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11institutetext: Departamento de Matematicas, IES Carrus, Elche 03205,
Alicante, Spain. 22institutetext: Department of Physics, National Defence
Academy, Khadakwasla, Pune-411023, India. 33institutetext: Department of
Mathematics, Jadavpur University, Kolkata-700032, West Bengal, India

# Testing of $\kappa(\mathcal{R},\mathcal{T})-$gravity through gravastar
configurations

G R P Teruele1,addr1  Ksh. Newton Singh e5,addr2  Tanmoy Chowdhury e4,addr3
Farook Rahaman e2,addr3  Monimala Mondal e3,addr3

(Received: date / Accepted: date)

###### Abstract

In this article, we are reporting for the first time the existence of
gravastar configurations in the framework of
$\kappa(\mathcal{R},\mathcal{T})-$ gravity, which can be treated as an
alternative to a black hole (Mazur and Mottola). This strengthens how much
this new gravity theory may be physically demanding to the gravity community
in the near future. We first develop the gravastar field equations for a
generic $\kappa(\mathcal{R},\mathcal{T})$ functional and then we study four
different models within this theory. We find that the solutions for the
interior region are regular everywhere regardless of the exact form of the
$\kappa(\mathcal{R},\mathcal{T})$ functional. The solutions for the shell
region indicate that two of the four models subjected to the study are
physically feasible. In addition, the junction conditions are considered at
each interface by using the Lanczos equations that yield the surface density
and pressure at the thin shell. We investigate various characteristics of the
gravastar structure such as the proper length, energy, and entropy of the
spherical distribution.

###### Keywords:

Gravastar $\kappa(\mathcal{R},\mathcal{T})-$ gravity Junction conditions

††journal: Eur. Phys. J. C

##  1 Introduction

The hierarchy of principles in physics is an old debate that should be revived
from time to time to ask ourselves if we are going in the right direction. On
the one hand, there are foundational physical principles, such as the
principle of inertia in classical mechanics, the uncertainty principle in
quantum mechanics, or the principle of relativity that directly founded or
contributed to establishing new scientific theories. On the other hand, there
are the auxiliary principles, which are not as fundamental as the first ones,
but help to strengthen our theories once they have been established by the
foundational principles. It is natural to wonder: To which of these two
categories does the principle of least action belong? At first glance, one
would answer that the least action principle should be an undisputed part of
the first group, i.e., a foundational one due to its ubiquitous presence in
our current theories. However, let’s carefully examine the history of physics.
We will appreciate that the variational principles, despite their enormous
success and importance, were not founding principles, in the sense that they
did not establish new scientific paradigms, although they turned out to be
versatile and flexible enough to know how to adapt to every paradigm shift in
theoretical physics.

In particular, if we examine the birth of the two major revolutions of the
twentieth century, the theories of relativity and quantum mechanics, we will
see that the least action principle had a secondary role reig ; Hei . In fact,
other principles were the foundational building blocks of these theories Ein ;
Hei2 . The action principle was adapted later to them and arrived to enrich
their formalism. Analogously, neither of the two great classical field
theories was discovered by means of the least action principle. Indeed,
Classical Electrodynamics (CE) was fulfilled with the inclusion of a source
term directly into the field equations (the Maxwell displacement current).
General relativity (GR) was also first derived in this way, after adding the
key trace term to the field equations hil15 ; ren07 . In both cases, the least
action principle came to strengthen the formalism after other more fundamental
principles had established the theories.

These historical lessons persuade us that the search for possible Non-
Lagrangian theories, that is, theories not based (at least initially) on the
variational method, should not be ruled out a priori.

In 2001, Mazur and Mottola Mazur01 ; Mazur04  proposed a very interesting and
encouraging stellar model, named gravitational vacuum star (or gravastar) as
an alternating black hole structure. In recent days, the study on gravastar
has become a considerably interesting topic as an alternating black hole
solution. Mazur and Mottola Mazur01 ; Mazur04  first introduced a novel
solution for the extreme point of gravitational collapse of cold and dark
matter, a neutral system, and compact stars. Therefore, this is an extended
form of Bose-Einstein condensation whose radius is similar to the
Schwarzschild radius. The total design of a gravastar can be defined by means
of 3 different zones with distinct EoS:

I. Interior: $0<r<r_{1}$ ( $p=-\rho$),

II. Shell: $r_{1}<r<r_{2}$ ($p=\rho$),

III. Exterior: $r_{2}<r$ ($p=\rho=0$).

Several researchers have already studied gravastar from different perspectives
and scooped out a new field as an alternative to the black hole paradigm.
Mazur01 ; Mazur04 ; visser04 ; cattoen05 ; Bilic06 ; Lobo07 ; Rocha08 ;
Horvat07 ; Nandi09 ; Usmani11 ; FR15 . Lack of observational evidence in
Einstein’s GR on the oscillating model of the universe alongside the presence
of dark matter has forced a conceptual challenge to this theory Chan ; C.F.C ;
R.CHAN ; D.J .

The repulsive nature of dark energy in GR requires considering an exotic fluid
with a negative pressure to explain the cosmic speed-up. Many modified
Lagrangian theories of gravity have been proposed to account for the
accelerating phase of the universe. These theories are based on the
generalization of the Einstein-Hilbert action viz $f(\mathbb{T},\mathcal{T})$
TH14 , $f(\mathbb{T})$ v.c , $f(\mathcal{R},\mathcal{T})$ Harko , EGB Lovelock
, $f(G)$ N.M.G , $f(G,\mathcal{T})$ MS , $f(Q,\mathcal{T})$ Y.X  gravity
theories, etc, which earned wide attention in recent years.

A major role of modified gravity theories is to test the validity of the
gravastar model by examining various realistic properties of these compact
objects. Barzegar et al. H.B  explored the 3D AdS gravitational vacuum stars
in the framework of gravity’s rainbow theory. Ghosh et al. S.G20  investigated
gravastars in $f(\mathbb{T},\mathcal{T})$ gravity and enlarged its research in
Rastall gravity as well S.G21 . Das et al. A.DAS17 , investigated the
formation of gravastars in $f(\mathcal{R},\mathcal{T})$ gravity and inspect
its theoretical efficiency. Enlarge the analysis Yousaf et al., recently
explored gravastars in $f(\mathcal{R},\mathcal{T},R_{\mu\nu}T^{\mu\nu})$
gravity Z.Y20 . Bhatti et al. M.Z20  reviewed charged gravastar for
spherically and cylindrically symmetric spacetimes and in $f(\mathcal{R},G)$
gravity M.Z.B2020 . Extending it, Yousaf ZY20  constructed charged cylindrical
gravastar-like structures in $f(\mathcal{R},\mathcal{T})$ gravity. Lobo and
Garattini F.S  analyzed gravastar solutions in the frame of non-invariant
geometry with their physical properties and characteristics. Bhatti et al. MZ
investigated the stability of this thin shell together with the
thermodynamical stability of locally isotropic gravastars with cylindrical
space-time. Bhatti et al. MZ21  talked over the symmetric gravastar model
within the framework of the modification of Gauss-Bonnet gravity. Das et al.
ad20  discussed the physically interesting and valid features within the
framework of the $f(\mathbb{T})$ theory of gravity.

Horvet et al. Horvat  used dominant force conditions to initialize the
stability of the gravastar after combining a sufficient external vacuum
solution with an internal geometry and meeting some possible constraints.
Also, they examined the formation of this type of compact object under the
influence of electromagnetism. Rahaman and others rahaman  analyzed the
electrically charged gravastar model considering the $(2+1)-$ dimensional
geometry. For a stable model of a spherically symmetric gravastar, they also
explored some viable properties as well as energy components, length, and
entropy of the gravastar. Ghosh et al. ghosh  investigated some new diagrams
in the formation of these matters with and without electric charge in high-
order manifolds. Ghosh et al. Ghosh19  discussed gravastar layout in the
framework of the Kuchowicz metric potential. Ghosh et al. Sg 19  figure out
the gravastar model under Einstein’s GR in ($3+1$) dimensions by incorporating
the Karmarkar condition. Sumita et al. sumita  studied gravastars under
Finslerian spacetime geometry. Sengupta et al. R.S20  investigated a gravastar
configuration in RS Brane gravity, while Ray et al. S.RAY20  reviewed a very
interesting paper on generalizations of gravastars in GR, including studies of
higher and lower dimensional GR with physical insights like the Randall-
Sundrum theory, and several modified gravity models such as
$f(\mathcal{R},\mathcal{T})$ theory or Rastall-Rainbow gravity.

Recently, a modified theory was invented gr18  as
$\kappa(\mathcal{R},\mathcal{T})-$ gravity. The conceptual framework of this
theory is not based on the standard modified gravity program. Rather, its
insight is inspired by Maxwell’s and Einstein’s ideas of including new
possible potential terms in the field equations. In this sense, we must recall
Maxwell’s addition to Ampere’s law of the displacement current term to
complete the Electromagnetic field equations, and the introduction of the key
term, $\frac{1}{2}\mathcal{R}\,g^{\mu\nu}$ by Einstein to consummate the GR
field equations hil15 ; ren07 . Despite the fact that the variational
principle is often regarded as our main tool to formulate a new physical
theory (and its generalizations), ought not to have been placed at the same
fundamental level as other truly first principles, like the equivalence
principle and the principle of general covariance, that are the two core
principles of GR. In spite of this, the variational method has reached the
pinnacle in our hierarchy of physical principles, becoming dominant to the
extent of being the starting point for any modification of GR. Indeed, the
vast majority of the modified gravity theories that are available in the
market, like the examples mentioned above, are Lagrangian in the sense that
they arise from the generalization of the Einstein–Hilbert action. The
overpopulation of alternatives suggests that some other fundamental principle
(beyond the equivalence principle and the principle of general covariance) is
required to guide us trough the deep jungle of possible Lagrangian theories.
The fact is, we have no reason to assure that general symmetries and general
conservation laws can always exist (in their current form) in the final
theories of nature, which moves us to think that the Non-Lagrangian proposals
should also be explored. In fact, according to Hojman Hoj , the use of
Lagrangian and Hamiltonian functions is not an essential requirement to derive
conservation theorems.

