# Concentrated sets in the Miller model

Valentin Haberl  Institut für Diskrete Mathematik und Geometrie, Technische
Universität Wien, Wiedner Hauptstrasse 8-10/104, 1040 Wien, Austria.
<EMAIL_ADDRESS>,  Piotr Szewczak  Institute of Mathematics,
Faculty of Mathematics and Natural Science, College of Sciences, Cardinal
Stefan Wyszyński University in Warsaw, Wóycickiego 1/3, 01–938 Warsaw, Poland
<EMAIL_ADDRESS>http://piotrszewczak.pl and  Lyubomyr Zdomskyy  Institut für
Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner
Hauptstrasse 8-10/104, 1040 Wien, Austria<EMAIL_ADDRESS>https://dmg.tuwien.ac.at/zdomskyy/

###### Abstract.

Using combinatorial covering properties, we show that there is no concentrated
set of reals of size $\omega_{2}$ in the Miller model. The main result refutes
a conjecture of Bartoszyński and Halbeisen.

###### Key words and phrases:

Concentrated set, Miller forcing, Hurewicz space, Rothberger space

###### 2020 Mathematics Subject Classification:

Primary: 03E35, 54D20. Secondary: 03E75.

The research of the first and the third authors was funded in whole by the
Austrian Science Fund (FWF) [I 5930]. The research of the second author was
funded by the National Science Centre, Poland Weave-UNISONO call in the Weave
programme Project: Set-theoretic aspects of topological selections
2021/03/Y/ST1/00122

##  1\. Introduction

We work in the Cantor space ${\operatorname{P}(\omega)}$. A set
$X\subseteq{\operatorname{P}(\omega)}$ is _concentrated on a set_
$D\subseteq{\operatorname{P}(\omega)}$ if the sets $X\setminus U$ are
countable for all open sets $U\subseteq{\operatorname{P}(\omega)}$ containing
$D$. A set $X\subseteq{\operatorname{P}(\omega)}$ is _concentrated_ if it is
uncountable and concentrated on some countable subset of $X$. One of the
remarkable features of concentrated sets is the following covering property
introduced by Rothberger [13] as a strengthening and a topological counterpart
of strong measure zero sets introduced by Borel. A set
$X\subseteq{\operatorname{P}(\omega)}$ is _Rothberger_ if for any sequence
$\mathcal{U}_{0},\allowbreak\mathcal{U}_{1},\allowbreak\dotsc$ of open covers
of $X$, there are sets
$U_{0}\in\mathcal{U}_{0},U_{1}\in\mathcal{U}_{1},\dotsc$ such that the family
$\\{\,U_{n}:n\in\omega\,\\}$ covers $X$.

Another motivation behind concentrated sets is their connection with so-called
$K$-Lusin sets, which in their turn are related to classical works of Banach
and Kuratowski on the existence of non-vanishing $\sigma$-additive measures
defined on all subsets of the real line, see, e.g., [3] and references
therein. Let ${[\omega]^{\omega}}$ and $\mathrm{Fin}$ be the families of all
infinite and finite subsets of $\omega$, respectively. Recall that a set
$X\subseteq{[\omega]^{\omega}}$ is _$K$ -Lusin_ if it is uncountable and the
sets $X\cap K$ are countable for all compact sets
$K\subseteq{[\omega]^{\omega}}$.

######  Observation 1.1.

There is a concentrated set in ${\operatorname{P}(\omega)}$ of cardinality
$\kappa$ if and only if there is a $K$-Lusin set in ${[\omega]^{\omega}}$ of
cardinality $\kappa$.

###### Proof.

($\Rightarrow$) Let $X\subseteq{\operatorname{P}(\omega)}$ be a concentrated
set of cardinality $\kappa$. Eventually, adding to the set $X$ countably many
points, we may assume that the set $X$ contains a countable dense set $D$ in
${\operatorname{P}(\omega)}$ such that $X$ is concentrated on $D$. The space
${\operatorname{P}(\omega)}$ (homeomorphic with ${2^{\omega}}$) is countable
dense homogeneous (see [2] and references therein), i.e., for any countable
dense subsets $D_{0},D_{1}$ of ${\operatorname{P}(\omega)}$ there exists a
homeomorphism $h\colon{\operatorname{P}(\omega)}\to{\operatorname{P}(\omega)}$
with $h[D_{0}]=D_{1}$. It follows that a homeomorphic copy $Y$ of $X$ in
${\operatorname{P}(\omega)}$ is concentrated on $\mathrm{Fin}$. For any
compact set $K\subseteq{[\omega]^{\omega}}$, the open set
${\operatorname{P}(\omega)}\setminus K$ contains $\mathrm{Fin}$, and thus the
set $Y\cap K$ is countable. It follows that $Y\setminus\mathrm{Fin}$ is a
$K$-Lusin set of cardinality $\kappa$.

($\Leftarrow$) Let $X\subseteq{[\omega]^{\omega}}$ be a $K$-Lusin set of
cardinality $\kappa$. For any open set $U\subseteq{\operatorname{P}(\omega)}$
containing $\mathrm{Fin}$, the set ${\operatorname{P}(\omega)}\setminus U$ is
compact in ${\operatorname{P}(\omega)}$, and thus the set $X\setminus
U=({\operatorname{P}(\omega)}\setminus U)\cap X$ must be countable. It follows
that the set $X\cup\mathrm{Fin}$ is concentrated on $\mathrm{Fin}$ and it has
cardinality $\kappa$. ∎

The main result of this paper is the non-existence of concentrated sets of
cardinality $\mathfrak{c}=\omega_{2}$ in the Miller model, i.e., the model
obtained by iterating the poset introduced in [11] with countable supports
$\omega_{2}$ many steps over a ground model of GCH.

######  Theorem 1.2.

In the Miller model, there is no $K$-Lusin set $X\subseteq{[\omega]^{\omega}}$
of size $\omega_{2}$. Equivalently, in this model there is no concentrated set
of size $\omega_{2}$.

This theorem refutes a conjecture indirectly stated in the proof of [3,
Proposition 3.2]. One of the main ingredients in the proof of Theorem 1.2 is
the fact [16] that in the Miller model every Rothberger set has the following
covering property of Hurewicz. A set $X\subseteq{\operatorname{P}(\omega)}$ is
_Hurewicz_ if for any sequence
$\mathcal{U}_{0},\allowbreak\mathcal{U}_{1},\allowbreak\dotsc$ of open covers
of $X$, there are finite families
$\mathcal{F}_{0}\subseteq\mathcal{U}_{0},\mathcal{F}_{1}\subseteq\mathcal{U}_{2},\dotsc$
such that the family $\\{\,\bigcup\mathcal{F}_{n}:n\in\omega\,\\}$ is a
$\gamma$-cover of $X$, i.e., it is infinite and each element $x\in X$ belongs
to all but finitely many sets $\bigcup\mathcal{F}_{n}$.

Since we never use cardinal characteristics of the continuum directly in our
proofs but rather some known combinatorial consequences of (in)equalities
between them, the knowledge of their definitions does not affect understanding
of this paper. Therefore we refer the reader to the survey of Blass [4]
instead of presenting the definitions of cardinal characteristics we mention.

##  2\. Proofs

We identify each infinite subset of $\omega$ with the increasing enumeration
of its elements, an element of the Baire space ${\omega^{\omega}}$. Then
${[\omega]^{\omega}}$ is a subset of ${\omega^{\omega}}$ and topologies in
${[\omega]^{\omega}}$ inherited from ${\operatorname{P}(\omega)}$ and
${\omega^{\omega}}$ coincide. Depending on context we refer to elements of
${[\omega]^{\omega}}$ as sets or functions. For natural numbers $n,m$ with
$n<m$, let $[n,m):=\\{\,i:n\leq i<m\,\\}$. For functions
$x,f\in{\omega^{\omega}}$ let $[f<g]:=\\{\,n:f(n)<g(n)\,\\}$ and we use this
convention also to another binary relations on $\omega$. Let
$h\in{[\omega]^{\omega}}$. A function $f\in{[\omega]^{\omega}}$ is _$h$
-unbounded over a set $M$_, if the sets
$\\{\,n\in\omega:[h(n),h(n+1))\subseteq[x<f]\,\\}$ are infinite for all
functions $x\in{\omega^{\omega}}\cap M$. Obviously, if $x\leq^{*}f$ for all
$x\in{\omega^{\omega}}\cap M$, then $f$ is $h$-unbounded over $M$ for any
$h\in{[\omega]^{\omega}}$.

######  Definition 2.1.

A poset $\mathbb{P}$ is _mild_ if for every sufficiently large $\theta$ and
countable elementary submodel $M$ of $H(\theta)$ with $\mathbb{P}\in M$, if a
function $f\in{[\omega]^{\omega}}$ is $h$-unbounded over $M$ for some
$h\in{[\omega]^{\omega}}$, then there is an $(M,\mathbb{P})$-generic condition
$q\in P$ with $q\leq p$ such that $q$ forces $f$ to be $h$-unbounded over
$M[\Gamma]$, where $\Gamma$ is the canonical name for a $\mathbb{P}$-generic
filter.

Obviously, each mild poset is proper and an iteration of finitely many mild
posets is again mild. Next, we shall prove that mild posets are preserved by
countable support iterations. As could be expected, the proof of the
aforementioned preservation is similar to a standard proof of the preservation
of properness by countable support iterations (see, e.g., non-vanishing finite
m [1, Lemma 2.8] for a very detailed presentation of the latter), with some
additional control on the sequence
${\langle\,\dot{p}_{i}:i\in\omega\,\rangle}$.

We follow conventions regarding notation related to iterations made in [12],
these are pretty standard and very similar to those in [1]. Let
${\langle\,\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha}:\alpha<\delta\,\rangle}$
be an iterated forcing construction for some ordinal number $\delta$. For
ordinal numbers $\alpha_{0},\alpha_{1}$ with
$\alpha_{0}\leq\alpha_{1}\leq\delta$, we denote by
$\mathbb{P}_{[\alpha_{0},\alpha_{1})}$ a $\mathbb{P}_{\alpha_{0}}$-name for
the quotient poset $\mathbb{P}_{\alpha_{1}}/\mathbb{P}_{\alpha_{0}}$, viewed
naturally as an iteration over the ordinals
$\xi\in\alpha_{1}\setminus\alpha_{0}$. For a $\mathbb{P}_{\alpha_{0}}$-generic
filter $G$ and a $\mathbb{P}_{\alpha_{1}}$-name $\tau$, where
$\alpha_{0}\leq\alpha_{1}\leq\delta$, we denote by $\tau^{G}$ the
$\mathbb{P}_{[\alpha_{0},\alpha_{1})}^{G}$-name in $V[G]$ obtained from $\tau$
by partially interpreting it with $G$. This allows us to speak about, e.g.,
$\mathbb{P}_{[\alpha_{1},\alpha_{2})}^{G}$ for
$\alpha_{0}\leq\alpha_{1}\leq\alpha_{2}\leq\delta$ and a
$\mathbb{P}_{\alpha_{0}}$-generic filter $G$. For a poset $\mathbb{P}$ we
shall denote the standard $\mathbb{P}$-name for a $\mathbb{P}$-generic filter
by $\Gamma_{\mathbb{P}}$. We shall write $\Gamma_{\alpha}$ instead of
$\Gamma_{\mathbb{P}_{\alpha}}$ whenever we work with an iterated forcing
construction which will be clear from the context. Also,
$\Gamma_{[\alpha_{0},\alpha_{1})}$ is a $\mathbb{P}_{\alpha_{1}}$-name whose
interpretation with respect to a $\mathbb{P}_{\alpha_{0}}$-generic filter $G$
is $\Gamma_{\mathbb{P}_{[\alpha_{0},\alpha_{1})}^{G}}$, which is an element of
$V[G]$.

######  Lemma 2.2.

If
${\langle\,\mathbb{P}_{\alpha},\dot{\mathbb{Q}}_{\alpha}:\alpha<\delta\,\rangle}$
is a countable support iteration of mild (hence proper) posets for some
ordinal number $\delta$, then $\mathbb{P}_{\delta}$ is also mild.

###### Proof.

The proof is by induction on $\delta$. Since the successor case is clear,
assume that $\delta$ is a limit ordinal number. Let $M$ be a countable
elementary submodel of $H(\theta)$, for a sufficiently large $\theta$ with
$\mathbb{P}_{\delta}\in M$ and $p\in\mathbb{P}_{\delta}$. Pick an increasing
sequence ${\langle\,\delta_{i}:i\in\omega\,\rangle}$ cofinal in $\delta\cap
M$, with $\delta_{i}\in M$ for all $i\in\omega$.

