# Odd-parity perturbations of the wormhole-like geometries and quasi-normal
modes in Einstein-Æther theory

Chao Zhanga,b,c,d<EMAIL_ADDRESS><EMAIL_ADDRESS>Anzhong
Wange111Corresponding author<EMAIL_ADDRESS>Corresponding author
Tao Zhua,b<EMAIL_ADDRESS>a Institute for theoretical physics and
cosmology, Zhejiang University of Technology, Hangzhou, 310023, China
b United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310023, China
c College of Information Engineering, Zhejiang University of Technology,
Hangzhou, 310023, China
d Department of Physics, Faculty of Science, Tokyo University of Science,
eGCAP-CASPER, Physics Department, Baylor University, Waco, Texas, 76798-7316,
USA

###### Abstract

The Einstein-$\AE$ther theory has drawn a lot of attentions in recent years.
As a representative case of gravitational theories that break the Lorentz
symmetry, it plays an important role in testing the Lorentz-violating effects
and shedding light on the attempts to construct quantum gravity. Since the
first detection to the gravitational wave, the event GW150914, a brand new
window has been opened to testing the theory of gravity with gravitational
wave observations. At the same time, the study of gravitational waves itself
also provides us a serendipity of accessing the nature of a theory. In this
paper, we focus on the odd-parity gravitational perturbations to a background
that describes a wormhole-like geometry under the Einstein-$\AE$ther theory.
Taking advantage of this set of analytic background solutions, we are able to
simplify the Lagrangian and construct a set of coupled single-parameter
dependent master equations, from which we solve for the quasi-normal modes
that carry the physical information of the emitted gravitational waves.
Basically, the results reflect a consistency between Einstein-$\AE$ther theory
and general relativity. More importantly, as long as the no-ghost condition
and the latest observational constraints are concerned, we notice that the
resultant quasi-normal mode solutions intimate a kind of dynamical
instability. Thus, the solutions are ruled out based on their stability
against small linear perturbations.

##  I Introduction

The detection of the first gravitational wave (GW) from the coalescence of two
massive black holes (BHs) by advanced LIGO/Virgo marked the beginning of a new
era — the GW astronomy Ref1 . Following this observation, about 90 GW events
have been identified by the LIGO/Virgo/KAGRA (LVK) scientific collaborations
(see, e.g., GWs ; GWs19a ; GWs19b ; GWsO3b ). In the future, more ground- and
space-based GW detectors will be constructed Moore2015 ; Gong:2021gvw , which
will enable us to probe signals with a much wider frequency band and larger
distances. This triggered the interest in the quasi-normal mode (QNM) of black
holes, as GWs emitted in the ringdown phase can be considered as the linear
combination of these individual modes Berti2009 ; Berti18 . Similarly,
attention has been paid in recent years to QNM originating from wormholes
Kono2018 ; Volkel2018 ; Kim2018 ; Franzin2022 .

From the classical point of view, QNMs are eigenmodes of dissipative systems.
The information contained in QNMs provides the keys to revealing whether BHs
are ubiquitous in our universe, and more important whether general relativity
(GR) is the correct theory to describe gravity even in the strong field regime
Berti18b . Basically, a QNM frequency $\omega$ contains two parts, the real
and imaginary parts. Its real part gives the vibration frequency while its
imaginary part provides the damping time.

In GR, according to the no-hair theorem, an isolated and stationary BH is
completely characterized by only three quantities, mass, angular momentum, and
electric charge. Astrophysically, we expect BHs to be neutral, so they are
uniquely described by the Kerr solution. Then, the QNM frequencies and damping
times will depend only on the mass and angular momentum of the finally formed
BH. Clearly, to extract physics from the ringdown phase, at least two QNMs are
needed. This will require the signal-to-noise ratio (SNR) to be of the order
100 Berti18 . Although such high SNRs are not achievable right now, it has
been shown that they may be achievable once the advanced LIGO, Virgo, and
KAGRA reach their fully designed sensitivities. In any case, it is certain
that they will be detected by the ground-based third-generation detectors,
such as Cosmic Explorer CE  and the Einstein Telescope ET , as well as the
space-based detectors, including LISA LISA , TianQin Liu2020 ; Shi2019 , Taiji
Taiji2 , and DECIGO DECIGO .

QNMs in GR have been studied extensively Chandra92 , including scalar, vector,
and tensor perturbations Iyer1987 . Such calculations have been extended from
the Schwarzschild BH to other more general cases, e.g., the Kerr BH Det1980 ;
Seidel1990 . In this procedure, several different techniques of computations
of QNMs were developed. For instance, the Wentzel-Kramers-Brillouin (WKB)
approach Will1985 ; Will1987 ; Konoplya2003 ; Jerzy2017 , the finite
difference method (FDM) XinLi2020 , the continued fraction method Leaver1985 ,
the shooting method Chandra1975 ; Doneva2010 , the matrix method Kai2017 , and
so on Kono2011 ; Gund1994 ; Bin2004 . Some of these methods have also been
applied to modified theories of gravity XinLi2020 ; Oliver2019 ; KZ07b . In
addition, some special approximations, e.g., the eikonal limit, have also been
extensively explored, see, for example, Ref. Huan2012  and references therein.

This paper focuses on the QNM problem in Einstein-Æther theory. A set of
analytic background solutions describing a throat geometry in the Einstein-
Æther theory will be considered, and the odd-parity perturbations will be
investigated. Such studies are well-motivated. In particular, the Einstein-
Æther theory is self-consistent, such as free of ghosts and instability
Jacobson , and satisfies all the experimental tests carried out so far OMW18 ;
Tsujikawa21 . Its Cauchy problem is also well-posed SBP19 , and energy is
always positive (as far as the hypersurface-orthogonal æther field is
concerned) GJ11 .

In comparison with other modified theories of gravity CFPS12 , including
scalar-tensor theories and their high-order corrections DL19 , Einstein-Æther
theory has the following distinguishable features: It is a particular vector-
tensor theory in which the vector field is always timelike. As a result, it
always defines a preferred frame and whereby violates locally the Lorentz
invariance (LI). Despite the facts that LI is the cornerstone of modern
physics, and all the experiments carried out so far are consistent with it
Bourgoin21 ; Bars19 ; Shao19 ; Bourgoin17 ; Flowers17 ; KR11 , violations of
LI have been well motivated and extensively studied in the past several
decades, especially from the point of view of quantum gravity Collin04 ;
Mattingly05 ; Liberati13 ; PT14 ; Wang17 . In Einstein-Æther theory there
exist three different species of gravitons, spin-0, spin-1, and spin-2, and
each of them propagates at different speeds JM04 . To avoid the vacuum gravi-
Čerenkov radiation, such as cosmic rays, each of these three species must move
with a speed that is at least no less than the speed of light EMS05 . As a
matter of fact, depending on the choice of the free coupling constants of the
theory, they can be arbitrarily large, and so far no upper limit of these
speeds are known KR11 .

With the above remarkable features of the Einstein-Æther theory, it is
interesting and important to find new predictions of the theory for the QNMs
mentioned above. The BH spectroscopy Berti18  has been extensively studied in
the last couple of years in terms of GWs emitted in the ringdown phase of
binary BHs (BBHs) (for example, see BLT21 ; LCV20 ; FBPF20 ; BFPF20 ; GST19 ;
IGFST19 ; CDV19 ; CN20 ; OC20 ; Shaik20 ; Uchi20 ; BCCK20 ; Cabero20 ;
Maselli20 ; Islam20  and references therein), and found that they are all
consistent with GR within the error bars allowed by the observations of the 90
GW events LVK2022 . And here, as mentioned earlier, we shift our insights to
the study of wormholes and throat geometries in the Einstein-Æther theory.

The rest of the paper is organized as follows. In Sec. II we provide a brief
introduction to the Einstein-Æther theory. Fundamental definitions will be
given and the background solutions of throat geometries will be discussed.
After that, a demonstration of simplifying the odd-parity perturbed Lagrangian
under the so-called isotropic coordinate is given in Sec. III. On top of that,
we are ready to process the derivations of QNM in Sec. IV. The basic steps of
calculations and main results are shown in there. With the results, some
concluding remarks are addressed in Sec. V.

We shall adopt the unit system so that $c=G_{N}=1$, where $c$ is the speed of
light and $G_{N}$ is the Newtonian gravitational constant [It’s worth
mentioning here that, after this, we still have one degree of freedom (d.o.f)
in choosing the unit system for $\\{time,length,mass\\}$]. We will also work
with the signature $(-,+,+,+)$. All the Greek letters run from 0 to 3.

##  II Einstein-Æther theory

In Einstein-Æther theory ($\ae$-theory), the fundamental variables of the
gravitational sector are Jacobson ,

$\left(g_{\mu\nu},u^{\mu},\lambda\right),$ (2.1)

where $g_{\mu\nu}$ is the four-dimension metric of the spacetime, $u^{\mu}$ is
the æther field, and $\lambda$ is a Lagrangian multiplier, which guarantees
that the æther field is always timelike and unity. Then, the general action of
the theory is given by,

$S=S_{\ae}+S_{m},$ (2.2)

where $S_{m}$ denotes the action of matter, and $S_{\ae}$ the gravitational
action of the $\ae$-theory, given, respectively, by

$\displaystyle S_{\ae}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G_{\ae}}\int{\sqrt{-g}\;d^{4}x\Big{[}{\cal{L}}_{\ae}\left(g_{\mu\nu},u^{\alpha},c_{i}\right)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+{\cal{L}}_{\lambda}\left(g_{\mu\nu},u^{\alpha},\lambda\right)\Big{]},$
$\displaystyle S_{m}$ $\displaystyle=$
$\displaystyle\int{\sqrt{-g}\;d^{4}x\Big{[}{\cal{L}}_{m}\left(g_{\mu\nu},u^{\alpha};\hat{\psi}\right)\Big{]}}.$
(2.3)

Here $\hat{\psi}$ collectively denotes the matter fields, and $g$ is the
determinant of $g_{\mu\nu}$, and

$\displaystyle{\cal{L}}_{\lambda}$ $\displaystyle\equiv$
$\displaystyle\lambda\left(g_{\alpha\beta}u^{\alpha}u^{\beta}+1\right),$
$\displaystyle{\cal{L}}_{\ae}$ $\displaystyle\equiv$ $\displaystyle
R(g_{\mu\nu})-M^{\alpha\beta}_{~{}~{}~{}~{}\mu\nu}\left(D_{\alpha}u^{\mu}\right)\left(D_{\beta}u^{\nu}\right),$
(2.4)

where $D_{\mu}$ denotes the covariant derivative with respect to $g_{\mu\nu}$,
$R$ is the Ricci scalar, and $M^{\alpha\beta}_{~{}~{}~{}~{}\mu\nu}$ is defined
as

$\displaystyle M^{\alpha\beta}_{~{}~{}~{}~{}\mu\nu}\equiv
c_{1}g^{\alpha\beta}g_{\mu\nu}+c_{2}\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}+c_{3}\delta^{\alpha}_{\nu}\delta^{\beta}_{\mu}-c_{4}u^{\alpha}u^{\beta}g_{\mu\nu},$

with $\delta_{\mu\nu}$ representing the Kronecker delta. Note that here we
assume that matter fields couple not only to $g_{\mu\nu}$ but also to the
æther field $u^{\mu}$. However, in order to satisfy the severe observational
constraints, such a coupling, in general, is assumed to be absent Jacobson .

The four coupling constants $c_{i}$’s are all dimensionless, and $G_{\ae}$ is
related to the Newtonian gravitational constant $G_{N}$ via the relation CL04
,

$G_{N}=\frac{G_{\ae}}{1-\frac{1}{2}c_{14}},$ (2.6)

where $c_{ij}\equiv c_{i}+c_{j}$.

