#### On stability of a neutron star system in Palatini gravity

Eur. Phys. J. C
On stability of a neutron star system in Palatini gravity
Aneta Wojnar 0
0 Institute of Physics, Maria Curie-Skłodowska University , Pl. Marii Curie-Skłodowskiej 1, Lublin 20-031 , Poland
We formulate the generalized Tolman-Oppenheimer-Volkoff equations for the f (Rˆ ) Palatini gravity in the case of static and spherical symmetric geometry. We also show that a neutron star can be a stable system independently of the form of the functional f (Rˆ ).
1 Introduction
Einstein by formulating his General Relativity [
1,2
] showed
that there is a relation between geometry of spacetime and
matter contribution. Until now, many astronomical
observations have tested GR giving us that it is the best
matching theory for explaining gravitational phenomena which we
have in hand. However, there are still many issues in
fundamental physics, astrophysics and cosmology (for review,
see e.g. [
3–7
]) indicating that GR is not a final theory of
gravitational interactions. There are such unsolved problems
as the dark matter puzzle [
8,9
], inflation [
10,11
], another
one is the late-time cosmic acceleration [
12,13
] which is
explained by the assumption that there exists an exotic fluid
called dark energy [
3–5
]. It is introduced to the dynamics
throughout adding the cosmological constant to the
standard Einstein’s field equations considered in Friedmann–
Robertson–Lemaitre–Walker (FRLW) spacetime. Due to the
above shortcomings one looks for different approaches in
order to find a good theory which will be able to answer the
above problems and to show us directions of future research.
The additional argument in a favor of searching
generalizations of gravity is non-renormalization of Einstein’s theory.
The renormalization problem seems to be solved if extra high
curvature terms are added [
14
]. One would also like to unify
gravity theory with the other ones (electromagnetism, weak
and strong interactions has been already unified into the
Standard Model) but no satisfactory result has been obtained so
far as e.g. string theory, supersymmetry, that could combine
particle physics and gravitation.
One of many ways to deal with the mentioned problems
are Extended Theories of Gravity (ETG) [
15,16
] which have
been gained a lot of attention. The main arguments which
advocate them are a possibility to explain problematic issues
without introducing exotic, immeasurable entities as well as
some approaches to quantum gravity [17] or the
geometrization of the Standard Model (plus gravity) as derived from
Chamseddine–Connes spectral action in noncommutative
geometry [
18,19
] which indicate that the effective action for
the classical gravity should be more complicated than the one
introduced by Einstein.
The geometric part of the action might be changed in many
different ways. One may assume that constant of Nature are
not really constant values [
20–22
]. A scalar field might be
added into Lagrangian and moreover, it can be minimally
or non-minimally coupled to gravity [
23,24
]. One proposes
much more complicated functionals than the simple linear
one used in GR, for example f (R) gravity [
10,25
]. The latter
approach has gained a lot of interest recently as the extra
geometric terms could explain not only dark matter issue
[
26,27
] but also dark energy problem because it produces
the accelerated late-time effect at low cosmic densities. The
field equations also differ from the Einstein’s ones so they
could provide different behavior of the early Universe.
The f (R) gravity is usually treated in two different ways:
in the metric approach [
6,7,28–30
] and Palatini one [
28,31–
33
]. The former arises to the fourth order differential
equations which are more difficult to handle than the second-order
field equations of GR1 Another point is that one believes that
physical equations of motion should be of the second order.
1 The theory possesses the scalar-tensor representation, that is, it is
mathematically equivalent to the Brans–Dicke gravity (with the
parameter ω = 0) with a potential of the scalar field. In this representation one
deals with the differential equations of the second order on the metric
components and a modified Klein–Gordon equation for the scalar field,
see the details e.g. [34].
In contrast to the metric formalism, the Palatini f (R) gravity
provides second order differential equations since the
connection and the metric are treated as independent objects.
