}(r)\sin^{2}\theta}{2}-\frac{l^{2}f(r)\sin^{2}\theta}{r^{2}+l^{2}},$
(19) $\displaystyle R^{2}_{020}=\frac{rf(r)f^{\prime}(r)}{2(r^{2}+l^{2})},$
(20) $\displaystyle R^{3}_{030}=\frac{rf(r)f^{\prime}(r)}{2(r^{2}+l^{2})},$
(21) $\displaystyle
R^{2}_{323}=\sin^{2}\theta-\frac{r^{2}f(r)\sin^{2}\theta}{r^{2}+l^{2}}.$ (22)

We see that substituting $l=0$ in the above equations yields the components
for Schwarzschild spacetime. For computing the Riemann curvature tensor
components in tetrad basis, we use the tetrad formalism as:

$R^{\hat{\mu}}_{\hat{\alpha}\hat{\beta}\hat{\gamma}}=R^{a}_{bcd}e^{\hat{\mu}}_{a}e^{b}_{\hat{\alpha}}e^{c}_{\hat{\beta}}e^{d}_{\hat{\gamma}}.$
(23)

Following the Eq. (20), the tidal tensor components are given by:

$\displaystyle
R^{\hat{1}}_{\hat{0}\hat{1}\hat{0}}=\frac{f^{\prime\prime}(r)}{2},$ (24)
$\displaystyle
R^{\hat{2}}_{\hat{0}\hat{2}\hat{0}}=\frac{rf^{\prime}(r)}{2(r^{2}+l^{2})}+\frac{l^{2}{(f(r)-E^{2})}}{(r^{2}+l^{2})^{2}},$
(25) $\displaystyle
R^{\hat{3}}_{\hat{0}\hat{3}\hat{0}}=\frac{rf^{\prime}(r)}{2(r^{2}+l^{2})}+\frac{l^{2}{(f(r)-E^{2})}}{(r^{2}+l^{2})^{2}}.$
(26)

##  IV TIDAL FORCE EQUATIONS

Upon obtaining the expressions from Eq. (21-23), we can obtain the relative
acceleration between two nearby particles as:

$\displaystyle\frac{D^{2}\eta^{\hat{r}}}{D\tau^{2}}=\frac{-f^{\prime\prime}(r)}{2}\eta^{\hat{r}},$
(27)
$\displaystyle\frac{D^{2}\eta^{\hat{\theta}}}{D\tau^{2}}=-\bigg{[}\frac{rf^{\prime}(r)}{2(r^{2}+l^{2})}+l^{2}\bigg{(}\frac{f(r)-E^{2}}{(r^{2}+l^{2})^{2}}\bigg{)}\bigg{]}\eta^{\hat{\theta}},$
(28)
$\displaystyle\frac{D^{2}\eta^{\hat{\phi}}}{D\tau^{2}}=-\bigg{[}\frac{rf^{\prime}(r)}{2(r^{2}+l^{2})}+l^{2}\bigg{(}\frac{f(r)-E^{2}}{(r^{2}+l^{2})^{2}}\bigg{)}\bigg{]}\eta^{\hat{\phi}}.$
(29)

By substituting the values of $f(r)$ and its higher derivatives in Eq. (24,
25, 26) we get:

$\displaystyle\frac{D^{2}\eta^{\hat{r}}}{D\tau^{2}}=-\frac{M(l^{2}-2r^{2})}{(l^{2}+r^{2})^{5/2}}\eta^{\hat{r}},$
(30)
$\displaystyle\frac{D^{2}\eta^{\hat{\theta}}}{D\tau^{2}}=\frac{-Mr^{2}+l^{2}(2M+(E^{2}-1)\sqrt{l^{2}+r^{2}})}{({l^{2}+r^{2})^{5/2}}}\eta^{\hat{\theta}},$
(31)
$\displaystyle\frac{D^{2}\eta^{\hat{\phi}}}{D\tau^{2}}=\frac{-Mr^{2}+l^{2}(2M+(E^{2}-1)\sqrt{l^{2}+r^{2}})}{({l^{2}+r^{2})^{5/2}}}\eta^{\hat{\phi}}.$
(32)

###  IV.1 RADIAL TIDAL FORCE

We notice from Eq. (30) that the radial tidal force reaches a finite value at
$r=0$, being equal to

$\displaystyle\frac{D^{2}\eta^{\hat{r}}}{D\tau^{2}}\bigg{|}_{r=0}=\frac{-M}{l^{3}}\eta^{\hat{r}}.$
(33)

This is in contradiction to the Schwarzschild case where the radial tidal
force tends to infinity (Fig 4). Also from Eq. (30) it can be shown that the
radial tidal force vanishes at a point $R=R_{0}^{rtf}$ given by:

$\displaystyle R_{0}^{rtf}=\pm\frac{l}{\sqrt{2}}.$ (34)

Radial tidal forces do not vanish in Schwarzschild spacetime, but are seen in
Reissner-Nordstorm and other regular black hole spacetimes. By comparing this
and Eq. (3) for the location of event horizon we find that at $l=\pm
2\sqrt{6}$ the radial tidal force inverts its direction and becomes
compressive at the event horizon. The radial tidal force takes a maximum value
at $R^{rtf}_{max}$ given as:

$\displaystyle R^{rtf}_{max}=\pm\sqrt{\frac{3}{2}}l.$ (35)

It can be observed from the plot in fig (4) that the peak value of force
shifts towards the left as the particle is made to radially fall from
infinity. This suggests that in case of a regular black hole geometry, maximum
radial stretching is achieved much closer to the horizon when compared with
one-way and two-way wormhole geometries.