Let ${\langle\,\dot{y}_{i}:i\in\omega\,\rangle}$ be an enumeration of all
$\mathbb{P}_{\delta}$-names for sequences in ${\omega^{\omega}}$ which are
elements of $M$, where every such a name appears infinitely often. Let also
${\langle\,D_{i}:i\in\omega\,\rangle}$ be an enumeration of all open dense
subsets of $\mathbb{P}_{\delta}$ which belong to $M$. Fix functions
$f,h\in{\omega^{\omega}}$ and assume that the function $f$ is $h$-unbounded
over $M$. By induction on $i\in\omega$ we will define a condition
$q_{i}\in\mathbb{P}_{\delta_{i}}$ and $\mathbb{P}_{\delta_{i}}$-names
$\dot{p}_{i},\dot{n}_{i}$ such that

  1. (i)

$\dot{p}_{i}$ is a name for an element of $\mathbb{P}_{\delta}$,
$q_{0}\Vdash_{\delta_{0}}\dot{p}_{0}\leq\check{p}$, and
$q_{i+1}\Vdash_{\delta_{i+1}}\dot{p}_{i+1}\leq\dot{p}_{i}$;

  2. (ii)

$q_{i+1}\upharpoonright\delta_{i}=q_{i}$;

  3. (iii)

$q_{i}$ is $(M,\mathbb{P}_{\delta_{i}})$-generic;

  4. (iv)

$\dot{n}_{i}$ is a $\mathbb{P}_{\delta_{i}}$-name for a natural number bigger
than $i$; and

  5. (v)

$q_{i}\Vdash_{\delta_{i}}\bigl{(}\dot{p}_{i}\upharpoonright\delta_{i}\in\Gamma_{\delta_{i}}\bigr{)}\wedge\bigl{(}\dot{p}_{i}\in
D_{i}\cap
M\bigr{)}\wedge\bigl{(}\dot{p}_{i}\Vdash_{\delta}[h(\dot{n}_{i}),h(\dot{n}_{i}+1))\subseteq[\dot{y}_{i}<f]\bigr{)}$.

Suppose now that we have constructed objects as above and set
$q:=\bigcup_{i\in\omega}q_{i}$. We have $q_{i}=q\upharpoonright\delta_{i}$ for
all $i\in\omega$. Since
$q_{i}\Vdash_{\delta_{i}}\dot{p}_{i}\upharpoonright\delta_{i}\in\Gamma_{\delta_{i}}$
and $q_{i+1}\Vdash_{\delta_{i+1}}\dot{p}_{i+1}\leq\dot{p}_{i}$ for all
$i\in\omega$, a standard argument yields that $q$ is
$(M,\mathbb{P}_{\delta})$-generic and
$q\Vdash_{\delta}\dot{p}_{i}\in\Gamma_{\delta}$ for all $i\in\omega$, see,
e.g., the proof of [1, Lemma 2.8] for details. Then

$q\Vdash_{\delta}\\{\,n\in\omega:[h(n),h(n+1))\subseteq[\dot{y}<f]\,\\}\text{
is ifninite}$

for any $\mathbb{P}_{\delta}$-name $\dot{y}\in M$ for an element of
$\omega^{\omega}$: For a $\mathbb{P}_{\delta}$-name $\dot{a}$, let
$a:=\dot{a}^{G}$. Fix a $\mathbb{P}_{\delta}$-generic filter $G$ with $q\in G$
and $i\in\omega$ with $\dot{y}=\dot{y_{i}}$. Then the condition $p_{i}$ lies
in $G$ and by (v), and therefore we have

$[h({n_{i}}),h(n_{i}+1))\subseteq[y<f].$

By (iv) and the fact that $\dot{y}$ apperas infinitely often in the sequence
${\langle\,\dot{y_{i}}:i\in\omega\,\rangle}$, we have infinitely many
$n_{i}\in\omega$ with the above property.

Returning now to the inductive construction, fix $i\in\omega$ and assume that
$q_{i}\in\mathbb{P}_{\delta_{i}}$ and $\mathbb{P}_{\delta_{i}}$-names
$\dot{p}_{i},\dot{n}_{i}$ satisfying relevant instances of $(i)$–$(v)$ have
already been constructed. For the remaining part of the proof, let
$G_{\delta_{i}}$ be a $\mathbb{P}_{\delta_{i}}$-generic filter containing
$q_{i}$ and for a $\mathbb{P}_{\delta_{i}}$-name $\dot{a}$ let
$a:=\dot{a}^{G_{\delta_{i}}}$. We have $p_{i}\in\mathbb{P}_{\delta}\cap M$. By
(v), we know that $p_{i}\upharpoonright\delta_{i}\in G_{\delta_{i}}$. In
$V[G_{\delta_{i}}]$, let $p_{i}^{\prime}\in M\cap D_{i+1}$ be a condition such
that $p_{i}^{\prime}\leq p_{i}$ and
$p_{i}^{\prime}\upharpoonright\delta_{i}\in G_{\delta_{i}}$. By the maximality
principle, we get a $\mathbb{P}_{\delta_{i}}$-name $\dot{p}^{\prime}_{i}$ for
a condition in $\mathbb{P}_{\delta}$ such that

$q_{i}\Vdash_{\delta_{i}}\bigl{(}\dot{p}^{\prime}_{i}\leq\dot{p}_{i}\bigr{)}\wedge\bigl{(}\dot{p}^{\prime}_{i}\in
M\cap
D_{i+1}\bigr{)}\wedge\bigl{(}\dot{p}^{\prime}_{i}\upharpoonright\delta_{i}\in\Gamma_{\delta_{i}}\bigr{)}.$

Given a $\mathbb{P}_{\delta_{i+1}}$-generic filter $R$, construct in $V[R]$ a
decreasing sequence ${\langle\,r_{m}:m\in\omega\,\rangle}\in M[R]$ of
conditions in $\mathbb{P}^{R}_{[\delta_{i+1},\delta)}$ below
$(\dot{p}^{\prime}_{i}\upharpoonright[\delta_{i+1},\delta))^{R}$ such that for
some $s_{m}\in\omega^{m}$ we have

$r_{m}\Vdash_{\mathbb{P}^{R}_{[\delta_{i+1},\delta)}}\dot{y}_{i+1}\upharpoonright
m=s_{m}.$

By the maximality principle, we get a sequence
${\langle\,\rho_{m}:m\in\omega\,\rangle}\in M$ of
$\mathbb{P}_{\delta_{i+1}}$-names for elements of
$\mathbb{P}_{[\delta_{i+1},\delta)}$ such that

$\Vdash_{\delta_{i+1}}\bigl{(}\rho_{m+1}\leq\rho_{m}\bigr{)}\wedge\bigl{(}\operatorname{\exists}\nu_{m}\in\omega^{m}\bigr{)}\bigl{(}\rho_{m}\Vdash_{\mathbb{P}_{[\delta_{i+1},\delta)}}\nu_{m}\mbox{=}\dot{y}_{i+1}\upharpoonright
m\bigr{)}.$

In the notation used above, let $\dot{z}\in M$ be a
$\mathbb{P}_{\delta_{i+1}}$-name for $\bigcup_{m\in\omega}\nu_{m}$, an element
of $\omega^{\omega}$.

We have $p^{\prime}_{i}\in\mathbb{P}_{\delta}\cap M\cap D_{i+1}$. It also
follows from the above that $p^{\prime}_{i}\upharpoonright\delta_{i}\in
G_{\delta_{i}}$. For a while we shall be working in $V[G_{\delta_{i}}]$. Since
by our inductive assumption the poset
$\mathbb{P}^{G_{\delta_{i}}}_{[\delta_{i},\delta_{i+1})}$ is mild in
$V[G_{\delta_{i}}]$, there exists an
$(M[G_{\delta_{i}}],\mathbb{P}^{G_{\delta_{i}}}_{[\delta_{i},\delta_{i+1})})$-generic
condition
$\pi\leq(p_{i}^{\prime}\upharpoonright[\delta_{i},\delta_{i+1}))^{G_{\delta_{i}}}$
such that

$\pi\Vdash_{\mathbb{P}^{G_{\delta_{i}}}_{[\delta_{i},\delta_{i+1})}}\tau:=\\{n\in\omega:[h(n),h(n+1))\subseteq[\dot{z}^{G_{\delta_{i}}}<f]\\}$

is infinite. Let $H$ be a
$\mathbb{P}^{G_{\delta_{i}}}_{[\delta_{i},\delta_{i+1})}$-generic filter over
$V[G_{\delta_{i}}]$ containing $\pi$, and fix
$n_{i+1}\in\tau^{H}\setminus(i+2)$. Thus in $V[G_{\delta_{i}}*H]$ we have

$\rho_{h(n_{i+1}+1)}^{G_{\delta_{i}}*H}\Vdash_{\mathbb{P}_{[\delta_{i+1},\delta)}^{G_{\delta_{i}}*H}}[h(n_{i+1}),h(n_{i+1})+1)\subseteq[\dot{y}_{i+1}^{G_{\delta_{i}}*H}<f].$

In $M[G_{\delta_{i}}]$ pick a condition $s\in M[G]\cap H$ below
$(p_{i}^{\prime}\upharpoonright[\delta_{i},\delta_{i+1}))^{G_{\delta_{i}}}$,
forcing the above properties of $n^{j}_{i+1}$, $\tau$, and
$\rho_{h(n_{i+1}+1)}$. By the maximality principle we obtain
$\mathbb{P}_{[\delta_{i},\delta_{i+1})}^{G_{\delta_{i}}}$-names $\dot{s}$ and
$\rho$ in $M[G_{\delta_{i}}]$ for some elements of
$\mathbb{P}_{[\delta_{i},\delta_{i+1})}^{G_{\delta_{i}}}$ and
$\mathbb{P}_{[\delta_{i+1},\delta)}^{G_{\delta_{i}}}$, respectively, and a
name $\dot{n}_{i+1}$ for a natural number such that

(2.2.1)
$\pi\Vdash_{{\mathbb{P}_{[\delta_{i},\delta_{i+1})}^{G_{\delta_{i}}}}}\bigl{(}\dot{s}\in
M[G_{\delta_{i}}]\cap\Gamma^{G_{\delta_{i}}}_{[\delta_{i},\delta_{i+1})}\bigr{)}\wedge\bigl{(}\dot{s}\leq(\dot{p}^{\prime}\upharpoonright[\delta_{i},\delta_{i+1})^{G_{\delta_{i}}}\bigr{)}\wedge\\\
\wedge\bigl{(}\dot{s}\Vdash_{{\mathbb{P}_{[\delta_{i},\delta_{i+1})}^{G_{\delta_{i}}}}}\rho\leq(\dot{p}^{\prime}\upharpoonright[\delta_{i+1},\delta))^{G_{\delta_{i}}}\bigr{)}\wedge\rho\Vdash_{{\mathbb{P}_{[\delta_{i+1},\delta)}^{G_{\delta_{i}}}}}[h(\dot{n}_{i+1}),h(\dot{n}_{i+1}+1))\subseteq[\dot{y}_{+1}<f].$

Using the maximality principle again, we can find
$\mathbb{P}_{\delta_{i}}$-names for the objects appearing in equation (2.2.1)
such that $q_{i}$ forces this equation. We shall use the same notation for
these names. It remains to set $q_{i+1}:=q_{i}\hat{\ \ }\pi$ and
$\dot{p}_{i+1}:=\dot{p}^{\prime}_{i}\upharpoonright\delta_{i}\hat{\ \
}\dot{s}\hat{\ \ }\rho$ and note that they together with the name
$\dot{n}_{i+1}$ satisfy $(i)$–$(v)$ for $i+1$. ∎

By a _Miller tree_ we understand a subtree $T$ of $\omega^{<\omega}$
consisting of increasing finite sequences such that the following conditions
are satisfied:

  * •

Every $t\in T$ has an extension $s\in T$ which is splitting in $T$, i.e.,
there are more than one immediate successors of $s$ in $T$;

  * •

If $s$ is splitting in $T$, then it has infinitely many immediate successors
in $T$.

The _Miller forcing_ is the collection $\mathbb{M}$ of all Miller trees
ordered by inclusion, i.e., smaller trees carry more information about the
generic. This poset was introduced in [11]. For a Miller tree $T$ we shall
denote the set of all splitting nodes of $T$ by $\mathsf{Split}(T)$. The set
$\mathsf{Split}(T)$ may be written in the form
$\bigcup_{i\in\omega}\mathsf{Split}_{i}(T)$, where

$\mathsf{Split}_{i}(T):=\\{t\in\mathsf{Split}(T)\>:\>|\\{s\in\mathsf{Split}(T):s\subsetneq
t\\}|=i\\}.$

If $T_{0},T_{1}\in\mathbb{M}$, then $T_{1}\leq_{i}T_{0}$ means that $T_{1}\leq
T_{0}$ and $\mathsf{Split}_{i}(T_{1})=\mathsf{Split}_{i}(T_{0})$. It is easy
to check that for any sequence
${\langle\,T_{i}:i\in\omega\,\rangle}\in\mathbb{M}^{\omega}$, if
$T_{i+1}\leq_{i}T_{i}$ for all $i\in\omega$, then
$\bigcap_{i\in\omega}T_{i}\in\mathbb{M}$.