The Riemann and Ricci tensors are constructed with the
connection while for building the Ricci scalar we also use the
physical metric in order to contract the indices. The
Palatini approach allows to get the modified Friedmann equation
[
35–37
] in a form that might be compared with the
observational data [
38–42
]. It shows the potential of the Palatini
formulation when it is applied to the gravitational problems
[
43,44
].
There also exist disadvantages of such an approach: being
in conflict with the Standard Model of particle physics
[
34,48–50
], the algebraic dependence of the post-Newtonian
metric on the density [
51,52
], and the complications with
the initial values problem in the presence of matter [
33,53
],
although the problem was already solved in [54]. Another
one happens at microscopic scales, that is, the theory
produces instabilities in atoms which disintegrate them. One
should also mention limitations which arrive when one treats
the extra terms as fluid-like [
45–47
]. However, it was shown
[
55
] that high curvature corrections do not cause such
problem. What is also very promising, some of the Palatini
Lagrangians avoid the Big Bang singularity. Moreover, the
effective dynamics of Loop Quantum Gravity can be
reproduced by the Palatini theory which gives the link to one of
approaches to Quantum Gravity [
56
]. High curvature
correction of the form changes the notion of the independent
connection: in the simple Palatini f (R) gravity the connection
is auxiliary field while in the more general Palatini theory it
is dynamical without making the equations of motion second
order in the fields.
The problem which is our interest in this work concerns
astrophysical aspects of Palatini gravity, for example black
holes [
57–60
], wormholes [61] and neutron stars [
62
]. We
would like to examine the last objects in the context of
Palatini gravity. In general, the maximal mass of neutron stars is
still an open problem: Einstein’s gravity gives us, together
with the recent observations, the limit as 2 M . The pulsar
PSR J1614-2230 is found to have the limit 1.97M [
63
] while
another one is Vela X-1 with the mass ∼ 1.8M [
64
]. One
also supposes the existence more massive neutron stars as
for instance the ones with masses around 2.1 M [
65
] and
2.4 M (B1957+20) [
66
]. For the recent neutron stars mass
determination see [
67
]. There also exists “hyperon puzzle”:
it turns out that equations of state including hyperons make
the maximal mass limit in the case of neutron stars without
magnetic field lower than 2 M [
68–70
]. Due to that fact it
seems that these equations of state could not be feasible for
massive neutron stars in the framework of General
Relativity. Some solutions were already proposed, like for example
hyperon-vector coupling, chiral quark-meson coupling and
the existence of strong magnetic fields inside the star. An
emissary for the latter idea are works indicating an influence
of magnetic fluid which increases masses of the stars [
71–
73
]. This arises to a question if neutron stars without strong
magnetic field with mass larger than 2 M can exist [
74–
76
] whose structure can be still described by the equations
derived from General Relativity.
Moreover, because of the very recent neutron stars’ merger
observation [
77
] the confrontation of gravitational theories
with the observational data is an appealing task. Neutron
stars are perfect objects for tests at high density regimes
thus one may perceive possible deviations from GR. It also
seems that using General Relativity as a description of strong
gravitational fields [
76,79
] can be an extrapolation for fields
sourced by neutron stars since they are many orders of
magnitude larger than the one we may probe by solar system
tests. Therefore, likely GR should be modified in a strong
gravity regime and in the case of large spacetime curvature
[80].
Since we are interested in Palatini gravity, one should
mention another problem which arose throughout the studies on
neutron stars in this formalism. In [
81,82
] it was shown that
there exist surface singularities of static spherically
symmetric objects in the case of polytropic equation of state. It would
seem that all Palatini f (Rˆ ) functionals besides
EinsteinHilbert one could be ruled out since they provide unphysical
behavior. However in [83] it was demonstrated that the
problem appears because of the particular equation of state rather
than the dynamics of the theory itself. Further approach to the
problem [
84
] indicates that polytropic equation of state is not
a fundamental issue but only an approximation of the
matter content in the star. Nevertheless, they examined the case
of problematic matter description according to the Ehlers–
Pirani–Schild (EPS) interpretation [
85–87
], which we also
follow in our work. Considering the conformal metric as the
one responsible for the free fall they revealed that in this case
the singularities are not generated in comparison to metric
which was used in [
81
].