$\kappa(\mathcal{R},\mathcal{T})-$gravity is a recent proposal, and so its
implications have not been explored well enough. However, for the past few
years, some works are dedicated to studying its cosmic aspects. In particular,
Pradhan and Ahmed ahm22  showed that the theory can deal with the current
scenario of an accelerating universe, claiming that the theory requires a very
small value of the cosmological constant, which agrees with observations.
Pradhan et al. arc22  further extended this study to include a more complete
cosmological scenario, while Dixit et al. arc23  studied the thermodynamics of
the expansion of the cosmos in the context of this theory. Sarkar et al.
sark19 , explored a model of wormhole in $\kappa(\mathcal{R},\mathcal{T})-$
gravity, while Teruel et al. GRP  introduced the first internal solutions
representing compact stars in $\kappa(\mathcal{R},\mathcal{T})$-gravity by
solving the field equations in isotropic coordinates. Also very recently,
Taṣer and Dogru Tas23  investigated whether the Krori-Barua model produces
valid results in this theory. All these papers have one remarkable common
feature: they were devoted to studying only a particular case of the theory
with
$\kappa(\mathcal{R},\mathcal{T})\equiv\kappa(\mathcal{T})=8\pi-\lambda\mathcal{T}$.

Up to now, nobody studied gravastar solutions in
$\kappa(\mathcal{R},\mathcal{T})-$ gravity, it would therefore be interesting
to investigate and discuss whether this theory can support a consistent
gravastar configuration from a physical point of view. In this sense, four
different models, i.e., four different choices of the running
$\kappa(\mathcal{R},\mathcal{T})-$ gravitational constant are analyzed in this
investigation. Therefore, we don’t restrict ourselves only to the
$\kappa(\mathcal{T})$ choice like all the previous works, but extend the study
to the more general possible case. Indeed, once derived the equations for a
generic $\kappa(\mathcal{R},\mathcal{T})$ functional, we see how the theory
would look for specific cases.

The structure of the paper is as follows. We start in section 2 with the basic
framework of the theory. Section 3 deals with the field equations derived from
the Morris-Thorne metric for the general $\kappa(\mathcal{R},\mathcal{T})$
case. The solutions for the interior region of the gravastar are discussed in
section 4. The shell region of the gravastar is analyzed in section 5, where
we study the physical acceptability of 4 different models within the theory.
Entropy within the shell is addressed in section 6. The junction interface and
surface stresses are the subjects of section 7. The brief section 8 deals with
the surface redshift within the thin shell. Finally, in section 9, we
summarize the findings and discuss the conclusions of the work.

##  2 Framework of $\kappa(\mathcal{R},\mathcal{T})-$ gravity

The structure of the theory depends on the following field equations

$R_{\mu\nu}-\frac{1}{2}\mathcal{R}~{}g_{\mu\nu}-\Lambda
g_{\mu\nu}=\kappa(\mathcal{R},\mathcal{T})\,T_{\mu\nu},$ (1)

where $R_{\mu\nu}$ is the Ricci tensor, $g_{\mu\nu}$ is the space-time metric,
$\Lambda$ corresponds to the cosmological constant, $T_{\mu\nu}$ is the
stress-energy tensor of the material content, while
$\kappa(\mathcal{R},\mathcal{T})$ is a generalization of Einstein’s
gravitational constant, that we extend to the status of a function of the
traces $\mathcal{T}\equiv g_{\mu\nu}T^{\mu\nu}$, and $\mathcal{R}\equiv
g_{\mu\nu}R^{\mu\nu}$. The introduction of the functional
$\kappa(\mathcal{R},\mathcal{T})$, means that we inspect the possibility of a
running gravitational constant, although not at the level of the variational
method. The possibility of considering a variable Einstein’s gravitational
constant in the action was studied many years ago by Brans and Dicke brans1 ;
brans2 , and the resulting theory is quite different from (1).

The field equations (1) imply the non-covariant conservation of $T_{\mu\nu}$.
Indeed, since the divergence of the left-hand side of Einstein’s field
equation vanishes, we have

$\nabla{{}^{\nu}}\Big{[}\kappa(\mathcal{R},\mathcal{T})\,T_{\mu\nu}\Big{]}=0.$
(2)

Then, this non-conservation of the $T_{\mu\nu}$ can be expressed, for
$\kappa(\mathcal{R},\mathcal{T})\neq 0$, as

$\nabla^{\nu}T_{\mu\nu}=-\frac{\nabla^{\nu}\kappa(\mathcal{R},\mathcal{T})}{\kappa(\mathcal{R},\mathcal{T})}~{}T_{\mu\nu},~{}~{}~{}~{}~{}\forall~{}~{}~{}~{}\kappa(\mathcal{R},\mathcal{T})\neq
0.$ (3)

When $\kappa(\mathcal{R},\mathcal{T})=0$, the right-hand side of Einstein’s
field equations vanishes. This interesting case probably may happen in some
specific models in the early universe for sufficiently high densities, and it
will imply an exponential expansion driven by a cosmological constant. Of
course, the theory can be recast into a conservative form by defining a new
(effective) stress-energy tensor

$S_{\mu\nu}=\kappa(\mathcal{R},\mathcal{T})\,T_{\mu\nu}.$ (4)

The field equations then take the form

$R_{\mu\nu}-\frac{1}{2}\mathcal{R}~{}g_{\mu\nu}-\Lambda\,g_{\mu\nu}=S_{\mu\nu},$
(5)

and the Bianchi identities imply that Eq. (2) is

$\nabla^{\nu}S_{\mu\nu}=0.$ (6)

Then, this theory has the same formal structure as GR, with a non-trivial
modification of the material content. Examples of famous non-conservative
gravitational theories are Rastall’s gravity Ras , or the Lagrangian
$f(\mathcal{R},\mathcal{T})$ theory by Harko et al. Har .

Lindblom and Hisock Lind , and also Visser Viss  more recently, criticized
some non-conservative gravity theories (in particular the Rastall case),
claiming that they are formally identical to Einstein’s gravity and that one
can always build a conserved effective stress-energy tensor
$T^{eff}_{\mu\nu}$, which is constructed solely from the matter sources.
Furthermore, they underline the fact that, if the effective stress-energy
tensor of the matter sources does not depend on the space-time curvature,
(such is the case of Rastall gravity), then these theories will not represent
truly alternative theories of gravity, reducing the question of the non-
conservation of the stress-energy tensor to the domain of special relativity.
Nevertheless, since the effective stress-energy tensor of
$\kappa(\mathcal{R},\mathcal{T})$ gravity is not in general determined by the
matter sources solely (it can have a dependence on the space-time curvature as
well via the trace $\mathcal{R}$), we conclude that
$\kappa(\mathcal{R},\mathcal{T})$ theory represents a true generalization of
GR and not only a redefinition of its matter sector. Some relevant features of
the theory are the following:

  * •

The modification procedure involves the right-hand side of Einstein’s field
equations, that is, only the material content sector is generalized.
Therefore, the equations will be second order in the metric coefficients, and
the theory will be free of typical instabilities that plagued many of the
higher-order gravitational theories.

  * •

Since the pure geometrical sector is the same as GR, the theory boils down to
GR in the absence of matter sources.

  * •

The dependence on $\mathcal{T}$ means that, for particular cases of the type
$\kappa(\mathcal{T})=8\pi-\lambda\mathcal{T}$, or for more general
functional111This is also true for a generic functional
$\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}f(\mathcal{T})$, with
$\gamma$ a real constant and $f(\mathcal{T})$ an arbitrary polynomial function
of $\mathcal{T}$., that directly couples the $\mathcal{R}$ and $\mathcal{T}$
traces, the theory would predict the same physics as GR when coupled to
standard (traceless) electromagnetic fields. In such cases, only significant
departures are expected for non-linear electrodynamics, where $\mathcal{T}\neq
0$.

##  3 Field Equations in $\kappa(\mathcal{R},\mathcal{T})-$ gravity

The interior of the gravastar is taken to be as Morris-Thorne Spacetime given
by

$\displaystyle
ds^{2}=-e^{2f}dt^{2}+\left[1-\frac{b}{r}\right]^{-1}dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2}).$
(7)

For $\Lambda=0$, the Einstein field equations for a general
$\kappa(\mathcal{R},\mathcal{T})$ model will be

$\displaystyle\rho(r)~{}\kappa(\mathcal{R},\mathcal{T})$ $\displaystyle=$
$\displaystyle\frac{b^{\prime}(r)}{r^{2}},$ (8) $\displaystyle
p(r)~{}\kappa(\mathcal{R},\mathcal{T})$ $\displaystyle=$
$\displaystyle\frac{2f^{\prime}(r)}{r}\left(1-\frac{b(r)}{r}\right)-\frac{b(r)}{r^{3}}.$
(9)

On the other hand, the non-covariant conservation of the stress-energy tensor
given by Eq. (2) acquires the form

$p(r)~{}\frac{d\kappa(\mathcal{R},\mathcal{T})}{dr}+\kappa(\mathcal{R},\mathcal{T})\left(\frac{df(r)}{dr}\\{\rho(r)+p(r)\\}+\frac{dp(r)}{dr}\right)=0.$
(10)

##  4 Regular Interior Solutions in $\kappa(\mathcal{R},\mathcal{T})-$ gravity
with $p=-\rho$

In this section, we show that the interior solutions for a gravastar
configuration are regular everywhere in $\kappa(\mathcal{R},\mathcal{T})$
theory, regardless of the specific choice of the running gravitational
constant. To see how this interesting property of the theory arises, notice
that for $p(r)=-\rho(r)$, Eq. (10) reduces to