###  II.1 Field Equations

The variations of the total action, respectively, with respect to
$g_{\mu\nu}$, $u^{\mu}$ and $\lambda$ yield respectively the field equations
Chao2020b ,

$\displaystyle R^{\mu\nu}-\frac{1}{2}g^{\mu\nu}R-S^{\mu\nu}$ $\displaystyle=$
$\displaystyle 8\pi G_{\ae}T^{\mu\nu},$ (2.7) $\displaystyle\AE_{\mu}$
$\displaystyle=$ $\displaystyle 8\pi G_{\ae}T_{\mu},$ (2.8) $\displaystyle
g_{\alpha\beta}u^{\alpha}u^{\beta}$ $\displaystyle=$ $\displaystyle-1,$ (2.9)

where $R^{\mu\nu}$ denotes the Ricci tensor, and

$\displaystyle S_{\alpha\beta}$ $\displaystyle\equiv$ $\displaystyle
D_{\mu}\Big{[}J^{\mu}_{\;\;\;(\alpha}u_{\beta)}+J_{(\alpha\beta)}u^{\mu}-u_{(\beta}J_{\alpha)}^{\;\;\;\mu}\Big{]}$
$\displaystyle+c_{1}\Big{[}\left(D_{\alpha}u_{\mu}\right)\left(D_{\beta}u^{\mu}\right)-\left(D_{\mu}u_{\alpha}\right)\left(D^{\mu}u_{\beta}\right)\Big{]}$
$\displaystyle+c_{4}a_{\alpha}a_{\beta}+\lambda
u_{\alpha}u_{\beta}-\frac{1}{2}g_{\alpha\beta}J^{\delta}_{\;\;\sigma}D_{\delta}u^{\sigma},$
$\displaystyle\AE_{\mu}$ $\displaystyle\equiv$ $\displaystyle
D_{\alpha}J^{\alpha}_{\;\;\;\mu}+c_{4}a_{\alpha}D_{\mu}u^{\alpha}+\lambda
u_{\mu},$ $\displaystyle T^{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle\frac{2}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}{\cal{L}}_{m}\right)}{\delta
g_{\mu\nu}},$ $\displaystyle T_{\mu}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{\sqrt{-g}}\frac{\delta\left(\sqrt{-g}{\cal{L}}_{m}\right)}{\delta
u^{\mu}},$ (2.10)

with

$J^{\alpha}_{\;\;\;\mu}\equiv
M^{\alpha\beta}_{~{}~{}~{}~{}\mu\nu}D_{\beta}u^{\nu}\,,\quad a^{\mu}\equiv
u^{\alpha}D_{\alpha}u^{\mu}.$ (2.11)

From Eq.(2.8), we find that

$\lambda=u_{\beta}D_{\alpha}J^{\alpha\beta}+c_{4}a^{2}-8\pi
G_{\ae}T_{\alpha}u^{\alpha},$ (2.12)

where $a^{2}\equiv a_{\lambda}a^{\lambda}$. Notice that, by considering only
the vacuum solutions (as what we will do later), the matter fields disappear,
leading to the absence of $T^{\mu\nu}$ as well as $T_{\mu}$ in Eqs. (2.7) and
(2.8).

It is easy to show that the Minkowski spacetime is a solution of $\ae$-theory,
in which the æther is aligned along the time direction,
${u}_{\mu}=\delta^{0}_{\mu}$. Then, the linear perturbations around the
Minkowski background show that the theory in general possesses three types of
excitations, scalar (spin-0), vector (spin-1), and tensor (spin-2) modes, with
their squared speeds given by JM04

$\displaystyle c_{S}^{2}$ $\displaystyle=$
$\displaystyle\frac{c_{123}(2-c_{14})}{c_{14}(1-c_{13})(2+c_{13}+3c_{2})}\,,$
$\displaystyle c_{V}^{2}$ $\displaystyle=$
$\displaystyle\frac{2c_{1}-c_{13}(2c_{1}-c_{13})}{2c_{14}(1-c_{13})}\,,$
$\displaystyle c_{T}^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{1-c_{13}},$
(2.13)

respectively. Here $c_{ijk}\equiv c_{i}+c_{j}+c_{k}$.

Requiring that the theory: 1) be self-consistent, such as free of ghosts; and
2) satisfies all the observational constraints obtained so far, it was found
that the parameter space of the theory is considerably restricted. In
particular, $c_{14}$, $c_{2}$ and $c_{13}$ are restricted to OMW18  222The
recent studies of the neutron binary systems showed that the PPN parameter
$\alpha_{1}$ is further restricted to $|\alpha_{1}|<10^{-5}$ GHBBCYY , which
is an order of magnitude stronger than the bounds from lunar laser ranging
experiments MWT08 . This will translate the constraint on $c_{14}$ given by
Eq.(2.14) to $0\lesssim c_{14}\lesssim\times 2.5\times 10^{-6}$, as one can
see clearly from Eq.(3.12) given in OMW18 .,

$\displaystyle 0\lesssim c_{14}\lesssim 2.5\times 10^{-5},$ (2.14)
$\displaystyle c_{14}\lesssim c_{2}\lesssim 0.095,$ (2.15)
$\displaystyle\left|c_{13}\right|\lesssim 10^{-15}.$ (2.16)

Taking $c_{13}=0$, the stability of the odd-parity perturbations of BHs
further requires $c_{4}=0$ Tsujikawa21 .

###  II.2 Background Geometry

From now on, we shall consider solely the vacuum solutions to the field
equations [cf., Eqs.(2.7)-(2.9)]. For later convenience, in this subsection,
we shall first adopt the isotropic coordinate, represented by
$x^{\mu}=(t,\rho,\theta,\phi)$ (see, e.g., §8.2 of Weinberg1972 ). Thus, for
the spherically symmetric time-independent case, the background line element
and the æther field are given by Jacob2021

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-e^{2\mu(\rho)}dt^{2}+e^{2\nu(\rho)}\left(d\rho^{2}+\rho^{2}d\Omega^{2}\right),$
$\displaystyle u^{\alpha}$ $\displaystyle=$ $\displaystyle
e^{-\mu(\rho)}\delta^{\alpha}_{0},$ (2.17)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$.

As pointed in Jacob2021 , substituting (II.2) into Eq.(2.7), and picking up
the $\rho\rho$\- 333Notice that, in Jacob2021  the coordinate “$\rho$” was
written as “$r$”. and $\theta\theta$-components, we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{1}{2}c_{14}\left(\mu^{\prime}\right)^{2}+2\mu^{\prime}\nu^{\prime}+\left(\nu^{\prime}\right)^{2}+\frac{2\mu^{\prime}}{r}+\frac{2\nu^{\prime}}{r},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\left(1-\frac{c_{14}}{2}\right)\left(\mu^{\prime}\right)^{2}+\mu^{\prime\prime}+\nu^{\prime\prime}+\frac{\mu^{\prime}}{r}+\frac{\nu^{\prime}}{r},$
(2.18)

where a prime in the superscript expediently stands for the derivative
respective to $\rho$.

Clearly, a master equation could be easily constructed from Eq.(II.2). As
shown in Jacob2021 , that leads to the solutions

$\displaystyle\mu$ $\displaystyle=$
$\displaystyle({\bar{q}}+1)\ln\left(\frac{r-r_{0}}{r+r_{0}}\right),$
$\displaystyle\nu$ $\displaystyle=$
$\displaystyle\ln\left[\left(1-\frac{r_{0}^{2}}{r^{2}}\right)\left(\frac{r-r_{0}}{r+r_{0}}\right){}^{-({\bar{q}}+1)}\right],$
(2.19)

where ${\bar{q}}\equiv\sqrt{1/(1-c_{14}/2)}-1\in[0,6.25\times 10^{-7})$[cf.,
Eq.(2.14) and footnote #1] and $r_{0}\equiv m/2$ Weinberg1972 ; Jacob2021 .
Note that, the solutions in (II.2) are characterized by a single parameter
$c_{14}$. In addition, they describe a throat geometry. As long as $c_{14}\neq
0$, a marginally trapped throat with a finite non-zero radius always exists,
and on one side of it the spacetime is asymptotically flat, while on the other
side, the spacetime becomes singular within a finite proper distance from the
throat Jacob2021 .

When moving to the Schwarzschild coordinate, represented by
$x^{\mu}=(t,r,\theta,\phi)$, the line element could be written as

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2m}{r}\right)^{{\bar{q}}+1}dt^{2}+\left(1-\frac{2m}{r}\right)^{-({\bar{q}}+1)}dr^{2}$
(2.20)
$\displaystyle+\left(1-\frac{2m}{r}\right)^{-{\bar{q}}}r^{2}d\Omega^{2},$

which will immediately go back to that of GR at the ${\bar{q}}\to 0$ limit
Jacob2021 .

Notice that, a simple relation between $r$ and $\rho$ is given by Weinberg1972

$\displaystyle r$ $\displaystyle=$
$\displaystyle\rho\left(1+\frac{r_{0}}{\rho}\right)^{2}.$ (2.21)

To make it more clear, we show the relation between $r$ and $\rho$ in Fig. 1,
in which we set $r_{0}=1/4$.

Figure 1: The relation between $r$ and $\rho$, where we have set $r_{0}=1/4$.

##  III The odd-parity perturbations

In this section, we move to the odd-parity perturbations to the background
described by Eq.(II.2). The full metric and æther field are given like

$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle{\bar{g}}_{\mu\nu}+\epsilon h_{\mu\nu},$ $\displaystyle u_{\mu}$
$\displaystyle=$ $\displaystyle{\bar{u}}_{\mu}+\epsilon w_{\mu}.$ (3.1)

Here, the ${\bar{g}}_{\mu\nu}$ and ${\bar{u}}_{\mu}$ denote the background
fields, described by Eq.(II.2), while $\epsilon$ stands for an infinitesimal
constant.

Firstly, we shall keep working in the isotropic coordinate. By mimicking
Thomp2017 , the perturbation terms could be written as

$\displaystyle h_{\mu\nu}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\begin{pmatrix}0&0&C_{lm}\csc\theta\partial_{\phi}&-C_{lm}\sin\theta\partial_{\theta}\\\
0&0&J_{lm}\csc\theta\partial_{\phi}&-J_{lm}\sin\theta\partial_{\theta}\\\
sym&sym&G_{lm}\csc\theta\big{(}\cot\theta\partial_{\phi}-\partial_{\theta}\partial_{\phi}\big{)}&sym\\\
sym&sym&\frac{1}{2}G_{lm}\big{(}\sin\theta\partial^{2}_{\theta}-\cos\theta\partial_{\theta}-\csc\theta\partial^{2}_{\phi}\big{)}&-G_{lm}\sin\theta\big{(}\cot\theta\partial_{\phi}-\partial_{\theta}\partial_{\phi}\big{)}\end{pmatrix}Y_{lm}(\theta,\phi),~{}~{}~{}$
(3.2)

and

$\displaystyle w_{\mu}$ $\displaystyle=$
$\displaystyle\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\begin{pmatrix}0\\\ 0\\\
a_{lm}\csc\theta\partial_{\phi}\\\
-a_{lm}\sin\theta\partial_{\theta}\end{pmatrix}Y_{lm}(\theta,\phi),$ (3.3)

where $a_{lm}$, $C_{lm}$, $J_{lm}$ and $G_{lm}$ are functions of $t$ and
$\rho$444 It must not be confused with the function $J_{lm}$ introduced here
and the tensor $J_{\alpha\beta}$ appearing in Eq.(II.1)., while
$Y_{lm}(\theta,\phi)$ denotes the spherical harmonics. Starting from now on,
we shall set $m=0$ in the above expressions so that
$\partial_{\phi}Y_{lm}(\theta,\phi)=0$, as now the background has the
spherical symmetry, and the corresponding linear perturbations do not depend
on $m$ Regge57 ; Thomp2017  555Notice that, since we are using the isotropic
coordinate and following some different conventions, the terms
$\\{C_{lm},J_{lm},G_{lm}\\}$ in (3.2) are not necessarily equal to their
counterparts given in Thomp2017 ..

###  III.1 Gauge Transformations and Gauge Fixing

For later convenience, we here investigate the infinitesimal gauge
transformations (Recall that we only consider the odd-parity perturbations and
$m=0$)

$\displaystyle x^{\alpha}\to
x^{\prime\alpha}=x^{\alpha}+\epsilon\xi^{\alpha},$ (3.4)

where

$\displaystyle\xi^{\alpha}$ $\displaystyle=$
$\displaystyle-\frac{\csc\theta\partial_{\theta}Y_{lm}(\theta,\phi)}{r^{2}}\left(0,0,0,1\right)e^{-2\nu}\Lambda,~{}~{}~{}~{}$
(3.5)

with $\Lambda$ being a function of $t$ and $\rho$. Under the transformation of
Eq.(3.4), we have

$\displaystyle\Delta
w_{\mu}\equiv\left(w_{\mu}\right)_{new}-\left(w_{\mu}\right)_{old}=-{\cal{L}}_{\xi}{\bar{u}}_{\mu},$
$\displaystyle\Delta
h_{\mu\nu}\equiv\left(h_{\mu\nu}\right)_{new}-\left(h_{\mu\nu}\right)_{old}=-{\cal{L}}_{\xi}{\bar{g}}_{\mu\nu},$
(3.6)

where $\cal{L}$ stands for the Lie derivative Invb . From Eq.(III.1) we find

$\displaystyle\Delta
C_{lm}\equiv\left(C_{lm}\right)_{old}-\left(C_{lm}\right)_{new}={\dot{\Lambda}},$
(3.7) $\displaystyle\Delta
G_{lm}\equiv\left(G_{lm}\right)_{old}-\left(G_{lm}\right)_{new}=-2\Lambda,$
$\displaystyle\Delta
J_{lm}\equiv\left(J_{lm}\right)_{old}-\left(J_{lm}\right)_{new}=\Lambda^{\prime}-\frac{2(1+r\nu^{\prime})}{r}\Lambda,$
$\displaystyle\Delta
a_{lm}\equiv\left(a_{lm}\right)_{old}-\left(a_{lm}\right)_{new}=0,$

where a prime and a dot stand for the derivatives with respect to $\rho$ and
$t$, respectively. With the above gauge transformations, we can construct the
gauge-invariant (GI) quantities, and due to the presence of the æther field,
three such independent quantities can be constructed, in contrast to the
relativistic case, in which only two such quantities can be constructed. These
three gauge invariants can be defined as

$\displaystyle{\cal{X}}_{lm}(t,\rho)$ $\displaystyle\equiv$
$\displaystyle{C}_{lm}+\frac{1}{2}{\dot{G}}_{lm},$
$\displaystyle{\cal{Y}}_{lm}(t,\rho)$ $\displaystyle\equiv$
$\displaystyle{J}_{lm}+\frac{1}{2}{G}_{lm}^{\prime}-\left(\frac{1}{r}+\nu^{\prime}\right){G}_{lm},$
$\displaystyle{\cal{Z}}_{lm}(t,\rho)$ $\displaystyle\equiv$ $\displaystyle
a_{lm}.$ (3.8)

Of course, any combination of these quantities is also GI. According to
Eq.(III.1), by simply choosing the gauge condition ${G}_{lm}=0$ (This could be
referred as the RW gauge Thomp2017 ), ${\cal{X}}_{lm}$ and ${\cal{Y}}_{lm}$
will reduce to $C_{lm}$ and $J_{lm}$, respectively. This will be the gauge
condition that we shall adopt in the following of this paper.