Due to the above brief considerations on neutron stars,
there are two main problems concerning these objects:
how to model dense matter inside the star, which one
would like to describe by an equation of state, and viable
equations which depict the macroscopic properties, that is,
mass and radius. The alternative equations, or rather
modified Tolman–Oppenheimer–Volkoff equations, describing
the macroscopic values of relativistic stars are actively
proposed [
88–94
] and tested with the observational data. In this
work we would also like to challenge this problem and hence
in the next section we will present briefly f (Rˆ ) gravity in
Palatini formalism and then we are going to focus on stars’
stability problem. We will also provide appropriate TOV
equations for this generalization of General Relativity. We
are using the Weinberg’s [
95
] signature convention, that is,
(−, +, +, +).
2 A brief description of Palatini gravity: frames
As we have already mentioned, we are interested in Palatini
gravity which is a particular case of EPS formalism [
85
].
That means that geometry of spacetime is described by two
structures which in Palatini gravity turns out to be the
metric g and the connection ˆ which are independent objects.
In addition, the connection is a Levi-Civita connection of a
metric conformally related to g. Due to this interpretation,
one considers motion of a mass particle appointed by the
connection while clocks and distances are ruled by the metric g.
Let us briefly, but in a quite detailed way, introduce the basic
of the Palatini f (Rˆ ) gravity. The action is given by
1
S = Sg + Sm = 2
√−g f (Rˆ )d4x + Sm,
where Rˆ = Rˆ μν gμν is the generalized Ricci scalar. The
variation of Eq. (1) with respect to the metric gμν provides
1
f (Rˆ )Rˆμν − 2 f (Rˆ )gμν = Tμν ,
where Tμν = − √2−g δgδμν Sm = (ρ + p)uμuν + pgμν is
energy momentum tensor, p pressure while ρ energy density
of the fluid. The vector uα is a 4-velocity of the observer
co-moving with the fluid (gμν uμuν = −1). Taking the trace
of the Eq. (2) with respect to gμν yields a structural equation,
which is given by
f (Rˆ )Rˆ − 2 f (Rˆ ) = T .
Assuming that we are able to solve Eq. (3) as Rˆ (T ) we see that
f (Rˆ ) is also a function of the trace of the energy momentum
tensor, where T = gμν Tμν ≡ 3 p − ρ.
The variation with respect to the independent connection
gives
∇ˆ α(
√−g f (Rˆ )gμν ) = 0,
from which we immediately notice that ∇ˆ α is the covariant
derivative calculated with respect to ˆ , that is, it is the
LeviCivita connection of the conformal metric hμν = f (Rˆ )gμν .
It is well known [
29
] that the action (1) is dynamically
equivalent to the constraint system with first order Palatini
gravitational Lagrangian:
1
S(gμν , ρλσ , χ ) = 2κ
d4x√−g f (χ )(Rˆ − χ ) + f (χ )
(6)
(7)
(8)
(9)
(10)
(11)
(12a)
(12b)
(13)
providing that f (Rˆ ) = 0.2 Introducing further a scalar field
= f (χ ) and taking into account the constraint equation
χ = Rˆ , one can rewrite the action in dynamically equivalent
way as a Palatini action
1
S(gμν , ρλσ , ) = 2k
d4x√−g
Rˆ − U ( ) +Sm (gμν , ψ),
where the potential U ( ) “remembers” the form of function
f (Rˆ ). It is defined as
U ( ) = χ ( )
− f (χ ( ))
where χ ≡ Rˆ = dUd( ) and = d df(χχ) .