For a node $t$ in a Miller tree $T$ we denote by $T_{t}$ the set $\\{s\in T:s$
is compatible with $t\\}$, which is also a Miller tree.

######  Lemma 2.3.

The Miller forcing $\mathbb{M}$ is mild.

###### Proof.

Let $N$ be a countable elementary submodel of $H(\theta)$ for a sufficiently
large $\theta$ with $\mathbb{M}\in N$ and $T\in\mathbb{M}\cap N$. Let
${\langle\,\dot{y}_{i}:i\in\omega\,\rangle}$ be an enumeration of all
$\mathbb{M}$-names for infinite subsets of $\omega$ which are elements of $N$,
in which every such a name appears infinitely often. Let also
${\langle\,D_{i}:i\in\omega\,\rangle}$ be an enumeration of all open dense
subsets of $\mathbb{M}$ which belong to $N$. Suppose that a function
$f\in{[\omega]^{\omega}}$ is $h$-unbounded over $N$ for some
$h\in{[\omega]^{\omega}}$. We shall inductively construct a sequence
${\langle\,T_{i}:i\in\omega\,\rangle}$ such that $T_{i+1}\leq_{i}T_{i}$ and
$T_{\infty}:=\bigcap_{i\in\omega}T_{i}$ is as required. Set $T_{0}:=T$ and
suppose that $T_{i}$ has already been constructed. Moreover, we shall assume
that $(T_{i})_{t}\in N$ for all $t\in\mathsf{Split}_{i}(T_{i})$. Let
${\langle\,t_{j}:j\in\omega\,\rangle}$ be a bijective enumeration of
$\mathsf{Split}_{i}(T_{i})$. For every $j,k\in\omega$ with $t_{j}\hat{\ \
}k\in T_{i}$ fix a decreasing sequence
${\langle\,S^{i,j,k}_{n}:n\in\omega\,\rangle}\in N$ of elements of $D_{i}$
below $(T_{i})_{t_{j}\hat{\ }k}$ such that each condition $S^{i,j,k}_{n}$
decides some $a^{i,j,k}_{n}\in\omega^{n}$ to be $\dot{y}_{i}\upharpoonright
n$. Thus $y^{i,j,k}:=\bigcup_{n\in\omega}a^{i,j,k}_{n}\in\omega^{\omega}\cap
N$, and hence there is a natural number $m^{i,j,k}\geq i$ such that

$[h(m^{i,j,k}),h(m^{i,j,k}+1))\subseteq[y^{i,j,k}<f].$

Set

$T_{i+1}:=\bigcup\\{S^{i,j,k}_{h(m^{i,j,k}+1)}:j\in\omega,t_{j}\hat{\ \ }k\in
T_{i}\\}.$

This completes our inductive construction of the fusion sequence
${\langle\,T_{i}:i\in\omega\,\rangle}$.

We claim that $T_{\infty}$ is as required. First of all, the condition
$T_{\infty}$ is $(N,\mathbb{M})$-generic because the collection
$\\{\,S^{i,j,k}_{h(m^{i,j,k}+1)}:j\in\omega,t_{j}\hat{\ \ }k\in T_{i}\,\\}$ is
a subset of $D_{i}\cap N$ and predense below $T_{i+1}$ (and hence also below
$T_{\infty}$). Now fix an $\mathbb{M}$-name $\dot{y}\in N$ for an element of
$\omega^{\omega}$ and suppose to the contrary, that there exist $i\in\omega$
and $R\leq T_{\infty}$ that forces $[h(m),h(m+1))\not\subseteq[\dot{y}<f]$ for
all $m\geq i$. Enlarging $i$, if necessary, we may assume that
$\dot{y}=\dot{y}_{i}$. Replacing $R$ with a stronger condition, if necessary,
we may assume that $R\leq(T_{i})_{t_{j}\hat{\ }k}$, where $t_{j}\hat{\ }k\in
T_{i}$ for some $i,j,k\in\omega$. But then $R\leq S^{i,j,k}_{h(m^{i,j,k}+1)}$,
and the latter condition forces

$[h(m^{i,j,k}),h(m^{i,j,k}+1))\subseteq[\dot{y}_{i}<f],$

which leads to a contradiction since $m^{i,j,k}$ has been chosen to be above
$i$. ∎

######  Corollary 2.4.

Let $G_{\omega_{2}}$ be an $\mathbb{M}_{\omega_{2}}$-generic filter over $V$,
$x\in({[\omega]^{\omega}})^{V[G_{\omega_{2}}]}$, and
$\psi:({[\omega]^{\omega}})^{V}\to({[\omega]^{\omega}})^{V}$ be a function
which is an element of $V$. Then there exists an element
$f\in{[\omega]^{\omega}}\cap V$ such that the set

$\\{\,n\in\omega:[\psi(f)(n),\psi(f)(n+1))\subseteq[x<f]\,\\}$

is infinite.

###### Proof.

Suppose, contrary to our claim, that there exists a condition $p\in
G_{\omega_{2}}$ and an $\mathbb{M}_{\omega_{2}}$-name $\dot{x}$ for an element
$x\in\omega^{\omega}\cap V[G_{\omega_{2}}]$ such that

(2.4.2) $p\Vdash\bigl{(}\operatorname{\forall}f\in{[\omega]^{\omega}}\cap
V\bigr{)}\bigl{(}\operatorname{\forall^{*}}n\in\omega\bigr{)}\bigl{(}\big{[}\psi(f)(n),\psi(f)(n+1)\big{)}\cap[\dot{x}\geq
f]\neq\emptyset\bigr{)}.$

We work in $V$ in what follows. Let $N$ be a countable elementary submodel of
$H(\omega_{3})$ with $p,\dot{x}\in N$. Fix $f\in{[\omega]^{\omega}}$ such that
$z\leq^{*}f$ for all $z\in{[\omega]^{\omega}}\cap N$. Then $f$ is
$\psi(f)$-unbounded over $N$, and hence Lemmata 2.3 and 2.2 imply that there
exists an $(N,\mathbb{M}_{\omega_{2}})$-generic condition $q\leq p$ which
forces that $f$ is $\psi(f)$-unbounded over
$N[\Gamma_{\mathbb{M}_{\omega_{2}}}]$. In particular, the condition $q$ forces
that the set

$\\{\,n\in\omega:[\psi(f)(n),\psi(f)(n+1))\subseteq[\dot{x}<f]\,\\}$

is infinite, which contradicts (2.4.2). ∎

Remark. As it was established in [8], in the Miller model there exists an
ultrafilter $\mathcal{F}$ having a base in $V$, i.e., there exists a family
$\\{F_{\alpha}:\alpha\in\omega_{1}\\}\in V$ such that

$\mathcal{F}=\\{X\subseteq\omega:\bigl{(}\operatorname{\exists}\alpha\in\omega_{1}\bigr{)}(F_{\alpha}\subseteq
X)\\}$

is an ultrafilter in $V[G_{\omega_{2}}]$. By [6, Theorem 3.1] there exists a
function $x\in{[\omega]^{\omega}}$ such that $[f<x]\in\mathcal{F}$ for all
functions $f\in{[\omega]^{\omega}}\cap V$. Thus, for every function
$f\in{[\omega]^{\omega}}\cap V$ there exists an ordinal number
$\alpha(f)<\omega_{1}$ such that $F_{\alpha(f)}\subseteq[f<x]$. Let
$\psi_{\mathcal{F}}(f)\in{[\omega]^{\omega}}\cap V$ be a function such that
$[\psi_{\mathcal{F}}(f)(n),\psi_{\mathcal{F}}(f)(n+1))\cap
F_{\alpha(f)}\neq\emptyset$ for all $n\in\omega$. Thus $\psi_{\mathcal{F}}$
does not satisfy the conclusion of Corollary 2.4, and hence is an example
showing that the condition $\psi\in V$ in the above-mentioned corollary cannot
be omitted. $\Box$

We are in a position now to present the proof of the main result.

Proof of Theorem 1.2. Assume that there is a $K$-Lusin set
$X\subseteq{[\omega]^{\omega}}$. For each subset $X^{\prime}$ of $X$, the set
$X^{\prime}\cup\mathrm{Fin}\subseteq{\operatorname{P}(\omega)}$ is Hurewicz:
If $X^{\prime}$ is countable, then the assertion is true. Assume that
$X^{\prime}$ is uncountable. Then $X^{\prime}$ is a $K$-Lusin set and as in
the proof of Observation 1.1, we have that $X^{\prime}\cup\mathrm{Fin}$ is
concentrated. Each concentrated set is obviously Rothberger and in the Miller
model each Rothberger set is Hurewicz. Indeed, each Rothberger set is Hurewicz
under $\mathfrak{u}<\mathfrak{g}$ by [16, Theorem 5] combined with [14,
Theorem 15]. The fact that $\mathfrak{u}<\mathfrak{g}$ holds in the Miller
model is a direct consequence of [5, Theorem 2] combined with the results of
[8].

Work in $V[G_{\omega_{2}}]$. Fix a function $f\in{[\omega]^{\omega}}$ and
define $X_{f}:=\\{\,x\in X:x\not\leq^{*}f\,\\}$. Then the set
$\\{\,x\in{[\omega]^{\omega}}:x\leq^{*}f\,\\}$ is $\sigma$-compact and hence
the set

$X\setminus X_{f}=\\{\,x\in
X:x\leq^{*}f\,\\}=X\cap\\{\,x\in{[\omega]^{\omega}}:x\leq^{*}f\,\\}$

is countable. Each finite subset of $\omega$ we identify with the increasing
enumeration of its elements, an element of $\omega^{<\omega}$. Let
$\phi_{f}\colon X_{f}\cup\mathrm{Fin}\to{[\omega]^{\omega}}$ be a function
such that

$\phi_{f}(y):=\begin{cases}[y\geq f],&\text{ if }y\in X_{f},\\\
\\{\,n<|y|:y(n)\geq f(n)\,\\}\cup\\{|y|,|y|+1,\dotsc\\},&\text{ if
}y\in\mathrm{Fin}.\end{cases}$

Since $X_{f}\subseteq\\{\,x\in X:[x\geq f]\text{ is infinite}\,\\}$, the map
$\phi_{f}$ is well defined. Moreover, the map $\phi_{f}$ is continuous.

By the above, the set $X_{f}\cup\mathrm{Fin}$ is Hurewicz, and thus, the
continuous image $\phi_{f}[X_{f}\cup\mathrm{Fin}]\subseteq[\omega]^{\omega}$
is Hurewicz as well. Any Hurewicz subspace of $[\omega]^{\omega}$ is bounded
[9, 10]. It is well-known and easy to check that for every bounded set
$B\subseteq[\omega]^{\omega}$ there exists an increasing function
$s\in{[\omega]^{\omega}}$ such that $b\cap[s(n),s(n+1))\neq\emptyset$ for
every $b\in B$ and all but finitely many $n\in\omega$ [15, Lemma 2.13]. Thus,
there exists a function $\psi\colon{[\omega]^{\omega}}\to{[\omega]^{\omega}}$
such that for each element $y\in X_{f}\cup\mathrm{Fin}$, we have
$[\psi(f)(n),\psi(f)(n+1))\cap\phi_{f}(y)\neq\emptyset$ for all but finitely
many $n\in\omega$.

By a standard argument [7, the proof of Lemma 5.10], there exists an ordinal
number $\alpha<\omega_{2}$ such that

  * •

$\psi[{[\omega]^{\omega}}\cap V[G_{\alpha}]]\subseteq V[G_{\alpha}]$;

  * •

$\psi\upharpoonright({[\omega]^{\omega}}\cap V[G_{\alpha}])\in V[G_{\alpha}]$;

  * •

$X\setminus X_{f}\in V[G_{\alpha}]$ for all $f\in V[G_{\alpha}]$; and

  * •

the map $V[G_{\alpha}]\ni f\mapsto X\setminus X_{f}$ belongs to
$V[G_{\alpha}]$.