###  III.2 Simplified Lagrangian

To derive the partial differential equations (PDEs) for the perturbations, we
need to first simplify the original Lagrangian, adopted in (2.2). Up to the
2nd order of $\epsilon$, the total action can be cast in the form

$\displaystyle{S}$ $\displaystyle=$
$\displaystyle{S}_{(0)}+\epsilon{S}_{(1)}+\epsilon^{2}{S}_{(2)}+{\cal
O}(\epsilon^{3}).$ (3.9)

Following Tsujikawa21 , we substitute (III) [together with Eqs.(II.2), (3.2)
and (3.3)] into (2.2), and then pick up the ${\cal O}(\epsilon^{2})$ terms. In
addition, from now on, we shall set

$\displaystyle c_{13}$ $\displaystyle=$ $\displaystyle 0,$ (3.10)

since it has been confined to an extremely narrow region [cf., (2.16)]. The
resultant 2nd-order action is given by

$\displaystyle{\cal S}_{(2)}=\int dtd\rho{\cal L}_{(2)},$ (3.11)
$\displaystyle{\cal L}_{(2)}={\upsilon}{\cal
L}_{\text{odd}},\quad{\upsilon}\equiv{\frac{r^{2}e^{\mu+3\nu}}{16\pi
G_{\ae}}}.$ (3.12)

Notice that, the Lagrangian ${\cal L}_{(2)}$ is only a function of $t$ and
$\rho$, as $\theta$ and $\phi$ have been integrated out, during which
procedure, the features of spherical harmonics have been used intensively (see
Appendix A).

With the help of integration by parts Arfken , plus the background field
equations [cf., (2.7)-(2.9)], the quantity ${\cal L}_{\text{odd}}$ is
simplified to

$\displaystyle{\cal L}_{\text{odd}}$ $\displaystyle=$
$\displaystyle\alpha_{1}{\dot{a}}_{lm}^{2}+\alpha_{2}a_{lm}^{\prime
2}+\alpha_{3}\chi{\dot{J}}_{lm}+\alpha_{4}\chi{C}_{lm}^{\prime}+\alpha_{5}a_{lm}^{2}+\alpha_{6}\chi^{2}+\alpha_{7}J_{lm}^{2}+\alpha_{8}C_{lm}^{2}+\alpha_{9}\chi
a_{lm}+\alpha_{10}\chi{C}_{lm}$ (3.13) $\displaystyle=$
$\displaystyle\alpha_{1}{\dot{a}}_{lm}^{2}+\alpha_{2}a_{lm}^{\prime
2}+\alpha_{3}\chi{\dot{J}}_{lm}+\alpha_{4}\chi{C}_{lm}^{\prime}+\alpha_{5}a_{lm}^{2}-\alpha_{6}\chi^{2}+\alpha_{7}J_{lm}^{2}+\alpha_{8}C_{lm}^{2}+\alpha_{9}\chi
a_{lm}+\alpha_{10}\chi{C}_{lm}$
$\displaystyle+2\alpha_{6}\chi\left[2c_{14}e^{\mu}\mu^{\prime}a_{lm}-C_{lm}^{\prime}+2\left(\nu^{\prime}+\frac{1}{\rho}\right)C_{lm}+{\dot{J}}_{lm}\right],$

where

$\displaystyle\chi(t,\rho)$ $\displaystyle\equiv$ $\displaystyle
2c_{14}e^{\mu}\mu^{\prime}a_{lm}-C_{lm}^{\prime}+2\left(\nu^{\prime}+\frac{1}{\rho}\right)C_{lm}+{\dot{J}}_{lm},$

which could be matched to its counterpart in Tsujikawa21 . Recall that
$C_{lm}$, $J_{lm}$ and $a_{lm}$ (as well as their combination $\chi$) are GI
under the chosen gauge condition, viz., $G_{lm}=0$ [cf., Eq.(III.1)]. Note
that, before stimulating any confusion, we shall omit the subscript $lm$ for
$\chi$. The coefficients $\alpha_{i}$’s are functions of $\rho$, $L$,
$c_{14}$, $c_{1}$, etc. We abbreviate their full expressions since they are
only for the narration of intermediate steps.

In (3.13), the apparent d.o.f for the odd-parity perturbations is 3, spanned
by $\\{J_{lm},C_{lm},a_{lm}\\}$. However, the true d.o.f is 2 Tsujikawa21 . To
get rid of that redundant d.o.f, we apply the Euler-Lagrange (E-L) equation
Taylor05  to ${\cal L}_{(2)}$ with respect to $J_{lm}$ and $C_{lm}$, and
obtain

$\displaystyle J_{lm}$ $\displaystyle=$ $\displaystyle
e^{2\nu}\rho^{2}{\dot{\chi}}$
$\displaystyle\times\Big{\\{}e^{2\mu}\Big{[}c_{14}\rho^{2}\left(\mu^{\prime}\right)^{2}-L+2\rho^{2}\left(\nu^{\prime}\right)^{2}$
$\displaystyle+4\rho\mu^{\prime}\left(\rho\nu^{\prime}+1\right)+4\rho\nu^{\prime}+2\Big{]}\Big{\\}}^{-1},$
$\displaystyle C_{lm}$ $\displaystyle=$
$\displaystyle\frac{\rho^{2}}{2-L}\chi^{\prime}+\frac{\rho\left(\rho\mu^{\prime}-\rho\nu^{\prime}-2\right)}{L-2}\chi,$
(3.15)

where we have defined $L\equiv l(l+1)$. Substitute (III.2) into the second
line of (3.13), together with some rearrangements, ${\cal L}_{\text{odd}}$
becomes

$\displaystyle{\cal L}_{\text{odd}}$ $\displaystyle=$
$\displaystyle\beta_{1}{\dot{\chi}}^{2}+\beta_{2}{\dot{a}}_{lm}^{2}+\beta_{3}{\chi}^{\prime
2}+\beta_{4}{a}_{lm}^{\prime 2}$ (3.16)
$\displaystyle+\beta_{5}{\chi}^{2}+\beta_{6}{a}_{lm}^{2}+\beta_{7}\chi{a}_{lm}.$

The coefficients $\beta_{i}$’s could be found in Appendix B. Clearly, as
promised earlier, the current apparent d.o.f for the 2nd-order Lagrangian is
2, spanned by $\\{\chi,a_{lm}\\}$.

##  IV Quasi-normal modes of the odd-parity perturbations

###  IV.1 Master Equations

Combining (3.12) and (3.16), we obtain the simplified 2nd-order Lagrangian for
the $c_{13}=0$ case. Now we are ready to derive the master equation for
calculating the QNMs. With this Lagrangian, we apply the E-L equation to it
with respect to $\chi$ and $a_{lm}$. That leads to a set of coupled PDEs

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-{\ddot{\chi}}+\gamma_{11}\chi^{\prime\prime}+\gamma_{12}\chi^{\prime}+\gamma_{13}\chi+\gamma_{14}a_{lm},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle-{\ddot{a}}_{lm}+\gamma_{21}a_{lm}^{\prime\prime}+\gamma_{22}a_{lm}^{\prime}+\gamma_{23}a_{lm}+\gamma_{24}\chi,~{}~{}~{}~{}$
(4.1)

where a prime in the superscript expediently denotes the derivative with
respect to $\rho$. The expressions of $\gamma_{ij}$’s could be found in
Appendix C.

In the next step, we shall apply the analytic solutions (II.2), and transfer
our PDEs from the isotropic coordinate to the Schwarzschild coordinate
$(t,r,\theta,\phi)$. In addition, taking advantage of the residual d.o.f for
choosing the unit system, we further set $m=1/2$ (so that the unit system of
$\\{time,length,mass\\}$ is totally fixed). By doing so, the PDEs are
translated to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-{\ddot{\chi}}+\left(\frac{r-1}{r}\right)^{\bar{q}+2}\chi^{\prime\prime}+\frac{2}{r}\left(\frac{r-1}{r}\right)^{\bar{q}+\frac{3}{2}}\chi^{\prime}$
$\displaystyle+\frac{1}{4}(r-1)^{\bar{q}}r^{-\bar{q}-4}\Big{[}(8r-4)\left(\bar{q}+2\right)-\left(\bar{q}+2\right)^{2}+4r(L(-r)+L+r-1)-4\sqrt{(r-1)r^{3}}+2\sqrt{(r-1)r}-3\Big{]}\chi$
$\displaystyle-\frac{4(L-2)\left[-2r+2\sqrt{(r-1)r}+1\right]\left[\left(\bar{q}+2\right)^{2}-4\right]}{[(r-1)r]^{3/2}\left(\bar{q}+2\right)}\left(\frac{r-1}{r}\right)^{\frac{5}{4}\left(\bar{q}+2\right)}a_{lm},$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle-{\ddot{a}}_{lm}+\frac{c_{1}\left(\bar{q}+2\right)^{2}}{2\left[\left(\bar{q}+2\right)^{2}-4\right]}\left(\frac{r-1}{r}\right)^{\bar{q}+2}a_{lm}^{\prime\prime}+\frac{c_{1}\left(\bar{q}+2\right)^{3}(r-1)^{\bar{q}+1}r^{-\bar{q}-3}}{4\left[\left(\bar{q}+2\right)^{2}-4\right]}a_{lm}^{\prime}$
(4.2)
$\displaystyle+\frac{(r-1)^{\bar{q}}r^{-\bar{q}-4}}{32\left[\left(\bar{q}+2\right)^{2}-4\right]}a_{lm}$
$\displaystyle\times\left\\{\left(\bar{q}+2\right)\left[\left(\bar{q}+2\right)\left(4\left(c_{1}-2\right)(2r-1)\left(\bar{q}+2\right)-3c_{1}\left(\bar{q}+2\right)^{2}-16c_{1}L(r-1)r+32\right)+64r-32\right]-128\right\\}$
$\displaystyle+\frac{1}{16}\left[2r+2\sqrt{(r-1)r}-1\right]\left(\bar{q}+2\right)(r-1)^{\frac{3\bar{q}}{4}}r^{\frac{1}{4}(-3)\left(\bar{q}+4\right)}\chi,~{}~{}~{}~{}$

where a prime in the superscript now stands for the derivative with respect to
$r$. A dot, as usual, stands for the derivative with respect to $t$.

In writing down Eq.(IV.1), the stability condition of BHs that found in
Tsujikawa21 , viz., $c_{4}=0$, needs to be considered [so that the only
theory-dependent coupling constant of Eq.(IV.1) is $c_{1}$ (or equivalently,
$\bar{q}$), since now we have $c_{1}=c_{14}$]. By performing suitable
coordinate transformations, the PDEs in (IV.1) could be further translated to
[the formulas in Appendix D were used in deriving Eqs.(4.3) and (4.4)]

$\displaystyle Q_{1}\Psi_{2}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}\Psi_{1}}{\partial
x^{2}}-\left[\frac{\partial^{2}}{\partial t^{2}}+V_{1}(r)\right]\Psi_{1},$
(4.3) $\displaystyle Q_{2}\Psi_{1}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}\Psi_{2}}{\partial
x^{2}}-\left[\frac{\partial^{2}}{\partial
t^{2}}+V_{2}(r)\right]\Psi_{2},~{}~{}$ (4.4)

where

$\displaystyle\frac{dr}{dx}$ $\displaystyle=$
$\displaystyle\left(\frac{r-1}{r}\right)^{\frac{1}{2}\left(\bar{q}+2\right)},$
$\displaystyle\Psi_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{-2r+2\sqrt{(r-1)r}+1}\left(\frac{r-1}{r}\right)^{\frac{1}{4}\left(-\bar{q}-2\right)}\chi,$
$\displaystyle\Psi_{2}$ $\displaystyle\equiv$ $\displaystyle a_{lm},$ (4.5)

and

$\displaystyle V_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{\left[-24(r-1)\bar{q}+3\bar{q}^{2}+16(r-1)(Lr-3)\right]}{16r^{4}}\left(\frac{r-1}{r}\right)^{\bar{q}},$
$\displaystyle V_{2}$ $\displaystyle\equiv$
$\displaystyle\Big{\\{}\left(\bar{q}+2\right)\left[\left(\bar{q}+2\right)\left(4\left(\bar{q}+2\right)\left(-2c_{1}r+c_{1}+4r-2\right)+3c_{1}\left(\bar{q}+2\right)^{2}+16c_{1}L(r-1)r-32\right)-64r+32\right]$
$\displaystyle~{}~{}~{}+128\Big{\\}}\left(\frac{r-1}{r}\right)^{\bar{q}}\frac{1}{32r^{4}\bar{q}\left(\bar{q}+4\right)},$
$\displaystyle Q_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{4(L-2)\bar{q}\left(\bar{q}+4\right)(r-1)^{\bar{q}+\frac{1}{2}}r^{-\bar{q}-\frac{7}{2}}}{\bar{q}+2},$
$\displaystyle Q_{2}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{16}\left(\bar{q}+2\right)(r-1)^{\bar{q}+\frac{1}{2}}r^{-\bar{q}-\frac{7}{2}}.$
(4.6)

It’s worth mentioning here that, at the ${\bar{q}}\to 0$ limit, Eq.(4.3) will
reduce precisely to that of GR (see, e.g., the Eq.(2.13) of Kono2011 ).

