$\textstyle{{\mathcal{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{const}}$$\scriptstyle{\mathrm{const}}$$\scriptstyle{\mathrm{const}}$$\textstyle{\textup{Pro}({\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{lim}}$$\scriptstyle{\gamma^{\omega}}$$\textstyle{\textup{Cond}^{\omega}\ignorespaces\ignorespaces\ignorespaces\ignorespaces({\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{can}}$$\scriptstyle{R}$$\textstyle{\textup{Cond}_{\kappa}({\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{can}}$$\textstyle{\mathcal{C}}$

where $R$ is the right Kan extension from Remark A.7. Explicitly, the functor
$\gamma^{\omega}$ sends the pro-object
$\operatorname*{``lim^{\prime\prime}}_{i\in I}X_{i}$ to the limit $\lim_{i\in
I}\underline{X_{i}}$.

###### Proof.

The commutativity of the upper right triangle is clear and the commutativity
of the lower right triangle follows since $R$ preserves limits. Since
${\mathcal{C}}$ admits small filtered limits, the category
$\textup{Cond}^{\omega}({\mathcal{C}})$ admits small filtered limits as well.
Hence restriction along the constant functor
${\mathcal{C}}\overset{}{\rightarrow}\textup{Pro}({\mathcal{C}})$ induces an
equivalence

$\operatorname{Fun}^{\prime}(\textup{Pro}({\mathcal{C}}),\textup{Cond}^{\omega}({\mathcal{C}}))\overset{\simeq}{\longrightarrow}\operatorname{Fun}({\mathcal{C}},\textup{Cond}^{\omega}({\mathcal{C}}))$

where the source denotes the full subcategory of all functors spanned by those
which preserve small filtered limits [Lur18, A.8.1.6]. Thus the constant
functor
${\mathcal{C}}\overset{}{\rightarrow}\textup{Cond}^{\omega}({\mathcal{C}})$
yields the functor $\gamma=\gamma^{\omega}$ and the upper left commutative
triangle.

The commutativity of the lower left triangle follows if $\gamma$ preserves all
limits, as both compositions $\lim\circ\mathrm{const}$ are equivalent to the
identity functor.

We are left to show that $\gamma$ preserves finite pullbacks, as then $\gamma$
preserves all limits by [Lur09, Prop. 4.4.2.7]. For this, we remark that
finite limits in $\textup{Pro}({\mathcal{C}})$ can be computed pointwise in
the sense of [KST19a, §2.1]. Thus, any diagram

${X}$${Y}$${Z}$

in $\textup{Pro}({\mathcal{C}})$ can be lifted to an element
$F\in\textup{Pro}(\operatorname{Fun}(K,{\mathcal{C}}))$, where $K$ is the set
representing the diagram above, such that its image under

$\textup{Pro}(\lim)\colon\textup{Pro}(\operatorname{Fun}(K,{\mathcal{C}}))\overset{}{\longrightarrow}\textup{Pro}({\mathcal{C}})$

is equivalent to $X\times_{Z}Y$. We also have a functor

$\textup{Cond}(\lim)\colon\textup{Cond}^{\omega}(\operatorname{Fun}(K,{\mathcal{C}}))\overset{}{\longrightarrow}\textup{Cond}^{\omega}({\mathcal{C}})$

that is induced by the following construction. We have a limit preserving
functor

$\gamma^{K}\colon\textup{Cond}^{\omega}(\operatorname{Fun}(K,{\mathcal{C}}))\overset{}{\hookrightarrow}\operatorname{Fun}(\textup{ProFin}^{\omega},\operatorname{Fun}(K,{\mathcal{C}}))\simeq\operatorname{Fun}(K,\operatorname{Fun}(\textup{ProFin}^{\omega},{\mathcal{C}})).$

Then $\textup{Cond}(\lim)$ is the composition of the above functor with the
limit functor and sheafification. Now we obtain an essentially commutative
square

${\textup{Pro}(\operatorname{Fun}(K,{\mathcal{C}}))}$${\textup{Cond}^{\omega}(\operatorname{Fun}(K,{\mathcal{C}}))}$${\textup{Pro}({\mathcal{C}})}$${\textup{Cond}^{\omega}({\mathcal{C}}).}$$\scriptstyle{\textup{Pro}(\lim)}$$\scriptstyle{\gamma^{K}}$$\scriptstyle{\textup{Cond}(\lim)}$$\scriptstyle{\gamma}$
($\clubsuit$)

Note that limits and colimits in functor categories can be computed pointwise,
thus we have an equivalence
$\textup{Cond}^{\omega}(\operatorname{Fun}(K,{\mathcal{C}}))\simeq\operatorname{Fun}(K,\textup{Cond}({\mathcal{C}}))$.
Under this equivalence $\textup{Cond}(\lim)$ corresponds to the limit functor
$\operatorname{Fun}(K,\textup{Cond}^{\omega}({\mathcal{C}}))\rightarrow\textup{Cond}^{\omega}({\mathcal{C}})$.
Therefore, commutativity of $(\clubsuit)$ shows that $\gamma$ preserves
$K$-indexed limits, i.e. pullbacks. ∎

######  Remark A.9.

An informal argument for the comparison functor $\gamma$ to commute with
cofiltered limits goes as follows: Let
$A\overset{}{\rightarrow}\textup{Pro}(\operatorname{Sp}),\alpha\mapsto
X_{\alpha}$ be a cofiltered diagram of pro-spectra
$X_{\alpha}=\operatorname*{``lim^{\prime\prime}}_{i\in
I_{\alpha}}X_{\alpha,i}$. We may assume that $A$ is cofinite and that the
diagram $A\overset{}{\rightarrow}\textup{Pro}(\operatorname{Sp})$ is given in
level-representation with cofiltered index set $I$ ($=I_{\alpha}$ for all
$\alpha\in A$) so that $A\times I$ is cofiltered again and
$\operatorname*{``lim^{\prime\prime}}_{A\times
I}X_{\alpha,i}\simeq\lim_{A}X_{\alpha}$ is the limit [Isa02, §4]. Then

$\displaystyle\gamma(\lim_{A}X_{\alpha})\simeq\gamma(\operatorname*{``lim^{\prime\prime}}_{A\times
I}X_{\alpha,i})=\lim_{A\times
I}\underline{X_{\alpha,i}}\simeq\lim_{A}\,\lim_{I}\underline{X_{\alpha,i}}=\lim_{A}\gamma(X_{\alpha}).$

######  Corollary A.10.

The comparison functor $\gamma$ preserves cofibre sequences and pushouts.

###### Proof.

This follows since both $\textup{Pro}(\operatorname{Sp})$ and
$\textup{Cond}(\operatorname{Sp})$ are stable categories. ∎

Next we want to show that the comparison functor is conservative. For this
purpose, we start with some preparations.

######  Lemma A.11.

Let $\bigl{(}(X_{i})_{i},(p_{i}\colon
X_{i}\overset{}{\rightarrow}X_{i-1})_{i}\bigr{)}$ be a tower of spectra and
$n\in\mathbb{Z}$. Then there is an exact sequence

$0\overset{}{\longrightarrow}{\lim_{i}}^{1}\pi_{n+1}(X_{i})\overset{}{\longrightarrow}\pi_{n}(\lim_{i}X_{i})\overset{}{\longrightarrow}\lim_{i}\pi_{n}(X_{i})\overset{}{\longrightarrow}0$

of condensed abelian groups, the so-called Milnor sequence.

###### Proof.

This is a standard proof.151515Cf.
https://ncatlab.org/nlab/show/lim^1+and+Milnor+sequences. The sequential limit
$\lim_{i}X_{i}$ fits into a fibre sequence

$\lim_{i}X_{i}\overset{}{\longrightarrow}\prod_{i}X_{i}\overset{\operatorname{id}-p_{*}}{\longrightarrow}\prod_{i}X_{i}$

where $p_{*}$ is the map induced by the maps $(p_{i}\colon
X_{i}\overset{}{\rightarrow}X_{i-1})_{i}$. Consider the associated long exact
sequence of homotopy groups

$\ldots\overset{}{\longrightarrow}\prod_{i}\pi_{n+1}(X_{i})\overset{\operatorname{id}-p_{*}}{\longrightarrow}\prod_{i}\pi_{n+1}(X_{i})\overset{\alpha}{\longrightarrow}\pi_{n}(\lim_{i}X_{i})\overset{\beta}{\longrightarrow}\prod_{i}\pi_{n}(X_{i})\overset{\operatorname{id}-p_{*}}{\longrightarrow}\prod_{i}\pi_{n}(X_{i})\overset{}{\longrightarrow}\ldots$

We have that

$\operatorname{im}(\beta)=\ker\bigl{(}\prod_{i}\pi_{n}(X_{i})\overset{\operatorname{id}-p_{*}}{\longrightarrow}\prod_{i}\pi_{n}(X_{i})\bigr{)}=\lim_{i}\pi_{n}(X_{i})$

and

$\ker(\alpha)=\operatorname{coker}\bigl{(}\prod_{i}\pi_{n+1}(X_{i})\overset{\operatorname{id}-p_{*}}{\longrightarrow}\prod_{i}\pi_{n+1}(X_{i})\bigr{)}={\lim_{i}}^{1}\pi_{n+1}(X_{i})$

which shows the claim. ∎

The following result and its proof was communicated to us by Peter Scholze and
we do not claim originality.

######  Theorem A.12 (Clausen-Scholze).

The comparison functor

$\gamma_{\kappa}:=R\circ\gamma^{\omega}\,\colon\,\textup{Pro}^{\omega}(\operatorname{Sp}^{+})\overset{}{\rightarrow}\textup{Cond}_{\kappa}(\operatorname{Sp})$

is conservative. In particular, the comparison functor $\gamma^{\omega}$ is
conservative, too.

###### Proof.

Since the categories in question are stable (Lemma 5.2 and Lemma A.4), it
suffices to check that the functor $\gamma$ reflects zero objects. Let
$\ldots\overset{}{\rightarrow}X_{2}\overset{}{\rightarrow}X_{1}\overset{}{\rightarrow}X_{0}$
be a system of spectra such that $\lim_{i}\underline{X_{i}}=0$ as condensed
spectra (where each $X_{i}$ is considered as a discrete/constant condensed
spectrum, and the limit is taken in condensed spectra). We have to see that
the pro-spectrum $\operatorname*{``lim^{\prime\prime}}_{i}X_{i}$ is zero; this
we can show on homotopy pro-groups $\lim_{i}\pi_{j}(X_{i})$ for all
$j\in\mathbb{Z}$ [KST19a, 2.8] and we may assume $j=0$ by shifting. The Milnor
sequences (Lemma A.11) for $n\in\\{-1,0\\}$

$0\overset{}{\longrightarrow}{\lim_{i}}^{1}\pi_{n+1}(X_{i})\overset{}{\longrightarrow}\pi_{n}(\lim_{i}X_{i})\overset{}{\longrightarrow}\lim_{i}\pi_{n}(X_{i})\overset{}{\longrightarrow}0$

in $\textup{Cond}(\textup{Ab})$ yield that
$\lim_{i}\pi_{0}X_{i}=\lim_{i}^{1}\pi_{0}X_{i}=0$ in this category. Thus, the
claim follows from the Lemma A.15 below. ∎

######  Remark A.13.

The comparison functor
$\gamma\colon\textup{Pro}^{\omega}(\textup{Ab})\overset{}{\rightarrow}\textup{Cond}_{\kappa}(\textup{Ab})$
is not conservative. Consider the tower $(p^{i}\mathbb{Z})_{i}$ whose
structure maps are the inclusions (for a prime number $p$). Then
$\operatorname*{``lim^{\prime\prime}}_{i}p^{i}\mathbb{Z}$ is not zero in
$\textup{Pro}(\textup{Ab})$ as for every $j\geq i$ the map
$p^{j}\mathbb{Z}\overset{}{\rightarrow}p^{i}\mathbb{Z}$ is not zero. But
$\gamma(\operatorname*{``lim^{\prime\prime}}_{i}p^{i}\mathbb{Z})=\lim_{i}\underline{p^{i}\mathbb{Z}}=\underline{\lim_{i}p^{i}\mathbb{Z}}=0$
since we can check this after passing to TopAb (Remark A.14) and since
$\lim_{i}p^{i}\mathbb{Z}=0$ in TopAb as its underlying abelian group is zero.