Without loss of generality, we may assume that $V[G_{\alpha}]=V$. Since
$|X|=\omega_{2}$ and $\left|{[\omega]^{\omega}}\cap V\right|\leq\omega_{1}$,
there is an element $x\in X\setminus V$. Let $f\in{[\omega]^{\omega}}\cap V$
be a function from Corollary 2.4, applied to
$\psi\restriction({[\omega]^{\omega}}\cap V)\in V$ and $x\in X$. Then the set

$\\{\,n\in\omega:[\psi(f)(n),\psi(f)(n+1))\subseteq[x<f]\,\\}$

is infinite. Since $x\not\in V$, we have $x\in X_{f}$. By the definition of
$\psi$, we have

$[\psi(f)(n),\psi(f)(n+1))\cap[x\geq
f]=[\psi(f)(n),\psi(f)(n+1))\cap\phi_{f}(x)\neq\emptyset$

for all but finitely many $n\in\omega$, a contradiction $\Box$

We leave the reader with the following question asking whether the kind of
concentrated sets whose non-existence in the Miller model we proved above, can
exist at all.

######  Problem 2.5.

Is it consistent that there exists a set $X\subseteq{[\omega]^{\omega}}$ of
size $|X|>\omega_{1}$ such that for every $f\in{[\omega]^{\omega}}$

  * •

the set $\\{x\in X:x\leq^{*}f\\}$ is countable (i.e., $X$ is $K$-Lusin), and

  * •

there exists $\psi(f)\in{[\omega]^{\omega}}$ such that for every $x\in X$, if
$x\not\leq^{*}f$, then

$[\psi(f)(n),\psi(f)(n+1))\cap[x>f]\neq\emptyset$

for all but finitely many $n\in\omega$?

## Acknowledgments

We would like to thank Professor Piotr Zakrzewski who draw our attention to
the paper of Bartoszyński and Halbeisen [3].

## References

  * [1] Abraham, U., _Proper Forcing_ , in: Handbook of Set Theory in: Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), Springer, 2010, 333–394.
  * [2] Arhangel’skii, A.; van Mill, J., Topological homogeneity, in: Recent Progress in General Topology III (KP Hart, J. van Mill, P. Simon, eds.), Springer 2014, 1–68.
  * [3] Bartoszyński, T.; Halbeisen, L., On a theorem of Banach and Kuratowski and $K$-Lusin sets, Rocky Mountain J. Math. 33 (2003), 1223–1231.
  * [4] Blass, A., Combinatorial cardinal characteristics of the continuum, in: Handbook of Set Theory (M. Foreman, A. Kanamori, and M. Magidor, eds.), Springer, 2010, 395–491.
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  * [6] Blass, A.; Mildenberger, H., On the cofinality of ultrapowers, J. Symbolic Logic 64 (1999), 627–636.
  * [7] Blass, A.; Shelah, S., There may be simple $P_{\aleph_{1}}$\- and $P_{\aleph_{2}}$-points and the Rudin–Keisler ordering may be downward directed, Ann. Pure Appl. Logic 33 (1987), 213–243.
  * [8] Blass, A.; Shelah, S., Near coherence of filters. III. A simplified consistency proof, Notre Dame J. Formal Logic 30 (1989), 530–538.
  * [9] Hurewicz, W., Über Folgen stetiger Funktionen, Fund. Math. 9 (1927), 193–204.
  * [10] Just, W.; Miller, A.W.; Scheepers, M.; Szeptycki, P.J., The combinatorics of open covers. II, Topology Appl. 73 (1996), 241–266.
  * [11] Miller, A., Rational perfect set forcing, in: Axiomatic Set Theory (J. Baumgartner, D. A. Martin, S. Shelah, Eds.), Contemporary Mathematics 31, American Mathematical Society, Providence, Rhode Island, 1984, 143–159.
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# Precession and Lense-Thirring effect of hairy Kerr spacetimes

Meng-He Wu<EMAIL_ADDRESS>School of Mathematics, Physics and Statistics,
Shanghai University of Engineering Science, Shanghai 201620, China  Center of
Application and Research of Computational Physics, Shanghai University of
Engineering Science, Shanghai 201620, China  Hong Guo<EMAIL_ADDRESS>School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai
200240, China  Xiao-Mei Kuang<EMAIL_ADDRESS>(corresponding author)
Center for Gravitation and Cosmology, College of Physical Science and
Technology, Yangzhou University, Yangzhou, 225009, China

###### Abstract

We investigate the frame-dragging effect of the hairy Kerr spacetimes on the
spin of a test gyro and accretion disk. Firstly, we analyze Lense-Thirring
(LT) precession frequency, geodetic precession frequency, and the general spin
precession frequency of a test gyro attached to a stationary observer in the
spacetime. We find that the black hole hair suppresses those precession
frequencies in comparison with that occurs in Kerr spacetime in general
relativity. Moreover, using those frequencies as probe, we differentiate the
hairy Kerr black hole (BH) from naked singularity (NS). Specifically, as the
observer approaches the central source along any direction, the frequencies
grow sharply for the hairy Kerr BH, while for the hairy NS they are finite
except at the ring singularity. Then, we investigate the quasiperiodic
oscillations (QPOs) phenomena as the accretion disk approaches the hairy Kerr
BH or NS. To this end, we analyze the bound circular orbits and their
perturbations. We find that as the orbits approach the corresponding inner-
most stable circular orbit (ISCO), both LT precession frequency and periastron
precession frequency behave differently in the hairy Kerr BH and NS.
Additionally, the hairy parameters have significant effects on the two
frequencies. We expect that our theoretical studies could shed light on
astrophysical observations in distinguishing hairy theories from Einstein’s
gravity, and also in distinguishing BH from NS in spacetime with hair.

###### Contents

  1. I Introduction
  2. II Hairy Kerr spacetime and the timelike geodesic equations
  3. III Spin precession of test gyroscope in hairy Kerr spacetime
    1. III.1 Case with $\Omega=0$: LT precession frequency
    2. III.2 Case with $a=0$: Geodetic precession frequency
    3. III.3 Spin precession frequency in hairy Kerr black hole and naked singularity
  4. IV LT precession and periastron precession of accretion disk physics around hairy Kerr spacetime
    1. IV.1 Bound orbit and ISCO
    2. IV.2 LT precession and periastron precession
  5. V Conclusion and discussion

##  I Introduction

Einstein’s general relativity (GR) is a successful theory in modern physics
and passes plenty of test in astrophysics as well as astronomy. The existence
of black holes is a prediction of GR, and black holes provide natural
laboratories to test gravity in the strong field regime. Recent observations
on gravitational waves Abbott et al. (2016, 2019, 2020) and the supermassive
black hole images Akiyama et al. (2019a, b, c, 2022a, 2022b) agree well with
the predictions of Kerr black hole described by the Kerr metric from GR. More
observations, such as those from Next Generation Very Large Array D. James et
al. (2019) and Thirty Meter Telescope Skidmore et al. (2015), also provide
significant properties in the regime of strong gravity of black hole
spacetimes. In particular, these observations open a valuable window to
explore, distinguish, or constrain physically viable black hole solutions that
exhibit small deviations from the Kerr metric. On the other hand, due to the
additional surrounding sources, the black holes in our Universe could obtain
an extra global charge dubbed ‘hair’ and the spacetime may deviate from the
Kerr metric. Recently, a rotating hairy Kerr black hole was constructed with
the use of the gravitational decoupling (GD) approach Ovalle et al. (2021);
Contreras et al. (2021), which is designed for describing deformations of
known solutions of GR due to the inclusion of additional sources. The hairy
Kerr black hole attracts lots of attentions. Plenty of theoretical and
observational investigations have been done in this hairy Kerr black hole
spacetime, for examples, thermodynamics Mahapatra and Banerjee (2022),
quasinormal modes and (in)stability Cavalcanti et al. (2022); Yang et al.
(2022); Li (2022), strong gravitational lensing and parameter constraint from
Event Horizon Telescope observations Islam and Ghosh (2021); Afrin et al.
(2021).

There are many theoretical scenarios in which a visible singularity exists,
especially after it was found in Joshi et al. (2011) that a spacetime with a
central naked singularity can be formed as an equilibrium end state of the
gravitational collapse of general matter cloud. However, whether naked
singularity (NS) really exists in nature and what is its physical signature
distinguished from black hole (BH) are still open questions. In Mathematics,
the Kerr spacetime metric in GR, as a solution of Einstein field equations,
describes Kerr BH and the Kerr singularity is contained in the event horizon,
otherwise, if the event horizon disappears, the metric describes a NS. In
shadow scenario, the shadow cast by NS spacetime was found to have similar
size to that an equally massive Schwarzschild black hole can cast Shaikh et
al. (2019). More elaborate discussions on shadow from NS have been carried out
in Joshi et al. (2020); Dey et al. (2020, 2021), which are important
theoretical studies and match recent observations on the shadow of M87* and
SgrA*. In addition, the orbital dynamics of particles or stars around a
horizonless ultra-compact object could be another important physical signature
which could be distinguishable from that around black holes, since the nature
of timelike geodesics in a spacetime closely depends upon the geometrical
essence of the spacetime. In this scenario, the timelike geodesics around
different types of compact objects and the trajectory of massive particles or
stars are widely studied Pugliese et al. (2011); Martínez et al. (2019);
Hackmann et al. (2014); Potashov et al. (2019); Bhattacharya et al. (2020);
Bambhaniya et al. (2019); Deng (2020); Lin and Deng (2021); Bambhaniya et al.
(2021a); Ota et al. (2022) and reference therein. People expect that the
nature of the precession of the timelike orbits can provide more information
about the central compact object, as GRAVITY Bauböck et al. (2020) and SINFONI
Eisenhauer et al. (2005) are continuously tracking the dynamics of ‘S’ stars
orbiting around SgrA*.

For the timelike orbits around a rotating compact body, the geodetic
precession de Sitter (1916) and Lense-Thirring (LT) precession Lense and
Thirring (1918) are two extraordinary effects predicted by GR. The former
effect is also known as de Sitter precessions and it is due to the spacetime
curvature of the central body, while the latter effect is due to the rotation
of the central body which causes the dragging of locally inertial frames. One
can examine the dragging effects by considering a test gyro based on the fact
that a gyroscope tends to keep its spinning axis rigidly pointed in a fixed
direction relative to a reference star. The LT precession and geodetic effects
have been measured in the Earth’s gravitational field by the Gravity Probe B
experiment, in which the satellite consists of four gyroscopes and a telescope
orbiting $642km$ above the Earth Everitt et al. (2011). The geodetic
precession in Schwarzschild BH and the Kerr BH have been studied in Sakina and
Chiba (1979). The LT precession is more complex and usually some
approximations should be involved. In the weak field approximations, the
magnitude of LT precession frequency is proportional to the spin parameter of
the central body and decreases in the order of $r^{-3}$ with $r$ the distance
between the test gyro from the central rotating body Hartle (2009). In the
strong gravity limit, the LT precession frequency were studied in various
rotating compact objects, for instances, in Kerr black hole Chakraborty and
Majumdar (2014); Bini et al. (2016) and its generalizations Chakraborty and
Pradhan (2013), in rotating traversable wormhole Chakraborty and Pradhan
(2017) and in rotating neutron star Chakraborty et al. (2014). Those studies
further show that the behavior of LT frequency in the strong gravity regime
closely depends on the nature of the central rotating bodies. In particular,
it was proposed in Chakraborty et al. (2017a) that the spin precession of a
test gyro can be used to distinguish Kerr naked singularities from black
holes, which was then extended in modified theories of gravity Rizwan et al.
(2018, 2019); Pradhan (2020); Solanki et al. (2022).

Thus, one of the main aims of this paper is to investigate the spin precession
frequency, including the LT frequency and geodetic frequency, of the test gyro
in the hairy Kerr spacetime, and differentiate the hairy Kerr BH from hairy
NS. We find a clear difference on the precession frequencies between hairy
rotating BH and NS, and the effects of the hairy parameters are also
systematically explored.