$\rho(r)~{}\frac{d\kappa(\mathcal{R},\mathcal{T})}{dr}+\kappa(\mathcal{R},\mathcal{T})~{}\frac{d\rho(r)}{dr}=0,$
(11)

which further takes the form

$\frac{d}{dr}\Big{[}\kappa(\mathcal{R},\mathcal{T})~{}\rho(r)\Big{]}=0.$ (12)

This implies the relation

$\kappa(\mathcal{R},\mathcal{T})\,\rho(r)=C,$ (13)

where $C$, the constant of integration. Therefore, field equations reduce to

$\displaystyle C$ $\displaystyle=$ $\displaystyle\frac{b^{\prime}(r)}{r^{2}},$
(14) $\displaystyle-C$ $\displaystyle=$
$\displaystyle\frac{2f^{\prime}(r)}{r}\left(1-\frac{b(r)}{r}\right)-\frac{b(r)}{r^{3}},$
(15)

from which we can obtain $b(r)$ and $f(r)$ as

$\displaystyle b(r)$ $\displaystyle=$ $\displaystyle\frac{C}{3}r^{3}+D,$ (16)
$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\ln\Big{(}1-\frac{C}{3}r^{2}-\frac{D}{r}\Big{)}.$
(17)

Regularity at $r=0$ implies that we set $D=0$. At $r=0$, the core of the
gravastar, we have that $f(r)=0$ and the metric functions become $g_{tt}=-1$,
and $g_{rr}=1$. The density should also be constant and finite at the
gravastar’s interior. At first we observed that the curvature scalar is
constant and non-divergent everywhere at the interior region. After a
cumbersome calculation, we have arrived at

$\displaystyle\mathcal{R}=-2\Big{(}1-\frac{b(r)}{r}\Big{)}\Big{[}f^{\prime\prime}(r)+\frac{2f^{\prime}(r)}{r}+f^{\prime}(r)^{2}\Big{]}+\frac{2b^{\prime}(r)}{r^{2}}$
$\displaystyle\hskip
28.45274pt+\frac{d}{dr}\left(\frac{b(r)}{r}\right)f^{\prime}(r)=4C.$ (18)

Therefore, an important conclusion of our results is that the interior
solution for the gravastar will be regular and well-behaved everywhere,
regardless of the specific functional forms of
$\kappa(\mathcal{R},\mathcal{T})$.

###  4.1 Model
$\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}-\alpha\mathcal{T}$

Let us consider the choice,
$\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}-\alpha\mathcal{T}$,
where the pre-multiplier are the free parameters. For this particular case we
have, using that $\rho(r)=-p(r)$ and the results above, that Eq. (13) has the
form

$\Big{[}8\pi+4\beta C-4\alpha\rho(r)\Big{]}\rho(r)=C.$ (19)

This algebraic equation implies, as we mentioned before, that the density (and
pressure) for the interior layer of the gravastar is uniform, and for
$\alpha\neq 0$, it has the following possible values

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha}\Big{(}\omega\pm\sqrt{\omega^{2}-\alpha
C}\Big{)}=\rho_{1}$ (20) $\displaystyle p$ $\displaystyle=$
$\displaystyle-\frac{1}{2\alpha}\Big{(}\omega\pm\sqrt{\omega^{2}-\alpha
C}\Big{)}=p_{1},$ (21)

where $\omega$ is a constant given by $\omega=2\pi+\beta C$ and
$\omega\geq\sqrt{\alpha C}$.

The gravitational mass at the interior can is found from

$\displaystyle M(C_{1})=\int_{0}^{R_{1}=C_{1}}4\pi
r^{2}\rho~{}dr=\frac{4}{3}\pi C_{1}^{3}\rho_{1}.$ (22)

###  4.2 Model
$\kappa(\mathcal{R},\mathcal{T})\equiv\kappa(\mathcal{T})=8\pi-\alpha\mathcal{T}$

The density and pressure for this $\kappa(\mathcal{R},\mathcal{T})$ choice can
be computed by setting $\beta=0$ in the previous case. Taking into account
that for this choice, the constant $\omega=2\pi+\beta C$ reduces to
$\omega=2\pi$, we obtain

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{2\alpha}\Big{(}2\pi\pm\sqrt{4\pi^{2}-\alpha
C}\Big{)}=\rho_{2}$ (23) $\displaystyle p$ $\displaystyle=$
$\displaystyle-\frac{1}{2\alpha}\Big{(}2\pi\pm\sqrt{4\pi^{2}-\alpha
C}\Big{)}=p_{2}.$ (24)

Thus, the active gravitational follows from the direct computation

$\displaystyle M(C_{1})=\int_{0}^{R_{1}=C_{1}}4\pi
r^{2}\rho\,dr=\frac{4}{3}\pi C_{1}^{3}\rho_{2}.$ (25)

###  4.3 Model
$\kappa(\mathcal{R},\mathcal{T})\equiv\kappa(\mathcal{R})=8\pi+\beta\mathcal{R}$

The density and pressure for this specific choice can be computed by setting
$\alpha=0$ in the case A. Then, Eq. (19) reduces to

$\Big{(}8\pi+4\beta C\Big{)}\rho=C.$ (26)

Therefore,

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{1}{4(\frac{2\pi}{C}+\beta)}=\rho_{3}$ (27) $\displaystyle
p$ $\displaystyle=$ $\displaystyle-\frac{1}{4(\frac{2\pi}{C}+\beta)}=p_{3}.$
(28)

The mass is obtained in the same fashion as models (4.1) and (4.2), i.e, by
direct integration of the density in the range $r=0$ and $r=R_{1}$

$\displaystyle M(C_{1})=\int_{0}^{R_{1}=C_{1}}4\pi
r^{2}\rho\,dr=\frac{4}{3}\pi C_{1}^{3}\rho_{3}.$ (29)

###  4.4 Model
$\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}\mathcal{T}$

For this direct coupling among matter and curvature terms, we have that the
density, pressure and active gravitational mass are given by

$\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{\pi}{4C\gamma}\Big{(}1\pm\sqrt{1-\frac{C^{2}\gamma}{{\pi}^{2}}}\Big{)}=\rho_{4}$
(30) $\displaystyle p$ $\displaystyle=$
$\displaystyle-\frac{\pi}{4C\gamma}\Big{(}1\pm\sqrt{1-\frac{C^{2}\gamma}{{\pi}^{2}}}\Big{)}=p_{4},$
(31) $\displaystyle M(C_{1})=\int_{0}^{R_{1}=C_{1}}4\pi
r^{2}\rho\,dr=\frac{4}{3}\pi C_{1}^{3}\rho_{4}.$ (32)

where, $\gamma$ is a parameter of the model that should satisfy the constraint
$\gamma\leq(\pi/C)^{2}$ in order to get positive density in the core.

##  5 Shell Region $p=\rho$

This region of the gravastar supposedly contains a stiff fluid that obeys EoS
$p=\rho$ or $v=\sqrt{dp/d\rho}=1$, which is the most stiff EoS known as
Zeldovich fluid. This condition implies that the non-covariant conservation of
the stress-energy tensor given by Eq. (10) acquires the form

$\rho(r)\,\kappa^{\prime}(\mathcal{R},\mathcal{T})+\kappa(\mathcal{R},\mathcal{T})\Big{[}2f^{\prime}(r)\rho(r)+\rho^{\prime}(r)\Big{]}=0.$
(33)

On the other hand, Einstein’s field equations become

$\displaystyle\rho(r)\,\kappa(\mathcal{R},\mathcal{T})$ $\displaystyle=$
$\displaystyle\frac{b^{\prime}(r)}{r^{2}},$ (34)
$\displaystyle\rho(r)\,\kappa(\mathcal{R},\mathcal{T})$ $\displaystyle=$
$\displaystyle\frac{2f^{\prime}(r)}{r}\left(1-\frac{b(r)}{r}\right)-\frac{b(r)}{r^{3}}.$
(35)

Since, in general, the running gravitational constant
$\kappa(\mathcal{R},\mathcal{T})$ depends on the scalars $\mathcal{R}$,
$\mathcal{T}$, which in turn are functions of $b(r)$, $f(r)$, $\rho(r)$. To
solve the unknown functions $b(r)$, $f(r)$ and $\rho(r)$ we need three
equations, two from field equations and one conservation equation. From the
last two equations we obtain

$\frac{b^{\prime}(r)}{r^{2}}=\frac{2f^{\prime}(r)}{r}\left(1-\frac{b(r)}{r}\right)-\frac{b(r)}{r^{3}}.$
(36)

###  5.1 The Shell Region Differential Equations in
$\kappa(\mathcal{R},\mathcal{T})-$ gravity

To find the general equations for the shell region in
$\kappa(\mathcal{R},\mathcal{T})$ gravity we can proceed in the following way:
using (34), we eliminate $\kappa(\mathcal{R},\mathcal{T})$, i.e. we have for
$\rho(r)\neq 0$, can always write

$\kappa(\mathcal{R},\mathcal{T})=\frac{1}{\rho(r)}\frac{b^{\prime}(r)}{r^{2}}.$
(37)

Substituting this into Eq. (33), we obtain the result

$\displaystyle\frac{d}{dr}\left[\frac{b^{\prime}}{r^{2}}\right]-\frac{b^{\prime}}{r^{2}}\frac{\rho^{\prime}}{\rho}+\frac{b^{\prime}}{r^{2}}\cdot
2f^{\prime}+\frac{b^{\prime}}{r^{2}}\frac{\rho^{\prime}}{\rho(r)}$
$\displaystyle=$ $\displaystyle 0$ (38)
$\displaystyle\mbox{Or}~{}~{}~{}~{}\frac{d}{dr}\Big{(}\frac{b^{\prime}}{r^{2}}\Big{)}+\frac{b^{\prime}}{r^{2}}\cdot
2f^{\prime}(r)$ $\displaystyle=$ $\displaystyle 0.$ (39)

This differential equation can be integrated to provide

$f(r)=-\frac{1}{2}\ln\bigg{(}\frac{b^{\prime}(r)}{r^{2}}\bigg{)}.$ (40)

This general relation between the metric functions $b(r)$, $f(r)$ should be
satisfied for any $\kappa(\mathcal{R},\mathcal{T})$. Inserting now this
outcome into Eq. (36), we obtain a differential equation for $b(r)$ as

$\left[\frac{b^{\prime}(r)}{r^{2}}\right]^{2}=-\frac{1}{r}\frac{d}{dr}\left(\frac{b^{\prime}(r)}{r^{2}}\right)\left(1-\frac{b(r)}{r}\right)-\frac{b(r)b^{\prime}(r)}{r^{5}},$
(41)

a non-linear second order differential equation very difficult to solve. To go
further, we consider the so-called thin shell approximation.