Figure 4: Radial tidal force as a function of the radial coordinate. We notice
that the force vanishes at a single point in contrast to what happens with
Schwarzschild spacetime. We take $M=1$ and $b=50M$. Dotted line shows the
location of horizon for Schwarzschild spacetime

###  IV.2 ANGULAR TIDAL FORCE

Just like the radial tidal force , the angular tidal force also reaches a
finite value at $r=0$ given by the solution of Eq.(31) at as:

$\displaystyle\frac{D^{2}\eta^{\hat{i}}}{D\tau^{2}}\bigg{|}_{r=0}=\frac{2M+(E^{2}-1)l}{l^{3}}\eta^{\hat{i}}.$
(36)

where $i=\\{\theta,\phi\\}$. This result differs from the Schwarzschild case
where the angular tidal forces diverge at $r=0$. Also, from Eq. (31) the
radial coordinate where the angular tidal force switches its sign can be
found. The expression obtained is a sixth order polynomial, but can be
simplified if we assume a numerical value of energy ($E$). In this study we
use $E=1$ which results in the following value of $R_{0}^{atf}$:

$\displaystyle R_{0}^{atf}=\pm\sqrt{2}l.$ (37)

It can be deduced from the graph that for a given value of energy, as the
value of $l$ is increased, there comes a point at which the angular tidal
force becomes entirely of stretching nature.Plot shown in fig(5) also shows
peak value of force shifting to left for different geometries, suggesting the
same conclusion as made in the case of radial tidal force.

Figure 5: Angular tidal force as a function of the radial coordinate. We
notice that the force vanishes at a single point in contrast to what happens
with Schwarzschild spacetime. We take $M=1$, $E=1$ and $b=50M$. Dotted line
shows the location of horizon for Schwarzschild spacetime

##  V GEODESIC DEVIATION VECTOR

In this section we discuss the variation of geodesic deviation vector with the
radial coordinate in Simpson-Visser spacetime. The deviation vector measures
the deformation of a body falling radially in the Simpson-Visser spacetime. We
can convert Eq. (30) and (31) in second derivatives w.r.t. $r$ by substituting
$dr/d\tau$ = $-\sqrt{E^{2}-f(r)}$ which results from Eq. (11). This gives us
the following second order differential equations in $r$ as:

$\displaystyle(E^{2}-f(r))\frac{D^{2}\eta^{\hat{r}}}{Dr^{2}}-\frac{f^{\prime}(r)}{2}\frac{D\eta^{\hat{r}}}{dr}+\frac{f^{\prime\prime}(r)}{2}\eta^{\hat{r}}=0,$
(38)
$\displaystyle\begin{split}(E^{2}-f(r))\frac{D^{2}\eta^{\hat{i}}}{Dr^{2}}-\frac{f^{\prime}(r)}{2}\frac{D\eta^{\hat{i}}}{dr}+\bigg{[}\frac{rf^{\prime}(r)}{2(r^{2}+l^{2})}+\\\
l^{2}\bigg{(}\frac{f(r)-E^{2}}{(r^{2}+l^{2})^{2}}\bigg{)}\bigg{]}\eta^{\hat{i}}=0.\end{split}$
(39)

where $i=\\{\theta,\phi\\}$. The analytic solution for Eq. (37) as pointed out
in Lima:2020wcb  can be given as:

$\displaystyle\eta^{\hat{r}}(r)=\sqrt{E^{2}-f(r)}\bigg{[}C_{1}+C_{2}\int\frac{dr}{(E^{2}-f(r))^{3/2}}\bigg{]}.$
(40)

where $C_{1}$ and $C_{2}$ are constants of integration. A similar solution can
also be obtained for Eq. (38). In order to find their value, we numerically
solve the differential equations by imposing some initial conditions. For the
purpose of this study we take the following initial conditions.

$\displaystyle\eta^{\hat{\beta}}(b)>0,\dot{\eta}^{\hat{\beta}}(b)=0,\quad(ICI)$
(41)
$\displaystyle\eta^{\hat{\beta}}(b)=0,\dot{\eta}^{\hat{\beta}}(b)>0.\quad(ICII)$
(42)

where $\beta=\\{r,\theta,\phi\\}$. $\eta^{\hat{\beta}}(b)$ represents the
separation between two nearby geodesics at $r=b$ in the radial and angular
directions. The initial condition in Eq. (40) represents a mass released from
rest at $r=b$ and the initial condition in Eq. (41) a body constituted of dust
to explode at $r=b$ . In the next sections we will discuss the components of
radial and the angular deviation vectors in detail.

###  V.1 RADIAL COMPONENT

In Fig.(6) we have shown the radial component of the geodesic deviation vector
after solving Eq. (37) with ICI. We notice that for non zero values for $l$,
the radial component of the geodesic deviation vector reaches a finite value
as $r\rightarrow 0$, unlike in the Schwarzschild black hole scenario, where it
goes till infinity because of the infinite stretching radial tidal force at
the singularity. This value keeps on decreasing as the value of $l$ is
increased and reaches zero for infinitely large distances. Since the Simpson-
Visser spacetime has values of $r$ from $\\{-\infty,\infty\\}$, the curve
shown in the graph is symmetric about the vertical axis.