Additionally, we also investigate the accretion disk physics in the hairy Kerr
spacetime, which is realized by studying the orbital precession of a test
timelike particle around the hairy Kerr BH and NS. The motivation stems from
the followings. Accretion in Low-Mass X-ray Binaries (LMXBs) occurs in the
strong gravity regime around compact bodies, in which the quasiperiodic
oscillations (QPOs) phenomena involves and their frequencies in the $Hz$ to
$kHz$ range have been detected Torok et al. (2011); Bambi (2012); Tripathi et
al. (2019). Plenty of efforts have been made to explain the QPOs phenomena,
see van der Klisin (2006) for a review. It shows that the geodesic models of
QPOs are related with three characteristic frequencies of the massive
particles orbiting around the compact bodies, namely the orbital epicyclic
frequency and the radial and vertical epicyclic frequencies. Therefore, since
the QPOs provide a way to testify the strong gravity, these three frequencies
could be used as a tool to study the crucial differences among the central
compact objects Rizwan et al. (2019) or test/constrain alternative theories of
gravity Aliev et al. (2013); Doneva et al. (2014); Azreg-Aïnou et al. (2020);
Jusufi et al. (2021); Ghasemi-Nodehi et al. (2020). In particular, more
recently it was proposed in Motta et al. (2018) that LT effect may have
connection with QPOs phenomena and perhaps be used to explain the QPOs of
accretion disks around rotating black holes. Thus, Along with using the spin
precession frequency and LT precession frequency of the test gyro to indicate
the differences between hairy Kerr BH and NS, we further study the three
characteristic frequencies of massive particles orbiting around the
corresponding central rotating objects. It is found that the LT precession and
periastron precession of massive particle behave crucial differences between
hairy Kerr BH and NS, as a support of the results from the test gyro in strong
gravity regime of compact bodies.

Our paper is organized as follows. In section II, we briefly review the hairy
Kerr spacetime derived from GD approach, and analyze the parameter spaces for
the corresponding black hole and naked singularity. Then we derive the
timelike geodesic equations in the spacetime. In section III, we derive the
spin precession frequency of a test gyro in the hairy Kerr spacetime. Then we
compare the LT frequency as well as the spin frequency between the hairy Kerr
BH and NS cases. In section IV, we study the accretion disk physics by
analyzing inner-most stable circular orbit (ISCO), and LT precession and
periastron precession in terms of the three characteristic frequencies, of a
timelike particle around the hairy Kerr BH and NS, respectively. The last
section is our conclusion and discussion. Throughout the paper, we use $G_{\rm
N}=c=\hbar=1$ and all quantities are rescaled to be dimensionless by the
parameter $M$.

##  II Hairy Kerr spacetime and the timelike geodesic equations

In this section, firstly, we will show a brief review on the idea of
gravitational decoupling (GD) approach and the hairy spacetime constructed
from GD approach by Ovalle Ovalle et al. (2021). Then we derive the timelike
geodesic equations in the hairy Kerr spacetime.

The no-hair theorem in classical general relativity states that black holes
are only described by mass, electric charge and spin Israel (1967); Hawking
(1972); Mazur (1982). But it is possible that the interaction between black
hole spacetime and matters brings in other charge, such that the black hole
could carry hairs. The physical effect of these hairs can modify the spacetime
of the background of black hole, namely hairy black holes may form. Recently,
Ovalle et.al used the GD approach to obtain a spherically symmetric metric
with hair Ovalle et al. (2021). The hairy black hole in this scenario has
great generality because there is no certain matter fields in the GD approach,
in which the corresponding Einstein equation is expressed by

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=8\pi\tilde{T}_{\mu\nu}.$ (1)

Here $\tilde{T}_{\mu\nu}$ is the total energy momentum tensor written as
$\tilde{T}_{\mu\nu}=T_{\mu\nu}+\vartheta_{\mu\nu}$ where $T_{\mu\nu}$ and
$\vartheta_{\mu\nu}$ are energy momentum tensor in GR and energy momentum
tensor introduced by matter fields or others, respectively.
$\nabla_{\mu}\tilde{T}_{\mu\nu}=0$ is satisfied because of the Bianchi
indentity. It is direct to prove that when $\vartheta_{\mu\nu}=0$, the
solution to (1) degenerates into Schwarzschild metric. The hairy solution with
proper treatment (strong energy condition) of $\vartheta_{\mu\nu}$ was
constructed and the detailed algebra calculations are shown in Ovalle et al.
(2021); Contreras et al. (2021). Here we will not re-show their steps, but
directly refer to the formula of the hairy metric

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})~{}~{}\mathrm{with}~{}~{}f(r)=1-\frac{2M}{r}+\alpha
e^{-r/(M-l_{0}/2)}.$ (2)

In this solution, $M$ is the black hole mass, $\alpha$ is the deformation
parameter due to the introduction of surrounding matters and it describes the
physics related with the strength of hairs, and $l_{0}=\alpha l$ with $l$ a
parameter with length dimension corresponds to primary hair which should
satisfy $l_{0}\leq 2M$ to guarantee the asymptotic flatness. The metric (2)
reproduces the Schwarzchild spacetime in the absence of the matters, i.e.,
$\alpha=0$.

Later, considering that astrophysical black holes in our universe usually have
rotation, the authors of Afrin et al. (2021) induced the rotating counterpart
of the static solution (2), which is stationary and axisymmetric, and in
Boyer-Lindquist coordinates it reads as

$\displaystyle ds^{2}=$ $\displaystyle
g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\theta\theta}d\theta^{2}+g_{\phi\phi}d\phi^{2}+2g_{t\phi}dtd\phi$
$\displaystyle=$
$\displaystyle-\left(\frac{\bigtriangleup-a^{2}\sin^{2}\theta}{\Sigma}\right)dt^{2}+\sin^{2}\theta\left(\Sigma+a^{2}\sin^{2}\theta\left(2-\frac{\bigtriangleup-a^{2}\sin^{2}\theta}{\Sigma}\right)\right)d\phi^{2}+\frac{\Sigma}{\bigtriangleup}dr^{2}+\Sigma
d\theta^{2}$ (3)
$\displaystyle-2a\sin^{2}\theta\left(1-\frac{\bigtriangleup-a^{2}\sin^{2}\theta}{\Sigma}\right)dtd\phi$

with

$\displaystyle\Sigma=r^{2}+a^{2}\cos^{2}\theta,~{}~{}~{}\bigtriangleup=r^{2}+a^{2}-2Mr+\alpha
r^{2}e^{-r/(M-\frac{l_{0}}{2})}.$ (4)

It is noticed that the above metric is also proved to satisfy the equations of
motion in the GD approach Ovalle et al. (2021). The metric describes certain
deformation of the Kerr solution due to the introduction of additional
material sources (such as dark energy or dark matter). In the metric, $a$ is
the spin parameter and $M$ is the black hole mass parameter. Similar to those
in static case, $\alpha$ is the deviation parameter from GR and $l_{0}$ is the
primary hair which is required to be $l_{0}\leq 2M$ for asymptotic flatness.
When $\alpha=0$, the metric reduces to standard Kerr metric in GR, namely no
surrounding matters.

The metric (II) could describe non-extremal hairy Kerr black hole, extremal
hairy Kerr black hole and hairy naked singularity which correspond to two
distinct, two equal and no real positive roots of $g^{rr}=\bigtriangleup=0$ as
well as $\Sigma\neq 0$, meaning they have two horizons, single horizon and no
horizon, respectively. The parameter space $(l_{0}-a)$ of the related
geometries for some discrete $\alpha$ has been show in Afrin et al. (2021).
Here in FIG. 1, we show a 3D plot in the parameters space $(l_{0},a,\alpha)$
for various geometries. In the figure, for the parameters in the white region,
the spacetime with the metric (II) is a non-extremal hairy Kerr black hole
with two horizons, while it is a naked singularity without horizon for the
parameters in the shaded region; and for the parameters on the orange surface,
it is an extremal hairy Kerr black hole with single horizon. Besides, how the
hairy parameters affect the static limit surface and the ergoregion of the
hairy Kerr black hole has also been explored in Afrin et al. (2021). Moreover,
as we aforementioned, some theoretical and observational properties of the
hairy Kerr black hole have been carried out, such as the thermodynamics
Mahapatra and Banerjee (2022), quasinormal modes and (in)stability Cavalcanti
et al. (2022); Yang et al. (2022); Li (2022), strong gravitational lensing and
parameter constraint from Event Horizon Telescope observations Islam and Ghosh
(2021); Afrin et al. (2021).

Figure 1: The parameters space $(l_{0},a,\alpha)$ for non-extremal hairy black
hole (the white region), extremal black hole (the orange surface) and naked
singularity (the shaded region).

In the following sections, we shall partly study the physical signatures in
the strong field regime of those different central objects in the hairy Kerr
spacetime (II). We will analyze the inertial frame dragging effects on a test
gyro and on the accretion physics respectively, which is expected to show the
difference between black hole and naked singularity. To proceed, we should
analyze the timelike geodesic equations in the hairy Kerr spacetime. Since for
the spacetime, we have two Killing vector fields $\partial_{t}$ and
$\partial_{\phi}$, so we can define the conserved energy $\mathscr{E}$ and
axial-component of the angular momentum $L$ of which the expressions are

$\mathscr{E}=-g_{tt}\dot{t}-g_{t\phi}\dot{\phi},~{}~{}~{}~{}~{}~{}L=g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi},$
(5)

where the dot represent the derivative with respect to the affine parameter
$\lambda$. We employ the Hamilton-Jacobi method Carter (1968) and introduce
the Hamilton-Jacobi equation for the particle with rest mass $\mu$

$\mathscr{H}=-\frac{\partial
S}{\partial\lambda}=\frac{1}{2}g_{\mu\nu}\frac{\partial S}{\partial
x^{\mu}}\frac{\partial S}{\partial x^{\nu}}=\mu^{2}$ (6)

where $\mathscr{H}$ and $S$ are the canonical Hamiltonian and Jacobi action,
respectively, and $x^{\mu}$ denote the coordinates $t,r,\theta,\phi$ in the
metric (II). Then after variation, we find that we can separate the Jacobi
action as
$S=\frac{1}{2}\mu^{2}\lambda-\mathscr{E}t+L\phi+S_{r}(r)+S_{\theta}(\theta)$
and define the constant $\mathscr{C}$ via

$\left(\frac{dS_{\theta}}{d\theta}\right)^{2}+\frac{(L-a\mathscr{E}\sin^{2}\theta)^{2}}{\sin^{2}\theta}=-\Delta\left(\frac{dS_{r}}{dr}\right)^{2}+\frac{\left((r^{2}+a^{2})\mathscr{E}^{2}-aL\right)^{2}}{\Delta}=\mathscr{C}$
(7)

to separate the outcome into four first-order differential equations
describing the geodesic motion

$\displaystyle\Sigma\dot{t}$ $\displaystyle=$ $\displaystyle
a(L-a\mathscr{E}\sin^{2}\theta)+\frac{r^{2}+a^{2}}{\Delta}\left((r^{2}+a^{2})\mathscr{E}-aL\right),$
(8) $\displaystyle\Sigma\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{L}{\sin^{2}\vartheta}-a\mathscr{E}+\frac{a}{\Delta}\left((r^{2}+a^{2})\mathscr{E}-aL\right),$
(9) $\displaystyle\Sigma\dot{r}$ $\displaystyle=$
$\displaystyle\Delta\left(\frac{dS_{r}}{dr}\right)=\sqrt{\left(r^{2}+a^{2})\mathscr{E}-aL\right)^{2}-\Delta\left(Q+(L-a\mathscr{E})^{2}+\mu^{2}r^{2}\right)},$
(10) $\displaystyle\Sigma\dot{\theta}$ $\displaystyle=$
$\displaystyle\frac{dS_{\theta}}{d\theta}=\sqrt{Q+\cos^{2}\theta\biggl{(}a^{2}(\mathscr{E}^{2}-\mu^{2})-\frac{L^{2}}{\sin^{2}\theta}\biggr{)}},$
(11)

where $Q\equiv\mathscr{C}-(L-a\mathscr{E})^{2}$ with $\mathscr{C}$ defined in
(7) is the Carter constant. Thus, the geodesics equations in the hairy Kerr
spacetime (II) are separable as in Kerr case, and the modification from the
hair appears in the metric function $\Delta$. Now we can proceed to
investigate the influence of hair on the orbital precession. For convenience,
we will assume that the gyros/stars/particles we will consider are minimally
coupled to the metric, and also there is no direct coupling between them and
the surrounding matters, i.e., the modified gravity effect or print of hair
would only be reflected in the metric functions.

##  III Spin precession of test gyroscope in hairy Kerr spacetime

In this section, we will study the spin precession frequency of a test
gyroscope attached to a stationary observer, dubbed stationary gyroscopes for
brevity in the hairy Kerr spacetime, for whom the $r$ and $\theta$ coordinates
are remaining fixed with respect to infinity. Such a stationary observer has a
four-velocity $u_{stationary}^{\mu}=u_{stationary}^{t}(1,0,0,\Omega)$, where
$t$ is the time coordinate and $\Omega=d\phi/dt$ is the angular velocity of
the observer.