###  IV.2 Calculate for the QNMs

As has been shown, taking $c_{13}=c_{4}=0$ brings us the coupled PDEs,
Eqs.(4.3) and (4.4). With the above master equations, we are now at the stage
of solving them for QNMs. To deal with this set of coupled PDEs, we shall
apply the finite difference method (FDM) XinLi2020 ; Habermanb . By working
with the FDM, we are expecting to solve (4.3) and (4.4) for $\Psi_{1,\;2}$.
The solved $\Psi_{1,\;2}$ will carry the information of QNM frequency $\omega$
(see, e.g., Berti2009  for a review). Notice that, $\Psi_{1,\;2}$ can only
reflect the comprehensive effects of all the existing $\omega$’s and it’s not
trivial to extract individual $\omega$’s from $\Psi_{1,\;2}$.

To perform FDM, we shall basically follow Habermanb ; Kai2016 , and obtain the
recursion formula

$\displaystyle\Psi_{1,\;2}(t+\delta t,x)$ $\displaystyle\cong$
$\displaystyle-\Psi_{1,\;2}(t-\delta
t,x)-\left[2\left(\mathfrak{v}_{1,2}^{2}\frac{\delta t^{2}}{\delta
x^{2}}-1\right)+\delta t^{2}V_{1,\;2}(r(x))\right]\Psi_{1,\;2}(t,x)$ (4.7)
$\displaystyle+\mathfrak{v}_{1,2}^{2}\frac{\delta t^{2}}{\delta
x^{2}}\left[\Psi_{1,\;2}(t,x-\delta x)+\Psi_{1,\;2}(t,x+\delta
x)\right]-\delta t^{2}Q_{1,\;2}(r(x))\Psi_{2,\;1}(t,x),$

where $\delta t$ and $\delta x$ are the step sizes for the $t$ and $x$
directions, respectively. They will be assigned suitable values in practice
according to our usage. Here, $\mathfrak{v}_{1,2}^{2}$ denote the speed factor
in front of the $\partial^{2}/\partial x^{2}$ terms. Of course, for PDEs like
Eqs.(4.3) and (4.4) we have $\mathfrak{v}_{1,2}=1$.

The calculations of $\Psi_{1,\;2}(t,x)$ will be performed on a isosceles
triangular lattice in the $t-x$ Cartesian coordinate system. The bottom side
of the triangle contains $2N+1$ points so that $x/\delta
x\in[-N,N]\cap\mathbb{Z}$, where $N$ is a positive integer that will be chosen
properly according to our usage. The height of the triangle contains $N+1$
points so that $t/\delta t\in[1,N+1]\cap\mathbb{Z}$. After $2N^{2}$
iterations, and with apt initial conditions, the functions
$\Psi_{1,\;2}(t,x=0)$ could be obtained numerically. In practice, the initial
conditions are chosen to be XinLi2020 ; Kai2016

$\displaystyle\Psi_{1,\;2}(t=0,x)$ $\displaystyle=$ $\displaystyle
e^{-(x-1)^{2}/2},$
$\displaystyle\left.\frac{d\Psi_{1,\;2}(t,x)}{dt}\right|_{t=0}$
$\displaystyle=$ $\displaystyle 0.$ (4.8)

|  |
---|---|---
(a) | (b) | (c)
|  |
(d) | (e) | (f)
|  |
(g) | (h) | (i)
Figure 2: The temporal evolution of the functions $\ln|\Psi_{1,2}(t,x=0)|$ for
the $l=2$ mode and $c_{13}=c_{4}=0$ case, together with the GR case as a
comparison ($\Psi_{1}$: green solid line; $\Psi_{2}$: purple dot-dashed line;
$\Psi_{\text{GR}}$: red dotted line). Panels (a)-(i) correspond to
${\bar{q}}=10^{-12},10^{-9},6\times 10^{-7},3\times 10^{-3},3\times
10^{-2},4\times 10^{-2},3\times 10^{-1},2\times 10^{-1},5\times 10^{-1}$,
respectively. Notice that, here we are adopting the unit system so that
$c=G_{N}=2m=1$. Here $N$ is chosen to be 250. Figure 3: The temporal evolution
of the functions $\ln|\Psi_{1,2}(t,x=0)|$ for the $l=2$ mode and
${\bar{q}}=2.848\times 10^{-1}$, where we have set $c_{13}=c_{4}=0$. Notice
that, here we are adopting the unit system so that $c=G_{N}=2m=1$. Here $N$ is
chosen to be 350.

Due to the observational significance of the $l=2$ mode Mag18 ; Shi2019 ;
Ghosh2021 ; LVK2022 , we shall mainly focus on this case (Of course, the
computations and analysis here could be easily extended to higher $l$’s). The
main results of the corresponding solutions are exhibited in Fig. 2. In there
we plot out the temporal evolution of $\ln|\Psi_{1,2}(t,x=0)|$, together with
the GR case as a comparison. Recall that we are adopting the unit system so
that $c=G_{N}=2m=1$. To show it more explicit how the magnitude of ${\bar{q}}$
influences the behaviors of $\Psi_{1,2}$, we considered both physically
allowed and forbidden ${\bar{q}}$’s. Based on Fig. 2, we have the following
comments:

  * •

From panel (a) to panel (i), the value of ${\bar{q}}$ is increasing. We
observed that, in general, the behavior of $\Psi_{1}$ is quite different from
that of $\Psi_{2}$, which makes sense since the former is mainly from the
gravitational perturbations [the contributions from the æther field to
$\Psi_{1}$ is suppressed by the small factor $c_{1}$, as seen from the
definition (III.2)] while the latter represents the contributions of the æther
field.

  * •

When $\bar{q}$ is small enough [e.g., panel (a)], the curve of $\Psi_{1}$ is
almost overlapped with that of GR. In contrast, when $\bar{q}$ is large [e.g.,
panel (i)], the curve of $\Psi_{1}$ will deviate a lot from that of GR. Of
course, these are what we expected.

  * •

As $\bar{q}$ approaching $0$ [cf., panels (a)-(d)], the deformation on the
curve of $\Psi_{2}$ tends to disappear. This makes sense since $V_{2}$ and
$Q_{2}$ are of the form ${\cal O}(1)+{\cal O}({\bar{q}})$ for a tiny $\bar{q}$
[cf., (IV.1)].

  * •

In panels (d)-(i), physically forbidden $\bar{q}$’s were used [cf.,
Eq.(2.14)]. Even though, from there we see clearly how the dynamical
instability arises. When $\bar{q}$ is large enough, $\Psi_{1,2}$ behaves in a
healthy way [e.g., panel (i)]. However, for small enough $\bar{q}$’s,
$\Psi_{1,2}$ will finally blow up, which reflects the existence of a dynamical
instability Konoplya2018  [e.g., panel (g)]. As could be estimated, the
critical point should occur at $\bar{q}\simeq 0.3$ [cf., panels (h) and (i)].

Since the position of the critical point has special significance, we carry
out a more careful investigation of that. By selecting a $\bar{q}$ near the
critical point, the curves of $\Psi_{1,2}$ are shown in Fig. 3 with a larger
scope of $r$. In there we observe the plateaux appearing on the curves of
$\Psi_{1,2}$, which, from the phenomenological point of view, indicates that
the current $\bar{q}$ is around the critical point Kai2016 .

  * •

In contrast, physically allowed $\bar{q}$’s [cf., Eq.(2.14)] were selected for
panels (a)-(c). Although $\Psi_{1,2}$ will blow up soon or later, we observe
that the curves of $\Psi_{1}$’s are almost overlapped with that of GR, before
blowing up. That makes sense since those physically allowed $\bar{q}$’s are
extremely small, and ${\bar{q}}=0$ means the GR limit.

  * •

From panels (a)-(h), we noticed that the times of blowing up for $\Psi_{1,2}$
are getting earlier and earlier simultaneously as $\bar{q}$ getting smaller
from a big enough value [cf., panel (h)]. This tendency vaporizes at
${\bar{q}}\approx 3\times 10^{-2}$, where $\Psi_{1}$ and $\Psi_{2}$ exchange
their chronological order of blowing up [cf., panels (e) and (f)]. Starting
from this point, the curve of $\Psi_{2}$ is getting frozen step by step [cf.,
panels (d) and (e)], and tends to lose its sensitivity on the magnitude of
$\bar{q}$ [cf., panels (a)-(d), as mentioned earlier]. While for the curve of
$\Psi_{1}$, the time of its blowing up is getting more and more postponed as
$\bar{q}$ approaches zero [cf., panels (a)-(e)].

Figure 4: The effective potential $V_{2}$ [cf., Eq.(IV.1)] as a function of
$r$ for the $l=2$ mode and different choices of $\bar{q}$’s, where we have set
$c_{13}=c_{4}=0$. Notice that, here we are adopting the unit system so that
$c=G_{N}=2m=1$.

Qualitatively speaking, the dynamical instability 666One should avoid
confusing this kind of instability with the ghost and Laplacian instabilities
discussed in Tsujikawa21 . In fact, by following the formalism of Tsujikawa21
and using the Lagrangian represented by (3.16), it’s straightforward to
exclude these two kinds of instabilities for the $c_{13}=c_{4}=0$ and
${\bar{q}}>0$ cases. mentioned above could mainly be attributed to the
features of the effective potential $V_{2}$. As seen from Fig. 4, the
effective potential $V_{2}$ can stay positive once $\bar{q}$ is large enough.
In contrast, a small $\bar{q}$ may drag a part of $V_{2}$ to the negative
regime. By following the formalism of Takahashi2013 ; Gannouji2022 , we notice
that being negative on $V_{2}$ (or $V_{1}$, although we abbreviated this part
of the discussion since $V_{1}$ was observed to stay positive for all the
reasonable $\bar{q}$’s, e.g., those considered in Fig. 2) can introduce
instability to our dynamical system. This explains the peculiar behaviors of
$\Psi_{1,2}$ shown in Fig. 2 and Fig. 3 777Notice that, although $V_{1}(r)$
stays positive (at least for the physically allowed values of $\bar{q}$’s), a
general analysis to the PDEs tells us that the blowing-up feature of
$\Psi_{2}$ could be conveyed to $\Psi_{1}$ Evans2010 , as now $\Psi_{2}$ is
acting as its source term [cf., Eq.(4.3)]. On the other hand, one should not
forget that $\Psi_{1}$ in fact carries the information of $\Psi_{2}$ by
definitions (III.2) and (IV.1).. Thus, although it sounds a little counter-
intuitive, we conclude that a relatively large $\bar{q}$ can guarantee the
stability of our dynamical system 888Similar analysis to this kind of
dynamical instability could be done for Kai2016 . It is also due to the
negative sign of the corresponding effective potential (under certain choices
of the coupling constant), that instability occurs. Nonetheless, different
from our case, in there, a smaller coupling constant could erase such an
instability., while on the other hand (as expected) enhancing the deviation
between $\Psi_{1}$ and $\Psi_{\text{GR}}$, and vice versa.

Another interesting phenomenon is that, the instability mentioned above could
be relaxed a little bit for a larger $l$. As seen from Fig.5, for same
$\bar{q}$, choosing a relatively larger $l$ could postpone the time of being
blow-up for $\Psi_{1,2}$ (what we observed here is different from that of
Konoplya2008 ). That is, higher multipoles tend to be more stable.

Figure 5: The temporal evolution of the functions $\ln|\Psi_{1}(t,x=0)|$
(upper panel) and $\ln|\Psi_{2}(t,x=0)|$ (lower panel) for the $l=2,4,6$ modes
(correspond to the green solid line, purple dot-dashed line and red dotted
line, respectively) and ${\bar{q}}=6\times 10^{-7}$, where we have set
$c_{13}=c_{4}=0$. Notice that, here we are adopting the unit system so that
$c=G_{N}=2m=1$. Here $N$ is chosen to be 250.