By constraining Eq. (37) with ICII, we obtain the plot of radial geodesic
deviation as shown in Fig.(7). It is observed that both the initial conditions
show similar trends, but the numerical value of deviation is significantly
higher in the case of second initial condition.

###  V.2 ANGULAR COMPONENT

In Fig.(8) we have shown the angular component of geodesic deviation vector by
constraining the Eq. (38) with ICI. The results are compared with the
Schwarzschild black hole. It is observed that for all values of $l>0$, the
value of angular deviation for a radially in-falling particle decreases and
reaches a minimum value as $r\rightarrow 0$, and then starts to increase. For
very large values of $l$, the angular deviation reaches a constant value. This
is in contrast with the Schwarzschild black hole case where the angular
deviation reaches $0$ value, as $r\rightarrow 0$. Also, in case of regular
black hole geometry ($l<2M$), the point of minima is reached inside the event
horizon as shown in fig (3).

For ICII, the angular geodesic deviation profile shows a parabolic trend. It
is seen in Fig (9) that when a body at rest at some position $r=b$ explodes,
the angular deviation starts to increase as the body approaches the center. It
reaches a point of maxima and starts to fall rapidly, only to encounter a
point of minima which is located outside the horizon for all values of $l<2M$.
After crossing this point the angular deviation starts to increase again till
$r=0$. The case for Schwarzschild black hole shows a very similar trend except
there is no point of minima, and the angular deviation falls to zero after
crossing the point of maximum value.

##  VI CONCLUSION

In this paper we investigated the tidal forces in Simpson-Visser spacetime,
which comes from the regularization of the Schwarzschild metric. The geometry
of this spacetime changes for different values of the regularization parameter
$l$, ranging from a regular black hole to a two-way wormhole. We noted that
the tidal forces are finite at $r=0$ unlike the Schwarzschild black holes. It
is shown that the radial tidal force equation is similar to that of any
spherically symmetric and static spacetime, but the angular tidal force
equation depends upon the value of the regularization parameter $l$ and
energy. We see that the tidal force become zero at a particular value of the
radial coordinate which, in the case of radial tidal force solely depends upon
the value of $l$. This implies that the radial tidal forces may become
compressing instead of stretching and the angular tidal forces may show
stretching instead of compressing.

The tidal forces can vanish outside the event horizon for some values of $l$,
giving us an opportunity to observe this phenomenon in some cases. We also
show that the event horizon for a one-way wormhole ($l=2M$) is formed at $r=0$
and no event horizon is formed in the case of two-way wormhole ($l>2M$).
Another important observation made from the radial and angular tidal force
curve is that it reaches its peak value later than that in the Schwarzschild
black hole case. This shows that the horizon in regular black hole geometries
forms after the event horizon in Schwarzschild black hole i.e. $r=2M$. It can
also be shown that the angular tidal forces switch their behavior from
compression to entirely stretching as the value of $l$ keeps on increasing.

In addition to this, we also examined that the geodesic deviation for a
radially falling particle under the influence of tidal forces produced by
static regular black hole geometry. Two type of initial conditions are
discussed where the first one is when a body is released from rest at a
faraway point and the second one is when a body at rest explodes. It is
observed that the radial component of geodesic deviation is qualitatively the
same for both the initial conditions, but the angular component shows a very
peculiar profile, in which it exhibits an oscillatory behavior i.e showing a
crest and trough when compared to the Schwarzschild black hole case for
different initial conditions, as we showed in section (V.2).

##  VII Acknowledgement

DD would like to acknowledge the support of the Atlantic Association for
Research in the Mathematical Sciences (AARMS) for funding the work.

Figure 6: Radial components of geodesic deviation vector as a function of $r$
($M=1$) using ICI ($b=50M$) for different values of $l$ ($l=1.8,l=2,l=5$
respectively). Dotted line represents event horizon in Schwarzschild
spacetime.

Figure 7: Radial components of geodesic deviation vector as a function of $r$
($M=1$) using ICII ($b=50M$) for different values of $l$ ($l=1.8,l=2,l=5$
respectively). Dotted line represents event horizon in Schwarzschild
spacetime.

Figure 8: Angular components of geodesic deviation vector as a function of $r$
($M=1$) using ICI ($b=50M$) for different values of $l$ ($l=1.8,l=2,l=5$
respectively). Dotted line represents event horizon in Schwarzschild
spacetime.

Figure 9: Angular components of geodesic deviation vector as a function of $r$
($M=1$) using ICII ($b=50M$) for different values of $l$ ($l=1.8,l=2,l=5$
respectively). Dotted line represents event horizon in Schwarzschild
spacetime.

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# Distortion elements in group of diffeomorphisms of the 2-sphere

Jonathan Conejeros

###### Abstract

We prove that every distortion element in the group of diffeomorphisms of the
2-sphere which has some recurrent point that is not fixed is an irrational
pseudo-rotation. Moreover we prove that the differential of a distortion
element in the group of diffeomorphisms of the 2-sphere having at least three
fixed points at a fixed point has a unique eigenvalue which is 1.