Palatini variation of this action provides
1
Rˆμν − 2 gμν Rˆ
1
+ 2 gμνU ( ) − κ Tμν = 0
∇ˆ λ(
√−g gμν ) = 0
Rˆ − U ( ) = 0
The last equation due to the constraint Rˆ = χ = U (φ) is
automatically satisfied. The middle Eq. (9) implies that the
connection ˆ is a metric connection for the new metric hμν =
gμν . The g-trace of the first equation − Rˆ + 2U ( ) =
κ T , can be recast as
2U ( ) − U ( )
= κ T .
which provides an analog of the structure Eq. (3).
Now the Eq. (8) can be rewritten as a dynamical equation
for the metric hμν [
41,42
]
1 1
R¯μν − 2 hμν R¯ = κ T¯μν − 2 hμνU¯ ( )
R¯ −
2 U¯ ( )
= 0
where we have introduced U¯ (φ) = U (φ)/ 2 and
appropriate energy momentum tensor T¯μν = −1Tμν . Moreover,
one notices that Rˆμν = R¯μν , R¯ = hμν R¯μν = −1 Rˆ and
hμν R¯ = gμν Rˆ . The last equation, together with the trace of
Eq. (12a), can be replaced by
(1)
(2)
(3)
(4)
U¯ ( ) + κ T¯ = 0 .
Thus the system Eqs. (12a)–(13) corresponds to a
scalartensor action for the metric hμν and (non-dynamical) scalar
field
+Sm (gμν , ψ), (5)
2 In that case the linear Einstein–Hilbert Lagrangian f (Rˆ) = Rˆ − 2
is excluded on that level.
1
S(hμν , ) = 2κ
where
T¯ μν = − √−h δhμν
2
δ
R¯ = 2U¯ ( ) − κ T¯ .
3 Stability condition
4 √
d x −h R¯ − U¯ ( ) + Sm ( −1hμν , ψ),
Sm = (ρ¯+ p¯)u¯μu¯ν + p¯hμν =
and u¯μ = − 21 uμ, ρ¯ = −2ρ , p¯ = −2 p, T¯μν =
−1Tμν , T¯ = −2T (see e.g. [
96
]). Further, the trace of
Eq. (12a), provides
Following [
95
] and [
98
] we wish to show that Palatini
neutron stars whose matter is modeled by perfect fluid are in
dynamical equilibrium and are described by modified TOV
equations. Moreover, we are going to demonstrate that this
equilibrium is stable which means that a considered
configuration is in thermodynamic equilibrium.
3.1 TOV equations in ETGs
It was shown in [
97
] how some modifications of the TOV
equations influence a mass and radius of a neutron star. The
introduced parameters to the equations not only allow to see
how the mass-radius diagram changes with respect to them
but also can be useful to give some clue on a kind of ETG
which we should look for in order to get the desired results
which are in agreement with the observations. The details
and discussion on gravitational theories which are related to
the parameters can be also found in [
97,99
].
The TOV equations can be generalized for a certain class
of theories after the generalization of the energy density and
pressure [
100
]:
Q(r ) := ρ(r ) +
(r ) := p(r ) +
σ (r )Wtt (r )
,
κ B(r )
σ (r )Wrr (r )
κ A(r )
where the extra terms above are provided by the modified
Einstein field equations of the particular form [
101–103
]
σ ( i )(Gμν − Wμν ) = κ Tμν .
As usually, the tensor Gμν = Rμν − 21 Rgμν is the Einstein
tensor, κ = −8π G, the factor σ ( i ) is a coupling to the
gravity while i represents for instance curvature invariants
or other fields, like scalar ones. The symmetric tensor Wμν
denotes additional geometrical terms which might appear in
considered ETG. It should be noted that Eq. (19) provides
a parameterization of gravitational theories at the level of
field equations. The energy–momentum tensor Tμν will be
considered as the one of a perfect fluid discussed in the
previous section. Because the tensor Wμν usually includes extra
fields like scalar or electromagnetic ones, apart the modified
Einstein’s field Eq. (19) we also deal with equations for the
additional fields.