Let us consider stationary gyroscopes moving along Killing trajectory, whose
spin undergoes Fermi-Walker transport along $u=(-K^{2})^{-1/2}K$ with $K$ the
timelike Killing vector field. Since the hairy Kerr spacetime (II) has two
Killing vectors which are the time translation Killing vector $\partial_{t}$
and the azimuthal Killing vector $\partial_{\phi}$, so a more general Killing
vector is $K=\partial_{t}+\Omega\partial_{\phi}$. In this case, the general
spin precession frequency of a test stationary gyroscope is the rescaled
vorticity filed of the observer congruence, and its one-form is given by
Straumann (2004)

$\widetilde{\Omega}_{p}=\frac{1}{2K^{2}}*(\widetilde{K}\wedge d\widetilde{K})$
(12)

where $\widetilde{K}$ is the covector of $K$, the denotations $*$ and $\wedge$
represent the Hodge dual and wedge product, respectively. According to the
pedagogical steps of Chakraborty et al. (2017a), the corresponding vector of
the covector $\widetilde{\Omega}_{p}$ is

$\vec{\Omega}_{p}=\frac{\varepsilon_{ckl}{}}{2\sqrt{-g}\left(1+2\Omega\frac{g_{0c}}{g_{00}}+\Omega^{2}\frac{g_{cc}}{g_{00}}\right)}\left[\left(g_{0c,k}-\frac{g_{0c}}{g_{00}}g_{00,k}\right)+\Omega\left(g_{cc,k}-\frac{g_{cc}}{g_{00}}g_{00,k}\right)+\Omega^{2}\left(\frac{g_{0c}}{g_{00}}g_{cc,k}-\frac{g_{cc}}{g_{00}}g_{0c,k}\right)\right]\partial_{l},$
(13)

where $g$ is the determinant of the metric $g_{\mu\nu}(\mu,\nu=0,1,2,3)$ and
$\varepsilon_{ckl}(c,k,l=1,2,3)$ is the Levi-Civita symbol. Then in the hairy
Kerr spacetime (II) which is stationary and axisymmetric spacetime, the above
expression can be reduced as

$\displaystyle\vec{\Omega}_{p}=$
$\displaystyle\frac{\left(C_{1}\hat{r}+C_{2}\hat{\theta}\right)}{2\sqrt{-g}\left(1+2\Omega\frac{g_{t\phi}}{g_{tt}}+\Omega^{2}\frac{g_{\phi\phi}}{g_{tt}}\right)},$
(14)

with

$\displaystyle C_{1}=$
$\displaystyle-\sqrt{g_{rr}}\left[\left(g_{t\phi,\theta}-\frac{g_{t\phi}}{g_{tt}}g_{tt,\theta}\right)+\Omega\left(g_{\phi\phi,\theta}-\frac{g_{\phi\phi}}{g_{tt}}g_{tt,\theta}\right)+\Omega^{2}\left(\frac{g_{t\phi}}{g_{tt}}g_{\phi\phi,\theta}-\frac{g_{\phi\phi}}{g_{tt}}g_{t\phi,\theta}\right)\right],$
(15) $\displaystyle C_{2}=$
$\displaystyle\sqrt{g_{\theta\theta}}\left[\left(g_{t\phi,r}-\frac{g_{t\phi}}{g_{tt}}g_{tt,r}\right)+\Omega\left(g_{\phi\phi,r}-\frac{g_{\phi\phi}}{g_{tt}}g_{tt,r}\right)+\Omega^{2}\left(\frac{g_{t\phi}}{g_{tt}}g_{\phi\phi,r}-\frac{g_{\phi\phi}}{g_{tt}}g_{t\phi,r}\right)\right].$

|
---|---
(a) | (b)
|
(c) | (d)

Figure 2: The vector field of the LT precession frequency for the hairy Kerr
spacetime. The plots in (a) with $a=0.9M,~{}\alpha=1,~{}l_{0}=1M$ and (b) with
$a=0.99M,~{}\alpha=1,~{}l_{0}=1.5M$ are for hairy Kerr black hole , while in
(c) with $a=2M,~{}\alpha=1,~{}l_{0}=1M$ and (d) with
$a=0.99M,~{}\alpha=1,~{}l_{0}=1M$ are for hairy naked singularity.

It is worth to emphasize that this expression is valid for observers both
inside and outside of the ergosphere for a restricted range of $\Omega$

$\Omega_{-}(r,\theta)<\Omega(r,\theta)<\Omega_{+}(r,\theta),$ (16)

with
$\Omega_{\pm}=(-g_{t\phi}\pm\sqrt{g^{2}_{t\phi}-g_{\phi\phi}g_{tt}})/g_{\phi\phi}$,
which could ensure that the observer at fixed $r$ and $\theta$ is timelike.

Since the gyroscope has an angular velocity $\Omega$, the general precession
frequency (14) stems from two aspects: spacetime rotation (LT precession) and
curvature (geodetic precession). To disclose the properties of precession of
gyroscope in hairy Kerr spacetime, we will study the influence of hairy
parameters on the LT precession, geodetic precession and the general spin
precession of the hairy Kerr black holes and naked singularities,
respectively.

###  III.1 Case with $\Omega=0$: LT precession frequency

When the angular velocity vanishes, saying $\Omega=0$, it means that the
gyroscope is attached to a static observer in a stationary spacetime. Such
static observers do not change their location with respect to infinity only
outside the ergoregion. It corresponds to a four-velocity
$u_{static}^{\mu}=u_{static}^{t}(1,0,0,0)$ and the Killing vector
$K=\partial_{t}$. Thus, $\vec{\Omega}_{p}$ reduces to the LT precession
frequency, $\vec{\Omega}_{LT}$, of the gyroscope attached to a static observer
outside the ergosphere Chakraborty and Majumdar (2014). Subsequently, the LT
precession frequency in the hairy Kerr spacetime (II) takes the form

$\displaystyle\vec{\Omega}_{LT}=$
$\displaystyle\frac{1}{2\sqrt{-g}}\Big{\\{}-\sqrt{g_{rr}}\left(g_{t\phi,\theta}-\frac{g_{t\phi}}{g_{tt}}g_{tt,\theta}\right)\hat{r}+\sqrt{g_{\theta\theta}}\left(g_{t\phi,r}-\frac{g_{t\phi}}{g_{tt}}g_{tt,r}\right)\hat{\theta}\Big{\\}}.$
(17)

Samples of the above LT-precession frequency vector for hairy Kerr black holes
and naked singularities are plotted in Cartesian plane shown in FIG.2. The
upper row exhibits the vector for hairy Kerr BH, giving that the LT precession
frequency is finite outside the ergoshpere and it diverges when the observer
approach the ergosphere. In the bottom row for hairy NS, the LT precession
frequency is regular in the entire spacetime region except the ring
singularity $r=0$ and $\theta=\pi/2$.

The magnitude of LT precession frequency is given by

$\displaystyle\Omega_{LT}=$
$\displaystyle\frac{1}{2\sqrt{-g}}\sqrt{g_{rr}\left(g_{t\phi,\theta}-\frac{g_{t\phi}}{g_{tt}}g_{tt,\theta}\right)^{2}+g_{\theta\theta}\left(g_{t\phi,r}-\frac{g_{t\phi}}{g_{tt}}g_{tt,r}\right)^{2}}$
(18)

which are shown in FIG. 3 and FIG. 4 for samples of parameters. In FIG. 3 for
hairy Kerr BH, the LT precession frequency diverges at the ergosphere. In the
region far away from the static limit, the LT precession frequency hardly
affects by various parameters as expected. When the observer goes near the
static limit, the LT precession frequency increases as the angle and spin
parameter increase, similar to that in the case of Kerr BH Chakraborty et al.
(2017b); in addition, it becomes smaller for larger $\alpha$ or smaller
$l_{0}$.

In FIG. 4 for hairy NS, the LT precession frequency is always finite and
possesses a peak. Similar to that in Kerr case Chakraborty et al. (2017b), as
the angle increases, both the peak and frequency near the ring singularity
increase, while the spin parameter has the opposite influence. Moreover, the
hairy parameter $l_{0}$ also enhances the peak but $\alpha$ suppresses it;
while the hairy parameters have no imprint on the frequency at the ring
singularity. Though the concrete expression of the LT precession frequency are
complex, we can analytically reduce the behaviors at $r\rightarrow 0$ as

$\Omega_{LT}|_{r\rightarrow 0}=\frac{\sec^{2}(\theta)\tan(\theta)}{a^{2}}.$
(19)

Therefore, it is obvious that LT precession frequency at $r\rightarrow 0$ is
independent of the hairy parameters $l_{0}$ and $\alpha$, and it is finite
unless $\theta=\pi/2$, which are consistent with what we see in FIG.4.

|
---|---
(a) | (b)
|
(c) | (d)

Figure 3: Lense-Thirling precession frequency as a function of radial
coordinate for hairy Kerr black hole. In plot (a), we focus the effect of the
angle with fixed $a=0.4M,~{}l_{0}=1M$ and $\alpha=1$; In plot (b), we focus on
the effect of the spining parameter $a$ with fixed $\alpha=1,~{}l_{0}=1.5M$
and $\theta=\pi/4$; In plot (c), we focus on the effect of $\alpha$ with fixed
$a=0.4M,~{}l_{0}=1M$ and $\theta=\pi/4$ ; In plot (d), we focus on the effect
of $l_{0}$ with fixed $a=0.4M,~{}\alpha=1$ and $\theta=\pi/4$.

|
---|---
(a) | (b)
|
(c) | (d)

Figure 4: Lense-Thirling precession frequency as a function of radial
coordinate for hairy naked singularity. In plot (a), we check the effect of
the angle with fixed $a=2M,~{}l_{0}=1M$ and $\alpha=1$; In plot (b), we check
the effect of the spining parameter $a$ with fixed $\alpha=1,~{}l_{0}=1.5M$
and $\theta=\pi/9$; In plot (c), we check the effect of $\alpha$ with fixed
$a=2M,~{}l_{0}=1M$ and $\theta=\pi/4$ ; In plot (d), we check the effect of
$l_{0}$ with fixed $a=2M,~{}\alpha=1$ and $\theta=\pi/4$.

###  III.2 Case with $a=0$: Geodetic precession frequency

The metric (II) with $a=0$ describes hairy Schwarzschild spacetime. In this
case, the precession frequency (14) does not vanish, though the LT-precession
frequency (17) vanishes. The non-vanishing sector of the spin precession is
known as the geodetic precession due to the curvature of the spacetime, and
its formula is

$\displaystyle\vec{\Omega}_{p}\mid_{a=0}=$
$\displaystyle\frac{1}{2\sqrt{-g}\left(1+\Omega^{2}\frac{g_{\phi\phi}}{g_{tt}}\right)}\left[-\Omega\sqrt{g_{rr}}\left(g_{\phi\phi,\theta}-\frac{g_{\phi\phi}}{g_{tt}}g_{tt,\theta}\right)\hat{r}+\Omega\sqrt{g_{\theta\theta}}\left(g_{\phi\phi,r}-\frac{g_{\phi\phi}}{g_{tt}}g_{tt,r}\right)\hat{\theta}\right].$
(20)

To proceed, we can choose the observer in the equatorial plane by setting
$\theta=\pi/2$ without loss of generality due to the spherical symmetry. Then
we have

$\displaystyle\Omega_{p}\mid_{a=0}=\frac{\Omega\sqrt{g_{\theta\theta}}\left(g_{\phi\phi,r}-\frac{g_{\phi\phi}}{g_{tt}}g_{tt,r}\right)}{2\sqrt{-g}\left(1+\Omega^{2}\frac{g_{\phi\phi}}{g_{tt}}\right)}=\frac{(3l_{0}M-6M^{2}+e^{\frac{2r}{l_{0}-2M}}r^{2}\alpha-
l_{0}r(1-e^{\frac{2r}{l_{0}-2M}}\alpha)+2Mr(1+e^{\frac{2r}{l_{0}-2M}}\alpha))\Omega}{(l_{0}-2M)(2M+r(-1-e^{\frac{2r}{l_{0}-2M}}\alpha+r^{2}\Omega^{2}))},$
(21)

meaning that the gyroscope moving in the hairy Schwarzschild spacetime will
also precess. Therefore, in this case, the frequency for the gyro moving along
a circular geodesic could be the Kepler frequency Glendenning and Weber (1994)
$\Omega_{Kep}=\sqrt{(M-\alpha r^{2}e^{-r/(M-l_{0}/2)}/(2M-l_{0}))/r^{3}}$ ,
with which we can induce

$\displaystyle\Omega_{p}\mid_{a=0,\Omega=\Omega_{Kep}}=\Omega_{Kep}=\sqrt{\frac{M}{r^{3}}-\alpha\frac{e^{-r/(M-l_{0}/2)}}{(2M-l_{0})r}}.$
(22)

The above expression gives the precession frequency in the Copernican frame,
computed with respect to the proper time $\tau$, which is related to the
coordinate time via

$d\tau=\sqrt{1-M/r^{3}+\alpha(1+r/(2M-l_{0}))e^{-r/(M-l_{0}/2)}}dt.$ (23)

Therefore, in the coordinate basis, the geodetic precession frequency is

$\displaystyle\Omega_{geodetic}=\left(1-\sqrt{1-\frac{M}{r^{3}}+\alpha(1+\frac{r}{2M-l_{0}})e^{-\frac{r}{M-l_{0}/2}}}\right)\sqrt{\frac{M}{r^{3}}-\alpha\frac{e^{-r/(M-l_{0}/2)}}{(2M-l_{0})r}}.$
(24)

It is obvious that when the deviation parameter, $\alpha$, vanishes, the above
expression reproduces the geodetic precession of the Schwarzschild black hole
found in Sakina and Chiba (1979). How the model parameters $\alpha$ and
$l_{0}$ affect the geodetic precession of the hairy Schwarzschild BH is shown
in FIG.5. We see that the geodetic precession is suppressed by the hair
comparing to that in GR, and its deviation from that in Schwarzschild black
hole is more significant as the $\alpha$ increases. Moreover, the geodetic
precession frequency is enhanced by larger $l_{0}$.