##  1 Introduction

We recall the concept of distortion in a group. We will say that $f$ in a
group $G$ is a distortion element in $G$ if it has infinite order and there
exists a finite subset $S$ of $G$ such that:

  * (i)

$f$ belongs to the group generated by $S$,

  * (ii)

If $\left|f^{n}\right|$ is the word length of $f^{n}$ in the generators of
$S$, then

$\lim_{n\to\infty}\frac{\left|f^{n}\right|}{n}=0.$

This limit exists because the sequence $(\left|f^{n}\right|)_{n\in\mathbb{N}}$
is sub-additive. The simplest examples of groups which contain a distortion
element of infinite order are the solvable Baumslag-Solitar groups:

$BS(1;p):=<f,g\,:\,gfg^{-1}=f^{p}>,\text{ where }p\geq 2.$

Indeed, for the group relation we have for every integer $n\geq 1$:
$g^{n}fg^{-n}=f^{p^{n}}$. Taking as generators the set $S=\\{f,g\\}$, we have
$\left|f^{p^{n}}\right|\leq 2n+1$ and the element $f$ is distorted in the
group $BS(1;p)$, with $p\geq 2$. Another example is the discrete Heisenberg
group $H_{3}$. It is the group of integer matrices of the form

$\begin{pmatrix}1&a&b\\\ 0&1&c\\\ 0&0&1\end{pmatrix}$

There are generators $g$ and $h$ of $H_{3}$ such that their commutator
$f=[g,h]$ has infinite order and generates the center of $H_{3}$. This implies
that $f^{n^{2}}=[g^{n},h^{n}]$ and hence that $f$ is a distortion element in
$H_{3}$.

Before stating our main result we recall some results of distortion elements
in subgroups of the group of homeomorphisms on manifolds. Given a manifold
$M$, we will denote by $\operatorname{Diff}^{r}_{0}\left(M\right)$ (resp.
$\operatorname{Homeo}_{0}\left(M\right)$) the connected component of the group
of $C^{r}$-diffeomorphisms (resp. homeomorphisms) of $M$ containing the
identity. The following result due to Calegari and Freedman (see [3]) shows
that every homeomorphism of the $d$-dimensional sphere is distorted.

######  Theorem 1.1 (Calegari and Freedman, [3]).

Let $d\geq 1$. Then every element in the group
$\operatorname{Homeo}_{0}\left(\mathbb{S}^{d}\right)$ is distorted.

In this paper we will consider the group of diffeomorphisms of the 2-sphere.
As it is cleverly noticed in [3], we have the following restrictions for a
distortion element $f$ in $\operatorname{Diff}^{1}_{0}\left(M\right)$.

  * ($P_{1}$)

If $x$ is a fixed point of $f$, then eigenvalues of $Df(x)$ have absolue value
$1$, and

  * ($P_{2}$)

$f$ has zero topological entropy

The authors of [3] use Oseledec’s theorem and Pesin-Ruelle inequality to
proving property 2. We note that recently Navas gives a most elementary proof
of property 2 which is valid for bi-Lipschitz homeomorphisms (see [14]). The
following result improves the above second property $(P_{2})$.

######  Theorem 1.2 (Franks and Handel, [6]).

Let $M$ be a closed oriented surface of genus at least two. Then there are no
distortion element in $\operatorname{Diff}_{0}\left(M,\mu\right)$ the subgroup
of the group $\operatorname{Diff}^{1}_{0}\left(M\right)$ which preserve a
Borel probability measure on $M$ with total support.

Also we can find infinitely many distortion elements in
$\operatorname{Diff}^{1}_{0}\left(M\right)$ for any manifold $M$. More
precisely we can see the solvable Baumslag-Solitar group (defined above)
$BS(1;p)$, with $p\geq 2$ as a subgroup of
$\operatorname{Diff}^{1}_{0}\left(M\right)$.

######  Example (Solvable Baumslag-Solitar group embeds in
$\operatorname{Diff}^{1}_{0}\left(M\right)$).

Choose a ball B in a manifold M. Let us consider a $C^{1}$-action of $BS(1;p)$
on $[0,1]$ such that the generators $f$, $g$ satisfy
$f^{\prime}(0)=g^{\prime}(0)=1$ and $f^{\prime}(1)=g^{\prime}(1)=1$ (see [2]).
Extend it to an action on $B$ which fixes an interior point and the boundary
$\partial B$ so that it can be extend as the identity outside $B$.

This example being an action of the line has no recurrence. Militon gives
another class of distorted element in
$\operatorname{Diff}^{\infty}_{0}\left(M\right)$. He uses Avila’s techniques
(see [1]) and a local perfection result (see [9]), he obtains that every
recurrent element in $\operatorname{Diff}^{\infty}_{0}\left(M\right)$, i.e.
for which arbitrarily large iterates are arbitrarily close to the identity in
the $C^{\infty}$-topology, is distorted.

######  Theorem 1.3 (Militon,[13]).

Let $M$ be a compact manifold. Then every recurrent element in the group
$\operatorname{Diff}^{\infty}_{0}\left(M\right)$ is distorted.