As shown [
97
], in the case of the spherical-symmetric
metric
ds2 = −B(r )dt 2 + A(r )dr 2 + r 2dθ 2 + r 2 sin2 θ dϕ2, (20)
the modified Einstain field equations of the form Eq. (19)
reproduces the generalized TOV equations
1 +
4πr 3 σ
M
1 −
2GM
r
−1
(21)
(22)
σ
GM
= − r 2
M (r ) =
Q
σ + σ
Wθθ
r 2 −
Wrr
A
2σ
+ κr
r
0
4πr˜2 Qσ((rr˜˜)) dr˜.
A specification of a gravitational theory allows us to study
stellar configurations. We would like to emphasize the
importance of extra field equations which may appear in the
considered theory of gravity, as we have already shown it in
the previous work as well as we will always take them into
account further. Moreover, let us notice that the modified
mass Eq. (22) comes directly from the solution of the field
Eqs. (19) with the metric ansatz (20) so the extra term
proportional to Wtt should not be skipped in the examination of
the star’s configuration. One may treat them as a contribution
from additional fields to the theory, for example
electromagnetic or scalar ones, as caused by the influence of modified
equation of state [
104–106
] or as just pure geometric
modifications.
3.2 Stability of Palatini stars
Let us focus our attention on the discussed stability problem
in the case of Palatini gravity. From the structural Eq. (3)
we expect that the crucial quantity in the Palatini formalism,
that is, the conformal factor φ = f (Rˆ ), will depend on the
trace of the energy momentum tensor which in the case of
the perfect fluid has a form T = 3 p − ρ. As we assumed
the spherical-symmetric geometry, all physical quantities
depend on the radial coordinate r only and hence φ = φ (r )
which makes the both metric spherical-symmetric. Another
important observation is that the “Palatini–Einstein” tensor
Gˆ μν = Rˆμν − 21 gμν Rˆ built of both, Palatini-Ricci tensor and
where
or, if one prefers to write the mass distribution for the physical
energy density
ρ r˜2U (r˜)
4πr˜2 φ (r˜)2 (r˜) + 4Gφ (r˜)2
dr˜.
Let us underline that we took into account the transformations
of Eq. (15) together with the conformal transformation of the
metric, that is, the diagonal element of the metric g will be
denoted by A˜(r ) = φ−1 A(r ) and so on, which will be also
transparent in the further considerations.
Due to the relations (17) and the Eq. (19) one has
We are dealing with two conserved quantities in Einstein
frame
u¯ν ∇ˆ μT¯eμffν = u¯ν ∇ˆ μ
T¯ μν
∇ˆ μ J¯μ = ∇ˆ μnu¯μ = 0,
1 hμνU¯
− 2
= 0,
where the first equation is the nucleon number current
conservation law while the second one is the Bianchi identity
contracted with the vector field u¯α, which naively could be
thought as an observer with respect to the metric h but she
is not the one responsible for the measurements. Since the
relation between the physical observer and the vector field
u¯α is as already mentioned
u¯α = φ− 21 uα
we may apply the vector field u¯α to the Eq. (30) in order
to simplify the calculations. The another observation is that
the proper nucleon number density n appearing in Eq. (29)
is frame independent, that is, n = −uα JNα = −u¯α J¯Nα . From
Eq. (30)
the metric gμν turns out to be equal to the Einstein tensor
considered in so-called Einstein frame G¯ μν = R¯μν − 21 hμν R¯ .
Due to that fact the equations in both frames give a solution
on the metric hμν while the metric gμν can be obtained from
the conformal relation Eq. (4). Therefore we will refer to the
metric Eq. (20) as the metric hμν which enters the Einstein
equations in the Einstein frame (12a). This is a metric
responsible for a geodesic motion, that is, it is a Levi-Civita metric
of the connection ∇ˆ μ. According to this interpretation [
85
]
motion of particles should be considered with respect to the
Palatini connection while the metric g is related to light cones
and distances which are measured by a normalized observer
uμ, that is, uμuν gμν = −1.