It is worth mentioning here that, the ratio between $\delta x$ and $\delta t$
plays an important role in controlling the numerical stability during the
calculations of $\Psi_{1,2}$. By defining $\mathfrak{k}\equiv\delta x/\delta
t$, a Courant-like stability condition is given by (see, e.g., the §6.5 of
Habermanb  and LeVeque2007 )

$\displaystyle\frac{2\mathfrak{v}_{1,2}^{2}}{\mathfrak{k}^{2}}{\bar{k}}-\delta
t^{2}\left(V_{1,\;2}+Q_{1,\;2}\right)\in(-4,0),$ (4.9)

where ${\bar{k}}\in(-2,0)$. Since the numerical values of $V_{1,\;2}$ and
$Q_{1,\;2}$ can be easily obtained, we can solve for the approximate valid
region of $\mathfrak{k}$ from (4.9) for any specific $\delta t$. We then
noticed that, for the $l=2$ case and a typical $\delta t$ [viz., $\delta
t\in(0,1)$], $\mathfrak{k}=1.2$ could guarantee the condition (4.9). This
could be further confirmed by monitoring the behavior of solved $\Psi_{1,2}$
with the varying $\mathfrak{k}$. Indeed, we observed that the deformation of
curves of $\Psi_{1,2}$ shows a convergent behavior for $\mathfrak{k}\gtrsim 1$
[In contrast, the curve of resultant $\Psi_{1}\setminus\Psi_{2}$ is sensitive
to the choice of $\mathfrak{k}$ for $\mathfrak{k}\in(0,1)$, which reflects a
numerical instability]. Notice that, such a convergent behavior also supports
the validity of the individual values of our chosen $\delta t$ and $\delta x$
999 To be specific, the emergence of such a convergent behavior means the
reasonable variations of $\delta t$ and $\delta x$ won’t change our numerical
results. Technically speaking, $\delta t$ and $\delta x$ are the most
significant factors in our numerical method. Other software-related factors
were found to have trivial influence on the numerical results, as long as we
are working under a hing enough precision and accuracy (this is exactly what
we did in practice). Therefore, by following some general protocols of judging
a numerical method, we are able to exclude the numerical instabilities for our
results. .

##  V Conclusions

In this paper, we investigated QNMs of odd-parity perturbations in Einstein-
Æther theory. Specially, here we pay attention to a kind of wormhole-like
backgrounds, described by (II.2). To find the corresponding QNMs, we first
work in the isotropic coordinate and simplify the 2nd-order Lagrangian to
Eq.(3.16) by following Tsujikawa21 . Then, the desired master equations in the
Schwarzschild coordinate could be obtained in terms of a set of coupled PDEs
[cf., Eqs.(4.3) and (4.4)], which depend only on the coupling constant $c_{1}$
(or equivalently, $\bar{q}$), after setting $c_{13}=c_{4}=0$ from the
requirement of no-ghost conditions Tsujikawa21  and constraints from
observations [cf., (2.16)].

As has been mentioned in Sec. I, there are many different techniques in
general for calculating QNMs. However, in reality, the FDM is identified as
one of the most apt ways for our case. By mainly focusing on the $l=2$ mode
(due to its physical significance Mag18 ; Shi2019 ; Ghosh2021 ; LVK2022 ),
that set of PDEs are solved numerically with the help of the recursion formula
(4.7). By varying the coupling constant $c_{1}$, or equivalently, $\bar{q}$
[cf., (II.2)], the corresponding solutions are shown in terms of $\Psi_{1,2}$
vs. $t$, and are exhibited in Figs. 2 and 3.

As shown in Sec. IV. B, we read off the features of $\Psi_{1,2}$, and their
dependence on $\bar{q}$ from Figs. 2 and 3. Specially, we found that choosing
the physically allowed values of $\bar{q}$’s [cf., (2.16)] will bring the
dynamical system to a kind of instability (which is different from that found
in Tsujikawa21 ), as observed from the panels (a)-(c) of Fig. 2. This
phenomenon is actually consistent with the behaviors of the effective
potentials [cf., Eq.(IV.1)], since a physically allowed value of $\bar{q}$
will drag a part of the potential $V_{2}$ to the negative regime [cf., Fig.
4]. Therefore, we conclude that the background solutions described by (II.2)
are not stable against the odd-parity perturbations and should be ruled out.

## Acknowledgements

This work is supported by the National Key Research and Development Program of
China under Grant No.2020YFC2201503, the National Natural Science Foundation
of China under Grant No. 12275238, No. 11975203, No. 11675143, No. 12205254,
the Zhejiang Provincial Natural Science Foundation of China under Grant No.
LR21A050001 and LY20A050002, and the Fundamental Research Funds for the
Provincial Universities of Zhejiang in China under Grant No. RF-A2019015.

## Appendix A: Integral formulas for spherical harmonics

The spherical harmonics $Y_{lm}(\theta,\phi)$ has the following integral
features Zettili ; Kase:2018voo

$\displaystyle\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta
Y_{l0}^{2}(\theta,\phi)\sin\theta=1,$ (A. 1)
$\displaystyle\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\left|\frac{\partial}{\partial\theta}Y_{l0}(\theta,\phi)\right|^{2}\sin\theta=L,$
(A. 2)
$\displaystyle\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\Bigg{[}\csc\theta\left|\frac{\partial}{\partial\theta}Y_{l0}(\theta,\phi)\right|^{2}+\sin\theta\left|\frac{\partial^{2}}{\partial\theta^{2}}Y_{l0}(\theta,\phi)\right|^{2}\Bigg{]}=L^{2}.~{}~{}~{}$
(A. 3)

## Appendix B: Expressions of $\beta_{i}$

The coefficients $\beta_{n}$ that appear in Eq.(3.16) are given by

$\displaystyle\beta_{1}$ $\displaystyle\equiv$
$\displaystyle-\frac{Le^{-2(2\mu+\nu)}}{2\left[c_{14}\rho^{2}\left(\mu^{\prime}\right)^{2}+4\rho\mu^{\prime}\left(\rho\nu^{\prime}+1\right)-L+2\rho^{2}\left(\nu^{\prime}\right)^{2}+4\rho\nu^{\prime}+2\right]},$
$\displaystyle\beta_{2}$ $\displaystyle\equiv$
$\displaystyle\frac{c_{14}Le^{-2(\mu+\nu)}}{\rho^{2}},\quad\beta_{3}\equiv\frac{Le^{-2(\mu+2\nu)}}{4-2L},\quad\beta_{4}\equiv-\frac{c_{1}Le^{-4\nu}}{\rho^{2}},$
$\displaystyle\beta_{5}$ $\displaystyle\equiv$
$\displaystyle-\frac{Le^{-2(\mu+2\nu)}\left[\left(c_{14}+4\right)\left(-\rho^{2}\right)\left(\mu^{\prime}\right)^{2}-4\rho\mu^{\prime}\left(3\rho\nu^{\prime}+2\right)+4L+10\rho^{2}\left(\nu^{\prime}\right)^{2}+24\rho\nu^{\prime}\right]}{8(L-2)\rho^{2}},$
$\displaystyle\beta_{6}$ $\displaystyle\equiv$
$\displaystyle-\frac{Le^{-4\nu}\left[-2\left(c_{1}-c_{14}\right)\rho\mu^{\prime}\left(\rho\nu^{\prime}+1\right)+\left(c_{1}+2\left(c_{14}-1\right)c_{14}\right)\rho^{2}\left(\mu^{\prime}\right)^{2}+c_{1}L\right]}{\rho^{4}},$
$\displaystyle\beta_{7}$ $\displaystyle\equiv$
$\displaystyle\frac{2c_{14}Le^{-\mu-4\nu}\mu^{\prime}}{\rho^{2}}.$ (B. 1)

Here, a prime in the superscript denotes the derivative with respect to
$\rho$.

## Appendix C: Expressions of $\gamma_{ij}$

The coefficients $\gamma_{ij}$ that appear in Eq.(IV.1) are given by

$\displaystyle\gamma_{11}$ $\displaystyle\equiv$
$\displaystyle-\frac{e^{2\mu-2\nu}\left(c_{14}\rho^{2}\left(\mu^{\prime}\right)^{2}+4\rho\mu^{\prime}\left(\rho\nu^{\prime}+1\right)-L+2\rho^{2}\left(\nu^{\prime}\right)^{2}+4\rho\nu^{\prime}+2\right)}{L-2},$
$\displaystyle\gamma_{12}$ $\displaystyle\equiv$
$\displaystyle\frac{e^{2\mu-2\nu}\left(\rho\mu^{\prime}+\rho\nu^{\prime}-2\right)\left(c_{14}\rho^{2}\left(\mu^{\prime}\right)^{2}+4\rho\mu^{\prime}\left(\rho\nu^{\prime}+1\right)-L+2\rho^{2}\left(\nu^{\prime}\right)^{2}+4\rho\nu^{\prime}+2\right)}{(L-2)\rho},$
$\displaystyle\gamma_{13}$ $\displaystyle\equiv$
$\displaystyle\frac{e^{2\mu-2\nu}\left(c_{14}\rho^{2}\left(\mu^{\prime}\right)^{2}+4\rho\mu^{\prime}\left(\rho\nu^{\prime}+1\right)-L+2\rho^{2}\left(\nu^{\prime}\right)^{2}+4\rho\nu^{\prime}+2\right)}{4(L-2)\rho^{2}}$
$\displaystyle\times\left(\left(c_{14}+4\right)\left(-\rho^{2}\right)\left(\mu^{\prime}\right)^{2}-4\rho\mu^{\prime}\left(3\rho\nu^{\prime}+2\right)+4L+10\rho^{2}\left(\nu^{\prime}\right)^{2}+24\rho\nu^{\prime}\right),$
$\displaystyle\gamma_{14}$ $\displaystyle\equiv$
$\displaystyle-\frac{2c_{14}e^{3\mu-2\nu}\mu^{\prime}\left(c_{14}\rho^{2}\left(\mu^{\prime}\right)^{2}+4\rho\mu^{\prime}\left(\rho\nu^{\prime}+1\right)-L+2\rho^{2}\left(\nu^{\prime}\right)^{2}+4\rho\nu^{\prime}+2\right)}{\rho^{2}},$
$\displaystyle\gamma_{21}$ $\displaystyle\equiv$
$\displaystyle\frac{c_{1}e^{2\mu-2\nu}}{c_{14}},\quad\gamma_{22}\equiv\frac{c_{1}e^{2\mu-2\nu}\left(\mu^{\prime}-\nu^{\prime}\right)}{c_{14}},$
$\displaystyle\gamma_{23}$ $\displaystyle\equiv$
$\displaystyle-\frac{e^{2\mu-2\nu}}{c_{14}\rho^{2}}\Big{[}-2\left(c_{1}-c_{14}\right)\rho\mu^{\prime}\left(\rho\nu^{\prime}+1\right)$
$\displaystyle+\left(c_{1}+2\left(c_{14}-1\right)c_{14}\right)\rho^{2}\left(\mu^{\prime}\right)^{2}+c_{1}L\Big{]},$
$\displaystyle\gamma_{24}$ $\displaystyle\equiv$ $\displaystyle
e^{\mu-2\nu}\mu^{\prime}.$ (C. 1)

Here, a prime in the superscript denotes the derivative with respect to
$\rho$.

## Appendix D: The formulas for obtaining Schrödinger-like PDEs

The procedures for obtaining Eqs.(4.3) and (4.4) may not be that
straightforward. To show it more clearly, here we introduce the formulas used
in this part of calculations. Suppose we have a set of linearly coupled 2nd-
order PDEs of the form

$\displaystyle\left[{\eta}_{1}(r)\frac{\partial^{2}}{\partial
r^{2}}+{\bar{\eta}}_{1}(r)\frac{\partial}{\partial
r}+{\tilde{\eta}}_{1}(r)-\frac{\partial^{2}}{\partial
t^{2}}\right]J_{1}=\xi_{1}(r)J_{2}(t,r),$
$\displaystyle\left[{\eta}_{2}(r)\frac{\partial^{2}}{\partial
r^{2}}+{\bar{\eta}}_{2}(r)\frac{\partial}{\partial
r}+{\tilde{\eta}}_{2}(r)-\frac{\partial^{2}}{\partial
t^{2}}\right]J_{2}=\xi_{2}(r)J_{1}(t,r).$

Let us introduce a set of new functions $\\{\Psi_{1},\Psi_{2}\\}$ and a new
variable $x$ by

$\displaystyle J_{1,2}=q_{1,2}(r)\Psi_{1,2}(t,x(r)),\quad\frac{dr}{dx}=p(r).$
(D. 2)

Then, choosing $q_{1,2}$ as

$\displaystyle q_{1,2}(r)$ $\displaystyle=$
$\displaystyle\sqrt{p(r)}\exp{\left(-\frac{1}{2}\int\frac{{\bar{\eta}}_{1,2}}{{\eta}_{1,2}}dr\right)},$
(D. 3)

we find that Eq.(Appendix D: The formulas for obtaining Schrödinger-like PDEs)
can be written as a set of coupled Schrödinger-like PDEs

$\displaystyle\left[\frac{\eta_{1,2}}{p^{2}}\frac{\partial^{2}}{\partial
x^{2}}-\left(\frac{\partial^{2}}{\partial
t^{2}}+V_{1,2}\right)\right]\Psi_{1,2}=Q_{1,2}\Psi_{2,1},$ (D. 4)

where

$\displaystyle V_{1,2}(r)$ $\displaystyle\equiv$
$\displaystyle-\Bigg{\\{}\frac{\eta_{1,2}}{4p^{2}}\bigg{[}2p\frac{d\left(p^{\prime}-p{\bar{\eta}}_{1,2}/\eta_{1,2}\right)}{dr}$
(D. 5)
$\displaystyle~{}~{}~{}~{}-\left(p^{\prime}-p{\bar{\eta}}_{1,2}/\eta_{1,2}\right)^{2}\bigg{]}+{\tilde{\eta}}_{1,2}\Bigg{\\}},~{}~{}~{}~{}~{}~{}~{}~{}$

and

$\displaystyle Q_{1,2}(r)$ $\displaystyle\equiv$
$\displaystyle\xi_{1,2}\frac{q_{2,1}}{q_{1,2}}.$ (D. 6)

Here a prime denotes the derivative with respect to $r$. In addition, a common
choice of $p(r)$ is $p(r)=\sqrt{\eta_{1}}$. Notice that, since we were
considering the case $c_{13}=c_{4}=0$ in deriving Eqs.(4.3) and (4.4), the
corresponding $\eta_{1}$ and $\eta_{2}$ are happen to be identical under this
condition.