For instance, irrational rotations of the 2-sphere are distorted (previously
showed by Calegari and Freedman (see [3])). More generally, using the Anosov-
Katok method (see [10] and [5]), we can build recurrent elements in the case
of the 2-sphere which are not conjugate to a rotation (such an element are
called irrational pseudo-rotation). We start by introducing some definitions.
Let $f$ be a homeomorphism of $\mathbb{S}^{2}$. We say that
$z\in\mathbb{S}^{2}$ is a (positively) recurrent point that is not fixed for
$f$ if there are arbitrarily large (positively) iterates $f^{n}(z)$ which are
arbitrarily close to $z$ and $f(z)\neq z$. We say that a homeomorphism $f$ of
$\mathbb{S}^{2}$ is an irrational pseudo-rotation if:

  * (i)

$f$ has exactly two periodic points, $z_{0}$ and $z_{1}$, which are both
fixed;

  * (ii)

there exists a point $z$ whose $\alpha$-limit set or $\omega$-limit set is not
included in $\\{z_{0},z_{1}\\}$; and

  * (iii)

if $\check{f}$ is a lift of $f|_{\mathbb{S}^{2}\setminus\\{z_{0},z_{1}\\}}$ to
the universal covering space of $\mathbb{S}^{2}\setminus\\{z_{0},z_{1}\\}$,
its unique rotation number is an irrational number.

Our main result is the following.

######  Theorem A.

Let $f$ be a distortion element in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$ such that has a
(positively) recurrent point that is not fixed. Then $f$ is an irrational
pseudo-rotation.

Clearly, by Theorem 1.1 the hypothesis of differentiability is necessary.
Moreover the solvable Baumslag-Solitar group embeds in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$ (above example) and
Heisenberg group $H_{3}$ acts smoothly on $\mathbb{S}^{2}$ by restrinting and
projectivizing the standard action of $GL(3,\mathbb{R})$ on $\mathbb{R}^{3}$.
These examples show that hypothesis of the existence of recurrence in Theorem
A is also necessary.

What about above property 1 $(P_{1})$ ? A priori this property can be improve
with an additional very strong hypothesis. If a distortion element $f$ belongs
to a finitely generated group which acts with a global fixed point $z$. Then
the differential at $z$ induces an action of the group on $GL(2,\mathbb{R})$,
and as it is noticed in [3] the eigenvalues of $Df(z)$ are roots of unity.

The property “eigenvalues are roots of unity” is again true in some cases for
our examples of groups having distortion elements without any additional
hypothesis.

Case of solvable Baumslag-Solitar group. Here Guelman and Liousse showed the
existence of a $g$-minimal set $\Lambda$ contained in $\operatorname{Fix}(f)$
(see [7]). Using the fact that every point in $\Lambda$ is recurrent, the
authors of [7] proved, in [8], that for every $z\in\Lambda$ the eigenvalues of
$Df(z)$ are roots of unity.

Case of Heisenberg group. If $H_{3}$ acts on $\mathbb{S}^{2}$ by
diffeomorphisms, then Ribón insures the existence of an orbit of cardinality
at most $2$ (see [15]). Considering power of the generating elements (if
necesary) we can suppose that $H_{3}$ acts with a global fixed point $z$. Then
Franks and Handel (see [6]) proved that $Df(z)=\pm Id$.

Our second result shows that for distortion elements in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$ any additional
hypothesis is necessary.

######  Theorem B.

Let $f$ be a distortion element in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$ which has at least
three fixed points. If $z$ is a fixed point of $f$, then $Df(z)$ has a unique
eigenvalue which is 1.

These results are part of a bigger plan which seeks to characterize dynamics
of distortion elements of the $2$-sphere.

We will proved Theorem A using the new criterion for the existence of
topological horseshoes for surface homeomorphisms which are isotopic to the
identity recently find for Le Calvez and Tal (see [11]).

### Acknowledgements

The preparation of this article was funded by the FONDECYT. I would like to
thanks Patrice Le Calvez and Frédéric Le Roux for suggesting some questions
which motivated Theorem A. I would also like thanks Andrés Navas for some
useful comments and remarks.

##  2 Preliminary results

In this section we recall some results of distortion elements in the group of
diffeomorphisms of the $2$-sphere and homeomorphisms of the $2$-sphere with no
topological horseshoes that we will use in the rest of the article.

###  2.1 Distortion elements in the group of diffeomorphisms of the $2$-sphere

The following result is Theorem 1.2 for the case of the $2$-sphere.

######  Theorem 2.1 (Theorem 1.3, [6]).

Let $f$ be a distortion element in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$ and let $\mu$ be an
$f$-invariant Borel probability measure. If $f^{n}$ has at least three fixed
points for some smaller integer $n$, then
$\operatorname{Supp}(\mu)\subset\operatorname{Fix}(f^{n})$.

We state the following corollary which will be used in the rest of the
article.

######  Corollary 2.2.

Let $f$ be a distortion element in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$. Suppose that $f$ has
at least three fixed points, then

  * (i)

the set of periodic points of $f$ coincides with that of its fixed points;

  * (ii)

if $X$ is a closed $f$-invariant set, then $X$ contains a fixed point; and

  * (iii)

if $h(f)$ denotes the topological entropy of $f$, then $h(f)=0$.