Writing the Eq. (12a) as
1
R¯μν = κ T¯μν − 2 hμν (T¯ + U¯ (φ))
and using the spherical-symmetric ansatz on the metric hμν
one may write
R¯rr R¯00 R¯θθ A 1 1
2 A + 2B + r 2 = − r A2 − r 2 + Ar 2 = κρ¯ + 21 r 2U¯ (24)
from which immediately we get that
A(r ) =
1 −
2GM (r ) −1
r
,
(29)
(30)
(31)
(32)
(33)
(34)
(35)
which we write as
n 1 n
∇ˆ μn = ρ¯ + p¯ ∇ˆ μρ¯ + 2κ ρ¯ + p¯ U¯ ∇ˆ μφ,
where we may also use the relation (13) in the last term. Thus
the infinitesimal changes with respect to the infinitesimal
changes of the energy density are
n 1 n
δn = ρ¯ + p¯ δρ¯ + 2κ ρ¯ + p¯ U¯ δφ,
which turns out to be
n
δn = Q¯ + ¯ δ Q¯
after applying the definitions (28) and δρ¯ = δ Q¯ − U2¯κ δφ.
In order to examine the stability condition in the case of
Palatini gravity, we will follow the approach presented, for
example, in [
95
], that is, the Lagrange multipliers method.
Following Weinberg, let us formulate the proposition:
Proposition 1 A particular stellar configuration in the
Palatini gravity with an arbitrary function f (Rˆ ), with uniform
entropy per nucleon and chemical composition, will satisfy
the equations (φ = f (Rˆ ))
we interchange the r and r integrals in the last term after
replacing it by the above to get
4πr˜2 φQ((r˜r˜))2 dr˜,
d (r )
dr φ2(r )
= −
A˜ GM
r 2
+ Q
φ (r )2
is stationary with respect to all variations of Q¯ (r ) that leave
unchanged the quantity (nuclear number)
0
R
N =
4πr 2 1 −
1
2GM (r ) − 2
r
n(r )dr
and that leave the entropy per nucleon and the chemical
composition uniform and unchanged. The equilibrium is stable
with respect to radial oscillations if M is a minimum with
respect to all such variations.
In order to demonstrate the validity of the proposition, let
us firstly notice that the nuclear number N also depends on
modified geometry via the relation Eqs. (26) or (27).
Moreover, the extra term appearing in Eq. (21) vanishes in our
particular theory. Then we see that M is stationary with respect
to all variations that leave the nuclear number fixed if there
exists a constant λ such that M − λN is stationary with
respect to all variations. It means that we are interested in
the case when M can be an extremum for the unchanged
nuclear number, together with uniform entropy per nucleon
mass and the chemical decomposition. Hence we deal with
0
∞
0
−λ
0 = δM − λδ N = 0∞ 4πr 2δ Q¯ dr
∞4πr 2 1 − 2G Mr (r) − 21 δn(r )dr
−λG
3
4πr 1 − 2G Mr (r) − 2 n(r )δM (r )dr.
Using the relation Eq. (35) and gravitational mass definition
(26)
δM (r ) =
4πr 2δ Q¯ (r )dr ,
(36)
(37)
(38)
(39)
(40)
(41)
(42)
where A˜ = φ−1 A(r ). Thus, δM vanishes for all δ Q with the
constraint δ N = 0 if and only if one deals with Eq. (42). We
see from Eq. (38) that one deals with a stationary point of M
if the expression on the right-hand side vanishes. It will be
so if we are equipped with the generalized TOV equations in
the given form since this expression can be transformed into
it.
∞
r
r
δM − λδ N =
∞
0
λn(r )
4πr 2 1 − ¯ (r ) + Q¯ (r ) 1 −
1
2GM (r ) − 2
r
−λG
4πr n(r ) 1 −
3
2GM (r ) − 2
r
⎞
dr ⎠ δ Q¯ (r )dr.