######  Remark A.14.

The faithful functor
$\textup{Top}\textup{Ab}\overset{}{\rightarrow}\textup{Cond}_{\kappa}(\textup{Ab}),M\mapsto\underline{M}:=C(-,M)$,
commutes with limits and hence admits a left adjoint
$\textup{L}\colon\textup{Cond}_{\kappa}(\textup{Ab})\overset{}{\rightarrow}\textup{Top}\textup{Ab}$.
Thus for a tower $(M_{i})_{i}$ of topological abelian groups with limit
$M:=\lim_{i}M_{i}$ in TopAb, the canonical morphism
$\underline{M}\overset{}{\rightarrow}\lim_{i}\underline{M_{i}}$ of associated
condensed abelian groups is an isomorphism. In
$\textup{Cond}_{\kappa}(\textup{Ab})$ we have an exact sequence

$0\overset{}{\longrightarrow}\lim_{i}\underline{M_{i}}\overset{}{\longrightarrow}\prod_{i}\underline{M_{i}}\overset{\delta}{\longrightarrow}\prod_{i}\underline{M_{i}}\overset{}{\longrightarrow}{\lim_{i}}^{1}\underline{M_{i}}\overset{}{\longrightarrow}0$

where $\delta$ is the difference of the identity and the structure map of the
tower. As the canonical morphism
$\underline{\prod_{i}M_{i}}\overset{}{\longrightarrow}\prod_{i}\underline{M_{i}}$
is an isomorphism and since for every $N\in\textup{Top}\textup{Ab}$ the counit
$\textup{L}(\underline{N})\overset{}{\rightarrow}N$ is an isomorphism, the
sequence $\textup{L}(\clubsuit)$ identifies with the exact sequence

$0\overset{}{\longrightarrow}\lim_{i}M_{i}\overset{}{\longrightarrow}\prod_{i}M_{i}\overset{\delta}{\longrightarrow}\prod_{i}M_{i}\overset{}{\longrightarrow}{\lim_{i}}^{1}M_{i}\overset{}{\longrightarrow}0$

of topological abelian groups, hence
$\textup{L}(\lim_{i}^{1}\underline{M_{i}})=\lim_{i}^{1}M_{i}$.

######  Lemma A.15.

For a tower $(M_{i})_{i}$ of abelian groups the following are equivalent:

  1. 1.

The tower is Mittag-Leffler and $\lim_{i}M_{i}=0$.

  2. 2.

The tower is pro-zero, i.e. $\operatorname*{``lim^{\prime\prime}}_{i}M_{i}=0$
in $\textup{Pro}(\textup{Ab})$.

  3. 3.

It holds $\lim_{i}\underline{M_{i}}=0=\lim^{1}_{i}\underline{M_{i}}$ in
$\textup{Cond}_{\kappa}(\textup{Ab})$.

###### Proof.

First we show (1)$\Rightarrow$(2). The tower $(M_{i})_{i}$ being Mittag-
Leffler means that for every $i\geq 0$ the tower
$(\operatorname{im}(M_{j}\overset{}{\rightarrow}M_{i}))_{j\geq i}$ stabilises.
Let $B_{i}\subset M_{i}$ be this stable image, i.e. $B_{i}=\bigcap_{j\geq
i}\operatorname{im}(M_{j}\overset{}{\rightarrow}M_{i})$. Then $(B_{i})_{i}$ is
a subtower of $(M_{i})_{i}$, hence $\lim_{i}B_{i}\subset\lim_{i}M_{i}=0$. Thus
for every $i$ there exists a $j\geq i$ such that the map
$M_{j}\overset{}{\rightarrow}M_{i}$ is zero, hence the pro-system
$\operatorname*{``lim^{\prime\prime}}_{i}M_{i}$ is zero. The direction
(2)$\Rightarrow$(1) works analogously.

Now we assume (3). For any profinite set $S$ we have that
$C(S,M_{i})=C(S,\mathbb{Z})\otimes_{\mathbb{Z}}M_{i}$ and $C(S,\mathbb{Z})$ is
some large free abelian group [Sch19, Thm. 5.4], and so has
$\bigoplus_{\mathbb{N}}$ $\mathbb{Z}$ as a direct factor if $S$ is large
enough. Thus there exists an extremally disconnected set $S$ such that

${\lim_{i}}^{1}\bigoplus_{\mathbb{N}}M_{i}\subset{\lim_{i}}^{1}C(S,M_{i})={\lim_{i}}^{1}\underline{M_{i}}(S)\overset{(\clubsuit)}{=}\bigl{(}{\lim_{i}}^{1}\underline{M_{i}}\bigr{)}(S)=0$

where $(\clubsuit)$ holds as $\operatorname{id}_{S}$ is cofinal among all
covers of $S$ (Remark A.6). The condition that
$\lim_{i}^{1}\bigoplus_{\mathbb{N}}M_{i}=0$ in Ab is equivalent to the tower
$(M_{i})_{i}$ being Mittag-Leffler [Emm96, Cor. 6], hence we have (1).

Assuming (1) it remains to show that $\lim_{i}^{1}\underline{M_{i}}$ in
$\textup{Cond}_{\kappa}(\textup{Ab})$. For any extremally disconnected set $S$
we have

$({\lim_{i}}^{1}\underline{M_{i}})(S)={\lim_{i}}^{1}\underline{M_{i}}(S)={\lim_{i}}^{1}C(S,M_{i})\cong{\lim_{i}}^{1}\bigoplus_{J}M_{i}$

for some set $J$ and the last term vanishes as the tower is Mittag-Leffler. ∎

######  Example A.16.

If a tower $(M_{i})_{i}$ of abelian groups satisfies
$\lim_{i}M_{i}=0=\lim_{i}^{1}M_{i}$, then it needs not to be pro-zero. For
instance, consider the tower $(p^{i}\mathbb{Z}_{p})_{i}$ for some prime number
$p$.

###  A.3 t-structures

In this subsection, we examine a t-structure on condensed spectra which is
induced by the canonical t-structure on spectra.

######  Lemma A.17.

There exists a t-structure on $\textup{Cond}_{\kappa}(\operatorname{Sp})$ with
connective part $\textup{Cond}(\operatorname{Sp})_{\geq
0}=\textup{Cond}(\operatorname{Sp}_{\geq 0})$ and coconnective part
$\textup{Cond}(\operatorname{Sp})_{\leq
0}=\textup{Cond}(\operatorname{Sp}_{\leq 0})$

###### Proof.

Since $\operatorname{Sp}_{\geq 0}$ is presentable, this inherits to the
presheaf category
$\operatorname{Fun}(\textup{EDS}_{<\kappa}^{\operatorname{op}},\operatorname{Sp}_{\geq
0})$ [Lur09, 5.5.3.6] and its Bousfield localisation
$\textup{Cond}_{\kappa}(\operatorname{Sp}_{\geq 0})$ [Lur09, 5.5.4.15]. The
category $\operatorname{Sp}_{\geq 0}$ is closed under small colimits in
$\operatorname{Sp}$ since the inclusion functor is left-adjoint. Since
colimits in presheaf categories can be computed pointwise,
$\operatorname{Fun}(\textup{EDS}_{<\kappa}^{\operatorname{op}},\operatorname{Sp}_{\geq
0})$ is closed under small colimits in
$\operatorname{Fun}(\textup{EDS}_{<\kappa}^{\operatorname{op}},\operatorname{Sp}_{\geq
0})$. Since a colimit of sheaves is given by the sheafification of the
colimits of the underlying presheaves,
$\textup{Cond}_{\kappa}(\operatorname{Sp}_{\geq 0})$ is closed under small
colimits in $\textup{Cond}_{\kappa}(\operatorname{Sp})$. From the long exact
sequences of homotopy groups one sees that
$\textup{Cond}_{\kappa}(\operatorname{Sp}_{\geq 0})$ is closed under extension
in $\textup{Cond}_{\kappa}(\operatorname{Sp})$. These three closure properties
imply the existence of a t-structure on
$\textup{Cond}_{\kappa}(\operatorname{Sp})$ whose connective part is
$\textup{Cond}_{\kappa}(\operatorname{Sp}_{\geq 0})$ [Lur17, 1.4.4.11].

The category $\textup{Cond}_{\kappa}(\operatorname{Sp})_{\geq 0}$ should be
the same as the full subcategory of 0-connective objects in the sense of
[Lur18, 1.3.2.5]. According to [Lur18, 1.3.2.7], the coconnective part of the
t-structure is given by the full subcategory of 0-truncated objects; these are
precisely those sheaves that take value in 0-truncated spectra [Lur18,
1.3.2.6] (since the inclusion functor $\operatorname{Sp}_{\leq
0}\overset{}{\hookrightarrow}\operatorname{Sp}$ preserves limits. ∎

######  Lemma A.18.

There exists a t-structure on $\textup{Pro}(\operatorname{Sp})$ with
$\textup{Pro}(\operatorname{Sp})_{\geq 0}$ given by the essential image of the
functor $\textup{Pro}(\operatorname{Sp}_{\geq
0})\overset{}{\rightarrow}\textup{Pro}(\operatorname{Sp})$ and with
$\textup{Pro}(\operatorname{Sp})_{\leq 0}$ given by the full subcategory
$\textup{Pro}(\operatorname{Sp}_{\leq
0})\overset{}{\hookrightarrow}\textup{Pro}(\operatorname{Sp})$.

For
$X=\operatorname*{``lim^{\prime\prime}}X_{i}\in\textup{Pro}(\operatorname{Sp})$
and $n\in\mathbb{Z}$ we can compute the $n$-truncation pointwise, i.e.
$\pi_{n}X\simeq\operatorname*{``lim^{\prime\prime}}\pi_{n}X_{i}$.

###### Proof.

First let us remark that $\textup{Pro}(\operatorname{Sp})$ is a stable
category [KST19a, Lem. 2.5]. The inclusion $\iota\colon\operatorname{Sp}_{\leq
0}\subseteq\operatorname{Sp}$ induces an adjunction

${\iota^{*}\colon\textup{Pro}(\operatorname{Sp})}$${\textup{Pro}(\operatorname{Sp}_{\leq
0})\colon\textup{Pro}(\iota)}$

[Lur18, Ex. A.8.1.8]. The functor $\textup{Pro}(\iota)$ is fully faithful
[Lur18, Prop. A.8.1.9].

We can apply the same argument to the truncation $\tau_{\geq
0}\colon\operatorname{Sp}\rightarrow\operatorname{Sp}_{\geq 0}$ and get an
adjunction

${\tau_{\geq 0}^{*}\colon\textup{Pro}(\operatorname{Sp}_{\geq
0})}$${\textup{Pro}(\operatorname{Sp})\colon\textup{Pro}(\tau_{\geq 0}).}$

Let us now define $\textup{Pro}(\operatorname{Sp})_{\geq 0}$ as the essential
image of $\tau_{\geq 0}^{*}$. We claim that the tuple

$(\textup{Pro}(\operatorname{Sp})_{\geq
0},\textup{Pro}(\operatorname{Sp}_{\leq 0}))$

defines a $t$-structure on $\textup{Pro}(\operatorname{Sp})$.

Indeed, finite limits and colimit can be computed levelwise in
$\textup{Pro}(\operatorname{Sp})$ [KST19a, lem. 2.1]. So, checking the axioms
of a $t$-structure reduces to the axioms for $(\operatorname{Sp}_{\geq
0},\operatorname{Sp}_{\leq 0})$, which holds by design.

For the homotopy groups note that $\tau_{\leq n}$ can be computed pointwise by
construction. For $\tau_{\geq n}$ we note that for any
$X\in\textup{Pro}(\operatorname{Sp})$, we have a fibre sequence

$\tau_{\geq n}X\overset{}{\rightarrow}X\overset{}{\rightarrow}\tau_{\leq
n-1}X$

by the proof of [Lur17, Prop. 1.2.1.5]. As finite limits in
$\textup{Pro}(\operatorname{Sp})$ can be computed pointwise, we see that also
$\tau_{\geq n}$ can be computed pointwise. ∎

######  Proposition A.19.

The canonical functor
$\gamma\colon\textup{Pro}(\operatorname{Sp})\overset{}{\rightarrow}\textup{Cond}(\operatorname{Sp})$
is left t-exact.

###### Proof.

Since the inclusion functor $\textup{Cond}(\operatorname{Sp}_{\leq
0})\overset{}{\hookrightarrow}\textup{Cond}(\operatorname{Sp})$ preserves
limits, the composition$\textup{Pro}(\operatorname{Sp}_{\leq
0})\overset{}{\rightarrow}\textup{Pro}(\operatorname{Sp})\overset{\gamma}{\rightarrow}\textup{Cond}(\operatorname{Sp})$
is equivalent to the composition $\textup{Pro}(\operatorname{Sp}_{\leq
0})\overset{}{\rightarrow}\textup{Cond}(\operatorname{Sp}_{\leq
0})\overset{}{\hookrightarrow}\textup{Cond}(\operatorname{Sp})$. Thus $\gamma$
is left t-exact. ∎

###  A.4 (Pre-)sheaves of condensed spaces and spectra

In this subsection, we examine the relation between sheafification and
stabilisation of presheaf categories. This will apply to the passage to
condensed objects. First, we start with a general lemma:

######  Lemma A.20.

Let $({\mathcal{C}},\tau)$ and $({\mathcal{D}},\sigma)$ be two sites. Then:

  1. 1.