Figure 5: The geodetic precession as a function of radial coordinate for the
hairy Schwarszchild black hole. In the left panel, we tune the deviation
parameter $\alpha$ with fixed $l_{0}=1M$, while in the right panel, we tune
$l_{0}$ with fixed $\alpha=1$.

###  III.3 Spin precession frequency in hairy Kerr black hole and naked
singularity

We move on to study the general spin precession frequency (14) in hairy Kerr
spacetime, and analyze its difference between hairy Kerr BH and NS. As we
aforementioned, the angular velocity has a restricted range for the timelike
stationary observers . So, we introduce a parameter $0<k<1$ so that the
angular velocity $\Omega$ can be rewritten in terms of $\Omega_{\pm}$ as

$\Omega=k\Omega_{+}+(1-k)\Omega_{-}=\frac{(2k-1)\sqrt{g^{2}_{t\phi}-g_{\phi\phi}g_{tt}}-g_{t\phi}}{g_{\phi\phi}}.$
(25)

It is obvious that for $k=1/2$, this expression can be reduced as

$\Omega\mid_{k=1/2}=-\frac{g_{t\phi}}{g_{\phi\phi}},$ (26)

with which the observer is called the zero-angular-momentum observer (ZAMO)
Bardeen et al. (1972). It was addressed in Chakraborty et al. (2017a) that the
precession frequency of the gyroscope attached to ZAMO in the Kerr black hole
spacetime has different behavior from a gyroscope attached to other observers
with other angular velocities, because these gyros has no rotation with
respect to the local geometry and stationary observer.

Substituting (25) into (14), we can obtain the general spin precession
frequency in terms of the parameter $k$, thus the magnitude of spin precession
frequency is written as

$\displaystyle\Omega_{p}=$
$\displaystyle\frac{\sqrt{C_{1}^{2}+C_{2}^{2}}}{2\sqrt{-g}\left(1+2\Omega\frac{g_{t\phi}}{g_{tt}}+\Omega^{2}\frac{g_{\phi\phi}}{g_{tt}}\right)},$
(27)

where $C_{1}$ and $C_{2}$ are defined in (15).

The magnitude of the general spin precession frequencies with different $k$
for the hairy Kerr BH and NS are depicted in FIG. 6, where $k=0.1,0.5,0.9$
from the left to right column. In the first row, we show the results for black
hole with $a=0.9M$, $\alpha=1$ and $l_{0}=1M$. It is obvious that for hairy
Kerr BH, the spin precession frequency of a gyroscope attached to any observer
beyond ZAMO, always diverges whenever it approaches the horizons along any
direction. However, $\Omega_{p}$ remains finite for ZAMO observer everywhere
including at the horizon. In the second row, we show the results for hairy
naked NS with $a=0.9M$, $\alpha=2$ and $l_{0}=1M$. It shows that the spin
precession frequency for NS is finite even as the observer approach $r=0$
along any direction except from the direction $\theta=\pi/2$. More results for
different model parameters are shown in FIG.7 where we have set $k=0.4$ and
$\theta=\pi/4$. Again, it is convenient to verity that

$\displaystyle\Omega_{p}|_{r\rightarrow
0,\theta=\pi/4}=\sqrt{\frac{a^{2}(1-2k)^{2}+(17+4(k-1)k(17+8(k-1)k))M^{2}+12\sqrt{2}(1-2k(2+k(2k-3)))M^{2}}{8a^{4}(k-1)^{2}k^{2}}}$
(28)

which is independent of $\alpha$ and $l_{0}$.

|  |
---|---|---
(a) | (b) | (c)
|  |
(d) | (e) | (f)

Figure 6: The effect of $\theta$ on the spin precession frequency $\Omega_{p}$
for different $k$ in hairy Kerr spacetime. Here we fix $a=0.9M$ and
$l_{0}=1M$, then $\alpha=1$ corresponds to hairy Kerr black hole (a-c) while
$\alpha=2$ corresponds to naked singularity (d-f). From left to right, $k=0.1$
(a, d), $0.5$ (b,e) and $0.9$ (c,f), respectively.

Figure 7: The dependence of the spin precession frequency $\Omega_{p}$ on the
model parameters. From left to right, we tune $l_{0}$, $\alpha$ and $a$,
respectively. In each plot, the dashed curves correspond to hairy Kerr black
hole while the solid curves correspond to naked singularity.

##  IV LT precession and periastron precession of accretion disk physics
around hairy Kerr spacetime

In this section, we will study the LT precession frequency and the periastron
precession frequency of accretion disk physics around different geometries in
the hairy Kerr spacetime, as the related physics could help to test the strong
gravity of the central compact bodies. We expect that this study could further
differentiate the hairy Kerr black hole and naked singularity as the central
objects. To this end, we should study the geodesic motion of a test massive
particle around the hairy Kerr spacetime. To study the accretion disk physic
via the orbits of test massive particle, we should fix the stable circular
orbit around the central bodies and then perturb it. One important stable
circular orbit is the inner-most stable circular orbit (ISCO) which is the
last or smallest stable circular orbit of the particle. Therefore, we will
first recall the procedure of deriving an orbit equation to introduce the
bound orbits, circular orbits and ISCO in the hairy Kerr spacetime. Then we
explore the LT precession frequency and the periastron precession frequency by
perturbing the circular orbit in equatorial plane. For convenience, we shall
focus on the orbits in the equatorial plane of the spacetime.

###  IV.1 Bound orbit and ISCO

For a test particle moving along the timelike geodesic with the four velocity
$p^{\mu}$ in hairy Kerr spacetime (II), we have two conserved quantities for
the massive particle defined in (5). So for the orbit in the equatorial plane
with $\theta=\pi/2$, we can solve out

$\displaystyle p^{t}$ $\displaystyle=$
$\displaystyle\frac{g_{\phi\phi}\mathscr{E}+g_{t\phi}L}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\mid_{\theta=\pi/2}=\frac{r^{3}\mathscr{E}+a^{2}\mathscr{E}\left(2M+r-e^{\frac{2r}{l_{0}-2M}}r\alpha\right)+aL\left(-2M+e^{\frac{2r}{l_{0}-2M}}r\alpha\right)}{r\left(a^{2}+r\left(-2M+r+e^{\frac{2r}{l_{0}-2M}}r\alpha\right)\right)},$
(29) $\displaystyle p^{\phi}$ $\displaystyle=$
$\displaystyle-\frac{g_{t\phi}\mathscr{E}+g_{tt}L}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}\mid_{\theta=\pi/2}=\frac{a\mathscr{E}\left(2M-e^{\frac{2r}{l_{0}-2M}}r\alpha\right)+L\left(-2M+r+e^{\frac{2r}{l_{0}-2M}}r\alpha\right)}{r\left(a^{2}+r\left(-2M+r+e^{\frac{2r}{l_{0}-2M}}r\alpha\right)\right)}.$
(30)

Inserting the above formulas into the normalization condition
$p^{\mu}p_{\mu}=-1$ for timelike geodesic, we can obtain the radial velocity
as

$\displaystyle p^{r}$ $\displaystyle=$
$\displaystyle\pm\sqrt{\frac{-1-g_{tt}(p^{t})^{2}-2g_{t\phi}p^{t}p^{\phi}-g_{\phi\phi}(p^{\phi})^{2}}{g_{rr}}}\mid_{\theta=\pi/2}$
(31) $\displaystyle=$
$\displaystyle\pm\sqrt{\left(\mathscr{E}^{2}-1-e^{\frac{2r}{l_{0}-2M}}\alpha\right)+\frac{2M}{r}-\frac{L^{2}-a^{2}(\mathscr{E}^{2}-1)+e^{\frac{2r}{l_{0}-2M}}\alpha(L-a\mathscr{E})^{2}}{r^{2}}+\frac{2M(L-a\mathscr{E})^{2}}{r^{3}}}$

where $\pm$ corresponds to the radially outgoing and incoming cases,
respectively.

Figure 8:  Samples of effective potential as the functions of $r$ with given
$E$ and $L$. The vertical lines indicate the positions of minimal effective
potential in each case.

Consequently, the effective potential of the equatorial timelike geodesics is

$\displaystyle V_{eff}(r)$ $\displaystyle=$ $\displaystyle
E-\frac{1}{2}(p^{r})^{2}$ (32) $\displaystyle=$
$\displaystyle-\frac{M}{r}+\frac{1}{2}e^{\frac{2r}{l_{0}-2M}}\alpha+\frac{L^{2}-a^{2}(\mathscr{E}^{2}-1)+e^{\frac{2r}{l_{0}-2M}}\alpha(L-a\mathscr{E})^{2}}{2r^{2}}-\frac{M(L-a\mathscr{E})^{2}}{r^{3}}$

where $E=\frac{1}{2}(\mathscr{E}^{2}-1)$ is the total relativistic energy of
the test particle. For the stable bound orbits, the total energy should not be
smaller than the minimal effective potential which is determined by

$\displaystyle\frac{dV_{eff}}{dr}\mid_{r_{m}}=0,~{}~{}~{}~{}\frac{d^{2}V_{eff}}{dr^{2}}\mid_{r_{m}}>0.$
(33)

It is difficult to give the analytical expression of $r_{m}$ and the minimal
effective potential $V_{min}$, so the we show a sample of $V_{eff}$ in FIG. 8
with given $E=-1/1000$ and various $L$. It is obvious that both $r_{m}$ and
$V_{min}$ becomes larger for larger $L$, similar to that found in Kerr
spacetime Bambhaniya et al. (2021b). The hairy parameters affect $V_{min}$ and
its location.

Using the bound orbit condition $V_{min}\leq E<0$ Bardeen et al. (1972), we
will explicitly show the shape of the orbit. The geodesic could give us how
$u=1/r$ changes with respect to $\phi$ in the way

$\displaystyle\frac{du}{d\phi}=-u^{2}\frac{p^{r}}{p^{\phi}}=\frac{1+e^{\frac{2}{(l_{0}-2M)u}}\alpha-2Mu+a^{2}u^{2}}{-L+e^{\frac{2}{(l_{0}-2M)u}}(a\mathscr{E}-L)\alpha+2(L-a\mathscr{E})Mu}X_{u},$
(34)

with

$\displaystyle X_{u}$ $\displaystyle=$
$\displaystyle\sqrt{\mathscr{E}^{2}-1-e^{\frac{2}{(l_{0}-2M)u}}\alpha+u(2M+Y_{u})},$
(35) $\displaystyle Y_{u}$ $\displaystyle=$ $\displaystyle
u\left(a^{2}(\mathscr{E}^{2}-1)-L^{2}-e^{\frac{2}{(l_{0}-2M)u}}(L-a\mathscr{E})^{2}\alpha+2(L-a\mathscr{E})^{2}Mu\right),$
(36)

such that

$\frac{d^{2}u}{d\phi^{2}}=\left(4(a\mathscr{E}-L)A_{u}C_{u}^{-}+4(C_{u}^{-}+a^{2}(l_{0}-2M)u^{3})B_{u}X_{u}^{2}+Z_{u}\right)\frac{A_{u}}{2(l_{0}-2M)B_{u}^{3}}$
(37)

with

$\displaystyle Z_{u}$ $\displaystyle=$ $\displaystyle
B_{u}A_{u}(2e^{\frac{2}{(l_{0}-2M)u}}\alpha+(l_{0}-2M)u^{2}(2M+Y_{u})+u^{2}((l_{0}-2M)Y_{u}+2(a\mathscr{E}-L)^{2}C_{u}^{+})),$
(38) $\displaystyle A_{u}$ $\displaystyle=$ $\displaystyle
1+e^{\frac{2}{(l_{0}-2M)u}}\alpha-2Mu+a^{2}u^{2},$ (39) $\displaystyle B_{u}$
$\displaystyle=$ $\displaystyle
L-e^{\frac{2}{(l_{0}-2M)u}}\alpha(a\mathscr{E}-L)+2(a\mathscr{E}-L)Mu,$ (40)
$\displaystyle C_{u}^{\pm}$ $\displaystyle=$ $\displaystyle\pm
e^{\frac{2}{(l_{0}-2M)u}}\alpha+(l_{0}-2M)Mu^{2}.$ (41)

Then by numerically integrating the above orbital equation, we can figure out
the shape of the bound orbit of a test particle freely falling in the hairy
Kerr BH and NS spacetime. We show the bound orbits in Kerr hairy black hole
($a=0.4M$) and naked singularity ($a=1.2M$) in FIG. 9, where we fix
$\alpha=1$, $l_{0}=1M$, $L=12$ and $E=-1/1000$. In the figure, we also
simultaneously plot the corresponding results for Kerr spacetime denoted by
black dotted curves, namely with $\alpha=0$. The differences between BH and NS
are slight both in hairy Kerr and Kerr spacetimes.