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# On Turán numbers for disconnected hypergraphs

Raffaella Mulas<EMAIL_ADDRESS>Max Planck Institute for Mathematics in the
Sciences, Leipzig, Germany  Vrije Universiteit Amsterdam, Amsterdam, The
Netherlands  Jiaxi Nie<EMAIL_ADDRESS>Max Planck Institute for
Mathematics in the Sciences, Leipzig, Germany  Shanghai Center for
Mathematical Sciences, Fudan University, Shanghai, China

###### Abstract

We introduce the following simpler variant of the Turán problem: Given
integers $n>k>r\geq 2$ and $m\geq 1$, what is the smallest integer $t$ for
which there exists an $r$-uniform hypergraph with $n$ vertices, $t$ edges and
$m$ connected components such that any $k$-subset of the vertex set contains
at least one edge? We prove some general estimates for this quantity and for
its limit, normalized by $\binom{n}{r}$, as $n\rightarrow\infty$. Moreover, we
give a complete solution of the problem for the particular case when $k=5$,
$r=3$ and $m\geq 2$.

Keywords: Turán numbers; Turán problem; Hypergraphs

##  1 Introduction

Given $r\geq 2$, an _$r$ -uniform hypergraph_, or _$r$ -graph_ for short, is a
pair $H=(V,E)$, where $V=V(H)$ is a finite set of _nodes_ or _vertices_ , and
$E=E(H)$ is a set of $r$-subsets of $V$, called _edges_. In particular, a
graph is a $2$-graph. Given an $r$-graph $F$, the _Turán number_
$\mathrm{ex}(n,{F})$ is the largest integer $t$ such that there exists an
$r$-graph on $n$ vertices and $t$ edges that does not contain $F$ as a sub-
hypergraph.
The _Turán problem_ consists of determining or estimating
$\mathrm{ex}(n,K_{k}^{(r)})$, where $K_{k}^{(r)}$ is a _complete_ $r$-graph on
$k$ vertices, i.e., the hypergraph consisting of all possible $r$-subsets of
$V$. Moreover, one is also interested in estimating the limit

$\pi_{r,k}:=\lim_{n\to\infty}\frac{\mathrm{ex}(n,K_{k}^{(r)})}{\binom{n}{r}},$

where $\binom{n}{r}$ is the number of edges of $K_{n}^{(r)}$. Note that the
edge density of an $n$-vertex $r$-graph equals the average edge density of its
$(n-1)$-vertex induced subgraphs. Hence
$\mathrm{ex}(n,K_{k}^{(r)})/\binom{n}{r}$ is non-increasing, which implies
that $\pi_{r,k}$ is well-defined. This simple averaging argument was first
introduced by Katona, Nemetz and Simonovits [14].
The Turán problem was introduced in 1941 by Paul Turán [28], who showed that
$\pi_{2,k}=1/(k-1)$. More specifically, let $\mathcal{T}_{n,k}$ be the graph
on $n$ vertices such that the vertex set can be written as
$V=V_{1}\sqcup\cdots\sqcup V_{k}$, each $V_{i}$ has size either $\lceil
n/k\rceil$ or $\lfloor n/k\rfloor$, and two vertices form an edge if and only
if they belong to different $V_{i}$’s. Turán showed that

$\mathrm{ex}(n,K_{k}^{(2)})=|E(\mathcal{T}_{n,k-1})|$

and $\mathcal{T}_{n,k-1}$ is the only $n$-vertex, $K_{k}^{(2)}$-free graph
that achieves this number of edges. Eighty years later, there are still very
few results on the Turán’s problem for $k>r\geq 3$. Turán conjectured that
$\pi_{3,4}=5/9$ and $\pi_{3,5}=3/4$. There are exponentially many
constructions achieving the conjectured densities, see [2, 9, 11, 18]. The
best upper bound so far for $\pi_{3,4}$ is $0.561666$ by Razborov [24] using
flag algebra method. Some classical surveys are [15, 25, 6, 12], and other
related results are presented, for instance, in [26, 22, 19, 13, 17, 10, 20,
1, 16, 7, 27, 21, 23].
Paul Erdős, who was a close collaborator of Turán, rarely offered prizes for
problems that were posed by others [3], offered 500 dollars for determining
$\pi_{r,k}$ for even one single pair $k>r>2$, and he offered 1000 dollars for
solving the whole set of problems [8]. After Erdős’ death, his close
collaborators Fan Chung and Ron Graham declared that they were willing to
offer these prizes, as a way to honor him [3]. At the time of writing, Fan
Chung is currently in charge of these rewards.

The Turán problem can also be reformulated in a dual way, as follows.
Given an $r$-graph $H=(V,E)$, its _complement_ is the hypergraph
$H^{\textrm{c}}:=(V,E^{\textrm{c}})$, where

$E^{\textrm{c}}:=\\{f\subseteq V:|f|=r\text{ and }f\notin E\\}.$

Clearly,

$|E^{\textrm{c}}|=\binom{n}{r}-|E|,$

therefore, given $t\in\mathbb{N}$,

$|E|\leq t\iff|E^{\textrm{c}}|\geq\binom{n}{r}-t.$

Also, $H$ is $K_{k}^{(r)}$-free if and only if $H^{\textrm{c}}$ satisfies the
condition

$\forall v_{1},\ldots,v_{k}\in V\Longrightarrow\exists
f\subseteq\\{v_{1},\ldots,v_{k}\\}:f\in E^{\textrm{c}}.$ (1)

Hence, $H$ is an _optimal solution_ for the Turán problem, i.e., it maximizes
the number of edges among all $K_{k}^{(r)}$-free $r$-graphs on $n$ nodes, if
and only if $H^{\textrm{c}}$ minimizes the number of edges among all
$r$-graphs on $n$ nodes that satisfy (1). Motivated by this dual (and
equivalent) formulation, we say that the _dual Turán problem_ consists of
determining or estimating

$T(n,K_{k}^{(r)}):=\binom{n}{r}-\mathrm{ex}(n,K_{k}^{(r)}),$

which is the smallest integer $t$ such that there exists an $r$-graph on $n$
vertices and $t$ edges satisfying (1). Further, we let

$t_{r,k}:=\lim_{n\to\infty}\frac{T(n,K_{k}^{(r)})}{\binom{n}{r}}.$

Clearly, we have $t_{r,k}=1-\pi_{r,k}$.

In this paper, we introduce and study a variant of the Turán number concerning
the number of connected components. Namely, we let $T(n,K_{k}^{(r)};m)$ be the
smallest integer $t$ such that there exists an $r$-graph with $n$ vertices,
$t$ edges and $m$ connected components that satisfies (1). We also consider
the limit

$t_{r,k}(m):=\lim_{n\to\infty}\frac{T(n,K_{k}^{(r)};m)}{\binom{n}{r}}.$

The existence of this limit can be proved by an argument similar to that of
$\pi_{r,k}$. One of our main results is the following theorem, that we shall
prove in Section 4.

######  Theorem 1.1.

Let $k>r\geq 2$ be integers. If $n\geq k+\binom{k-2}{r-1}$ and
$T(n,K_{k}^{(r)})=T(n,K_{k}^{(r)};m)$, then

$m\leq\left\lfloor\frac{k-1}{r-1}\right\rfloor.$

In particular, for $k\leq 2r-2$, Theorem 1.1 implies that the optimal solution
of $T(n,K_{k}^{(r)})$ must be connected when $n$ is large enough.

Note that in the case of graphs, the optimal solutions of $T(n,K_{k}^{(2)})$
always have $k-1$ connected components. That is,
$T(n,K_{k}^{(2)};k-1)<T(n,K_{k}^{(2)};m)$ for any $m<k-1$. It is natural to
ask whether this phenomenon extends to $r$-graphs, i.e., if the optimal
solutions of $T(n,K_{k}^{(r)})$ always have $\lfloor{(k-1)}/{(r-1)}\rfloor$
connected components. We show that the answer to this question is “no” for
$r$-graphs when $k-1$ is a multiple of $r-1$:

######  Theorem 1.2.

For integers $r\geq 3$, $k\geq 1$ and $n\geq(r-1)k+1+\binom{(r-1)k-1}{r-1}$
such that $k|n$, there exists an integer $m<k$ such that

$T(n,K_{(r-1)k+1}^{(r)};k)\geq T(n,K_{(r-1)k+1}^{(r)};m).$

Moreover, we determine $t_{3,2m+1}(m)$ and $t_{3,2m+2}(m)$ in terms of
$t_{3,4}$:

######  Theorem 1.3.

For integers $m\geq 1$,

$t_{3,2m+1}(m)=\frac{1}{m^{2}},$
$t_{3,2m+2}(m)=(m-1+t_{3,4}^{-\frac{1}{2}})^{-2}.$

We point out that it is unclear whether a similar result also holds for $r\geq
4$. For example, we do not know if $t_{4,7}(2)=1/8$. This is because the
current best lower bound for $t_{4,6}$, to the best of our knowledge, is
$1/10$, which is smaller than $1/8$.

Structure of the paper. In Section 2 we give further definitions and we prove
some general results that will be needed throughout the paper. In Section 3 we
give a complete solution of the problem for the case when $k=5$, $r=3$ and
$m\geq 2$. In Section 4 we prove Theorem 1.1, and in Section 5 we prove
Theorem 1.2 and Theorem 1.3. Finally, in Section 6 we propose some open
questions.

##  2 Basic definitions and general results

In the introduction we gave several definitions regarding hypergraphs and the
Turán problem. In this section, we give further definitions and we prove some
general results that hold for any $k>r\geq 3$.

######  Definition 2.1.

Given a hypergraph $H=(V,E)$ and a vertex $v\in V$, we let
$H\setminus\\{v\\}:=(V\setminus\\{v\\},E\setminus\\{v\\})$, where
$E\setminus\\{v\\}:=\\{e\in E\,:\,v\notin e\\}$.

######  Definition 2.2.

Given an $r$-graph $H=(V,E)$, an $r$-subset of $V$ is a _non-edge_ of $H$ if
it does not belong to $E$, or equivalently, if it belongs to $E^{\textrm{c}}$.

Given an hypergraph $H$, we let $C(H)$ denote the number of its connected
components.

######  Definition 2.3.

Let $H=(V,E)$ be an hypergraph. An _independent set_ of $H$ is a subset of $V$
which does not contain any edge of $H$. The _independence number_ of $H$,
denoted $\alpha(H)$, is the maximum size of an independent set.
The independence sequence of $H$, denoted $S(H)$, is the multiset of size
$C(H)$ whose members are the independence numbers of each component.

Given a multiset $S$ whose elements are positive integers, we let $|S|$ denote
the number of entries in $S$, and we let $\|S\|$ denote the sum of all entries
of $S$. That is, if $S=\\{s_{1},s_{2},\ldots,s_{t}\\}$, then $|S|:=t$ and
$\|S\|:=s_{1}+s_{2}+\cdots+s_{t}$.
Moreover, given two multisets $S$ and $S^{\prime}$, we let $S\uplus
S^{\prime}$ denote the union of $S$ and $S^{\prime}$. For instance,
$\\{3,1\\}\uplus\\{2,2,1\\}=\\{3,2,2,1,1\\}$. Given a positive integer $m$, we
let $m\cdot S$ denote the multiset union of $m$ copies of $S$. For example,
$3\cdot\\{2,1\\}=\\{2,2,2,1,1,1\\}$.