$\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},\textup{Sh}_{\sigma}({\mathcal{D}}))\simeq\textup{Sh}_{\sigma}({\mathcal{D}},\textup{PSh}({\mathcal{C}}))$

  2. 2.

$\textup{Sh}_{\tau}({\mathcal{C}},\textup{Sh}_{\sigma}({\mathcal{D}}))\simeq\textup{Sh}_{\tau\times\sigma}({\mathcal{C}}\times{\mathcal{D}})$

###### Proof.

(i) Both categories are canonically subcategories of
$\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}}\times{\mathcal{D}}^{\operatorname{op}},\textup{Spc})$.
Since equivalences of presheaves can be computed pointwise, the condition for
lying in both sides amounts to the sheaf condition for $\sigma$, hence the
subcategories agree.
(ii) This holds since one can check the respective sheaf conditions in each
factor independently. ∎

######  Corollary A.21.

Given a site $({\mathcal{C}},\tau)$, there is a canonical equivalence

$\textup{Sh}_{\tau}({\mathcal{C}},\textup{Cond}^{\omega}(\textup{Spc}))\simeq\textup{Cond}^{\omega}(\textup{Sh}_{\tau}({\mathcal{C}}))$

between the categories of $\tau$-sheaves of condensed spaces and condensed
$\tau$-sheaves. The analogous result for $\textup{Cond}_{\kappa}(-)$ is true
as well.

######  Lemma A.22.

Let $({\mathcal{C}},\tau)$ be a site and ${\mathcal{D}}$ be a category. There
is a canonical equivalence

$\textup{Sh}_{\tau}({\mathcal{C}},\operatorname{Sp}({\mathcal{D}}))\simeq\operatorname{Sp}\bigl{(}\textup{Sh}({\mathcal{C}},{\mathcal{D}})\bigr{)}$

between $\tau$-sheaves of spectra and spectrum objects in $\tau$-sheaves.

###### Proof.

This is [Lur18, 1.3.2.2]. ∎

######  Corollary A.23.

For any category ${\mathcal{D}}$ there is a canonical equivalence

$\textup{Cond}^{\omega}(\operatorname{Sp}({\mathcal{D}}))\simeq\operatorname{Sp}(\textup{Cond}^{\omega}({\mathcal{D}})).$

The analogous result for $\textup{Cond}_{\kappa}(-)$ is true as well.

###### Proof.

This follows from Lemma A.22. ∎

### Internal mapping objects

In order to render the functor $\textup{K}^{\textup{an}}_{\geq
0}\colon\textup{Rig}_{k}\overset{}{\rightarrow}\textup{Pro}(\textup{Spc})\overset{}{\rightarrow}\textup{Cond}^{\omega}(\textup{Spc})$
representable, we need to have internal condensed mapping spectra in
$\textup{RigH}(k,\textup{Cond}^{\omega}(\textup{Spc}))$. ByRemark 3.17 it is
enough to see that the $\otimes$-product in
$\textup{Cond}^{\omega}(\textup{Spc})$ given by Lemma A.5 preserves colimits
in each variable.

######  Lemma A.24.

The cartesian symmetric monoidal structure on $\textup{Cond}(\textup{Spc})$ is
closed, i.e. for any objects $Y,Z\in\textup{Cond}(\textup{Spc})$ there exists
an internal mapping object
$\underline{\textup{Map}}(Y,Z)\in\textup{Cond}(\textup{Spc})$ such that for
any $X\in\textup{Cond}(\textup{Spc})$ there is a canonical equivalence

$\textup{Map}_{\textup{Cond}(\textup{Spc})}(X,\underline{\textup{Map}}(Y,Z))\simeq\textup{Map}_{\textup{Cond}(\textup{Spc})}(X\times
Y,Z).$

Setting $X=y(S)$ for $S\in\textup{EDS}$ we get that

$\underline{\textup{Map}}(Y,Z)(S)\simeq\textup{Map}_{\textup{Cond}(\textup{Spc})}(y(S),\underline{\textup{Map}}(Y,Z))\simeq\textup{Map}_{\textup{Cond}(\textup{Spc})}(y(S)\times
Y,Z).$

Moreover, for any ${\mathcal{V}}\in\textup{CAlg}(\textup{Pr}^{L,\otimes})$, we
have $\textup{Cond}^{\omega}({\mathcal{V}})$ admits a closed monoidal
structure.

###### Proof.

This follows immediately from the definition, since
$\textup{Cond}^{\omega}(\textup{Spc})$ as an $\infty$-topos and thus colimits
are universal. If ${\mathcal{V}}$ is a symmetric monoidal presentable
category, then we have
$\textup{Cond}^{\omega}(\textup{Spc})\otimes{\mathcal{V}}\simeq\textup{Cond}^{\omega}({\mathcal{V}})$
[Dre18, Prop. 2.4]. ∎

##  Appendix B Enriched Yoneda

In this section, we want to recall the enriched Yoneda lemma in the case that
is of interest for us, following Hinich [Hin20]. We will give a short summary
of the constructions.

In the following, we will fix a category ${\mathcal{C}}$ and a symmetric
monoidal presentable category ${\mathcal{E}}$. Note that the work of Hinich
works in greater generality but we restrict to this case.

######  Notation B.1.

We denote by
$Y\colon{\mathcal{C}}^{\operatorname{op}}\times{\mathcal{C}}\rightarrow\textup{Spc}$
the functor $(x,y)\mapsto\operatorname{Hom}(x,y)$. We denote the composition
of this functor with the unit $\textup{Spc}\rightarrow{\mathcal{E}}$ by
$\tilde{Y}$.

In [Hin20, §3.1.3] Hinich defines a category of quivers
$\textup{Quiv}_{{\mathcal{C}}}^{\textup{BM}}({\mathcal{E}})$ as an
$\infty$-operad161616In loc. cit. Hinich uses a different notion of operads,
based on strong approximations of $\infty$-operads. We will ignore this detail
as this section stays valid, having [Hin20, §2.7] in mind. fibred over the
$\infty$-operad of bimodules BM (the $\infty$-operad with colours
$\\{a_{+},m,a_{-}\\}$). The colours of
$\textup{Quiv}_{{\mathcal{C}}}^{\textup{BM}}({\mathcal{E}})$ can be described
as

  1. $\bullet$

$\textup{Quiv}_{{\mathcal{C}}}({\mathcal{E}})\coloneqq\textup{Quiv}_{{\mathcal{C}}}({\mathcal{E}})_{a_{+}}\simeq\operatorname{Fun}({\mathcal{C}}^{op}\times{\mathcal{C}},{\mathcal{E}})$,

  2. $\bullet$

$\textup{Quiv}_{{\mathcal{C}}}({\mathcal{E}})_{m}\simeq\operatorname{Fun}({\mathcal{C}},{\mathcal{E}})$,
and

  3. $\bullet$

$\textup{Quiv}_{{\mathcal{C}}}({\mathcal{E}})_{a_{-}}\simeq{\mathcal{E}}$.

Let us remark that since ${\mathcal{E}}$ is symmetric monoidal, we have an
equivalence categories

$\textup{Quiv}_{{\mathcal{C}}}({\mathcal{E}})\simeq\textup{Quiv}_{{\mathcal{C}}^{\operatorname{op}}}({\mathcal{E}}).$

In particular, the functor $\tilde{Y}$ defines an associative algebra object
$\tilde{Y}^{\operatorname{op}}$ inside
$\textup{Quiv}_{{\mathcal{C}}^{\operatorname{op}}}({\mathcal{E}})$ via the
above equivalence in the sense of [Hin20]. Thus,
$\tilde{Y}^{\operatorname{op}}$ acts on
$\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})$ from the
left.

######  Definition B.2 ([Hin20, §6.2.2]).

We define the category of ${\mathcal{E}}$-presheaves on ${\mathcal{C}}$ as

$P_{{\mathcal{E}}}(\tilde{Y})\coloneqq\textup{LMod}_{\tilde{Y}^{\operatorname{op}}}(\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})).$

There is a lot to digest in this definition of ${\mathcal{E}}$-presheaves. But
in our context we can greatly simplify the definition.

First note
$\textup{Quiv}_{{\mathcal{C}}^{\operatorname{op}}}({\mathcal{E}})\simeq\textup{Quiv}_{{\mathcal{C}}}({\mathcal{E}})^{\textup{rev}}$,
where the superscript rev denotes the reversed monoidal category in the sense
of [Lur17, Rem. 4.1.1.7] (this follows from the construction [Hin20, §6.2.1]).
Further, by [Hin20, §4.7.3], we have that $\tilde{Y}$ is the unit object
inside $\textup{Quiv}_{{\mathcal{C}}}({\mathcal{E}})$ and thus
$Y^{\operatorname{op}}$ is the unit object in
$\textup{Quiv}_{{\mathcal{C}}}({\mathcal{E}})^{\textup{rev}}$. In particular,
we obtain

$\textup{LMod}_{Y^{\operatorname{op}}}(\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}}))\simeq\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})$
($\clubsuit$)

under the above identifications [Lur17, Prop. 4.2.4.9]. By design of
$\textup{Quiv}_{{\mathcal{C}}^{\operatorname{op}}}^{\textup{BM}}({\mathcal{E}})$,
we get an induced right ${\mathcal{E}}^{\textup{rev}}$ action on
$\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})$, which
is the same as a left ${\mathcal{E}}$-action. This action is induced by the
pointwise monoidal structure.

The left ${\mathcal{E}}$-module structure on
$\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})$ induces
an enrichment as the $\otimes$-product of ${\mathcal{E}}$ commutes with
arbitrary colimits, i.e. we obtain an internal mapping
object$\underline{\operatorname{Hom}}_{{\mathcal{E}}}(F,G)\in{\mathcal{E}}$
for all
$F,G\in\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})$.
By [Hin20, §6.2.4], we obtain a map
$Y^{\prime}\colon{\mathcal{C}}\rightarrow\textup{LMod}_{Y^{\operatorname{op}}}(\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}}))$
with a compatible system of maps

$\tilde{Y}(x,y)\otimes Y^{\prime}(x)\rightarrow Y^{\prime}(y)$

for all $x,y\in{\mathcal{C}}$. Under the equivalence ($\clubsuit$), we have
$Y^{\prime}\simeq\tilde{Y}$ [Hin20, §4.7.4] (see also the proof of [Hin20,
6.2.6 Cor.]). This allows us to formulate an enriched Yoneda lemma.

######  Lemma B.3 (Enriched Yoneda [Hin20, §6.2.7]).

Let $F\in\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})$
be an ${\mathcal{E}}$-presheaf on ${\mathcal{C}}$. Then, we have an
equivalence

$F(x)\simeq\underline{\operatorname{Hom}}_{{\mathcal{E}}}(\tilde{Y}(x),F)$

for all $x\in{\mathcal{C}}$. Moreover, the induced functor
$\tilde{Y}\colon{\mathcal{C}}\rightarrow\operatorname{Fun}({\mathcal{C}}^{\operatorname{op}},{\mathcal{E}})$
is fully faithful.