Figure 9: The shape of the bound orbit of a test particle freely falling in
the hairy Kerr black hole (left panel) and naked singularity spacetime (right
panel), respectively. The black dotted curves indicate the Kerr case with
$\alpha=0$.

An interesting type of bound orbit is the circular or spherical orbit which
satisfies

$\displaystyle
V_{eff}\mid_{r_{c}}=0,~{}~{}~{}~{}\frac{dV_{eff}}{dr}\mid_{r_{c}}=0,$ (42)

with $r_{c}$ the radius of the circular orbit. The bound circular orbit could
either be stable or unstable depending on the sign of
$\frac{d^{2}V_{eff}}{dr^{2}}$. $\frac{d^{2}V_{eff}}{dr^{2}}>0$ means that the
orbit is stable while for $\frac{d^{2}V_{eff}}{dr^{2}}<0$ it is unstable.
Thus, one can define the innermost stable circular orbit (ISCO), known as the
smallest stable marginally bound circular orbit which satisfies the above
conditions accompanying with the vanishing second order derivative Bardeen et
al. (1972), meaning that the orbital radius, $r_{ISCO}$, is determined by

$\displaystyle
V_{eff}\mid_{r_{ISCO}}=0,~{}~{}~{}\frac{dV_{eff}}{dr}\mid_{r_{ISCO}}=0,~{}~{}~{}~{}\frac{d^{2}V_{eff}}{dr^{2}}\mid_{r_{ISCO}}=0.$
(43)

Figure 10:  The ISCO of the hairy Kerr spacetime as a function of the spinning
parameter. In the left plot, we fix $\alpha=1$ and focus on the effect of
$l_{0}$, while in the right plot, we fix $l_{0}=1M$ and tune $\alpha$. In the
plots, the solid curves with different shapes are for hairy Kerr black hole,
while the dashed, dotted and dotdashed curves are all for naked singularity.

The exact formula of $r_{ISCO}$ for Kerr spacetime was solved out in Bardeen
et al. (1972), and it was $r_{ISCO}=6M$ as $a=0$ for Schwarzschild spacetime.
Again, due to the existence of exponential term, the expression of the
$r_{ISCO}$ for hairy Kerr spacetime becomes difficult, so we numerically
obtain $r_{ISCO}$ of the hairy Kerr BH and NS. The values of $r_{ISCO}$ as a
function of spin parameter are depicted in FIG.10, from which we see that
$r_{ISCO}$ is smaller for faster spinning hairy Kerr BH while it is larger for
fast spinning hairy Kerr NS, which is similar to those occur in Kerr spacetime
indicated by black curve in each plot. In addition, $r_{ISCO}$ for hairy Kerr
BH increases as $l_{0}$ increases, but decreases as $\alpha$ increases; while
the dependence of $r_{ISCO}$ for the hairy NS on the hairy parameters is
closely determined by the spinning of the central object.

###  IV.2 LT precession and periastron precession

We move on to study the LT precession frequency and the periastron precession
of the circular orbit by perturbing the geodesic equation of the massive
particle, which could disclose important features of accretion disk around the
central bodies, such that testify the strong gravity of the hairy Kerr
spacetime.

To proceed, we have to model the three fundamental frequencies which are very
important for accretion disk physics around the hairy Kerr spacetime. For a
test massive particle moving along a circle in the equatorial plane of the
metric (II), the orbital angular frequency, $\Omega_{\phi}$, is

$\Omega_{\phi}=\frac{d\phi}{dt}=\frac{-g_{t\phi,r}+\sqrt{(g_{t\phi,r})^{2}-g_{tt,r}g_{\phi\phi,r}}}{g_{\phi\phi,r}}.$
(44)

If the particle is perturbed, then it oscillates with some characteristic
epicyclic frequencies $\Omega_{r}$ and $\Omega_{\theta}$ in the radial or
vertical direction, respectively, which can be obtained by perturbing the
geodesic equation as Ryan (1995); Doneva et al. (2014)

$\displaystyle\Omega_{r}=\left\\{\frac{1}{2g_{rr}}\left[X^{2}\partial_{r}^{2}\left(\frac{g_{\phi\phi}}{g_{tt}g_{\phi\phi}-g_{t\phi}^{2}}\right)-2XY\partial_{r}^{2}\left(\frac{g_{t\phi}}{g_{tt}g_{\phi\phi}-g_{t\phi}^{2}}\right)+Y^{2}\partial_{r}^{2}\left(\frac{g_{tt}}{g_{tt}g_{\phi\phi}-g_{t\phi}^{2}}\right)\right]\right\\}^{1/2},$
(45)
$\displaystyle\Omega_{\theta}=\left\\{\frac{1}{2g_{\theta\theta}}\left[X^{2}\partial_{\theta}^{2}\left(\frac{g_{\phi\phi}}{g_{tt}g_{\phi\phi}-g_{t\phi}^{2}}\right)-2XY\partial_{\theta}^{2}\left(\frac{g_{t\phi}}{g_{tt}g_{\phi\phi}-g_{t\phi}^{2}}\right)+Y^{2}\partial_{\theta}^{2}\left(\frac{g_{tt}}{g_{tt}g_{\phi\phi}-g_{t\phi}^{2}}\right)\right]\right\\}^{1/2}$
(46)

with

$\displaystyle
X=g_{tt}+g_{t\phi}\Omega_{\phi},~{}~{}~{}~{}~{}Y=g_{t\phi}+g_{\phi\phi}\Omega_{\phi}.$
(47)

Subsequently, we can extract the nodal precession frequency $\Omega_{nod}$ and
periastron precession frequency $\Omega_{per}$ as Motta et al. (2014)

$\displaystyle\Omega_{nod}=\Omega_{\phi}-\Omega_{\theta},$ (48)
$\displaystyle\Omega_{per}=\Omega_{\phi}-\Omega_{r}$ (49)

which measure the precession of orbital plane and orbit of the accretion disk,
respectively. The nodal precession frequency is also known as LT precession
frequency.

Focusing on the equatorial plane with $\theta=\pi/2$, we plot $\Omega_{nod}$
and $\Omega_{pre}$ for hairy Kerr BH and NS with samples of parameters in
FIG.11, in which the red curves are for NS while the blue curves are for BH,
and the vertical lines correspond to the corresponding location of ISCO. It is
obvious that the LT precession frequency increases monotonously as the orbit
approaches the ISCO of the hairy Kerr BH. While in the hairy NS spacetime, as
the orbit moves towards the ISCO, $\Omega_{nod}$ first increases to certain
peak and then decreases. In addition, $\Omega_{nod}$ at the ISCO of NS is
always smaller than that at ISCO of BH, and it even can be negative indicating
a reversion of the precession direction. The periastron precession frequency
$\Omega_{per}$ in hairy Kerr BH and NS spacetimes has similar behavior that is
increasing with the decrease of $r$. The value of $\Omega_{per}$ for hairy
Kerr BH is always larger than that for NS, and their difference is more
significant as the orbit becomes smaller. The effect of different model
parameters on the LT precession frequency are shown in FIG.12 which indicates
that the parameters indeed have significant imprint on the LT frequencies.

Figure 11: The behavior of $\Omega_{nod}$ (upper row) and $\Omega_{pre}$
(bottom row) as a function of $r$ in hairy Kerr black hole and naked
singularity, respectively. In all plots, the blue curve is for naked
singularity case while the red curve is for black hole case, and the vertical
lines indicate the location of the corresponding ISCO. From left to right, the
hairy spacetime is either black hole or naked singularity, depending on
$l_{0}$, $\alpha$ and $a$, respectively.

Figure 12: The $\Omega_{nod}$ as a function of $r$ for various model
parameters. From the plots in the upper row, we can see the effect of $l_{0}$
(left), $\alpha$ (middle) and $a$ (right) on $\Omega_{nod}$ in the hairy Kerr
black hole. In the bottom row, we see their effects on $\Omega_{nod}$ in hairy
naked singularity.

##  V Conclusion and discussion

Gravitational wave and shadow are two important observations to test strong
gravity physics, such that they allow us to probe the structure of black
holes. Thus, they could also be used to reveal the scalar fields provided that
they leave an imprint on BH. The well known no-hair theorem in Einstein’s GR
predicts that the rotating black holes are described by the Kerr metric.
However, beyond GR with additional surrounding sources, the hairy rotating
black holes should be described by a deformed Kerr metric, including extra
hairy parameter. A hairy Kerr black hole was recently constructed using the
gravitational decoupling approach, describing deformations of Kerr black hole
due to including additional sources Contreras et al. (2021), Observational
investigations related with gravitational wave and shadow of the hairy Kerr
black hole have been studied in Cavalcanti et al. (2022); Yang et al. (2022);
Li (2022); Islam and Ghosh (2021); Afrin et al. (2021), which indicates that
the hairy Kerr black hole could not be ruled out by the current observations.
In this paper, we focused on the Lense-Thirring effect, another important
observable effect, to differentiate the hairy Kerr black hole from naked
singularity.

Firstly, we analyzed the spin precession of a test gyro attached to a
stationary observer in the hairy Kerr spacetime. When the observer is static
with respect to a fixed star, i.e, the the angular velocity vanishes, we
calculated the LT precession frequency. It was found that the LT precession
frequency diverges as the observer approaches the ergosphere of the hairy Kerr
BH along any direction, while it keeps finite in the whole region of the hairy
NS, except at the ring singularity. Then, we parameterized the range of
angular velocity of a stationary observer by $k$, and systematically studied
the general spin precession frequency. The general spin precession frequency
diverges as the observer approaches the horizon of the hairy Kerr BH, but it
is finite when $k=0.5$ defining ZAMO observer because in this case the test
gyro has no rotation with respect to the local geometry. For the hairy NS, it
is always finite unless the observer reaches the ring along the direction
$\theta=\pi/2$. We also obtained the geodetic precession for observers in a
hairy static black hole. The general spin frequency, LT frequency and geodetic
frequency all decrease as the parameter $\alpha~{}(l_{0})$ increases
(decreases) in hairy Kerr black hole. And $\alpha$ and $l_{0}$ have similar
effect on the LT frequency in hairy NS as that in BH case, but their effects
on general spin frequency in NS depend on the spinning.

Then, we investigated the quasiperiodic oscillations (QPOs) phenomena as the
accretion disk approaches the hairy Kerr BH and NS, which also show
difference. To this end, we first analyzed the orbital precession of bound
orbits, and ISCO of a test massive particle orbiting in the equatorial plane
of the hairy Kerr BH and NS spacetime, respectively. Then we perturbed the
stable circular orbit and computed the three fundamental frequencies related
with QPOs phenomena. Accordingly, our results show that as the orbit moves
towards the ISCO, the LT frequency increases monotonously in hairy Kerr BH,
while it first increases to certain peak and then decreases in hair NS; the
periastron frequency increases in both hairy Kerr BH and NS as the orbit
approaches the corresponding ISCO. Moreover, the hairy parameters indeed have
effects on the LT and periastron frequencies.

In conclusion, we do theoretical evaluation on various precession frequencies
caused by the frame-dragging effect of the central sources, which
differentiate the hairy Kerr BH from NS spacetime. We expect that our
theoretical studies could shed light on astrophysical observations on
distinguishing hairy theory from GR, distinguishing BH from NS and even
further constraining the hairy parameters.

###### Acknowledgements.

This work is partly supported by Natural Science Foundation of Jiangsu
Province under Grant No.BK20211601, Fok Ying Tung Education Foundation under
Grant No. 171006.

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