###  5.2 The Thin Shell Approximation

For this EoS, it is not an easy task to obtain exact analytic solutions. One
successful strategy usually employed in the literature is the so-called thin
shell approximation, $0<1-b(r)/r\equiv h<<1$. This approximation allow us to
set $h\approx 0$ so that one can consider only the lower order terms and we
get from (41)

$\frac{b^{\prime}(r)}{r^{2}}\left(\frac{b^{\prime}(r)}{r^{2}}+\frac{b(r)}{r^{3}}\right)=0.$
(42)

The non-trivial solution for $b(r)$ arises by imposing that

$\frac{b^{\prime}(r)}{r^{2}}+\frac{b(r)}{r^{3}}=0.$ (43)

The last equation can also be obtained more directly by applying the thin
shell approximation to Eq. (36). Solving for $b(r)$, and using (40), we obtain
that the metric potentials are given by

$\displaystyle b(r)=\frac{D}{r},~{}~{}~{}f(r)=\ln(C_{0}~{}r^{2}),$ (44)

where $D$, $C_{0}$ are constants.

In the thin shell approximation, the curvature scalar is given by

$\mathcal{R}=\frac{2b^{\prime}(r)}{r^{2}}+\frac{d}{dr}\left(\frac{b(r)}{r}\right)f^{\prime}(r)=-\frac{6D}{r^{4}}.$
(45)

Regarding the proper length of the thin shell, $e^{\lambda}$ assumes the
following form as

$e^{\lambda(r)}=\left(1-\frac{b(r)}{r}\right)^{-1}=\left(1-\frac{D}{r^{2}}\right)^{-1}.$
(46)

we get the value of the proper length

$\displaystyle l$ $\displaystyle=$
$\displaystyle\int_{R}^{R+\epsilon}dr\,e^{\lambda/2}=\int_{R}^{R+\epsilon}{dr\over\sqrt{1-D/r^{2}}}$
(47) $\displaystyle=$
$\displaystyle\int_{R}^{R+\epsilon}{\sqrt{\frac{r^{2}}{r^{2}-D}}}\,dr=\sqrt{r^{2}-D}\bigg{|}_{R}^{R+\epsilon}$
$\displaystyle=$ $\displaystyle\sqrt{(R+\epsilon)^{2}-D}-\sqrt{R^{2}-D}.$

This can be rewritten in the form

$l=(R+\epsilon)\sqrt{1-\frac{D}{(R+\epsilon)^{2}}}-R\sqrt{1-\frac{D}{R^{2}}}.$
(48)

If the constant $D$ is small compared to the radius $R$, i.e, $D<<R$ we can
approximate the proper length as

$l\approx\epsilon+\frac{D}{2}\Big{(}\frac{1}{R}-\frac{1}{R+\epsilon}\Big{)},$
(49)

this further can approximated if $\epsilon<<R$, we finally obtain

$l\approx\epsilon\Big{(}1+\frac{D}{2R^{2}}\Big{)}.$ (50)

This result means that the proper length of the thin shell is proportional to
the thickness $\epsilon$ of the shell. The variation of the proper length as a
function of the thickness can be seen in Fig. 1. Another important conclusion
is that the proper length of the thin shell does not depend on the particular
choice of the $\kappa(\mathcal{R},\mathcal{T})$ running gravitational
constant. Now, we discuss the solutions for some specific models.

###  5.3 Study of the Thin Shell Solutions for Some Specific Models

####  5.3.1 Model
$\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}-\alpha\mathcal{T}$

The density and pressure in the shell for this particular model can be found
analytically. Indeed, substituting
$\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}-\alpha\mathcal{T}$ in
(34), we find after straightforward manipulations the result

$p(r)=\rho(r)=\frac{1}{4a}\left[\frac{6bD}{r^{4}}-1+\sqrt{\Big{(}1-\frac{6bD}{r^{4}}\Big{)}^{2}-\frac{8ad}{r^{4}}}\right],$
(51)

where we have defined the constants $a=\alpha/8\pi$, $b=\beta/8\pi$ and
$d=D/8\pi$. In order to represent a feasible physical gravastar, namely, to
get positive density in the shell, we have to severely constrain the
parameters of the model, i.e, the condition $(1-6bD/r^{4})^{2}>aD/\pi r^{4}$
has to be satisfied, possibly together with $6bD/r^{4}>1$, We consider that
with these constraints, the model will become a restrictive an unrealistic
one. Therefore, we choose not to pursue a further study of this particular
choice, and proceed to discuss now other possible forms of
$\kappa(\mathcal{R},\mathcal{T})$.

Figure 1: Variation of proper length as a function of the thickness within the
thin shell.

####  5.3.2 Model $\kappa(\mathcal{T})=8\pi-\alpha\mathcal{T}$

Setting $\beta=0$ in the former case, we obtain

$p(r)=\rho(r)=\frac{1}{4a}\left(\sqrt{1-\frac{8ad}{r^{4}}}-1\right).$ (52)

This solution seems problematic, i.e., since $a$ is a positive constant,
$\rho$ seems to be only positive for negative values of $d$, namely, negative
values of the constant $D$, and this is not consistent with the thin shell
approximation. Indeed, notice that substituting the result $b(r)=D/r$ in the
approximation $1-b(r)/r<<1$, it is obtained that $1\approx D/r^{2}$. Hence, if
we denote $R$ as the thin shell radius, the last relation implies that $R$ is
of order $R\approx\sqrt{D}$, being $D$ necessarily a positive constant. Thus,
we conclude that the $\kappa(\mathcal{T})$ model cannot support a gravastar,
at least as long as such approximation is valid.

####  5.3.3 Model $\kappa(\mathcal{R})=8\pi+\beta\mathcal{R}$

The matter density in the model is given by

$p(r)=\rho(r)=\frac{d}{6bD-r^{4}}.$ (53)

The trend of the $p=\rho$ with respect to $r$ is shown in Fig. 2 (left). To be
a feasible physical model, i.e, to have positive density, we have to impose
the constraint

$6bD-r^{4}>0.$ (54)

Then, if we denote $R$ as the radius of the thin shell, we have the following
upper bound for the thin shell radius

$R<\sqrt[4]{6bD}.$ (55)

The energy within the shell for this model is therefore given by

$\displaystyle E(r)$ $\displaystyle=$
$\displaystyle\int_{R}^{R+\epsilon}4\pi\rho\,r^{2}\,dr=\frac{2\pi
d}{\sqrt[4]{6bD}}\Bigg{[}\tanh^{-1}\left(\frac{r}{\sqrt[4]{6bD}}\right)$ (56)
$\displaystyle\hskip
56.9055pt-\tan^{-1}\left(\frac{r}{\sqrt[4]{6bD}}\right)\Bigg{]}_{R}^{R+\epsilon}.$

The variation of energy in shown in Fig. 2 (right).

Figure 2: Variations of $p=\rho$ and energy within the thin shell for
$\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}$ by choosing $D=2$.

Figure 3: Variations of $p=\rho$ and energy within the thin shell for
$\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}\mathcal{T}$ by
choosing $D=0.9$.

####  5.3.4 Model
$\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}\mathcal{T}$

The matter density and pressure for this specific model that directly couples
the matter and curvature trace terms are given by the following expression

$p(r)=\rho(r)=ar^{4}+\frac{1}{2}\sqrt{(2ar^{4})^{2}+b}.$ (57)

Where we have defined the constants $a=\pi/3D\gamma$, $b=1/3\gamma$. Thus, the
model $\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}\mathcal{T}$,
admits positive density without the need to constrain any free parameter.
Again, the variation of $p=\rho$ with respect to $r$ in Fig. 3 (left). On the
other hand, the energy within the shell for this model can be computed by
means of the integral

$\displaystyle E(r)$ $\displaystyle=$
$\displaystyle\int_{R}^{R+\epsilon}4\pi\rho\,r^{2}\,dr=\frac{2\pi}{3\gamma
D}\Bigg{[}\frac{2\pi r^{7}}{7}+\frac{\sqrt{\gamma}\,Dr^{3}}{\sqrt{3}}$ (58)
$\hskip
42.67912pt{}_{2}F_{1}\left(-\frac{1}{2},\frac{3}{8};\frac{11}{8};-\frac{4\pi^{2}r^{8}}{3D^{2}\gamma}\right)\Bigg{]}_{R}^{R+\epsilon}.$

where ${}_{2}F_{1}$ is the ordinary hypergeometric function. The variations of
energy for $\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}$ and
$\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}\mathcal{T}$ are shown
in Fig. 3 (right).