## References

  * [Abb10] Ahmed Abbes.  Éléments de géométrie rigide. Volume I, volume 286 of Progress in Mathematics.  Birkhäuser/Springer Basel AG, Basel, 2010.  Construction et étude géométrique des espaces rigides.
  * [AGV22] Joseph Ayoub, Martin Gallauer, and Alberto Vezzani.  The six-functor formalism for rigid analytic motives.  Forum Math. Sigma, 10:Paper No. e61, 182, 2022.
  * [And23] Grigory Andreychev.  K-Theorie adischer Räume.  PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, September 2023.  https://nbn-resolving.org/urn:nbn:de:hbz:5-72174.
  * [Ayo07] Joseph Ayoub.  Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique. (I, II).  Astérisque, (314, 315), 2007.
  * [Ayo15] Joseph Ayoub.  Motifs des variétés analytiques rigides.  Mém. Soc. Math. Fr. (N.S.), (140-141):vi+386, 2015.
  * [CD19] Denis-Charles Cisinski and Frédéric Déglise.  Triangulated categories of mixed motives.  Springer Monographs in Mathematics. Springer, Cham, [2019] ©2019.
  * [Cis13] Denis-Charles Cisinski.  Descente par éclatements en $K$-théorie invariante par homotopie.  Ann. of Math. (2), 177(2):425–448, 2013.
  * [Con07] Brian Conrad.  Deligne’s notes on Nagata compactifications.  J. Ramanujan Math. Soc., 22(3):205–257, 2007.
  * [Dah24] Christian Dahlhausen.  Continuous K-theory and cohomology of rigid spaces.  Manuscripta Math., 173(1-2):119–153, 2024.
  * [DLØ+07] Bjørn Dundas, Marc Levine, Paul Østvær, Oliver Röndigs, and Vladimir Voevodsky.  Motivic Homotopy Theory: Lectures at a Summer School in Nordfjordeid, Norway, August 2002.  01 2007.
  * [Dre18] Brad Drew.  Motivic hodge modules, 2018.
  * [Emm96] Ioannis Emmanouil.  Mittag-Leffler condition and the vanishing of $\varprojlim^{1}$.  Topology, 35(1):267–271, 1996.
  * [FK18] Kazuhiro Fujiwara and Fumiharo Kato.  Foundations of Rigid Geometry I, volume 7 of Monographs in Mathematics.  EMS, 2018.
  * [GW10] Ulrich Görtz and Torsten Wedhorn.  Algebraic Geometry I.  Advanced Lectures in Mathematics. Vieweg + Teubner, Wiesbaden, 2010.  Schemes with examples and exercises.
  * [Hin20] Vladimir Hinich.  Yoneda lemma for enriched $\infty$-categories.  Advances in Mathematics, 367:107129, 2020.
  * [Hov01] Mark Hovey.  Spectra and symmetric spectra in general model categories.  J. Pure Appl. Algebra, 165(1):63–127, 2001.
  * [Hoy17] Marc Hoyois.  The six operations in equivariant motivic homotopy theory.  Adv. Math., 305:197–279, 2017.
  * [Hü24] Katharina Hübner.  Adic spaces.  Lectures at the spring school ‘Nonarchimedean Geometry and Eigenvarieties’ in March 2023 in Heidelberg. arXiv:2405.06435, 2024.
  * [Isa02] Daniel C. Isaksen.  Calculating limits and colimits in pro-categories.  Fund. Math., 175(2):175–194, 2002.
  * [KST19a] Moritz Kerz, Shuji Saito, and Georg Tamme.  K-theory of non-archimedean rings. I.  Doc. Math., 24:1365–1411, 2019.
  * [KST19b] Moritz Kerz, Shuji Saito, and Georg Tamme.  Towards a non-archimedean analytic analog of the Bass-Quillen conjecture.  Journal of the Institute of Mathematics of Jussieu, pages 1–16, 2019.
  * [KST23] Moritz Kerz, Shuji Saito, and Georg Tamme.  $K$-theory of non-Archimedean rings II.  Nagoya Math. J., 251:669–685, 2023.
  * [Lur09] Jacob Lurie.  Higher topos theory, volume 170 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2009.
  * [Lur17] Jacob Lurie.  Higher algebra.  https://www.math.ias.edu/~lurie/papers/HA.pdf, 2017.
  * [Lur18] Jacob Lurie.  Spectral algebraic geometry.  https://www.math.ias.edu/~lurie/papers/SAG-rootfile.pdf, 2018.
  * [LZ12] Yifeng Liu and Weizhe Zheng.  Enhanced six operations and base change theorem for higher artin stacks, 2012.  arXiv:1211.5948.
  * [Man22] Lucas Mann.  A $p$-adic 6-functor formalism in rigid-analytic geometry, 2022.
  * [Mor16] Matthew Morrow.  A historical overview of pro cdh descent in algebraic $k$-theory and its relation to rigid analytic varieties, 2016.
  * [MV99] Fabien Morel and Vladimir Voevodsky.  ${\bf A}^{1}$-homotopy theory of schemes.  Inst. Hautes Études Sci. Publ. Math., (90):45–143 (2001), 1999\.
  * [PPR09] Ivan Panin, Konstantin Pimenov, and Oliver Röndigs.  On Voevodsky’s algebraic $K$-theory spectrum.  In Algebraic topology, volume 4 of Abel Symp., pages 279–330. Springer, Berlin, 2009.
  * [Ray74] Michel Raynaud.  Géométrie analytique rigide d’après Tate, Kiehl,$\cdots$.  In Table Ronde d’Analyse Non Archimédienne (Paris, 1972), volume Tome 102 of Supplément au Bull. Soc. Math. France, pages 319–327. Soc. Math. France, Paris, 1974.
  * [Rob14] Marco Robalo.  Théorie homotopique motivique des espaces non-commutatifs.  2014\.  PhD-thesis, University of Montpellier.
  * [Rob15] Marco Robalo.  $K$-theory and the bridge from motives to noncommutative motives.  Adv. Math., 269:399–550, 2015.
  * [Sch19] Peter Scholze.  Lectures on Condensed Mathematics.  available online: https://www.math.uni-bonn.de/people/scholze/Condensed.pdf, 2019\.
  * [Sig16] Helene Sigloch.  Homotopy Theory for Rigid Analytic Varieties.  PhD thesis, Albert-Ludwigs-Universit¨at Freiburg im Breisgau, March 2016\.  https://freidok.uni-freiburg.de/data/11742.
  * [Sta23] The Stacks Project Authors.  Stacks Project.  https://stacks.math.columbia.edu, 2023.
  * [SW20] Peter Scholze and Jared Weinstein.  Berkeley lectures on $p$-adic geometry, volume 207 of Annals of Mathematics Studies.  Princeton University Press, Princeton, NJ, 2020.
  * [TT90] Robert W. Thomason and Thomas Trobaugh.  Higher algebraic $K$-theory of schemes and of derived categories.  In The Grothendieck Festschrift, Vol. III, volume 88 of Progr. Math., pages 247–435. Birkhäuser Boston, Boston, MA, 1990.
  * [Voe98] Vladimir Voevodsky.  ${\bf A}^{1}$-homotopy theory.  In Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), pages 579–604, 1998.
  * [Wei13] Charles A. Weibel.  The $K$-book, volume 145 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 2013.  An Introduction to Algebraic $K$-theory.

Christian Dahlhausen:
Institut für Mathematik, Universtität Heidelberg, Im Neuenheimer Feld 205,
69120 Heidelberg, Germany

E-mail address<EMAIL_ADDRESS>
Can Yaylali:
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7,
64289 Darmstadt, Germany

E-mail address<EMAIL_ADDRESS>


# (3+1)-D gravastar in de Rham–Gabadadze–Tolley like massive gravity

Piyali Bhar<EMAIL_ADDRESS>Department of Mathematics,Government
General Degree College Singur, Hooghly, West Bengal 712 409, India

###### Abstract

The physical properties of a (3+1)-D gravastar in the context of massive
gravity are discussed in this work. In present investigation, the field
equations have been solved for a static, uncharged sphere in order to achieve
the gravastar model as proposed by Mazur and Mottola [Mazur and Mottola in
Report No. LA-UR-01-5067,(2001); Mazur and Mottola, Proc Natl Acad Sci USA
101:9545, (2004)]. We address length of thin shell, energy, and entropy for
the thin shell containing an ultra-relativistic stiff fluid. Israel matching
criteria are used to ensure that the inner and outside geometries join
smoothly. It turns out that the behavior of the gravastar is entirely altered
by the existence of the graviton mass. Particularly, when $m\rightarrow 0$,
our findings precisely matched the outcomes of general relativity.

##  I Introduction

Mazur and Mottola Mazur:2004fk ; Mazur:2001fv  hypothesized the model of
gravastar as an alternate structure to a black hole. The internal and external
domains along with the shell region acting as an intermediary between them
constitute the three sectors of the gravastar geometry. For each of these
three zones, three different forms of the equation of state parameters are
defined. According to Horvat and Ilijic Horvat:2007qa  the gravastar model
contains a single spherical delta-shell at which the vacuum phase transition
occurs between the de Sitter interior and the Schwarzschild or Schwarzschild-
de Sitter outer geometries. By assuming that the dominant energy condition
(DEC) will hold at the shell, they are able to estimate sharp analytic bounds
on the surface compactness $2m/r$. The previous work was generalized by Horvat
et al. Horvat:2008ch  by introducing an electrically charged component. The
Einstein-Maxwell field equations are solved in the asymptotically de Sitter
interior, where the fluid energy density is connected to the source of the
electric field. Two distinct levels of complexity are applied to the linked
nonlinear issue in order to demonstrate the possibility of finding solutions
that possess all the characteristics of a gravastar in circumstances when the
prerequisites for their existence are being pushed to their limit. In an
another paper, they used the traditional Chandrasekhar method to study the
radial stability of the continuous pressure gravastar Horvat:2011ar . The
authors solve Einstein equations for small perturbations around the
equilibrium as an eigenvalue problem for radial pulsations and extract the
equation of state for the static gravastar solutions. Authors have
demonstrated that a splitting point between stable and unstable gravastar
configurations is represented by the extremum of the $\rho_{c}-R$ curve for
the centre energy density. The work of Chan et al. Chan:2009hp  demonstrated
that depending on the total mass of the system, cosmological constants, the
equation of state of the thin shell, and the initial position of the dynamical
shell, the final output can be a black hole, stable gravastar, “bounded
excursion” stable gravastar, or de Sitter spacetime. The authors have
discovered that the gravastar formation is constrained by the exterior
cosmological constant; that is, the exterior cosmological constant needs to be
less than the interior cosmological constant. Furthermore, it has been
demonstrated that a black hole can still arise in the specific scenario where
the Schwarzschild mass disappears notwithstanding the inability to build a
stable gravastar. Recently, Ghosh et al. Ghosh:2023wps  presented a typical
gravastar model within the context of UV corrected Loop Quantum Cosmology.
According to the authors’ findings, a bounce mechanism can prevent the centre
singularity of a self-gravitating object, causing the gravastar’s inner
density to achieve a maximum critical density and then remain there because of
an acting repulsive force. In agreement with the conjecture made by Mazur and
Mottola, Nojiri and Nashed Nojiri:2023zlp  provide a stellar model with two
scalar fields within the context of Einstein’s gravity. To prevent ghosts, the
two scalar fields in the model impose constraints that cause them to become
non-dynamical. The authors of this model postulated that the Schwarzschild
spacetime makes up the outside region and the de Sitter spacetime makes up the
inside. They make the assumption that a polynomial function of the radial
coordinate provides the metric on the shell that connects the two sections to
one another. The concept of a gravitational Bose-Einstein condensate star in
the presence of charge has been studied by Bhattacharjee and Chattopadhyay
Bhattacharjee:2023apw  who have investigated the function of charge in the
formation of gravastars and their characteristics in cylindrically symmetric
space-time. With this method, thin shell approximation takes the place of the
event horizon and eliminates the possibility of singularity at the gravastar’s
centre. Within the context of energy-momentum squared gravity, Sharif and Naz
Sharif:2022ibt  examined the impact of charge on the physical characteristics
of gravastars. Specifically, they found that the charge within the inner core
of gravastars maintains equilibrium by counteracting the inward gravitational
force. The impact of the Kuchowicz metric function on the gravastar model
within the context of braneworld gravity was explained by Ray et al.
Ray:2022lxl . As an alternative for the General Relativity (GR), Sengupta et
al. Sengupta:2020lhw  and Arbañil et al. Arbanil:2019xfi  derived stable
gravastar models within the framework of the Randall-Sundrum (RS) braneworld
model which is particularly relevant in the context of cosmology and
astrophysics. Within the context of the Mazur and Mottola model, Usmani et al.
Usmani:2010ac  have developed a charged (3 + 1)-dimensional gravastar that
admits conformal motion. The authors offer a substitute for static black holes
in this study. It is discovered that the energy density in the interior region
of the gravastar diverges. Regretfully, this leads to the model being singular
at r = 0, Bhar’s work Bhar:2014vra  provides an extension of Usmani et al.
Usmani:2010ac  on charged gravastar, admitting conformal motion with higher
dimensional space-time. The four-dimensional work on gravastar by Usmani et
al. Usmani:2010ac  was generalized to higher-dimensional space-time by Ghosh
et al. Ghosh:2015ohi , although conformal motion was not permitted in their
studies. The authors’ primary goal was to develop gravastars in Einstein-
Maxwell geometry and observe the higher-dimensional impacts. They also
demonstrated that there is a significant amount of quantitative change in the
physical parameter profiles using graphical analysis. As one progresses
towards higher dimensions, notable outcomes become apparent. Further papers on
gravastar in Einstein gravity as well as in modified gravity are available in
Errehymy:2017wpc ; Pani:2015tga ; Chirenti:2016hzd ; Rutkowski:2020fvm ;
Das:2020lqy ; Yousaf:2020nza ; Das:2017rhi ; Barzegar:2023ueo ; Rahaman:2012wc
; Ghosh:2019nsi ; Ghosh:2019upa ; Rahaman:2012xx ; DeBenedictis:2005vp .