###  2.2 Homeomorphisms of the 2-sphere with no topological horseshoes

In this section we state some results which were proved in [11], where the
authors study some properties of homeomorphisms of the 2-sphere with no
topological horseshoes. A compact subset $Y$ of a manifold $M$ is a
topological horseshoe if it is invariant by an iterated $f^{q}$ of $f$,
$f^{q}|_{Y}$ is semi-conjugated to $g:Z\to Z$ on a Hausdorff compact space, so
that the fibers by the factor map are all finite with an uniform bound $m$ in
their cardinality, and $g$ is semi-conjugated to the Bernoulli shift
$\sigma:\\{1,\cdots,N\\}\to\\{1,\cdots,N\\}$, where $N\geq 2$ is such that the
preimage of every $s$-periodic sequence of $\\{1,\cdots,N\\}$ by the factor
map contains a $s$-periodic point of $g$.

######  Remark 1.

Note that, if $h(f)$ denotes the topological entropy of $f$, then

$qh(f)=h(f^{q})\geq h(f^{q}|_{Y})=h(g)\geq h(\sigma)=\log(N),$

and that $f^{rn}$ has at least $q^{n}/m$ fixed points for every $n\geq 1$.

The first result is about rotation numbers for annular homeomorphisms with no
topological horseshoes. We will write
$\mathbb{A}=(\mathbb{R}/\mathbb{Z})\times\mathbb{R}$ the open annulus and
denote by $\pi:\mathbb{R}^{2}\to\mathbb{A}$ the universal covering projection
of $\mathbb{A}$ and by $\pi_{1}:\mathbb{R}^{2}\to\mathbb{R}$ the projection in
the first coordinate.

######  Theorem 2.3 (Theorem A, [11]).

Let $f$ be a homeomorphism of $\mathbb{A}$ which is isotopic to the identity
and let $\check{f}$ be a lift of $f$ to $\mathbb{R}^{2}$. We suppose that $f$
has no topological horseshoe. Then each point $z$ such that its $\omega$-limit
set is non empty has a well defined rotation number
$\operatorname{Rot}_{\check{f}}(z)$, i.e. for every compact set
$K\subset\mathbb{A}$ and every increasing sequence of integers
$(n_{k})_{k\in\mathbb{N}}$ such that $z$ and $f^{n_{k}}(z)$ belong to $K$, we
have

$\lim_{k\to\infty}\frac{1}{n_{k}}(\pi_{1}(\check{f}^{n_{k}}(\check{z})-\check{z}))=\operatorname{Rot}_{\check{f}}(z),$

where $\check{z}$ is a lift of $z$.

######  Theorem 2.4 (Theorem F, [11]).

Let $f$ be a homeomorphism of $\mathbb{A}$ which is isotopic to the identity
and let $\check{f}$ be a lift of f to $\mathbb{R}^{2}$. We suppose that
$\check{f}$ is fixed point free and that there exists a positively recurrent
point $z$ such that $\operatorname{Rot}_{\check{f}}(z)$ is well defined and
equal to $0$. Then $f$ has a topological horseshoe.

######  Theorem 2.5 (Theorem G, [11]).

Let $f$ be an orientation-preserving homeomorphism of $\mathbb{S}^{2}$ that
has no topological horseshoe. Then every non-wandering point $z$ whose
$\alpha$-limit set or $\omega$-limit set in not included in the set of fixed
points of $f$ lies in an open annulus $A$ which is a fixed point free and
$f|_{A}$ is isotopic to the identity.

##  3 Main Proposition

In this section, we will prove the following proposition which is key in the
demonstrations of Theorems A and B.

######  Proposition 3.1.

Let $f$ be a distortion element in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$ and let $z_{0}$ and
$z_{1}$ be two distinct fixed points of $f$. Let $f_{ann}$ be the
homeomorphism of the closed annulus $\overline{A}$ obtained from
$\mathbb{S}^{2}$ by blowing up both points $z_{0}$ and $z_{1}$. If
$\check{f}_{ann}$ is a lift of $f_{ann}$ to the universal covering space of
$\overline{A}$, then it has a unique rotation number, i.e. there exists $\rho$
such that

$\lim_{n\to\infty}\frac{1}{n}(\pi_{1}(\check{f}_{ann}^{n}(\check{z})-\check{z}))=\rho,$

where $\check{z}$ is a lift of $z\in\overline{A}$.

To prove this proposition we need the concept of spread introduced by Franks
and Handel in [6].

Spread.- Let $\gamma$ be a smooth curve with endpoints $z_{0}$ and $z_{1}$
(smooth at the endpoints) and $\beta$ be a simple nullhomotopic closed curve
on $\mathbb{S}^{1}$ such that the point $z_{1}$ is contained in the disk
bounded by $\beta$. For any curve $\alpha$, the spread
$L_{\gamma,\beta}(\alpha)$ is going to measure how many times $\alpha$ rotates
around $\beta$ with respect to $\gamma$. More formally the definition of
spread is the following: let $\overline{A}$ be the compact annulus obtained
from $\mathbb{S}^{2}$ by blowing up both points $z_{0}$ and $z_{1}$. In this
case the universal covering space of $\overline{A}$ can be identified with
$\mathbb{R}\times[0,1]$ and the covering translation can be identified by
$T(x,y)=(x+1,y)$. For each lift $\check{\alpha}$ of $\alpha$ and
$\check{\gamma}$ of $\gamma$ to $\mathbb{R}\times[0,1]$, there exist integers
$a<b$ such that $\check{\alpha}\cap T^{i}(\check{\gamma})\neq\emptyset$ if and
only if $a<i<b$. Define