##  6 ENTROPY WITHIN THE SHELL

The core region of a gravastar has vanishing entropy density Mazur01 ; Mazur04
. However, entropy within the shell is generally not vanishing. For a non-
collapsing gravatar, the entropy at the shell is defined as

$S=\int_{R}^{R+\epsilon}{4\pi
r^{2}~{}s(r)~{}dr\over\sqrt{1-b/r}}~{}~{}~{}~{}~{}\mbox{with}~{}~{}~{}~{}~{}s(r)=\chi\sqrt{{p\over
2\pi}},$ (59)

in the unit $\hbar=\kappa_{B}=1$ and $\chi$, a dimensionless parameter. The
thickness of the shell is $\epsilon$. Since (59) is non-integrable, we can
approximate as follows: consider the primitive integral of (59) as

$S=\int_{R}^{R+\epsilon}z^{\prime}(r)dr=z(r)\Big{|}_{R}^{R+\epsilon}=z(R+\epsilon)-z(R)\approx\epsilon\,z^{\prime}(R)$
(60)

at $\epsilon\rightarrow 0$. Hence, (59) can be written as

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int_{R}^{R+\epsilon}{4\pi
r^{2}~{}s(r)~{}dr\over\sqrt{1-b/r}}\approx{4\pi
R^{2}\,\epsilon~{}s(R)\over\sqrt{1-b(R)/R}}$ (61) $\displaystyle=$
$\displaystyle{4\pi
R^{2}\,\epsilon\,\alpha\over\sqrt{1-b(R)/R}}\,\sqrt{{p(R)\over 2\pi}}~{}.$

The variations of entropy for
$\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}$ and
$\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}\mathcal{T}$ can be
seen in Figs. 4. Entropy in both cases increases linearly with the thickness
of the shell $\epsilon$.

Figure 4: Variations of entropy within the thin shell for the two cases ($D=2$
and $D=0.9$ respectively).

Figure 5: Variations of surface pressure and energy density.

##  7 JUNCTION INTERFACE AND SURFACE STRESSES

Gravastar has three regions, the interior ($p=-\rho$) filled with dark energy,
the shell filled with Zeldovich fluid ($p=\rho$), and an exterior where both
pressure and density vanishes ($p=\rho=0$) i.e. a vacuum. The exterior
solution is generally accepted as the Schwarzschild solution given by

$ds^{2}=\left[1-{2M\over r}\right]\,dt^{2}-{dr^{2}\over
1-{2M/r}}-r^{2}(d\theta^{2}+\sin\theta\,d\phi^{2}).$ (62)

At the junctions $r=R$, the interior and the exterior connect smoothly.
However, due to the slight mismatch in their derivatives, there arises stress
at the junction. The stress tensor is given by the Lanczos equation as

$\mathcal{S}^{i}_{j}=-{1\over
8\pi}\big{(}K^{i}_{j}-\delta^{i}_{j}\,K^{q}_{q}\big{)},$ (63)

where $K^{ij}=\mathcal{K}^{+}_{ij}-\mathcal{K}^{-}_{ij}$ is the discontinuity
in the second fundamental form. Here the signs “$+$” and “$-$” correspond to
the exterior and the interior regions respectively. The second fundamental at
the junction is given by

$\displaystyle\mathcal{K}^{\pm}_{ij}=-n_{\nu}^{\pm}\left[{\partial^{2}x_{\nu}\over\partial\xi^{i}\partial\xi^{j}}+\Gamma^{\nu}_{\alpha\beta}\,{\partial
x^{\alpha}\over\partial\xi^{i}}{\partial
x^{\beta}\over\partial\xi^{i}}\right]_{\Sigma}.$ (64)

Figure 6: Variations of equation of state parameter and surface redshift
$C_{0}=0.895$ and $M=0.1$. Figure 7: Variations of shell mass with respect to
shell thickness $C_{0}=0.895$ and $M=0.1$.

Here, $\xi^{i}$ is the intrinsic coordinates on the shell, and $n_{\nu}^{\pm}$
is the unit normals to the surface $\Sigma$. For the exterior spacetime (62),
the unit normal to the surface is given by

$n_{\nu}^{\pm}=\pm\left|g^{\alpha\beta}{\partial F\over\partial
x^{\alpha}}{\partial F\over\partial x^{\beta}}\right|^{-1/2}{\partial
F\over\partial
x^{\nu}}~{},~{}~{}~{}~{}~{}\forall~{}~{}~{}~{}n^{\mu}n_{\mu}=1.$ (65)

and $F=1-2M/R$. If the surface stress-energy tensor is taken as
$\mathcal{S}^{i}_{j}=\text{diag}(\sigma,-\mathcal{P},-\mathcal{P},-\mathcal{P})$,
where $\sigma$ is the surface energy density and $\mathcal{P}$, the surface
pressure, they can be determined as

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle-{\sqrt{F}\over 4\pi
R}\bigg{|}^{+}_{-}~{}~{},~{}~{}~{}~{}~{}~{}\mathcal{P}=-{\sigma\over
2}+{1\over 16\pi}\left.{F^{\prime}\over\sqrt{F}}\right|^{+}_{-}.$ (66)

For the shell, the surface energy density and surface pressure take the form

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle-{1\over 4\pi
R}\left[\sqrt{1-{2M\over R}}-C_{0}R^{2}\right],$ (67)
$\displaystyle\mathcal{P}$ $\displaystyle=$
$\displaystyle\frac{2-2M/R-3C_{0}R^{2}\sqrt{1-2M/R}}{16\pi R\sqrt{1-2M/R}}.$
(68)

The trends of surface energy and surface pressure are shown in Fig. 5. Now, we
can determine the mass of the shell ($m_{s}$) as

$\displaystyle m_{s}=4\pi R^{2}\sigma=R\left[C_{0}R^{2}-\sqrt{1-{2M\over
R}}\right],$ (69)

from which we can determine the total mass of the gravastar as

$\displaystyle M=\frac{2C_{0}R^{3}m_{s}-C_{0}^{2}R^{6}-m_{s}^{2}+R^{2}}{2R}.$
(70)

Further, one can also determine the equation of state parameter at the
interface as

$\displaystyle\omega={\mathcal{P}\over\sigma}=\frac{3C_{0}R^{3}\sqrt{1-2M/R}+2M-2R}{-4C_{0}R^{3}\sqrt{1-2M/R}-8M+4R},$
(71)

which must have a real value, requires $2M/R<1$ and to avoid singularity
$C_{0}\neq{\sqrt{1-2M/R}/R^{2}}$.
The variations of the equation of state parameter $\omega$, and shell mass
$m_{s}$ can be seen in Fig. 6 (left), and Fig. 7.

##  8 SURFACE REDSHIFT WITHIN THE THIN SHELL

One of the most important physical parameters in gravastar structure analysis
is a ratio of wavelengths known as surface redshift $\mathcal{Z}_{s}$. The
study of this dimensionless quantity provides valuable information
corresponding to the stability and detection of these compact objects. The
surface gravitational redshift is defined as
$\mathcal{Z}_{s}=\frac{\Delta\lambda}{\lambda_{e}}=\frac{\lambda_{0}}{\lambda_{e}}$,
where $\Delta$ represents the fractional change wavelength among the emitted
$\lambda_{e}$ and received signal $\lambda_{0}$. For static isotropic matter
distribution, Buchdahl Buch  stated that the value of redshift parameter
$\mathcal{Z}_{s}$ should not exceed 2, i.e. $\mathcal{Z}_{s}<2$. The
expression of surface redshift is presented by

$\mathcal{Z}_{s}=|g_{tt}\rvert^{-1/2}-1,$ (72)

inserting into (72) the value of the metric potential $g_{tt}$, for the shell
region given by Eq. (44), we get the surface redshift as

$\mathcal{Z}_{s}=\frac{1}{C_{0}r^{2}}-1~{}~{}~{}~{}\text{with}~{}~{}~{}R\leq
r\leq R+\epsilon.$ (73)

The variation of surface redshift $\mathcal{Z}_{s}$ as a function of the
thickness of the shell is shown in Fig. 6 (right).

##  9 RESULTS AND DISCUSSION

In this current work, we have derived and explored the Einstein field
equations describing a gravastar for a generic
$\kappa(\mathcal{R},\mathcal{T})$ functional, and then we discussed several
special cases corresponding to specific choices of this functional, such as
$\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}-\alpha\mathcal{T}$,
$\kappa(\mathcal{R},\mathcal{T})$
$\equiv\kappa(\mathcal{T})=8\pi-\alpha\mathcal{T}$,
$\kappa(\mathcal{R},\mathcal{T})\equiv\kappa(\mathcal{R})=8\pi+\beta\mathcal{R}$
and $\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}\mathcal{T}$. A
summary of our findings is the following:

  * •

In the interior region with EoS $\rho=-p$, the solutions are regular
everywhere and, in particular, in the center regardless of the specific form
of the $\kappa(\mathcal{R},\mathcal{T})$ functional.

  * •

For the shell region with EoS $\rho=p$, the specific models,
$\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}-\alpha\mathcal{T}$, and
$\kappa(\mathcal{T})=8\pi-\alpha\mathcal{T}$ provide negative density,
therefore they are unphysical solutions and cannot support a gravastar
configuration. The viability of the model
$\kappa(\mathcal{R})=8\pi+\beta\mathcal{R}$ requires an upper bound for the
thin shell radius to get positive density. Nevertheless, the nonlinear model
that directly couples the $\mathcal{R}$ and $\mathcal{T}$ traces, i.e.
$\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}\mathcal{T}$ supports a
gravastar configuration without resorting to any constraint or fine-tuning of
the free parameters.

  * •

Inside the shell, the $p=\rho$ and energy are slightly increasing linearly
outward with respect to the shell thickness in the case of
$\kappa(\mathcal{R},\mathcal{T})=8\pi+\beta\mathcal{R}$. Both these parameters
decrease when the $\kappa(\mathcal{R})-$coupling parameter $\beta$ increases.
On the other hand, $p=\rho$ and energy for the case
$\kappa(\mathcal{R},\mathcal{T})=8\pi-\gamma\mathcal{R}\mathcal{T}$ are
constants throughout the shell and they too decrease when
$\kappa(\mathcal{R},\mathcal{T})-$coupling strength $\gamma$ increases.