We also let $\tilde{T}(n,r;S)$ be the smallest number of edges in an
$n$-vertex $r$-graph $H$ such that $S(H)=S$, and we let

$\tilde{t}_{r}(S):=\lim_{n\to\infty}\frac{\tilde{T}(n,r;S)}{\binom{n}{r}}.$

The existence of this limit can be proved by a simple averaging argument
similar to that for $\pi_{r,k}$. When $S=\\{s\\}$, we write
$\tilde{T}(n,r;s)=\tilde{T}(n,r;\\{s\\})$ and
$\tilde{t}_{r}(s)=\tilde{t}_{r}(\\{s\\})$ for short. One can then check that

$t_{r,k}(m)=\min_{S\,:\,|S|=m,\,\|S\|=k-1}\tilde{t}_{r}(S).$

The following lemmas will be needed throughout the paper.

######  Lemma 2.4.

Let $S^{(1)}$, $S^{(2)}$ and $S$ be multisets. If

$\tilde{t}_{r}(S^{(1)})\geq\tilde{t}_{r}(S^{(2)}),$

then

$\tilde{t}_{r}(S^{(1)}\uplus S)\geq\tilde{t}_{r}(S^{(2)}\uplus S).$

###### Proof.

It suffices to show that $\tilde{T}(n,r;S^{(1)}\uplus
S)\geq\tilde{T}(n,r;S^{(2)}\uplus S)+o(n^{r})$. For any optimal solution $H$
of $\tilde{T}(n,r;S^{(1)}\uplus S)$, let $H_{1}$ be the induced subgraph of
$H$ such that $S(H_{1})=S^{(1)}$. Since

$\tilde{T}(n,r;S^{(1)})\geq\tilde{T}(n,r;S^{(2)})+o(n^{r}),$

there is an $r$-graph $H_{2}$ on the same vertex set as $H_{1}$ such that
$S(H_{2})=S^{(2)}$ and $e(H_{1})\geq e(H_{2})+o(n^{r})$. Let $H^{\prime}$ be
the $r$-graph obtained from $H$ by replacing $H_{1}$ with $H_{2}$. Then, we
have $S({H^{\prime}})=S^{(2)}\uplus S$ and $e(H^{\prime})\leq e(H)+o(n^{r})$.
Therefore, $\tilde{T}(n,r;S^{(1)}\uplus S)\geq\tilde{T}(n,r;S^{(2)}\uplus
S)+o(n^{r})$. ∎

######  Lemma 2.5.

For integers $b\geq a>r>1$,

$\binom{a}{r}+\binom{b}{r}<\binom{a-1}{r}+\binom{b+1}{r}.$

###### Proof.

We use the fact that

$\binom{a}{r}=\binom{a-1}{r}+\binom{a-1}{r-1}$

and, similarly,

$\binom{b+1}{r}=\binom{b}{r}+\binom{b}{r-1}.$

This implies that

$\displaystyle\binom{a}{r}+\binom{b}{r}<\binom{a-1}{r}+\binom{b+1}{r}$
$\displaystyle\iff\binom{a-1}{r-1}<\binom{b}{r-1},$

which is true since $b>a-1$. ∎

######  Lemma 2.6.

For $i\in\\{1,\ldots,m\\}$, let $t_{i}>0$, and $p_{i}\in(0,1]$ be real numbers
such that $\sum_{i=1}^{m}p_{i}=1$. Then, for any integer $r\geq 2$,

$\sum_{i=1}^{m}t_{i}\cdot
p_{i}^{r}\geq\left(\sum_{i=1}^{m}t_{i}^{-\frac{1}{r-1}}\right)^{-r+1}.$ (2)

Equality holds if and only if
$p_{i}=t_{i}^{-\frac{1}{r-1}}/\sum_{j=1}^{m}t_{j}^{-\frac{1}{r-1}}$.

###### Proof.

Let $c_{i}:=t_{i}^{-\frac{1}{r-1}}$, and let $C:=\sum_{i=1}^{m}c_{i}$. Then,

$(*):=\sum_{i=1}^{m}t_{i}\cdot
p_{i}^{r}=\sum_{i=1}^{m}c_{i}\cdot\left(\frac{p_{i}}{c_{i}}\right)^{r}=C\sum_{i=1}^{m}\frac{c_{i}}{C}\cdot\left(\frac{p_{i}}{c_{i}}\right)^{r}.$

Hence, by Jensen’s inequality, we have

$(*)\geq
C\left(\sum_{i=1}^{m}\frac{c_{i}}{C}\cdot\frac{p_{i}}{c_{i}}\right)^{r}=C^{-r+1},$

and equality holds if and only if

$\frac{p_{1}}{c_{1}}=\frac{p_{2}}{c_{2}}=\cdots=\frac{p_{m}}{c_{m}}.$

Together with $\sum_{i=1}^{m}p_{i}=1$, this implies that $p_{i}=c_{i}/C$ for
all $i\in\\{1,\ldots,m\\}$. ∎

The next lemma establishes an equation for $\tilde{t}_{r}(S)$.

######  Lemma 2.7.

Let $r\geq 3$ be an integer, and let $S=\\{s_{1},s_{2},\dots,s_{m}\\}$ be a
multiset such that $s_{i}\geq r-1$ for all $i\in\\{1,\ldots,m\\}$. Then,

$\tilde{t}_{r}(S)=\left(\sum_{i=1}^{m}\tilde{t}_{r}(s_{i})^{-\frac{1}{r-1}}\right)^{-r+1}.$

###### Proof.

Let $G_{n}$ be an $r$-graph on $n$ vertices whose independence sequence is $S$
such that $e(G_{n})=\tilde{T}(n,r;S)$, and let $n_{i}$ be the number of
vertices in the component of $G_{n}$ corresponding to $s_{i}$, for $1\leq
i\leq m$. Then, by definition,

$\frac{e(G_{n})}{\binom{n}{r}}\geq\sum_{i=1}^{m}\frac{\tilde{T}(n_{i},r;s_{i})}{\binom{n}{r}}=\sum_{i=1}^{m}\frac{\binom{n_{i}}{r}}{\binom{n}{r}}\frac{\tilde{T}(n_{i},r;s_{i})}{\binom{n_{i}}{r}}$

It is not hard to check that

$\liminf_{n\to\infty}\frac{\binom{n_{i}}{r}}{\binom{n}{r}}\frac{\tilde{T}(n_{i},r;s_{i})}{\binom{n_{i}}{r}}=\liminf_{n\to\infty}\left(\frac{n_{i}}{n}\right)^{r}\tilde{t}_{r}(s_{i});$

when $\liminf(n_{i}/n)>0$, this is clearly true; and when
$\liminf(n_{i}/n)=0$, both sides are zero. Hence, by Lemma 2.6,

$\tilde{t}_{r}(S)=\liminf_{n\to\infty}\frac{e(G_{n})}{\binom{n}{r}}\geq\liminf_{n\to\infty}\sum^{m}_{i=1}\left(\frac{n_{i}}{n}\right)\tilde{t}_{r}(s_{i})\geq\left(\sum_{i=1}^{m}\tilde{t}_{r}(s_{i})^{-\frac{1}{r-1}}\right)^{-r+1}.$

On the other hand, we can construct a sequence of $r$-graphs whose density
converges to
$\left(\sum_{i=1}^{m}\tilde{t}_{r}(s_{i})^{-\frac{1}{r-1}}\right)^{-r+1}$ by
taking the union of the optimal solutions of $\tilde{T}(\lfloor
np_{i}\rfloor,r;{s_{i}})$, for $1\leq i\leq m$, where

$p_{i}=\frac{\tilde{t}_{r}(s_{i})^{-\frac{1}{r-1}}}{\sum_{j=1}^{m}\tilde{t}_{r}(s_{j})^{-\frac{1}{r-1}}}.$

We can therefore conclude that
$\tilde{t}_{r}(S)=\left(\sum_{i=1}^{m}\tilde{t}_{r}(s_{i})^{-\frac{1}{r-1}}\right)^{-r+1}.$
∎

##  3 Solution for $r=3$, $k=5$ and $m\geq 2$

In this section we solve the problem for the case when $r=3$, $k=5$, and
$m\geq 2$. This will serve as a motivation for our next results.

######  Lemma 3.1.

Let $H=(V,E)$ be a $K_{5}^{(3)}$-free $3$-graph on $n\geq 6$ nodes. Then,
$H^{\textrm{c}}$ has at most three connected components. Moreover,

  * •

If $H^{\textrm{c}}$ has three connected components, then $H^{\textrm{c}}$ is
given by two isolated vertices together with a complete $3$-graph on $n-2$
nodes.

  * •

If $H^{\textrm{c}}$ has two connected components and it has no isolated
vertices, then each connected component of $H^{\textrm{c}}$ is a complete
$3$-graph.

  * •

If $H^{\textrm{c}}$ has two connected components and one of them is an
isolated vertex, then the other one is $K_{4}^{(3)}$-free in $H$.

###### Proof.

We first assume, for the sake of a contradiction, that $H^{\textrm{c}}$ has at
least four connected components $V_{1},\ldots,V_{4}$. Let $v_{1}\in
V_{1},\ldots,v_{4}\in V_{4}$, and let $v_{5}\in V$ belong to any of the
connected components. Since $H$ is $K_{5}^{(3)}$-free, $H^{\textrm{c}}$
satisfies (1) and therefore there exists $f\subseteq\\{v_{1},\ldots,v_{5}\\}$
such that $f\in E^{\textrm{c}}$. Since $|f|=3$ and since $V_{1},\ldots,V_{4}$
are not joined by any edge, this gives a contradiction. Hence,
$H^{\textrm{c}}$ has at most three connected components.

If $H^{\textrm{c}}$ has three connected components, then clearly one component
has at least one edge and therefore at least $3$ vertices. Assume, for the
sake of a contradiction, that $H^{\textrm{c}}$ has three connected components
$V_{1},V_{2},V_{3}$ and at least two of them, say $V_{1}$ and $V_{2}$, have
cardinality larger than $1$. Then, we can pick $v_{1},v_{2}\in V_{1}$,
$v_{3},v_{4}\in V_{2}$ and $v_{5}\in V_{3}$. By (1), there exists
$f^{\prime}\subseteq\\{v_{1},\ldots,v_{5}\\}$ such that $f^{\prime}\in
E^{\textrm{c}}$. Since $f^{\prime}$ has cardinality $3$, this gives a
contradiction. Therefore, if $H^{\textrm{c}}$ has three connected component,
then two of them are given by isolated vertices. Now, assume that
$H^{\textrm{c}}$ has three connected components $V_{1},V_{2},V_{3}$, where
$V_{2}$ and $V_{3}$ are given by isolated vertices. Given
$v_{1},v_{2},v_{3}\in V_{1}$, $v_{4}\in V_{2}$, $v_{5}\in V_{3}$, (1) implies
that $\\{v_{1},v_{2},v_{3}\\}\in E^{\textrm{c}}$. Since this holds for all
$v_{1},v_{2},v_{3}\in V_{1}$, $V_{1}$ is a complete $3$-graph on $n-2$ nodes.

Now, assume that $H^{\textrm{c}}$ has two connected components $V_{1},V_{2}$,
and it has no isolated vertices. Since edges have cardinality $3$, there
cannot be connected components of cardinality $2$, therefore $|V_{i}|\geq 3$
for $i=1,2$. Let $v_{1},v_{2},v_{3}\in V_{1}$, and $v_{4},v_{5}\in V_{2}$. By
(1), there exists $f^{\prime\prime}\subseteq\\{v_{1},\ldots,v_{5}\\}$ such
that $f^{\prime\prime}\in E^{\textrm{c}}$. Since there are no edges between
$V_{1}$ and $V_{2}$, this implies that $f=\\{v_{1},v_{2},v_{3}\\}$. This is
true for all $v_{1},v_{2},v_{3}\in V_{1}$, and the same reasoning can be
applied to $V_{2}$. Hence, both $V_{1}$ and $V_{2}$ are complete $3$-graphs.

Finally, assume that $H^{\textrm{c}}$ has two connected components and one of
them is an isolated vertex $v_{1}$. Let $v_{2},\ldots,v_{5}\in
V\setminus\\{v_{1}\\}$. If $v_{2},\ldots,v_{5}$ form a complete $3$-graph on
$4$ vertices in $H$, then $v_{1},\ldots,v_{5}$ form a complete $3$-graph on
$5$ vertices in $H$, which leads to a contradiction. Hence,
$V\setminus\\{v_{1}\\}$ is $K_{4}^{(3)}$-free in $H$. ∎

Lemma 3.1 allows us to prove the following

######  Theorem 3.2.

Let $H=(V,E)$ be a $K_{5}^{(3)}$-free $3$-graph on $n\geq 6$ nodes such that
$H^{\textrm{c}}$ has more than one connected component.

  * •

If $n=2m$, then

$|E^{\textrm{c}}|\geq 2\cdot\binom{m}{3},$

or equivalently

$|E|\leq m^{2}(m-1),$

with equality if and only if $H^{\textrm{c}}$ is the disjoint union of two
copies of $K_{m}^{(3)}$.