Fierz and Pauli Fierz:1939ix  successfully generated massive gravity for the
first time by taking non-zero mass gravitons into account in 1939. Subsequent
research such as van Dam-Veltman-Zakharov discontinuityVanNieuwenhuizen:1973fi
; vanDam:1970vg ; Zakharov:1970cc , Vainshtein mechanism Vainshtein:1972sx
and Boulware-Deser ghost Boulware:1972yco , etc, however, revealed that this
modified gravity theory contains several difficulties. The de-Rham-Gabadadze-
Tolley (dRGT) massive gravity theory was presented in 2010 deRham:2010ik ;
deRham:2010kj , and it effectively resolved the issues mentioned above. Since
the dRGT massive gravity gives the graviton mass, it is a logical extension of
Einstein’s theory of general relativity and a fantastic field for theoretical
physics study. The dRGT massive gravity admits general relativity as a special
case, explaining nonlinear interaction terms as a correction of the Einstein-
Hilbert action. It is hypothesized that dRGT massive gravity may provide a
plausible explanation for the accelerated expansion of the Universe that
doesn’t need any dark energy or cosmological constant. Without taking into
account any extra dark matter, the dRGT model is quite satisfactorily fitted
with observational data for LSB galaxies and Milky Way rotation curves in
Panpanich:2018cxo . In this context, it has also been observed that the dRGT
model is consistent with the NFW profile. In dRGT massive gravity, the
spherically symmetric solutions were covered in Nieuwenhuizen:2011sq ;
Brito:2013xaa ,The associated charged black hole solution was discovered in
Berezhiani:2011mt , and its bi-gravity extension was discovered in
Babichev:2014fka , including as specific cases the previously identified
spherically symmetric black hole solutions. The massive gravity theory has
acquired substantial interest in the theoretical realm due to the detection of
gravitational waves caused by the merging of two Black holes and massive
stars, as detected by LIGO and VIRGO LIGOScientific:2016aoc ;
LIGOScientific:2017vwq .

In this work, we shall examine gravastar solutions in dRGT like massive
gravity, and investigate different physical properties. The structure of the
paper is as follows: We define the action and underlying field equations in
dRGT like massive gravity in Sec II. In Sec III, we have obtained the model of
the gravastar by solving the field equations in three different regions of the
gravastar. The purpose of Sec IV is to explain the junction condition. The
impact of the mass of the graviton on the physical quantities of the gravastar
model is discussed via graphical analysis in the next section. Sec VI
describes the stability of the gravastar. Contour plots are used to illustrate
some physical characteristics of the gravastar model in Sec VII, and the last
section provides some closing remarks.

##  II action and field equation in massive gravity

In dRGT like massive gravity, the action $\mathcal{I}$ is defined by,

$\mathcal{I}=\frac{1}{16\pi G}\int
d^{4}x\sqrt{-g}\left[\mathcal{R}+m^{2}\sum_{i}^{4}c_{i}\mathcal{U}_{i}(g,f)\right]+I_{\text{matter}},$
(1)

where $I_{\text{matter}}$ is the matter Lagrangian, $g$ and $f$ are a fixed
symmetric tensor and a metric tensor, respectively, and $\mathcal{R}$ is the
Ricci scalar curvature. The parameter $m$ is associated with the gravitational
mass. $\mathcal{U}_{i}$ represents the effective potential for graviton,and it
has the following expression in four dimensions of spacetime:

$\displaystyle\mathcal{U}_{1}$ $\displaystyle=$
$\displaystyle\left[\mathcal{K}\right],\;\;\;\;\;\;\;\mathcal{U}_{2}=\left[\mathcal{K}\right]^{2}-\left[\mathcal{K}^{2}\right],$
$\displaystyle\mathcal{U}_{3}$ $\displaystyle=$
$\displaystyle\left[\mathcal{K}\right]^{3}-3\left[\mathcal{K}\right]\left[\mathcal{K}^{2}\right]+2\left[\mathcal{K}^{3}\right],$
$\displaystyle\mathcal{U}_{4}$ $\displaystyle=$
$\displaystyle\left[\mathcal{K}\right]^{4}-6\left[\mathcal{K}^{2}\right]\left[\mathcal{K}\right]^{2}+8\left[\mathcal{K}^{3}\right]\left[\mathcal{K}\right]+3\left[\mathcal{K}^{2}\right]^{2}-6\left[\mathcal{K}^{4}\right].$

where [$\mathcal{K}$] represents the trace of the metric matrix
$\mathcal{K_{\nu}^{\mu}}=\sqrt{g^{\mu\alpha}f_{\alpha\nu}}$.
The equation of motion for this gravity can be found by varying the action (1)
with respect to $g_{\nu}^{\mu}$ as follows:

$R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+m^{2}\chi_{\mu\nu}=\frac{8\pi
G}{c^{4}}T_{\mu\nu},$ (2)

where the expression for $\chi_{\mu\nu}$ is given by,

$\displaystyle\chi_{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{c_{1}}{2}\left(\mathcal{U}_{1}g_{\mu\nu}-\mathcal{K}_{\mu\nu}\right)$
(3)
$\displaystyle-\frac{c_{2}}{2}\left(\mathcal{U}_{2}g_{\mu\nu}-2\mathcal{U}_{1}\mathcal{K}_{\mu\nu}+2\mathcal{K}_{\mu\nu}^{2}\right)$
$\displaystyle-\frac{c_{3}}{2}(\mathcal{U}_{3}g_{\mu\nu}-3\mathcal{U}_{2}\mathcal{K}_{\mu\nu}+6\mathcal{U}_{1}\mathcal{K}_{\mu\nu}^{2}-6\mathcal{K}_{\mu\nu}^{3})$
$\displaystyle-\frac{c_{4}}{2}\left(\mathcal{U}_{4}g_{\mu\nu}-4\mathcal{U}_{3}\mathcal{K}_{\mu\nu}+12\mathcal{U}_{2}\mathcal{K}_{\mu\nu}^{2}\right.$
$\displaystyle\left.-24\mathcal{U}_{1}\mathcal{K}_{\mu\nu}^{3}+24\mathcal{K}_{\mu\nu}^{4}\right).$

Now to describe a gravastar spacetime in $(3+1)$-dimension, let us consider
the following line element,

$ds^{2}=-e^{2\phi(r)}dt^{2}+e^{2\lambda(r)}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (4)

where $\phi$ and $\lambda$ are purely radial, i.e., they only depend on ‘r’.
We use the following reference metric, generalizing and adhering to the ansatz
in Cai:2014znn :

$f_{\mu\nu}=\text{diag}(0,0,C^{2},C^{2}\sin^{2}\theta),$ (5)

where $C$ is a positive constant. With the reference metric (5), we have
Cai:2014znn ,

$\mathcal{U}_{1}=\frac{2C}{r},\text{ \ \
}\mathcal{U}_{2}=\frac{2C^{2}}{r^{2}},\,\mathcal{U}_{3}=0,\,\mathcal{U}_{4}=0.$

Here, we consider the gravastar model having perfect fluid distribution whose
energy-momentum tensor is given by,

$T_{\mu\nu}=\left(c^{2}\rho+p\right)U_{\mu}U_{\nu}-pg_{\mu\nu},$ (6)

where $U_{\mu}$ is the fluid four-velocity. $p$ and $\rho$ are the fluid
pressure and density, respectively, as measured by the local observer.
Furthermore, considering equations (6) and (4), it is also straightforward to
obtain the subsequent nonzero field equation components:

$\displaystyle\frac{8\pi G}{c^{2}}r^{2}\rho$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}(2r\lambda^{\prime}-1)+1+m^{2}C(c_{2}C+c_{1}r),$ (7)
$\displaystyle\frac{8\pi G}{c^{4}}r^{2}p$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}(2r\phi^{\prime}+1)-1-m^{2}C(c_{2}C+c_{1}r),$ (8)
$\displaystyle\frac{8\pi G}{c^{4}}rp$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}(\phi^{\prime}+r\phi^{\prime
2}-\lambda^{\prime}-r\lambda^{\prime}\phi^{\prime}+r\phi^{\prime\prime})-\frac{m^{2}c_{1}C}{2}.$

Here the the first and the second derivatives of a function with respect to
$r$ respectively denoted by prime and double prime.
The conservation equation can be obtained from $T_{\mu;\nu}^{\nu}$=0 as
follows:

$\frac{dp}{dr}+\left(c^{2}\rho+p\right)\frac{d\phi}{dr}=0.$ (10)

To find the expression of $e^{-2\lambda}$, we can use Eq. (7) which leads to

$e^{-2\lambda}=1-\frac{2G}{c^{2}r}\int 4\pi r^{2}\rho
dr+m^{2}C\left(\frac{c_{1}r}{2}+c_{2}C\right).$ (11)

From eqn.(8), the expression for $\phi^{\prime}$ can be obtained as,

$\displaystyle\frac{d\phi}{dr}$ $\displaystyle=$
$\displaystyle\left[\frac{4\pi
G}{c^{4}}rp+\frac{1+m^{2}C(c_{2}C+c_{1}r)}{2r}-e^{-2\lambda}\right]e^{2\lambda}.$

We will solve the field equations in the upcoming section to obtain the
gravastar model.

##  III The geometry of a Gravastar

In case of gravastar, there are three distinct zones, and each has a specific
width. These zones are represented by the geometrical regions
$b=r_{1}<r<r_{2}=b+\epsilon$, where $\epsilon$ is a very small positive
quantity and $r_{1}$ and $r_{2}$ stand for the radii of the interior and
exterior regions, respectively.

  * •

Interior de Sitter region: $\rho=-p$, $r\in[0,\,r_{1})$

  * •

Thin Shell: $\rho=p$, $r\in(r_{1},\,r_{2})$

  * •

Exterior Schwarzschild geometry: $\rho=p=0$, $r\in(r_{2},\,\infty)$

The metric functions $e^{2\phi}$ and $e^{2\lambda}$ must be continuous at the
interfaces $r=r_{1}$ and $r=r_{2}$, but the first derivatives of $e^{2\phi}$
and $e^{2\lambda}$, and $p$ must be discontinuous Mazur:2001fv .

###  III.1 Interior region

Following Mazur-Mottola Mazur:2001fv , we now take into consideration the
following equation of state (EoS)

$p=-\rho,$ (13)

in order to solve the above field eqns. (7)-(II) within the interior region of
the gravastar. It is the generalized version of the dark energy EoS with
$\omega=-1$. The inward gravitational attraction of the shell is balanced by
this negative pressure, which operates radially outward from the center of the
spherically symmetric gravitating system.
Solving eqn.(II) with the help of (13), we obtain,

$\displaystyle\rho=-p=\rho_{c},$ (14)

where $\rho_{c}$ is a constant and this constant is denoted by
$\frac{3H^{2}}{8\pi G_{N}}$ Mazur:2001fv .
By choosing $G=c=1$ and $8\pi=k$ (from here and onwards), from eqns. (7) and
(14) we get the following equation:

$kr^{2}\rho_{c}=e^{-2\lambda}(2r\lambda^{\prime}-1)+1+m^{2}C(c_{2}C+c_{1}r).$
(15)

On integrating the above equation we obtain the expression for $e^{-2\lambda}$
as,

$\displaystyle
e^{-2\lambda}=1+m^{2}C(c_{2}C+\frac{c_{1}r}{2})-k\rho_{c}\frac{r^{2}}{3}+\frac{D}{r},$
(16)

where $D$ is the constant of integration. One can require that $D=0$ in order
to make the solution regular at the origin, which provides the expression of
the metric coefficients as,

$\displaystyle
e^{-2\lambda}=1+m^{2}C(c_{2}C+\frac{c_{1}r}{2})-k\rho_{c}\frac{r^{2}}{3}.$
(17)

Adding eqns. (7) and (8) we get,

$\displaystyle
k(p+\rho)r^{2}=e^{-2\lambda}2r(\lambda^{\prime}+\phi^{\prime}),$ (18)

Solving (18) with the help of (13) we obtain,

$\displaystyle\lambda^{\prime}+\phi^{\prime}=0.$ (19)

Integrating the eqn.(19), the expression for $e^{2\phi}$ can be obtained as
follows:

$\displaystyle e^{2\phi}$ $\displaystyle=$ $\displaystyle Ae^{-2\lambda},$
$\displaystyle\Rightarrow~{}e^{2\phi}$ $\displaystyle=$ $\displaystyle
A\Big{[}1+m^{2}C(c_{2}C+\frac{c_{1}r}{2})-k\rho_{c}\frac{r^{2}}{3}\Big{]},$
(20)

where $A$ is constant of integration. Here, one can note that both the metric
potentials depend on the massive gravity parameters $m^{2}c_{1}$ and
$m^{2}c_{2}$.