  * •

The entropy within the shell increases when the coupling strength $\beta$
increases in $\kappa(\mathcal{R})-$gravity while the same parameter decreases
when the coupling strength increases between geometry and matter in nonlinear
$\kappa(\mathcal{R},\mathcal{T})-$gravity.

  * •

The surface redshift $\mathcal{Z}_{s}$ is less than 2, and hence physically
inspired.

At the end, it can be concluded that the existence of gravastar configurations
in $\kappa(\mathcal{R},\mathcal{T})-$gravity strongly depends on the chosen
form of the $\kappa-$function. Similar to the parallel competing modified
theories of gravity, $\kappa(\mathcal{R},\mathcal{T})$ had already given the
wormhole solutions sark19 , compact star configurations GRP ; Tas23  and now
gravastar. Hence, this new theory is becoming a promising new theory of
gravity.

## Acknowledgments

FR, KNS would like to thank the authorities of the Inter-University Centre for
Astronomy and Astrophysics, Pune, India for providing research facilities.

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# Accelerating Secure and Verifiable Data Deletion in Cloud Storage via SGX
and Blockchain

Xiangman Li and Jianbing Ni X. Li and J. Ni are with the Department of
Electrical and Computer Engineering and Ingenuity Labs Research Institute,
Queen’s University, Kingston, Ontario, Canada K7L 3N6. Email:
<EMAIL_ADDRESS>
###### Abstract

Secure data deletion enables data owners to fully control the erasure of their
data stored on local or cloud data centers and is essential for preventing
data leakage, especially for cloud storage. However, traditional data deletion
based on unlinking, overwriting, and cryptographic key management either
ineffectiveness in cloud storage or rely on unpractical assumption. In this
paper, we present SevDel, a secure and verifiable data deletion scheme, which
leverages the zero-knowledge proof to achieve the verification of the
encryption of the outsourced data without retrieving the ciphertexts, while
the deletion of the encryption keys are guaranteed based on Intel SGX. SevDel
implements secure interfaces to perform data encryption and decryption for
secure cloud storage. It also utilizes smart contract to enforce the
operations of the cloud service provider to follow service level agreements
with data owners and the penalty over the service provider, who discloses the
cloud data on its servers. Evaluation on real-world workload demonstrates that
SevDel achieves efficient data deletion verification and maintain high
bandwidth savings.

###### Index Terms:

Cloud storage, secure data deletion, Intel SGX, data outsourcing,
verifiability.

##  I Introduction

Outsourcing data to the cloud storage is a common practice for data owners to
save the burden of self-managing massive data [1]. The data owners can on-
demand rent the storage spaces provided by the cloud service providers. The
outsourcing data management enables the data owners to access their data at
anytime and from anywhere. Due to the appealing features, the cloud storage
services, such as Amazon S3 [2], Google Drive [3], Dropbox [4], Apple iCloud
[5], and Microsoft OneDrive [6], have attracted a large number of stable and
loyal users. Data security is one of the primary concerns for data owners.
After data owners outsource their data to the cloud data centers, they lose
their physical control over their data. Thus, the data owners have no choice,
but relying on the cloud service providers to protect their data.
Unfortunately, due to the frequently happened data leakage or breach
accidents, security are always ranked as the top threats in cloud storage [7],
although the cloud service providers have great efforts for guaranteeing the
confidentiality, integrity, and availability of the outsourced data on their
cloud servers.

Data deletion [8], as one of important security technologies, has not received
sufficient attentions from both data owners or cloud service providers in
cloud storage. It provides methods to securely erase data from local storage
medium or remote cloud servers, which can significantly reduce the probability
of data leakage. Moreover, privacy regulations, such as GDPR [9], CPPA [10],
and PIPL [11], have clearly defined the principles of data deletion, called
right to be forgotten. The data centers should delete the personal data if
they are no longer necessary to the purpose for which it was collected, and
the data owners have the right to request the data centers to delete their
personal data stored on the data centers. Therefore, data deletion becomes
increasingly critical for the service providers, which should provide
effective ways to guarantee secure data deletion.

###  I-A Related Work

It is not a trivial problem to securely delete data. It is well recognized
that there is no existing software-based solution that can provide complete
data removal from storage medium. Existing deletion methods can be summarized
in the following categories:

Deletion by unlinking. This method is widely deployed on the file management
system in operating systems, such as Windows, IOS, and Linux. When the user
would like to delete a file (i.e., press the ”delete” button), the operating
system delete the link of the file from the file systems, and returns
”success” to the user. The file is no longer accessible because the link of
the file is removed. Nevertheless, this is not the real file deletion, as the
file content still remains on the disk. An adversary can simply use a file
recovery tool to access the deleted file by scanning the disk [12].

Deletion by block erasure. This method is utilized by storage mediym, such as
solid-state drive (SSD) to securely clean the data. It applies a voltage spike
to all available flash memory blocks in unison. Each block is altered with a
vendor-specific value and SSD become ”clean” [13]. However, this method erases
all the data on the drive and does cause a small amount of wear.

Deletion by overwriting. Overwriting is an important tool to delete the data
by overwriting the data with new, insensitive data, e.g., all zeros. There are
multiple tools that can offer 35-pass overwriting times. However, one inherent
limitation with the overwriting methods is that they cannot guarantee the
complete removal of data. It is effectively impossible to sanitize storage
locations by simply overwriting them, no matter how many overwrite passes are
made or what data patterns are written [14]. The conclusion holds for not only
magnetic drives, but also tapes, optical disks, and flash-based solid state
drives. In all these cases, an attacker, equipped with advanced microsoping
tools, may recover overwritten data based on the physical remanence of the
deleted data left on the storage medium. Therefore, although overwriting data
makes the recovery harder, it does not change the basic one-bit-return
protocol.

Deletion by encryption. Boneh and Lipton [15] proposed the first cryptography-
based method for secure date deletion by encrypting data before saving it to
the disk, and deleting the data by discarding the decryption key after
encryption. This method is desirable when duplicate copies of data are backed
up in distributed. However, this method essentially change the problem of
deleting a large amount of data to the problem of deleting a short key.
However, forgetting a decryption key is non-trivial. The key can be stored on
a hard disk is not easy to be permanently deleted, i.e., never be recoverable
for an adversary even it obtain the storage medium [16].

However, the problem of key deletion becomes dramatically difficult as the
cloud server performs the encryption of the outsourced data in traditional
secure cloud storage. Although some cloud storage services enable user-side
encryption, i.e., the data owners also can encrypt their data before
outsourcing, the server-side encryption is more general. The cloud server
encrypts the data after receiving them from the data owners with an encryption
key and decrypts the data that the owners would like to access before
returning them to the data owners. In this model, the encryption is fully
controlled by the cloud servers, which brings the worries of the data owners
about the encryption of their outsourced data and secure deletion of the
decryption keys.

###  I-B Contributions

In this paper, we propose a novel secure and verifiable data deletion scheme,
named SevDel, for cloud storage. To reduce the concern that whether the cloud
server honestly encrypts the outsourced, we utilize the randomly sampling
method and the zero-knowledge proof [17] to verify the encryption without
retrieving the ciphertexts of the outsourced data. The encryption is also
performed based on the Intel SGX [18] to prevent the possible data leakage.
The enclave is created for each file for the encryption and the management of
the keys. Thus, the operation of the deletion of the decryption becomes the
destroy of the enclave. In addition, to enforce the cloud servers to protect
the outsourced data, the smart contract is designed based on the service-level
agreements between the data owners and the cloud service providers. We
demonstrate the properties of confidentiality, verifiability, erasability, and
auditability of SevDel through security analysis and show that the proposed
SevDel has outstanding performance for deployment.

##  II System and Security Models

In this section, we introduce the system model and security model of our
SevDel.

###  II-A System Model

We present the system model of SevDel, that comprises three kinds of entities:
1) a data owner that outsources the data to the cloud and requests to delete
them after the data is processed or used; 2) a cloud service provider that
offers secure cloud storage services (i.e., the outsourced data of data owners
are encrypted by the service provider with its chosen secret keys or by the
data owners before outsourcing) to data owners with its storage servers in the
cloud data center, and each server has high-performance hard disks for data
storage and has the Intel Core that supports for SGX [18]; and 3) a blockchain
node [19] that participates the blockchain network to maintain transactions
happened between two parties. The blockchain can be the public blockchain,
e.g., Bitcoin blockchain, Ethereum blockchain, or Hyperledger. It maintains an
automatically executable smart contract that enforces the penalty on the cloud
service provider if it leaks the outsourced data of users.

Intel SGX [18], a suite of security-related instructions built into modern
Intel CPUs, can create a hardware-protected environment, enclave, for
shielding the execution of code and data. An enclave resides in a hardware-
guarded memory region called the enclave page cache (EPC) for hosting any
protected code and data. In enclave, SGX performs the encryption of the
outsourced data with a secret key stored on the EPC. The deletion of the
encrypted data for the data owner is the deletion of the secret key in
enclave. More specifically, the secret in enclave is erased after the enclave
is destroyed.

###  II-B Threat Model

The security threats are mainly from the outsider attackers or the data thief.
An outsider attacker or a data thief may compromise the cloud server to steal
the data on the hard disks. The frequently happened data leakage incidents on
cloud have witnessed the risks of cloud storage services. This risk is high
because of potential code vulnerability, and the damage is severe as the data
leakage incidents significantly affect reputation. Moreover, the employees in
cloud service may steal the data on cloud servers. We have witnessed many data
corruption or leakage incidents that occur due to the operation errors or
misbehavior of the employees. The main security objective is to protect the
cloud data for users against data leakage incidents.