  * •

If $n=2m+1$, then

$|E^{\textrm{c}}|\geq\binom{m}{3}+\binom{m+1}{3},$

or equivalently

$|E|\leq m^{3}+\frac{m^{2}}{2}-\frac{m}{2},$

with equality if and only if $H^{\textrm{c}}$ is the disjoint union of
$K_{m}^{(3)}$ and $K_{m+1}^{(3)}$.

###### Proof.

We consider three cases.

  1. 1.

Case 1: $H^{\textrm{c}}$ has more than two connected components. By Lemma 3.1,
$H^{\textrm{c}}$ is given by two isolated vertices together with a complete
$3$-graph on $n-2$ nodes. Therefore,

$|E^{\textrm{c}}|=\binom{n-2}{3}.$

It is easy to check that, for $n=2m$,

$|E^{\textrm{c}}|=\binom{n-2}{3}>2\cdot\binom{m}{3}$

and, for $n=2m+1$,

$|E^{\textrm{c}}|=\binom{n-2}{3}>\binom{m}{3}+\binom{m+1}{3}.$

  2. 2.

Case 2: $H^{\textrm{c}}$ has two connected components and no isolated
vertices. By Lemma 3.1, each connected component of $H^{\textrm{c}}$ is a
complete $3$-graph. Hence, if they have cardinality $c_{1}$ and $c_{2}$,
respectively, then

$|E^{\textrm{c}}|=\binom{c_{1}}{3}+\binom{c_{2}}{3}.$

By Lemma 2.5, for $n=2m$ this is minimized precisely by $c_{1}=c_{2}=m$, while
for $n=2m+1$ this is minimized by $c_{1}=m$ and $c_{2}=m+1$.

  3. 3.

Case 3: $H^{\textrm{c}}$ has two connected components and one of them is an
isolated vertex $\hat{v}$. By Lemma 3.1, $H\setminus\\{\hat{v}\\}$ is
$K_{4}^{(3)}$-free. Therefore, using the bound
$\mathrm{ex}(n-1,K^{(3)}_{4})\leq(0.63)\cdot\binom{n-1}{3}$ in [6, 4],

$\displaystyle|E|$ $\displaystyle=\deg\hat{v}+|E(H\setminus\\{\hat{v}\\})|$
$\displaystyle\leq\binom{n-1}{2}+\mathrm{ex}(n-1,K^{(3)}_{4})$
$\displaystyle<\binom{n-1}{2}+(0.63)\cdot\binom{n-1}{3}.$

If $n=2m$, then one can check that

$|E|<\binom{2m-1}{2}+(0.63)\cdot\binom{2m-1}{3}<m^{2}(m-1).$

Similarly, if $n=2m+1$, then

$|E|<\binom{2m}{2}+(0.63)\cdot\binom{2m}{3}<m^{3}+\frac{m^{2}}{2}-\frac{m}{2}.$

This proves the claim. ∎

Note that the last result implies that $3/4=t_{3,5}(2)\leq t_{3,5}(m)$ for
$m\geq 2$.
Moreover, Turán conjectured that the disjoint unions of two complete
$r$-graphs are optimal for $r=3$ and $k=5$ (in the general case when $m\geq
1$). However, as discussed in [25], counterexamples are known for any odd
$n\geq 9$. For $n$ even, no counterexample was found yet, and the conjecture
has proved to be true up to $n=12$.

##  4 Number of connected components

In this section we prove Theorem 1.1. As a preliminary result, we first prove
a theorem showing that the optimal solutions of $T(n,K_{k}^{(r)})$ have no
isolated vertices, if $n$ is big enough.

######  Lemma 4.1.

For integers $k>r\geq 2$ and $n\geq k+\binom{k-2}{r-1}$, let $H=(V,E)$ be a
$K_{k}^{(r)}$-free $r$-graph with $n$ vertices and
$\mathrm{ex}(n,K_{k}^{(r)})$ edges. Then, $H^{\textrm{c}}$ has no isolated
vertices.

###### Proof.

Suppose, for the sake of a contradiction, that $H^{\textrm{c}}$ contains an
isolated vertex $u$, and let $H^{\prime}:=H\setminus\\{u\\}$. Then,
$H^{\prime}$ is a $K_{k-1}^{(r)}$-free $r$-graph on $n-1$ vertices. Moreover,
since $H$ is an optimal solution for the Turán problem, if any edge is added
to $H$, then $H$ is not $K_{k}^{(r)}$-free anymore. Therefore, if any edge is
added to $H^{\prime}$, then $H^{\prime}$ is not $K_{k-1}^{(r)}$-free anymore.
Hence, given $f=\\{v_{1},\ldots,v_{r}\\}\in E({H^{\prime}}^{\textrm{c}})$,
there exist $v_{r+1},\ldots,v_{k-1}\in V\setminus\\{u\\}$ such that there are
no other edges among $v_{1},\ldots,v_{k-1}$ in ${H^{\prime}}^{\textrm{c}}$
other than $f$. Therefore, in particular, there are no edges in
${H^{\prime}}^{\textrm{c}}$ among $v_{2},\ldots,v_{k-1}$. Let now
$w_{1},\ldots,w_{l}\in V\setminus\\{u,v_{2},\ldots,v_{k-1}\\}$, for some
$l\geq 1$. Then, for each $i\in\\{1,\ldots,l\\}$, there exists $e_{i}\in
E({H^{\prime}}^{\textrm{c}})$ such that
$e_{i}\subset\\{w_{i},v_{2},\ldots,v_{k-1}\\}$ and $w_{i}\in e_{i}$.
If $l>\binom{k-2}{r-1}$, which is possible since $n\geq k+\binom{k-2}{r-1}$,
there exist $i,j\in\\{1,\ldots,l\\}$ such that $i\neq j$ and
$e_{i}\setminus\\{w_{i}\\}=e_{j}\setminus\\{w_{j}\\}=:U$.
Let now $H^{\prime\prime}$ be the $r$-graph obtained from $H$ by deleting
$U\cup\\{u\\}$ and adding $U\cup\\{w_{i}\\}$ and $U\cup\\{w_{j}\\}$ as edges.
Then, $H^{\prime\prime}$ is $K_{k}^{(r)}$-free. In fact,

  * •

Any $k$-set not containing $U$ is not $K_{k}^{(r)}$ since $H$ is
$K_{k}^{(r)}$-free;

  * •

Any $k$-set containing $U\cup\\{u\\}$ contains the non-edge $U\cup\\{u\\}$;

  * •

For any $k$-set $W$ containing $U$ but not $\\{u\\}$, fix a vertex $v\in U$.
Note that $W\setminus\\{v\\}$ is a $(k-1)$-set . This implies that there
exists a non-edge of $H^{\prime}$ contained in $W\setminus\\{v\\}$, which is
neither $U\cup\\{w_{i}\\}$ nor $U\cup\\{w_{j}\\}$, and which is also a non-
edge of $H^{\prime\prime}$.

Therefore, $H^{\prime\prime}$ is an $K_{k}^{(r)}$-free $r$-graph with $n$
nodes and more edges than $H$, which is a contradiction. ∎

Now we are ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Let $H$ be an optimal solution of $T(n,K_{k}^{(r)})$. By Lemma 4.1, $H$ has no
isolated verteces. Hence, the independence number of each connected component
of $H$ is at least $r-1$. Suppose, for the sake of a contradiction, that
$m>\left\lfloor{(k-1)}/{(r-1)}\right\rfloor$. Then $\alpha(H)\geq(r-1)m>k-1$,
which contradicts to $\alpha(H)\leq k-1$. ∎

##  5 Turán numbers for disconnected $3$-graphs

This section is dedicated to the proofs of Theorem 1.2 and Theorem 1.3.

###### Proof of Theorem 1.2.

The proof is divided into two cases.

Case 1: The optimal solution of $T(n,K_{(r-1)k+1}^{(r)};k)$ has an isolated
vertex. In this case, by Lemma 4.1, we have

$T(n,K_{(r-1)k+1}^{(r)};k)>T(n,K_{(r-1)k+1}^{(r)};k-1).$

Case 2: The optimal solution of $T(n,K_{(r-1)k+1}^{(r)};k)$ does not contain
isolated vertices. In this case, the $k$ components are all complete
$r$-graphs. Hence, by Lemma 2.5, these components all have size $n/k$,
implying that

$T(n,K_{(r-1)k+1}^{(r)};k)=k\binom{n/k}{r}.$

But we also know that $T(n,K_{(r-1)k+1}^{(r)};1)\leq k\binom{n/k}{r}$, by the
construction of Sidorenko (Construction 4 in Section 4.2 of [25]). ∎

For the proof of Theorem 1.3, we need the following key lemma.

######  Lemma 5.1.

For any integer $l\geq 2$,

$\tilde{t}_{3}(\\{l+1\\}\uplus(l-1)\cdot\\{1\\})>\tilde{t}_{3}(l\cdot\\{2\\}),$
$\tilde{t}_{3}(\\{l+2\\}\uplus(l-1)\cdot\\{1\\})>\tilde{t}_{3}(\\{3\\}\uplus(l-1)\cdot\\{2\\}).$

###### Proof.

Due to de Caen [5], we have

$t_{r,l+1}\geq\frac{1}{\binom{l}{r-1}}.$

On the other hand, the construction of Sidorenko (Construction 4 in Section
4.2 of [25]) gives

$\tilde{t}_{r}(l)\leq\frac{(r-1)^{2}}{l^{2}}.$

Hence for any integer $l\geq 2$,

$\frac{1}{\binom{l}{2}}\leq t_{3,l+1}\leq\tilde{t}_{3}(l)\leq\frac{4}{l^{2}}.$
(3)

Therefore, by definition and inequality (3)

$\tilde{t}_{3}(\\{l+1\\}\uplus(l-1)\cdot\\{1\\})=\tilde{t}_{3}(l+1)\geq\frac{1}{\binom{l+1}{2}},$
$\tilde{t}_{3}(\\{l+2\\}\uplus(l-1)\cdot\\{1\\})=\tilde{t}_{3}(l+2)\geq\frac{1}{\binom{l+2}{2}}.$

On the other hand, by Lemma 2.7 and inequality (3)

$\tilde{t}_{3}(l\cdot\\{2\\})=l^{-2}<\frac{1}{\binom{l+1}{2}},$
$\tilde{t}_{3}(\\{3\\}\uplus(l-1)\cdot\\{2\\})=(l-1+(\tilde{t}_{3}(3))^{-\frac{1}{2}})^{-2}\leq\left(\frac{2l+1}{2}\right)^{-2}<\frac{1}{\binom{l+2}{2}}.$

∎

We are now ready to prove Theorem 1.3.

###### Proof of Theorem 1.3.

For any multiset $S$ such that $|S|=m$ and $\|S\|=2m$, if there exists an
element of $S$ which is larger than 2, then there exists an integer $l>2$ such
that $\\{l\\}\uplus(l-2)\cdot\\{1\\}$ is a subset of $S$. Let $S^{\prime}$ be
a multiset obtained from $S$ by replacing $\\{l\\}\uplus(l-2)\cdot\\{1\\}$ by
$(l-1)\cdot\\{2\\}$. Then, by Lemma 2.4 and Lemma 5.1,
$\tilde{t}_{3}(S^{\prime})<\tilde{t}_{3}(S)$. Also, note that the number of
$2$’s in $S$ is smaller than that in $S^{\prime}$. Hence, by repeating the
above argument finitely many times and Lemma 2.7, we obtain
$\tilde{t}_{3}(S)\geq\tilde{t}_{3}(m\cdot\\{2\\})=1/m^{2}$. Therefore,

$t_{3,2m+1}(m)=\min_{S\,:\,|S|=m,\,\|S\|=2m}\tilde{t}_{3}(S)=\tilde{t}_{3}(m\cdot\\{2\\})=\frac{1}{m^{2}}.$

Similarly, we have

$t_{3,2m+2}(m)=\min_{S\,:\,|S|=m,\,\|S\|=2m+1}\tilde{t}_{3}(S)=\tilde{t}_{3}(\\{3\\}\uplus(m-1)\cdot\\{2\\})=(m-1+\tilde{t}_{3}(3)^{-\frac{1}{2}})^{-2}.$

By Theorem 1.1, the optimal solution of $T(n,K_{4}^{(3)})$ must be connected
when $n$ is large enough. This implies $t_{3,4}=\tilde{t}_{3}(3)$. Therefore,

$t_{3,2m+2}(m)=(m-1+t_{3,4}^{-\frac{1}{2}})^{-2}.$

∎

##  6 Open questions

We conclude by formulating some open questions.

######  Question 1.

Can we improve the bound $n\geq k+\binom{k-2}{r-1}$ in Theorem 1.1?

######  Question 2.

For $m<k$, can we explicitly express $t_{3,2k+1}(m)$ or $t_{3,2k+2}(m)$ in
terms of the $t_{3,l}$’s?

######  Question 3.

Can we prove results similar to Theorem 1.3 for $r\geq 4$?

## Funding

Raffaella Mulas was supported by the Max Planck Society’s Minerva Grant.

## Acknowledgments

The authors are grateful to the anonymous referees for the comments and
suggestions that have greatly improved the first version of this paper.

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