One can avoid the issue of the core singularity of a classical black hole by
observing that the interior solutions of the aforementioned solutions (17) &
(III.1) do not have a singularity.
One can derive the active gravitational mass $M(r)$ using the following
formula:

$\displaystyle M(r)$ $\displaystyle=$ $\displaystyle
4\pi\int_{0}^{r}\eta^{2}\rho(\eta)d\eta$ (21) $\displaystyle=$
$\displaystyle\frac{4\pi r^{3}}{3}\rho_{c}.$

###  III.2 The Intermediate Thin Shell

According to the hypothesis proposed by Mazur & Mottola Mazur:2001fv , the
relationship between the pressure and the energy density in the shell of the
gravastar is,

$p=\rho.$ (22)

Getting a general solution to the field equations in the non-vacuum region, or
inside the shell, is challenging. We search for an analytical solution between
$0<(e^{-2\lambda}\equiv h)<<1$ in the thin shell limit. We can set $h$ to zero
to the leading order as a benefit of it.Using this approximation, the field
equations (8)-(10) with the aforementioned EoS can be reformulated in the
following way:

$\displaystyle kr^{2}\rho$ $\displaystyle=$ $\displaystyle
e^{-2\lambda}(2r\lambda^{\prime})+1+m^{2}C(c_{2}C+c_{1}r),$ (23)
$\displaystyle kr^{2}p$ $\displaystyle=$
$\displaystyle-1-m^{2}C(c_{2}C+c_{1}r),$ (24) $\displaystyle krp$
$\displaystyle=$
$\displaystyle-e^{-2\lambda}\lambda^{\prime}-r\lambda^{\prime}\phi^{\prime}e^{-2\lambda}-\frac{m^{2}c_{1}C}{2}$
(25)

Solving the eqns. (23)-(24) with the help of eqn. (22), we obtain the
expression for $e^{-2\lambda}$ as,

$\displaystyle e^{-2\lambda}=2\ln r+2m^{2}C(c_{2}C\ln r+c_{1}r)+B,$ (26)

From eqns. (24) and (25), with the help of the expression $e^{-2\lambda}$, we
obtain the expression of the another metric coefficient as,

$\displaystyle e^{2\phi}$ $\displaystyle=$ $\displaystyle
E\left(\frac{1+m^{2}C^{2}c_{2}+m^{2}Cc_{1}r}{r^{4}}\right).$ (27)

From the TOV equation, with the help of (22), we get,

$\displaystyle\frac{dp}{dr}$ $\displaystyle=$
$\displaystyle-2p\frac{d\phi}{dr},$ (28)

Solving (28), we obtain the expression of pressure inside the thin shell as,

$\displaystyle p$ $\displaystyle=$ $\displaystyle Fe^{-2\phi},$ (29)
$\displaystyle=$
$\displaystyle\frac{Gr^{4}}{1+m^{2}C^{2}c_{2}+m^{2}Cc_{1}r}=\rho,$ (30)

where, both $F$ and $G$ are constants of integration.

Figure 1: Pressure and density variations within the thin shell

###  III.3 Exterior Spacetime and matching

We take into consideration $p=\rho=0$ for this region, which guarantees that
the Schwarzschild line element describes the outer space-time which is
described by,

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle\left(1-\frac{2\mathcal{M}}{r}\right)dt^{2}-\left(1-\frac{2\mathcal{M}}{r}\right)^{-1}dr^{2}$
(31) $\displaystyle-r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),$

where, $\mathcal{M}$ being the mass of the gravastar.

The gravastar structure, which resembles a hollow sphere, features two
junction surfaces as opposed to the single junction surface found in a compact
star. One is at $r=r_{1}$, between the internal area and the intermediate thin
shell, while the other is at $r=r_{2}$, between the shell and external
spacetime. By using the continuity of the metric potentials $g_{rr}$ and
$g_{tt}$ for our current gravastar model at the junction of the core and shell
at $r=r_{1}$ the following relationships are obtained.

$\displaystyle
1+m^{2}C(c_{2}C+\frac{c_{1}r_{1}}{2})-k\rho_{c}\frac{r_{1}^{2}}{3}$
$\displaystyle=$ $\displaystyle 2\ln r_{1}+2m^{2}C(c_{2}C\ln
r_{1}+c_{1}r_{1})+B,$ (32) $\displaystyle
A\left[1+m^{2}C(c_{2}C+\frac{c_{1}r_{1}}{2})-k\rho_{c}\frac{r_{1}^{2}}{3}\right]$
$\displaystyle=$ $\displaystyle
E\left(\frac{1+m^{2}C^{2}c_{2}+m^{2}Cc_{1}r_{1}}{r_{1}^{4}}\right)$ (33)

Now using the matching condition between the shell and the exterior region at
$r=r_{2}$ (exterior radius) yields the following relationships:

$\displaystyle 2\ln r_{2}+2m^{2}C(c_{2}C\ln r_{2}+c_{1}r_{2})+B$
$\displaystyle=$ $\displaystyle 1-\frac{2\mathcal{M}}{r_{2}},$ $\displaystyle
E\left(\frac{1+m^{2}C^{2}c_{2}+m^{2}Cc_{1}r_{2}}{r_{2}^{4}}\right)$
$\displaystyle=$ $\displaystyle 1-\frac{2\mathcal{M}}{r_{2}}.$

In the current work, in order to examine the physical behavior of various
model parameters for the gravastar model, we have to choose values for the
unknown parameters. To determine the numerical values of three different
constants $B,\,E,\,A$, and the mass of the gravastar, we choose interior
radius $r_{1}=10$ km., while the outer radius is $10.1$ km. i.e., the
thickness of the thin shell is equal to $0.1$ km.
Using a specific choice of $r_{1}=10$ km., $r_{2}=10.1$ km, $m^{2}c_{1}=0.1$,
$C=0.5$, $G=0.1\times 10^{-5}$, and $\rho_{c}=0.001~{}km^{-2}$, we
simultaneously solved equations (32)-(III.3) to determine the values of the
constants $B,\,E,\,A$, and mass of the gravastar for four distinct values of
$m^{2}c_{2}$. The results are shown in Table 1.
On the other hand, by simultaneously solving equations (32)-(III.3) for a
specific option of $r_{1}=10$ km., $r_{2}=10.1$ km, $m^{2}c_{2}=-0.5$,
$C=0.5$, $G=0.1\times 10^{-5}$, and $\rho_{c}=0.001~{}km^{-2}$, we were able
to calculate the values of the constants $B,\,E,\,A$, and mass of the
gravastar for four distinct values of $m^{2}c_{1}$. The result has been
presented in table 2.
From both the tables, one can note that the ratio of obtain mass of the
gravastar ($\mathcal{M}$) and outer radius $r_{2}$ for all the combinations of
$\\{m^{2}c_{1},\,m^{2}c_{2}\\}$ satisfy the ratio
$\frac{\mathcal{M}}{r_{2}}<\frac{4}{9}$, i.e., respect the buchdahl limit.

Table 1: Values of $B,\,E,\,A$ and mass of the gravastar have been obtained for four distinct values of $m^{2}c_{2}$ with $r_{1}=10$ km., $r_{2}=10.1$ km, $m^{2}c_{1}=0.1$, $C=0.5$, $G=0.1\times 10^{-5}$, and $\rho_{c}=0.001~{}km^{-2}$ $m^{2}c_{2}$ |  | $B$ | $M$ | $E$ | $A$
---|---|---|---|---|---
$-0.25$ |  | $-4.96761$ | $3.13909$ | $2729.73$ | $0.137238$
$-0.10$ |  | $-5.1028$ | $2.94594$ | $2929.48$ | $0.167326$
$0.05$ |  | $-5.23799$ | $2.7528$ | $3119.35$ | $0.200394$
$0.20$ |  | $-5.37319$ | $2.55965$ | $3300.07$ | $0.236442$
Table 2: Values of $B,\,E,\,A$ and mass of the gravastar have been obtained for four distinct values of $m^{2}c_{1}$ with $r_{1}=10$ km., $r_{2}=10.1$ km, $m^{2}c_{2}=-0.5$, $C=0.5$, $G=0.1\times 10^{-5}$, and $\rho_{c}=0.001~{}km^{-2}$ $m^{2}c_{1}$ |  | $B$ | $M$ | $E$ | $A$
---|---|---|---|---|---
$0.10$ |  | $-4.74228$ | $3.46099$ | $2372.69$ | $0.0937112$
$0.15$ |  | $-5.11728$ | $2.80449$ | $2834.36$ | $0.189872$
$0.20$ |  | $-5.49228$ | $2.14799$ | $3172.35$ | $0.31956$
$0.25$ |  | $-5.86728$ | $1.49149$ | $3430.49$ | $0.48276$

##  IV Junction condition

According to Mazur and Mottola, the shell of the gravastar is confined with
the cold radiation fluid by the surface tensions at the timelike interfaces
$r_{1}$ and $r_{2}$. These result from discontinuities in pressure and can be
computed using the Lanczos-Israel junction conditions Israel:1966rt ;
Berezin:1987bc . The expression for surface stress energy tensor is described
as Israel:1966rt ,

$\mathcal{S}^{\xi}_{\zeta}=-\frac{1}{8\pi}(\kappa^{\xi}_{\zeta}-\delta^{\xi}_{\zeta}\kappa^{\varphi}_{\varphi}),$
(36)

where $\xi,\,\zeta$ take the value $t,\,\theta,\,\phi$. $\kappa_{\xi\zeta}$
has the expression
$\kappa_{\xi\zeta}=\left[K_{\xi\zeta}\right]^{+}-\left[K_{\xi\zeta}\right]^{-}$,
and it represents the discontinuity in the extrinsic curvature $K_{\xi\zeta}$.
The expressions for $K_{\xi\zeta}^{\pm}$ is represented as,

$K_{\xi\zeta}^{\pm}=-n_{\nu}^{\pm}\left(\partial_{\zeta}e_{\xi}^{\nu}+\Gamma^{\nu}_{\alpha\beta}e_{\xi}^{\alpha}e_{\zeta}^{\beta}\right),$
(37)

where the unit normal vectors on the surface is described by $n_{\nu}^{\pm}$
with $n^{\nu}n_{\nu}=1$, and $e_{\xi}^{\alpha}=\frac{\partial
x^{\alpha}}{\partial\chi^{\xi}}$, $\chi^{\xi}$ describes the coordinate on the
shell. $\Gamma^{\nu}_{\alpha\beta}$ represents the Christoffel symbols. The
signs “$-$” and “$+$” denote the interior spacetime, and exterior spacetime,
i.e., Schwarzschild spacetime, respectively.

The surface energy density ($\sigma$) and surface pressure ($\mathcal{P}$) for
our present model of gravastar in massive gravity can be calculated as,

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle-\frac{1}{4\pi
b}\left[\sqrt{f}\right]_{-}^{+}$ $\displaystyle=$ $\displaystyle-\frac{1}{4\pi
b}\Big{[}\sqrt{1-\frac{2\mathcal{M}}{b}}-\sqrt{1+m^{2}C(c_{2}C+\frac{c_{1}b}{2})-k\rho_{c}\frac{b^{2}}{3}}\Big{]},$
$\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle-\frac{\sigma}{2}+\frac{1}{16\pi}\left[\frac{f^{\prime}}{\sqrt{f}}\right]_{-}^{+}$
(39) $\displaystyle=$ $\displaystyle\frac{1}{8\pi
b}\left[\frac{1-\frac{\mathcal{M}}{b}}{\sqrt{1-\frac{2\mathcal{M}}{b}}}-\frac{12+3Cm^{2}(c_{1}+4Cc_{2}+2c_{1}b)-4kb(1+b)\rho_{c}}{12\sqrt{1+m^{2}C(c_{2}C+\frac{c_{1}b}{2})-k\rho_{c}\frac{b^{2}}{3}}}\right].$

With the surface energy density equation, determining the mass of the surface
of the thin shell ($m_{\text{s}}$) is now straightforward, which is given by,

$\displaystyle m_{\text{s}}$ $\displaystyle=$ $\displaystyle 4\pi
b^{2}\sigma,$ $\displaystyle=$ $\displaystyle
b\bigg{[}\sqrt{1+m^{2}C(c_{2}C+\frac{c_{1}b}{2})-k\rho_{c}\frac{b^{2}}{3}}-\sqrt{1-\frac{2\mathcal{M}}{b}}\bigg{]}.$

From the above expression of $m_{\text{s}}$, we get a suitable range of the
radius of the gravastar as,

$\displaystyle
2\mathcal{M}<b<\frac{3c_{1}Cm^{2}+\sqrt{9c_{1}^{2}C^{2}m^{4}+48k(1+C^{2}c_{2}m^{2})\rho_{c}}}{4k\rho_{c}}.$

It is interesting to note that, twice the mass of the gravastar acts as a
lower limit of ‘b’.
Performing some mathematical computations, we determine the mass of the
gravastar in terms of the thin shell mass as follows:

$\displaystyle\mathcal{M}$ $\displaystyle=$
$\displaystyle\frac{b}{2}\Big{[}-\frac{1}{2}Cm^{2}(2Cc_{2}+c_{1}b)+\frac{1}{3}kb^{2}\rho_{c}-\left(\frac{m_{s}}{b}\right)^{2}$
(42)
$\displaystyle+\frac{2m_{s}}{b}\sqrt{1+m^{2}C(c_{2}C+\frac{c_{1}b}{2})-k\rho_{c}\frac{b^{2}}{3}}\Big{]}.$

##  V Some physical properties of our present model

The purpose of this section is to investigate how the massive parameter
affects the physical characteristics of a 4D gravastar in the context of
massive gravity. We investigate here the proper length of the thin shell, the
energy of the relativistic structure of 4D gravastars inside the shell, the
entropy inside the shell, and the EoS parameter. Furthermore, we will use
graphical representations to demonstrate our results.