A cloud service provider is the legitimate processor of the Intel SGX and
holds the service level agreements with the data owners for maintaining
outsourced data. It is expected that the cloud service provider stores the
encrypted outsourced data of data owners on the hard disks of cloud servers
and deletes the data under the requests of data owners or based on the
principles of privacy regulations, like GDPR and CPPA, and PIPEDA. It is
assumed that the cloud service provider may not deviate from the expectation
due to the agreement with data owners, that is, the cloud service provider is
rational. It follows the service level agreements to honestly offer data
storage services. Undoubtedly, regulating the implementation of the agreement
between the users and the cloud service provider become necessary.

A data owner is an honest party to rent storage spaces from the cloud storage
services and outsources the data to the cloud servers in the data center. The
data owner chooses the reliable service providers for data outsourcing.
According to the modes for protecting cloud data in cloud storage, e.g.,
Amazon S3 of Amazon Web Service, the owners can determine whether to encrypt
their data before outsourcing. The data owners can use secret keys to encrypt
their data before outsourcing. If the owners do not encrypt the data, the
cloud server chooses the secret keys for data encryption. In this paper, we
study secure data erasure for the latter case because it is trivial to achieve
data deletion if the data owners encrypt their data by themselves, as they can
delete their keys and then no one can read the cloud data.

##  III Proposed SevDel

In this section, we propose the overview and the detailed construction of our
SevDel.

##  IV Overview

Our SevDel accelerates security and verifiability of cloud data erasure in
cloud storage. It can serve as the central element of secure cloud storage and
erasure in cloud storage services, such as Amazon S3 Find and Forget, the
solution to selectively erase records from data lakes stored on Amazon S3. To
prevent data leakage, the received file from the data owner is encrypted by
the cloud server with a randomly selected private key with additive
homomorphic encryption, such as lifted ElGamal encryption [20]. The encryption
operation is performed in the enclave of Intel SGX. The encryption of the file
is audited by the data owner to ensure that the file is correctly encrypted as
claimed by the cloud service provider. The random sampling is utilized to
enable probabilistic auditing of the encrypted data and the ciphertexts are
aggregated to compress auditing messages. The cloud server proves to the data
owner that the entire file is encrypted with lifted ElGamal encryption by a
randomly chosen key with a large probability, without retrieving the encrypted
file. The challenge here is to ensure that the proved ciphertext is really the
encryption of the correct outsourced file. To blind the file and its
ciphertext during auditing, the cloud server should prove that the plaintext
of the ciphertext is the outsourced file in the homomorphic authentication
tags, which are produced by the data owner and outsourced along with the file.
Meanwhile, they can also used to verify the integrity of the outsourced file
based on provable data possession [21] or proof of retrievability [22].

The deletion of the oursourced file on the cloud server is enabled by the
deletion of the secret key of the file. If the secret key is permanently
deleted, no one is able to decrypt the ciphertext. The secret key deletion is
realized by the Intel SGX. An enclave is created when the cloud server
receives the file and the encryption is performed in the enclave. Also, the
encryption key is stored in the enclave. In order to permanently forget the
key, the simple way is to destroy the corresponding enclave.

To ensure the cloud service provider to honestly maintain and encrypt
outsourced files of data owners, a smart contract is created based on the
service level agreement between the service provider and data owners. the
deposits of the service provider are made when the cloud storage service is
bootstrapped. The deposits are paid to the data owner if the file of the data
owner is found on the Internet, which means that the file is leaked during
storage. The condition to trigger the payment is the key point of the smart
contract. We convert this data leakage problem to be the provable data
possession. If a data owner is succeed to giving a proof that she possesses
the encrypted version of her outsourced files, the penalty is performed over
the service provider and a certain amount of the deposits is transferred to
the data owner. The conversion is valid because only the cloud server has the
encrypted version of the outsourced file of the data owner. The cloud server
performs encryption after receiving the outsourced file and decryption before
returning it to the data owner. The ciphertext of the outsourced file should
be only known by the cloud server. Although the data owner knows the cleartext
of the file, the data owner obtains the same ciphertext, as the data
encryption on the side of the cloud server is probabilistic.

Our SecDel consists of the following algorithms.

Setup: This algorithm is run by the cloud service provider to bootstrap the
cloud storage systems. With the input of the security parameter, the algorithm
outputs the system parameters and the public-private key pairs of the cloud
servers.

Contract: This algorithm is run by the cloud service provider to initialize a
smart contract that implements the service level agreement with the data
owners. The smart contract is maintained by the blockchain nodes.

KeyGen: This algorithm is run by the data owner. With the input of the system
parameters, the algorithm takes the input of the system parameter and generate
the public-private key pair of the data owner for data outsourcing.

Outsource: This algorithm is run by the data owner to outsource the file to
the cloud server. With the input of the security parameters, the private key
of the data owner, and the outsourcing file, the algorithm produces the
homomorphic authentication tags of the data blocks of the file and outsource
the file, along with the generated tags.

Encrypt: This algorithm is run by the cloud server that encrypts the received
file with a randomly chosen private key. With the input of the file, the
private key of the cloud server, and the chosen private key, the algorithm
outputs the encrypted file, the corresponding public key, and the homomorphic
authentication tags of the data blocks of the encrypted file.

Verify: This is an interactive protocol between the cloud server and the data
owner to audit the encryption of the outsourced file. The data owner randomly
samples the data blocks, and the cloud server generates a proof that proves
the encryption of the sampled data blocks. The data owner finally verifies the
proof to learn whether the file has been encrypted by the cloud server.

Delete: This algorithm is run by the cloud server who deletes the file under
the request of the data owner or the data is no longer needed for data
analysis.

Audit: This is an interactive protocol between the data owner and the
blockchain nodes. The blockchain nodes randomly samples the data blocks owned
by the data owners and the data owner responds the proof that proves the
ownership of the encrypted data block. Then, the blockchain nodes verify the
proof to learn whether the file has been disclosed. If the proof is valid, the
smart contract is executed to give penalty to the cloud service provider.

The correctness of SevDel has the following aspects: 1) The encryption of the
outsourced file should be correctly recovered by the cloud server with the
corresponding secret key; 2) the data owner can identify that the cloud server
does not encrypt the oursourced file on hard disks as agreed with the service
level agreement; 3) the deleted outsourced file can be no longer recovered;
and 4) the blockchain node can execute the penalty if the data owners find the
leaked outsourced data.

###  IV-A Detailed SevDel

Setup: Let $q$ be a large prime and $\mathbb{G}_{1}$, $\mathbb{G}_{2}$ and
$\mathbb{G}_{T}$ be three multiplicative cyclic groups of the same prime order
$p$. $g_{1}$ and $g_{2}$ are the generators of $\mathbb{G}_{1}$ and
$\mathbb{G}_{2}$, respectively.
${e}:\mathbb{G}_{1}\times\mathbb{G}_{2}\rightarrow\mathbb{G}_{T}$ denotes an
admissible bilinear pairing.

The file $M$ to be outsourced is divided into $n$ blocks and each block is
further split into $s$ sectors. Thus, the fiel is denoted as
${M}={\\{m_{ij}\\}}_{i\in[1,n],j\in[1,s]}$ and the abstract information of $M$
is denoted as ${\mathbb{I}}_{M}$. $H:\\{0,1\\}^{*}\rightarrow{\mathbb{G}_{1}}$
is a cryptographic hash function that maps the ${\mathbb{I}}_{M}$ to a point
in $\mathbb{G}_{1}$.

The cloud service provider chooses a random number $a\in{\mathbb{Z}}_{p}$ and
calculates $A=g_{2}^{a}\in{\mathbb{G}_{2}}$. The private key of the data owner
is $a$, and the corresponding public key is $A$.

Contract: The service provider creates the smart contract CS-SevDel to provide
cloud storage services to data owners. To provide the service, the service
provider initiates CS-SevDel.Init to setup the smart contract and deposits an
amount of money on the blockchain as insurance in CS-SevDel.Service. The a
part of the deposit would be sent to the data owner if the outsourced data is
leaked and the remainder would be re-fund to the service provider.

KeyGen: An data owner chooses a random number $w\in{\mathbb{Z}}_{p}$ and
calculates $W=g_{2}^{w}\in{\mathbb{G}_{2}}$. The private key of the data owner
is $w$, and the corresponding public key is $W$.

Outsource: The data owner chooses $s$ random values
$x_{1},\cdots,x_{s}\in{\mathbb{Z}}_{p}$ and computes
$u_{j}={g_{1}}^{x_{j}}\in{\mathbb{G}_{1}}$ for $j\in[1,s]$. Then, for each
block $m_{i}$ ($i\in[1,n]$), it computes a tag $t_{i}$ as

$\phi_{i}=(H({\mathbb{I}}_{M}||i)\cdot\prod_{j=1}^{s}{u_{j}}^{m_{ij}})^{w}$.

The data owner outputs the set of homomorphic authentication tags
$T={\\{\phi_{i}\\}}_{i\in[1,n]}$. The tag set $\Phi$, the file index
${\mathbb{I}}_{M}$, and the file $M$ are sent to the cloud server.

Encrypt: After receiving $({\mathbb{I}}_{M},M,\Phi)$ from a data owner, the
cloud server first randomly selects a private key $v\in{\mathbb{Z}}_{p}$ and
computes $V=g_{1}^{v}\in{\mathbb{G}_{1}}$. The cloud server uses the random
private key $v$ to encrypt each data block of the received file $m_{ij}$ as
$E_{ij}=(E^{\prime}_{ij},E^{\prime\prime}_{ij})=(g_{1}^{m_{ij}}V^{r_{ij}},g_{1}^{r_{ij}})$,
where $r_{ij}$ is a random number chosen from ${\mathbb{Z}}_{p}$. The set of
the encrypted blocks is denoted as $E={\\{E_{i}\\}}_{i\in[1,n]}$. Then, for
each encrypted block $E_{i}$ ($i\in[1,n]$), the cloud server computes a
homomorphic authentication tag $\sigma_{i}$ for the encrypted block as

$\sigma_{i}=(H({\mathbb{I}}_{M}||i)\cdot\prod_{j=1}^{s}{u_{j}}^{E^{\prime}_{ij}}{v_{j}}^{E^{\prime\prime}_{ij}})^{a}$.