From the above expression one sees that δM −λδ N vanishes
for all δ Q¯ (r ) if and only if
1 n(r )
λ = ¯ + Q¯
∞
1 −
1
2GM (r ) − 2
r
+ G
4πr n(r ) 1 −
3
2GM (r ) − 2
r
dr
for some the Lagrange multiplier λ. That means that the
righthand side the above result should be independent of r and
hence
0 =
n n( ¯ + Q¯ )
¯ + Q¯ − ( ¯ + Q¯ )2
Gn
+ ¯ + Q¯
− 4π Gr n 1 −
M
4πr Q¯ − r 2
3
2GM (r ) − 2
r
.
1 −
1 −
1
2GM (r ) − 2
r
3
2GM (r ) − 2
r
Since the entropy per nucleon is uniform, thus using
n (r ) =
n(r )Q¯ (r )
¯ + Q¯
we write Eq. (40) in the form of the TOV equations in Palatini
gravity with an arbitrary Lagrangian function as
φ (r )2
= −
G A˜M
r 2
Q +
φ (r )2
M (r ) =
0
r
4πr˜2 φQ((r˜r˜))2 dr˜,
1 +
4πr 3 φ(r)2
M
We have just shown that the mass M appearing in the
TOV equations is stationary, so the necessary condition for
the star to pass from stability to instability with respect to
some radial perturbation is satisfied. In order to have the
TOV equations which describe a stable star’s configuration,
the constrained extremum must be a minimum: therefore one
deals with δ2M > 0. We see from the form of (27) that
the second order in δ Q¯ has to be positive-definite for all
perturbations, analogously to the δρ in General Relativity.
2
It has to be so since if any of the squared frequencies ωn
from the all perturbation modes takes negative values then
the frequency is purely imaginary and the time variation of
this mode would grow exponentially leading to an unstable
system.
4 Conclusions
We have examined the stability condition of neutron stars in
the case of the Palatini gravity with an arbitrary Lagrangian
functional f ( Rˆ ). We immediately notice that the obtained
result Eq. (42) reduces to the General Relativity equations
describing equilibrium of the relativistic stars, specifically,
when f ( Rˆ ) = Rˆ since φ ≡ f ( Rˆ ) = 1, where prime denotes
here the derivative with respect to the Palatini scalar Rˆ . Then
also the term proportional to U (φ), which appears in the
definitions of generalized energy density and pressure, vanishes.
The performed analysis on the stability criterion for
Palatini gravity, which is the simplest representation of EPS
formulation, shows that a stability condition of a neutron star is
very similar to the one in General Relativity. Let us notice
that in the case of General Relativity one gives the final
statement if the star’s system is stable or not after the examination
of the stability conditions with respect to an equation of state.
In Palatini gravity one has to do even more: we see that the
generalized energy density Q depends on ρ and the form of
the f ( Rˆ ) via the potential U and coupling φ. Thus, besides
the equation of state we need to choose a model which we
wish to study in Palatini formulation. We would like to add
here that in the case when we are able to solve the master
equation to get Rˆ = Rˆ (T ), the generalized energy density
appears finally to depend on the equation of state only.
With the all discussed points above, and together with
the previous result presented in [
100
], where we
considered scalar-tensor gravity with a minimally coupled scalar
field and an arbitrary potential, we conclude that there are
other gravitational theories besides General Relativity which
allow us to consider possible stable stars’ configurations with
respect to radial oscillations. Future and more detailed work
along these lines is currently underway.
Acknowledgements AW is grateful to the Mainz Institute for
Theoretical Physics (MITP) for its hospitality and its partial support during
the completion of this work. The work is supported by the NCN Grant
D EC − 2014/15/B/ST 2/00089. The author would also like to thank
Andrzej Borowiec and Diego Rubiera-Garcia for their comments and
helpful discussions.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
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Commons license, and indicate if changes were made.
Funded by SCOAP3.
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