###  V.1 Proper length of the shell

Mazur and Mottola postulated that the stiff fluid satisfying the EoS $p=\rho$
is located in between the lower and upper boundaries of the shell, where the
boundaries are given at $r=b$ and $r=b+\epsilon$ respectively with
$0~{}<\epsilon\ll 1$. One can use the following formula to find the proper
length $\mathcal{L}$ of the shell as,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\int_{b}^{b+\epsilon}\frac{1}{\sqrt{e^{-2\lambda}}}dr,$ (43)
$\displaystyle=$ $\displaystyle\int_{b}^{b+\epsilon}\frac{1}{\sqrt{2\ln
r+2m^{2}C(c_{2}C\ln r+c_{1}r)+B}}.$

To perform the above integral analytically is quite difficult due to its
expression. Let us assume that there exists a function $g(r)$ such that,
$\frac{d}{dr}g(r)=\frac{1}{\sqrt{e^{-2\lambda}}}$.
Using the above assumption,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\int_{b}^{b+\epsilon}\frac{d}{dr}g(r)dr=g(b+\epsilon)-g(b),$
(44)

Using the Taylor series expansion of $g(b+\epsilon)$ about the point ‘b’ and
retaining upto the second order term of $\epsilon$, (since $\epsilon<<1$),the
expression of $\mathcal{L}$ can be obtained as,

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{\epsilon}{\sqrt{2\ln b+2m^{2}C(c_{2}C\ln b+c_{1}b)+B}}$
$\displaystyle-\frac{\epsilon^{2}}{2}\frac{Cm^{2}(c_{1}+\frac{Cc_{2}}{b})+\frac{1}{b}}{\left\\{B+2\log(b)+2Cm^{2}\big{(}c_{1}b+Cc_{2}\log(b)\big{)}\right\\}^{\frac{3}{2}}}.$

Figure 2: Variation between the gravastar’s proper length inside its shell and
its shell thickness.

By retaining a fixed value for $m^{2}c_{2}$, the behavior of the proper
thickness of the shell with respect to its thickness $\epsilon$ is depicted in
Fig. 2 for different values of $m^{2}c_{1}$ and vice versa. From the figures
it is clear that proper length increases with thickness but for higher values
of massive gravity parameters $m^{2}c_{2}$ and $m^{2}c_{1}$, it takes lower
values.

###  V.2 Entropy

The measurement of entropy is correlated with the degree of disorder in a
geometric structure. Thin-shell gravastar entropy is examined to comprehend
the chaotic nature of gravastar geometry. Using the equation of state,
$p=\rho=\frac{\alpha^{2}}{8\pi}\left(\frac{K_{B}}{\hbar}\right)^{2}$, one may
get the entropy of the thin shell Mazur:2001fv . According to the conventional
thermodynamic Gibbs relation, $Ts=p+\rho$, for a relativistic fluid with zero
chemical potential, the local specific entropy density is

$s(r)=\frac{2\alpha^{2}K_{B}^{2}T(r)}{\kappa\hbar^{2}}=2\alpha\left(\frac{K_{B}}{\hbar}\right)\sqrt{\frac{p(r)}{8\pi}}$

at the temperature $T(r)$. In the above equation, $\alpha$ is a dimensionless
constant, $K_{B}$ denotes the Boltzmann constant, and $h$ is the Planck
constant and $\hbar=\frac{h}{2\pi}$.
We determine the entropy of the model as follows:

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\int_{b}^{b+\epsilon}4\pi r^{2}s(r)\sqrt{e^{2\lambda}}dr,$ (46)
$\displaystyle=$
$\displaystyle\sqrt{8\pi}\alpha\left(\frac{K_{B}}{\hbar}\right)\int_{b}^{b+\epsilon}r^{2}\sqrt{\frac{p(r)}{e^{-2\lambda}}}dr,$

By employing the expressions for $p$ and $e^{2\lambda}$ in eqn. (46) we get,

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\sqrt{8\pi}\alpha\left(\frac{K_{B}}{\hbar}\right)\int_{b}^{b+\epsilon}r^{2}\sqrt{\frac{Gr^{4}}{\Big{(}1+Cm^{2}(Cc_{2}+c_{1}r)\Big{)}\left\\{B+2\log(r)+2Cm^{2}\Big{(}c_{1}r+Cc_{2}\log(r)\Big{)}\right\\}}}dr,$
(47) $\displaystyle=$
$\displaystyle\sqrt{8\pi}\alpha\left(\frac{K_{B}}{\hbar}\right)\int_{b}^{b+\epsilon}\mathcal{I}(r)~{}dr~{}\text{(say)}.$
(48)

Figure 3:  Variation of the gravastar’s internal entropy in relation to shell
thickness.

To find the expression of the entropy explicitly, we have to perform the
integral given in Eqn. (47), but it is highly challenging to execute the
integral given in eqn.(47) due to the intricacy of the formulation of
$\mathcal{I}$.
let us assume that $h(r)$ be a function so that
$\frac{dh(r)}{dr}=\mathcal{I}(r)$. After mathematical calculations, the
expression of $\mathcal{I}$ becomes,

$\mathcal{I}=h(b+\epsilon)-h(b),$ (49)

Taking into account the Taylor series expansion of $h(b+\epsilon)$ about ‘b’
and taking up to second order of $\epsilon$, we finally obtain the expression
of entropy as,

$\displaystyle\mathcal{S}=\sqrt{8\pi}\alpha\left(\frac{K_{B}}{\hbar}\right)\left[\epsilon
b^{4}\sqrt{\frac{G}{\big{\\{}1+C(bc_{1}+Cc_{2})m^{2}\big{\\}}\big{[}B+2\log(b)+2Cm^{2}\big{\\{}bc_{1}+Cc_{2}\log(b)\big{\\}}\big{]}}}+\frac{\epsilon^{2}}{2}\mathcal{J}(b)\right],$
(50)

where the expression of $\mathcal{J}(b)$ is given by,

$\displaystyle\mathcal{J}(b)$ $\displaystyle=$
$\displaystyle\frac{1}{2b^{3}G}\Big{[}\frac{b^{4}G}{\big{\\{}1+C(bc1+Cc_{2})m^{2}\big{\\}}\big{\\{}B+2bCc_{1}m^{2}+2(1+C^{2}c_{2}m^{2})\log(b)\big{\\}}}\Big{]}^{\frac{3}{2}}\times\bigg{[}12b^{2}C^{2}c_{1}^{2}m^{4}$
(51)
$\displaystyle+2\big{(}4B-C^{2}c_{2}m^{2}-1\big{)}(1+C^{2}c_{2}m^{2})+bCc_{1}m^{2}(12+7B+12C^{2}c_{2}m^{2})$
$\displaystyle+2(1+C^{2}c_{2}m^{2})\big{\\{}8+C(7bc_{1}+8Cc_{2})m^{2}\big{\\}}\log(b)\bigg{]}.$

We have shown the behavior of the entropy of the gravastar with respect to its
thickness $\epsilon$ by keeping a fixed value for $m^{2}c_{2}$ in Fig. 3 for
different values of $m^{2}c_{1}$ and vice versa. It is observed that as
massive gravity parameters $m^{2}c_{1}$ and $m^{2}c_{2}$ grow, the entropy
goes down but increases as $\epsilon$ increases.

###  V.3 The EoS parameter

At $r=b$, the equation of state parameter can be calculated as,

$\displaystyle\omega(b)=\frac{\mathcal{P}(b)}{\sigma(b)},$ (52)

Now using the eqns. (IV) and (39), we obtain the expression for $\omega(a)$
as,

$\displaystyle\omega(a)=\frac{1}{2}\left[\frac{\frac{1-\frac{\mathcal{M}}{b}}{\sqrt{1-\frac{2\mathcal{M}}{b}}}-\frac{12+3Cm^{2}(c_{1}+4Cc_{2}+2c_{1}b)-4kb(1+b)\rho_{c}}{12\sqrt{1+m^{2}C(c_{2}C+\frac{c_{1}b}{2})-k\rho_{c}\frac{b^{2}}{3}}}}{\sqrt{1+m^{2}C(c_{2}C+\frac{c_{1}b}{2})-k\rho_{c}\frac{b^{2}}{3}}-\sqrt{1-\frac{2\mathcal{M}}{b}}}\right].$

###  V.4 Energy within the thin shell

The energy within the thin shell can be calculated as,

$\displaystyle\mathcal{E}$ $\displaystyle=$
$\displaystyle\int_{b}^{b+\epsilon}4\pi r^{2}\rho dr,$ (54) $\displaystyle=$
$\displaystyle\frac{G\pi}{15C^{7}c_{1}^{7}m^{14}}\Big{[}Cc_{1}m^{2}r\Big{\\{}-60(1+C^{2}c_{2}m^{2})^{5}+30Cc_{1}m^{2}(1+C^{2}c2m^{2})^{4}r-20C^{2}c_{1}^{2}m^{4}(1+C^{2}c2m^{2})^{3}r^{2}$
$\displaystyle+15C^{3}c_{1}^{3}m^{6}(1+C^{2}c2m^{2})^{2}r^{3}-12C^{4}c_{1}^{4}m^{8}(1+C^{2}c_{2}m^{2})r^{4}+10C^{5}c_{1}^{5}m^{10}r^{5}\Big{\\}}$
$\displaystyle+60(1+C^{2}c_{2}m^{2})^{6}\log\big{(}1+Cm^{2}(Cc_{2}+c_{1}r)\big{)}\Big{]}_{b}^{b+\epsilon}$

The graphical representation of the energy within the thin shell is shown in
Fig. 4. Our findings in the aforementioned figures demonstrate that the energy
within the shell and shell thickness have a linear connection, and that the
energy tends to decrease as the associated $m^{2}c_{2}$ and $m^{2}c_{1}$
values increase, respectively in both the figures.

Figure 4: Variation of the gravastar’s internal energy versus shell thickness.

##  VI Stability of the gravastar

The stability of the gravastar is investigated here by using a parameter
$\eta$ that can be described as the ratio of the surface pressure and surface
density differentials, respectively, determined at the hypersurface $r=b$.
Symbolically,

$\displaystyle\eta(b)=\frac{\mathcal{P^{\prime}}(b)}{\sigma^{\prime}(b)}.$
(55)

Analysis of the stability of the thin shell wormhole under small radial
perturbations has been reported by Poisson and Visser Poisson:1995sv . They
have also obtained the stability region. In the implementation of
noncommutative geometry, Övgün et al. Ovgun:2017jzt  employed the
aforementioned parameter, which is essential in identifying the stability
regions of the corresponding solutions of the charged thin-shell gravastar
model. By incorporating eta, Debnath also examined the stability of charged
gravastars under Rastall-Rainbow gravity in ref. Debnath:2019eor . Using the
same method, Yousaf et al. Yousaf:2019zcb  also investigated the stability of
the charged gravastar in $f(R,\,T)$ gravity. This criterion was also applied
by Bhar and Rej Bhar:2021oag  to investigate the stability of a charged
gravastar in the presence of conformal motion within the context of $f(R,\,T)$
gravity. $\eta$ may be supposed to be understood as the speed of sound in
normal scenario. In addition, normally, eta would have to fall within the
interval $\eta~{}\in(0,\,1)$. However, the range of eta might be outside the
previously stated range on the surface layer, as suggested by Poisson & Visser
Poisson:1995sv  and Lobo & Crawford Lobo:2003xd . For our present model, the
profile of $\eta$ is shown in Fig. 5. The stability region lies on the upper
side of the curve as proposed by Yousaf:2019zcb ; Ovgun:2017jzt .

Figure 5: Stability regions of the charged gravastar.

Figure 6: The contour plots of different physical properties.

##  VII Some physical properties of Gravastar through contour plots

In this section we are interested to study some physical properties namely
proper length ($\mathcal{L}$), energy content ($\mathcal{E}$) and entropy
($\mathcal{S}$) of the gravastar model with the help of contour plots in Fig.
6 in $\left\\{m^{2}c_{1},\,m^{2}c_{2}\right\\}$ plane. For all the three
physical parameters, one can check that for a fixed value of $m^{2}c_{2}$ all
of the physical quantities proper length ($\mathcal{L}$), energy content
($\mathcal{E}$) and entropy ($\mathcal{S}$) decrease with the increasing value
of $m^{2}c_{1}$ and the same situation arises when we vary $m^{2}c_{1}$ by
keeping $m^{2}c_{2}$ as fixed.