The set of the tags of the encrypted blocks is denoted as
$\Sigma={\\{\sigma_{i}\\}}_{i\in[1,n]}$. Finally, the cloud server stores
$({\mathbb{I}}_{M},E,\Sigma)$ on the hard disks and uploads
$({\mathbb{I}}_{M},\Sigma)$ to the blockchain.

Verify: To verify the encryption of the outsourced file $M$, the data owner
takes the abstract information ${\mathbb{I}}_{M}$ as inputs. It selects some
data blocks to construct a _challenge set $Q$_ and picks a random
$l_{i}\in{\mathbb{Z}}_{p}^{*}$ for each $m_{i}$ $(i\in Q)$. The challenge
$(i,l_{i})_{i\in Q}$ is sent to the cloud server.

To respond the challenge, the cloud server generates $P_{1}$ as

$P_{1}=\prod_{i\in Q}g_{1}^{l_{i}m_{ij}}V^{l_{i}r_{ij}}.$

The cloud server computes $Q_{j}=\sum_{i\in Q}{l_{i}}\cdot{m_{ij}}$ for each
$j\in[1,s]$. Then, it computes $Q_{2}$ as

$P_{2}=\prod_{j=1}^{s}\phi_{i}^{l_{i}}.$

$\pi\leftarrow NIZK\\{(Q_{j},r_{ij}):P_{1}=\prod_{i\in
Q}g_{1}^{l_{i}m_{ij}}V^{l_{i}r_{ij}},P_{2}=\prod_{j=1}^{s}\phi_{i}^{l_{i}}\\}$.

The data owner verifies the validity of the zero-knowledge proof $\pi$ to
determine whether the outsourced file has been encrypted or not.

Delete: The cloud server deletes the random private key $v$ that is used to
encrypt the file $F$ by destroying the enclave that used to store $v$. The
cloud server creates an enclave for each file received and use the enclave to
maintain the private key.

Audit: If the data owner obtains the leaked encrypted file $F$, the data owner
can prove to the blockchain nodes that the cloud server has data leakage. The
blockchain node selects some data blocks to construct a _challenge set $R$_
and picks a random $\gamma_{i}\in{\mathbb{Z}}_{p}^{*}$ for each $E_{i}$ $(i\in
R)$. The challenge $(i,\gamma_{i})_{i\in R}$ is sent to the data owner.

To respond the challenge, the data owner generates $Q_{1}$ as

$Q_{1}=\prod_{i\in R}\gamma_{i}E_{ij}.$

Then, it computes $Q_{2}$ as

$Q_{2}=\prod_{j=1}^{s}\sigma_{i}^{\gamma_{i}}.$

The data owner returns $(Q_{1},Q_{2})$ to the blockchain node. The blockchain
node verifies $(Q_{1},Q_{2})$ to determine whether the cloud server has
disclosed the file $F$. If yes, the blockchain node performs CS-SevDel.Penalty
to give penalty to the cloud service provider.

The correctness of SevDel can be check that 1) the encryption of the
outsourced file is correctly recovered; 2) the verification equation can pass;
3) the security of Intel SGX; and 4) the blockchain node can execute the
penalty.

##  V Security of SevDel

The security of SevDel should capture the properties of confidentiality,
verifiability, erasability, and auditability.

The confidentiality of the outsourced file relies on the semantic security of
the data encryption scheme used by the cloud server. SevDel utilizes the
lifted ElGamal encryption scheme to encrypt each block of the outsourced file
$M$. Here, each block is independently encrypted with the key $V$. As the
lifted ElGamal encryption scheme can be proved semantic security under the
Decisional Diffie-Hellman (DDH) assumption, the confidentiality of the
outsourced file is achieve as long as the DDH assumption holds.

Smart Contract CS-SevDelInit: Set state:=INIT, File$:=\\{\\}$,
Onwer$:=\\{\\}$, RU$:=\\{\\}$,Tags$:=\\{\\}$,
Param:=SerDel$(1^{\lambda})$.Service: Upon receiving (“Create”, $N$, $file$,
$A$, Deposit,$T_{1},T_{2},T_{3},T_{4}$) from a service provider
$\mathcal{S}$:Assert state=INT.Assert current time $T\leq T_{1}$.Assert
ledger$\mid\mathcal{S}\mid\geq$ $Deposit.ledger
$\mid\mathcal{S}\mid$:=ledger$\mid\mathcal{S}\mid$–$Deposit.Set
state:=CREATED.Set
Accept:=0.File:=File$\cup${$\mathcal{S},N$,$A$,Deposit,Accept,$T_{j=1-4}$}.Agree:
Upon receiving (“Accept”, $\mathcal{U}_{i},N,R_{i}$) from adata owner
$\mathcal{U}_{i}$:Assert state=CREATED.Assert $T_{1}\leq T\leq T_{2}$.Assert
$R${}_{i}>$0.Assert ledger$\mid\mathcal{U}_{i}\mid\geq$ $Ri.ledger
$\mid\mathcal{U}_{i}\mid$:=ledger$\mid\mathcal{U}_{i}\mid$–$Ri.Set
Accept:=Accept+1.Set
statei:=ACCEPTED.OwnerN:=Owner${}_{N}\cup\\{\mathcal{U}_{i}$}.Claim: Current
time $T=T_{2}$:Assert statei=ACCEPTED.Assert the data outsourcing $N$.Set
state:=CLAIMED.Audit: Upon receiving (“Audit”,
$\mathcal{U}_{i},N,c_{i},d_{i},\sigma_{i},e_{i},rk_{i},$$\mathcal{PK}_{i}$)
from $\mathcal{U}_{i}$:Assert state=CLAIMED.Assert $T_{2}\leq T\leq
T_{3}$.Assert $\mathcal{U}_{i}\in$AUN.Assert $\mathcal{PK}_{i}=1$.Set
statei:=UPLOADED.Set ledger
$\mid\mathcal{U}_{i}\mid$:=ledger$\mid\mathcal{U}_{i}\mid$+$Ri.OwnerN:=owner${}_{N}\cup\\{\mathcal{U}_{i}$}.FileN:=File${}_{N}\cup\\{(\mathcal{U}_{i},N,\sigma_{i},e_{i},rk_{i})$}.Refund:
$T_{3}\leq T\leq T_{4}$ and OwnerN=FileN:Set state:=FULFILLED.Set ledger
$\mid\mathcal{U}_{i}\mid:=$ledger$\mid\mathcal{U}_{i}\mid$+$Depositi.Assert
$Deposit=$\sum_{i=1}^{n}$$Depositi.Set state:=FINISHED.Penalty: $T_{3}\leq
T\leq T_{4}$ and AU${}_{N}\supset$RUN:Set
state:=UNFULFILLED.ledger$\mid\mathcal{U}_{i}\mid:=$ledger$\mid\mathcal{U}_{i}\mid$+$R${}^{*}_{i}$,
for $\mathcal{U}_{i}\in$RUN.Assert
$\sum\limits_{i\in\\{{\texttt{AU}}_{N}-{\texttt{RU}}_{N}\\}}$$Ri
=$\sum\limits_{i\in\\{{\texttt{RU}}_{N}\\}}$$R${}^{*}_{i}$.Set
state:=ABORTED.Timer: If state=ABORTED and $T>T_{4}$;Set ledger
$\mid\mathcal{S}\mid:=$ledger$\mid\mathcal{S}\mid$+$Deposit.Set
state:=ABORTED.Alg. 1. Smart Contract CS-SevDel

The verifiability of the data encryption is achieved based on provable data
possession and zero-knowledge proofs. The data owners are able to audit the
encrypted data by randomly sampling the encrypted blocks. The homomorphic
authentication tags guarantee the authentication of data blocks in the
aggregated way. First, the homomorphic authentication tags are created in the
way of digital signatures. They are not forgeable under the assumption of
computational Diffie-Hellman assumption. Second, it is impossible to generate
a proof if the cloud server does not encrypt the sampled data blocks because
the proof is the linear aggregation of the tags. Therefore, the verifiability
of the data encryption is realized.

The erasability of the data is achieved based on the Intel SGX. The enclave is
created for the file when the cloud server receives the file. The enclave is
used to maintain the decryption key. The deletion of the data is achieved when
the enclave is destroyed. The destroy of the enclave would permanently lose
the information in the enclave. According to this feature, the decryption key
is lost after the destroy of the enclave. Thus, the encrypted file can never
be decrypted, so the file is permanently deleted.

The auditability of data leakage is achieved based on the smart contract. The
smart contract makes sure the automatic execution of the service-level
agreement between the data owners and the cloud service providers. The
condition that triggers penalty is the data leakage incident, so the data
owner needs to prove to the blockchain node that they have the leaked data.
This proof generation method is the same as the method for data encryption
proof, so they are based on the same assumption.

##  VI Conclusion

In this paper, present a secure and verifiable data deletion scheme that
leverages the zero-knowledge proof to achieve the verification of the
encryption of the outsourced data without retrieving the ciphertexts. The
deletion of the encryption keys are guaranteed based on Intel SGX. The
proposed scheme implements secure interfaces to perform data encryption and
decryption for secure cloud storage and utilizes smart contract to enforce the
operations of the cloud service provider to follow service level agreements
with data owners and the penalty over the service provider, who discloses the
cloud data on its servers. As the proposed scheme enables the cloud server to
handle the service-side encryption, which make the scheme particularly
suitable for the popular secure cloud storage services.

## References

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