$\check{L}_{\check{\beta},\check{\gamma}}(\check{\alpha})=\max\\{0,b-a-2\\}.$

and

$L_{\beta,\gamma}(\alpha)=\max_{\check{\alpha}}\\{\check{L}_{\check{\beta},\check{\gamma}}(\check{\alpha})\\}.$

Define the spread of $\alpha$ with respect to $f$, $\beta$ and $\gamma$ as

$\sigma_{f,\beta,\gamma}(\alpha):=\liminf_{n\to\infty}\frac{L_{\beta,\gamma}(f^{n}(\alpha))}{n}.$

In [6] the authors give some criteria for undistortion, one of them is
positive spread.

######  Lemma 3.2.

Let $g_{i}$, $1\leq i\leq k$ be a finite set of elements of
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$. There exists a
constant $C>0$ such that the following property holds: If $f$ belongs to the
group generated by the $g_{i}$ and if $\left|f^{n}\right|$ is the word length
of $f^{n}$ in the generators $g_{i}$, then for all curves $\alpha$, $\beta$,
$\gamma$ and all integer $n>0$,

$L_{\beta,\gamma}(f^{n}(\alpha))\leq
L_{\beta,\gamma}(\alpha)+C\left|f^{n}\right|.$

It follows the following proposition.

######  Proposition 3.3.

Let $f$ be a distortion element in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$. Then for all curves
$\alpha$, $\beta$, $\gamma$ we have

$\sigma_{f,\beta,\gamma}(\alpha)=0.$

###### Proof.

Since $f$ is a distortion element in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$, there exists
$g_{i}$, $1\leq i\leq k$ a finite set of elements of
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$ such that $f$ belongs
to the group generated by the $g_{i}$. Moreover if $\left|f^{n}\right|$ is the
word length of $f^{n}$ in the generators $g_{i}$,

$\lim_{n\to\infty}\frac{\left|f^{n}\right|}{n}=0.$

According to the definition of spread and above lemma we have

$\sigma_{f,\beta,\gamma}(\alpha)=\liminf_{n\to\infty}\frac{L_{\beta,\gamma}(f^{n}(\alpha))}{n}\leq\liminf_{n\to\infty}\frac{L_{\beta,\gamma}(\alpha)+C\left|f^{n}\right|}{n}=0.$

∎

The following result is about rotation numbers for homeomorphisms of the
closed annulus with no topological horseshoe.

######  Lemma 3.4.

Let $f$ be a homeomorphism of $(\mathbb{R}/\mathbb{Z})\times[0,1]$ which is
isotopic to the identity and let $\check{f}$ be a lift of $f$ to
$\mathbb{R}\times[0,1]$. We suppose that $f$ has no topological horseshoe.
Then each point $z\in(\mathbb{R}/\mathbb{Z})\times[0,1]$ has a well defined
rotation number

$\operatorname{Rot}_{\check{f}}(z)=\lim_{n\to\infty}\frac{\pi_{1}(\check{f}^{n}(\check{z}))-\pi_{1}(\check{z})}{n},$

where $\check{z}$ is a lift to $z$.

###### Proof.

Write $f(x,y)=(f_{1}(x,y),f_{2}(x,y))$, we can extend $f$ to a homeomorphism
$f^{\ast}$ of the open annulus $\mathbb{A}$ such that
$f^{\ast}(x,y)=(f_{1}(x,1),y)$ if $y\geq 1$ and $f^{\ast}(x,y)=(f_{1}(x,0),y)$
if $y\leq 0$. Note that $f^{\ast}$ has no horseshoe, because $f$ has no. As
$(\mathbb{R}/\mathbb{Z})\times[0,1]$ is included in a compact set of
$\mathbb{A}$, the result follows from Theorem 2.3. ∎

###  3.1 Proof of Proposition 3.1

In this section, we prove Proposition 3.1. Suppose by contradiction that there
exist $z$ and $z^{\prime}$ (in the interior of $\overline{A}$) such that
$\operatorname{Rot}_{\check{f}_{ann}}(z)=\rho$ and
$\operatorname{Rot}_{\check{f}_{ann}}(z^{\prime})=\rho^{\prime}$, with
$\rho<\rho^{\prime}$. Conjugation we can suppose that $\check{f}_{ann}$ is a
homeomorphism of $\mathbb{R}\times[0,1]$ and a lift of $\gamma$ to
$\mathbb{R}\times[0,1]$ is $\\{0\\}\times[0,1]$. Let $\alpha$ be a smooth
curve joining $z$ and $z^{\prime}$ and let $\check{\alpha}$ be a lift of
$\alpha$ with endpoints $\check{z}$ and $\check{z^{\prime}}$. For
$\epsilon<(\rho^{\prime}-\rho)/2$ and for every $n$ large enought we have

$\pi_{1}(\check{f}^{n}_{ann}(\check{z}))<n\epsilon+n\rho+\pi_{1}(\check{z})$

and

$-n\epsilon+n\rho^{\prime}+\pi_{1}(\check{z^{\prime}})<\pi_{1}(\check{f}^{n}_{ann}(\check{z^{\prime}}))$

Therefore we have

$-2\epsilon
n+(\rho^{\prime}-\rho)n<\pi_{1}(\check{f}^{n}_{ann}(\check{z^{\prime}}))-\pi_{1}(\check{f}^{n}_{ann}(\check{z}))\leq
L_{\beta,\gamma}(f^{n}(\alpha))+2.$

And so

$0<\sigma_{f,\beta,\gamma}(\alpha).$

This contradicts Proposition 3.3.