##  VIII Discussions and final remarks

In this work, we looked into the fundamental physical characteristics of an
uncharged gravastar in the circumstances of dRGT like massive gravity. The
field equations of dRGT like massive gravity are solved in three distinct
regions of the gravastar by using three separate equations of state. Our
primary objective is to examine how different parameters
($C,\,m^{2}c_{1},\,m^{2}c_{2}$) affect the gravastar structure. It has been
demonstrated both analytically and graphically that the presence of massive
graviton modifies the physical characteristics of the gravastar. Our study
shows that in the interior region of the gravastar, the metric coefficients
depend on both $m^{2}c_{1}$ and $m^{2}c_{2}$, but the massive gravity
parameters have no effect on pressure and density. Conversely, in the thin
shell region, the massive gravity factors influence the density, pressure, and
metric coefficients. Interestingly, every solution is positive inside the
gravastar’s interior region and regular at the centre $r=0$. The mathematical
results demonstrate the maximum mass decrease when both $m^{2}c_{1}$ and
$m^{2}c_{2}$ values increase. The highest mass might exceed $2M_{\odot}$. From
the tables 1-2 one can note that the maximal masses are respectively
$2.13~{}M_{\odot}$ and $2.346~{}M_{\odot}$ when
$C=0.5,\,m^{2}c_{1}=0.1,\,m^{2}c_{2}=-0.25$, and
$C=0.5,\,m^{2}c_{2}=-0.5,\,m^{2}c_{1}=0.1$.
We would like to draw attention to Barzegar et al.’s work in this regard. The
authors of this research examine the AdS scenario for 3D gravastars within the
framework of dRGT-like massive gravity. Our obtained result closely resembles
that of Barzegar et al. In the current work, we have taken the thin shell
approximation up to the second order, which yields findings for the physical
properties of the shell that are more accurate. Interestingly, the forms of
various physical quantities, such as energy, entropy, proper length, etc.,
have changed under the framework of dRGT like massive gravity, compared to
Einstein’s General Relativity. In particular, the gravastar structure reverts
to the general relativity scenario when the graviton mass is zero and we
restrict ourselves to the first-order approximation of the thickness parameter
$\epsilon$ of the shell. From all of the aforementioned investigations, it is
clear that the gravastar may exist in the background of dRGT like massive
gravity.

## Acknowledgments

P.B. is thankful to the Inter University Centre for Astronomy and Astrophysics
(IUCAA), Government of India, for providing visiting associateship.

## References

  * (1) P. O. Mazur and E. Mottola, Proc. Nat. Acad. Sci. 101 (2004), 9545-9550 doi:10.1073/pnas.0402717101 [arXiv:gr-qc/0407075 [gr-qc]].
  * (2) P. O. Mazur and E. Mottola, Universe 9 (2023) no.2, 88 doi:10.3390/universe9020088 [arXiv:gr-qc/0109035 [gr-qc]].
  * (3) D. Horvat and S. Ilijic, Class. Quant. Grav. 24 (2007), 5637-5649 doi:10.1088/0264-9381/24/22/022 [arXiv:0707.1636 [gr-qc]].
  * (4) D. Horvat, S. Ilijic and A. Marunovic, Class. Quant. Grav. 26 (2009), 025003 doi:10.1088/0264-9381/26/2/025003 [arXiv:0807.2051 [gr-qc]].
  * (5) R. Chan, M. F. A. da Silva and P. Rocha, JCAP 12 (2009), 017 doi:10.1088/1475-7516/2009/12/017 [arXiv:0910.2054 [gr-qc]].
  * (6) D. Horvat, S. Ilijic and A. Marunovic, Class. Quant. Grav. 28 (2011), 195008 doi:10.1088/0264-9381/28/19/195008 [arXiv:1104.3537 [gr-qc]].
  * (7) S. Ghosh, R. Sengupta and M. Kalam, [arXiv:2311.00692 [gr-qc]].
  * (8) S. Nojiri and G. G. L. Nashed, [arXiv:2310.16068 [gr-qc]].
  * (9) D. Bhattacharjee and P. K. Chattopadhyay, Phys. Scripta 98 (2023) no.8, 085013 doi:10.1088/1402-4896/ace483 [arXiv:2303.02592 [gr-qc]].
  * (10) M. Sharif and S. Naz, Eur. Phys. J. Plus 137 (2022) no.4, 421 doi:10.1140/epjp/s13360-022-02636-4 [arXiv:2204.01523 [gr-qc]].
  * (11) S. Ray, S. Ghosh and R. Sengupta, Mod. Phys. Lett. A 37 (2022) no.29, 2250195 doi:10.1142/S0217732322501954 [arXiv:2203.12027 [gr-qc]].
  * (12) R. Sengupta, S. Ghosh, S. Ray, B. Mishra and S. K. Tripathy, Phys. Rev. D 102 (2020) no.2, 024037 doi:10.1103/PhysRevD.102.024037 [arXiv:2004.01519 [gr-qc]].
  * (13) J. D. V. Arbañil, P. H. R. S. Moraes and M. Malheiro, Class. Quant. Grav. 36 (2019) no.23, 235012 doi:10.1088/1361-6382/ab47d6 [arXiv:1909.11502 [gr-qc]].
  * (14) A. A. Usmani, F. Rahaman, S. A. Rakib, S. Ray, K. K. Nandi, P. K. F. Kuhfittig and Z. Hasan, Phys. Lett. B 701 (2011), 388-392 doi:10.1016/j.physletb.2011.06.001 [arXiv:1012.5605 [gr-qc]].
  * (15) P. Bhar, Astrophys. Space Sci. 354 (2014) no.2, 2109 doi:10.1007/s10509-014-2109-2
  * (16) S. Ghosh, F. Rahaman, B. K. Guha and S. Ray, Phys. Lett. B 767 (2017), 380-385 doi:10.1016/j.physletb.2017.02.014 [arXiv:1511.05417 [physics.gen-ph]].
  * (17) A. Errehymy, M. Daoud and M. K. Jammari, Eur. Phys. J. Plus 132 (2017) no.11, 497 doi:10.1140/epjp/i2017-11777-0
  * (18) P. Pani, Phys. Rev. D 92 (2015) no.12, 124030 [erratum: Phys. Rev. D 95 (2017) no.4, 049902] doi:10.1103/PhysRevD.95.049902 [arXiv:1506.06050 [gr-qc]].
  * (19) C. Chirenti and L. Rezzolla, Phys. Rev. D 94 (2016) no.8, 084016 doi:10.1103/PhysRevD.94.084016 [arXiv:1602.08759 [gr-qc]].
  * (20) M. Rutkowski and A. Rostworowski, Phys. Rev. D 104 (2021) no.8, 084041 doi:10.1103/PhysRevD.104.084041 [arXiv:2009.12504 [gr-qc]].
  * (21) A. Das, S. Ghosh, D. Deb, F. Rahaman and S. Ray, Nucl. Phys. B 954 (2020), 114986 doi:10.1016/j.nuclphysb.2020.114986 [arXiv:2003.10842 [gr-qc]].
  * (22) Z. Yousaf, M. Z. Bhatti and H. Asad, Phys. Dark Univ. 28 (2020), 100527 doi:10.1016/j.dark.2020.100527 [arXiv:2003.04895 [gr-qc]].
  * (23) A. Das, S. Ghosh, B. K. Guha, S. Das, F. Rahaman and S. Ray, Phys. Rev. D 95 (2017) no.12, 124011 doi:10.1103/PhysRevD.95.124011 [arXiv:1702.08873 [gr-qc]].
  * (24) H. Barzegar, M. Bigdeli, G. H. Bordbar and B. Eslam Panah, Eur. Phys. J. C 83 (2023) no.2, 151 doi:10.1140/epjc/s10052-023-11295-3 [arXiv:2302.02433 [gr-qc]].
  * (25) F. Rahaman, A. A. Usmani, S. Ray and S. Islam, Phys. Lett. B 717 (2012), 1-5 doi:10.1016/j.physletb.2012.09.010 [arXiv:1205.6796 [physics.gen-ph]].
  * (26) S. Ghosh, S. Biswas, F. Rahaman, B. K. Guha and S. Ray, Annals Phys. 411 (2019), 167968 doi:10.1016/j.aop.2019.167968
  * (27) S. Ghosh, D. Shee, S. Ray, F. Rahaman and B. K. Guha, Results Phys. 14 (2019), 102473 doi:10.1016/j.rinp.2019.102473
  * (28) F. Rahaman, S. Chakraborty, S. Ray, A. A. Usmani and S. Islam, Int. J. Theor. Phys. 54 (2015) no.1, 50-61 doi:10.1007/s10773-014-2198-2 [arXiv:1209.6291 [physics.gen-ph]].
  * (29) A. DeBenedictis, D. Horvat, S. Ilijic, S. Kloster and K. S. Viswanathan, Class. Quant. Grav. 23 (2006), 2303-2316 doi:10.1088/0264-9381/23/7/007 [arXiv:gr-qc/0511097 [gr-qc]].
  * (30) M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173 (1939), 211-232 doi:10.1098/rspa.1939.0140
  * (31) P. Van Nieuwenhuizen, Nucl. Phys. B 60 (1973), 478-492 doi:10.1016/0550-3213(73)90194-6
  * (32) H. van Dam and M. J. G. Veltman, Nucl. Phys. B 22 (1970), 397-411 doi:10.1016/0550-3213(70)90416-5
  * (33) V. I. Zakharov, JETP Lett. 12 (1970), 312
  * (34) A. I. Vainshtein, Phys. Lett. B 39 (1972), 393-394 doi:10.1016/0370-2693(72)90147-5
  * (35) D. G. Boulware and S. Deser, Phys. Rev. D 6 (1972), 3368-3382 doi:10.1103/PhysRevD.6.3368
  * (36) C. de Rham and G. Gabadadze, Phys. Rev. D 82 (2010), 044020 doi:10.1103/PhysRevD.82.044020 [arXiv:1007.0443 [hep-th]].
  * (37) C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. Lett. 106 (2011), 231101 doi:10.1103/PhysRevLett.106.231101 [arXiv:1011.1232 [hep-th]].
  * (38) S. Panpanich and P. Burikham, Phys. Rev. D 98 (2018) no.6, 064008 doi:10.1103/PhysRevD.98.064008 [arXiv:1806.06271 [gr-qc]].
  * (39) T. M. Nieuwenhuizen, Phys. Rev. D 84 (2011), 024038 doi:10.1103/PhysRevD.84.024038 [arXiv:1103.5912 [gr-qc]].
  * (40) R. Brito, V. Cardoso and P. Pani, Phys. Rev. D 88 (2013), 064006 doi:10.1103/PhysRevD.88.064006 [arXiv:1309.0818 [gr-qc]].
  * (41) L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. D 85 (2012), 044024 doi:10.1103/PhysRevD.85.044024 [arXiv:1111.3613 [hep-th]].
  * (42) E. Babichev and A. Fabbri, JHEP 07 (2014), 016 doi:10.1007/JHEP07(2014)016 [arXiv:1405.0581 [gr-qc]].
  * (43) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 116 (2016) no.6, 061102 doi:10.1103/PhysRevLett.116.061102 [arXiv:1602.03837 [gr-qc]].
  * (44) B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 119 (2017) no.16, 161101 doi:10.1103/PhysRevLett.119.161101 [arXiv:1710.05832 [gr-qc]].
  * (45) R. G. Cai, Y. P. Hu, Q. Y. Pan and Y. L. Zhang, Phys. Rev. D 91 (2015) no.2, 024032 doi:10.1103/PhysRevD.91.024032 [arXiv:1409.2369 [hep-th]].
  * (46) W. Israel, Nuovo Cim. B 44S10 (1966), 1 [erratum: Nuovo Cim. B 48 (1967), 463] doi:10.1007/BF02710419
  * (47) V. A. Berezin, V. A. Kuzmin and I. I. Tkachev, Phys. Rev. D 36 (1987), 2919 doi:10.1103/PhysRevD.36.2919
  * (48) E. Poisson and M. Visser, Phys. Rev. D 52 (1995), 7318-7321 doi:10.1103/PhysRevD.52.7318 [arXiv:gr-qc/9506083 [gr-qc]].
  * (49) A. Övgün, A. Banerjee and K. Jusufi, Eur. Phys. J. C 77 (2017) no.8, 566 doi:10.1140/epjc/s10052-017-5139-4 [arXiv:1704.00603 [gr-qc]].
  * (50) U. Debnath, Eur. Phys. J. Plus 136 (2021) no.4, 442 doi:10.1140/epjp/s13360-021-01460-6 [arXiv:1909.01139 [physics.gen-ph]].
  * (51) Z. Yousaf, K. Bamba, M. Z. Bhatti and U. Ghafoor, Phys. Rev. D 100 (2019) no.2, 024062 doi:10.1103/PhysRevD.100.024062 [arXiv:1907.05233 [gr-qc]].
  * (52) P. Bhar and P. Rej, Eur. Phys. J. C 81 (2021) no.8, 763 doi:10.1140/epjc/s10052-021-09548-0 [arXiv:2108.06989 [gr-qc]].
  * (53) F. S. N. Lobo and P. Crawford, Class. Quant. Grav. 21 (2004), 391-404 doi:10.1088/0264-9381/21/2/004 [arXiv:gr-qc/0311002 [gr-qc]].