Suppose now that there exists $z$ (in the interior $\overline{A}$) such that
$\operatorname{Rot}_{\check{f}_{ann}}(z)=\rho$ and that the rotation number of
the boundary component associated to $z_{0}$ is $\rho_{0}$, with
$\rho<\rho_{0}$. We have two cases.

Case 1. The local rotation set at $z_{0}$ is empty. In this case following
[12] we have one of the next three possibilities

  * (i)

$z_{0}$ is a global attractor,

  * (ii)

$z_{0}$ is a global repeller,

  * (iii)

$f$ is locally conjugate to the application $z\mapsto e^{2\pi
ip/q}z(1+z^{qr})$, for some integers $p$, $q$ and $r$.

If possibility (i) or (iii) is valid, then there exists a point $z^{\prime}$
(in the interior of $\overline{A}$) whose $\omega$-limit set is contained in
the boundary component of $\overline{A}$ associated to $z_{0}$ and so
$\operatorname{Rot}_{\check{f}_{ann}}(z^{\prime})=\rho_{0}$. This contradicts
the first part. If possibility (ii) is valid for both boundary components of
$\overline{A}$ the result follows by the first part.

Case 2. The local rotation set at $z_{0}$ is not empty. In this case following
[4], we know that the rotation set of the open annulus is an interval. Take a
rational number written in an irreducible way $p/q$ such that
$\rho<p/q<\rho_{0}$. We have again three possibilities (see [4]). Again, there
exists a point $z^{\prime}$ (in the interior of $\overline{A}$) such that
$\rho<p/q\leq\operatorname{Rot}_{\check{f}_{ann}}(z^{\prime})$. This is again
a contradiction with the first part.

This proves the result.

##  4 Proof of main results

###  4.1 Proof of Theorem A

In this section, we prove Theorem $A$. Let $f$ be a distortion element in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$. We will consider two
cases:

Case 1: f has at least three fixed points.
Let $z$ be a (positively) recurrent point of $f$ which is not fixed. By
Theorem 2.5 there exists an open annulus A which is maximal fixed point free
and that contains $z$. As $f|_{A}$ is isotopic to the identity, by Corollary
2.2, we can fix two fixed points $z_{0}$ and $z_{1}$ one in each connected
component of the complement of $A$. Let $z_{2}$ be a third fixed point and let
$\check{f}$ be a lift of $f|_{\mathbb{S}^{2}\setminus\\{z_{0},z_{1}\\}}$ to
the universal covering space of $\mathbb{S}^{2}\setminus\\{z_{0},z_{1}\\}$
that fixes a lift of $z_{2}$. Then by Proposition 3.1 it has a unique rotation
number which must be $\\{0\\}$. In this case the positively recurrent point
$z$ in $A$ has a well defined rotation number
$\operatorname{Rot}_{\check{f}}(z)$ and it must be equal to 0. From Theorem
2.4, $f|_{A}$ has a topological horseshoe. This contradicts Corollary 2.2 (see
Remark 1).

Case 2: $f$ has exactly two fixed points.
Let $z_{0}$ and $z_{1}$ be the two fixed points of $f$ and let $\check{f}$ be
a lift of $f|_{\mathbb{S}^{2}\setminus\\{z_{0},z_{1}\\}}$ to the universal
covering space of $\mathbb{S}^{2}\setminus\\{z_{0},z_{1}\\}$. Then as above it
has a unique rotation number. If this is rational as above we can show that
$f$ has a topological horseshoe which contradicts Corollary 2.2 (see Remark
1). Hence the unique rotation number of $\check{f}$ is irrational and so $f$
is an irrational pseudo-rotation.

###  4.2 Proof of Theorem B

In this section, we prove Theorem $B$. Let $f$ be a distortion element in
$\operatorname{Diff}^{1}_{0}\left(\mathbb{S}^{2}\right)$. Suppose that $f$ has
at least three fixed points. Fix us three distinct fixed points $z_{0}$,
$z_{1}$ and $z$ of $f$. Let $f_{ann}$ be the homeomorphism of the closed
annulus $\overline{A}$ obtained from $\mathbb{S}^{2}$ by blowing up the fixed
points $z_{0}$ and $z$. If $\check{f}_{ann}$ is a lift of $f_{ann}$ to the
universal covering space of $\overline{A}$ which fixes a lift of $z_{2}$, then
by Proposition 3.1 it has a unique rotation number which must be $\\{0\\}$.
Therefore the rotation number of both boundary components of $\overline{A}$ is
zero. Hence $Df(z)$ has a unique eigenvalue which is $1$.

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Departamento de Matemática y Ciencia de la Computación,
Universidad de Santiago de Chile.
Avenida Libertador Bernardo O”Higgins 3363,
Estación Central, Santiago, Chile.
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