Phases of 4D Scalar-tensor black holes coupled to Born-Infeld nonlinear electrodynamics
ABSTRACT Recent results show that when non-linear electrodynamics is considered the no-scalar-hair theorems in the scalar-tensor theories (STT) of gravity, which are valid for the cases of neutral black holes and charged black holes in the Maxwell electrodynamics, can be circumvented. What is even more, in the present work, we find new non-unique, numerical solutions describing charged black holes coupled to non-linear electrodynamics in a special class of scalar-tensor theories. One of the phases has a trivial scalar field and coincides with the corresponding solution in General Relativity. The other four phases that we find are characterized by the value of the scalar field charge. The causal structure and some aspects of the stability of the solutions have also been studied. For the scalar-tensor theories considered, the black holes have a single, non-degenerate horizon, i.e., their causal structure resembles that of the Schwarzschild black hole. The thermodynamic analysis of the stability of the solutions indicates that a phase transition may occur.
arXiv:0708.4141v3 [gr-qc] 20 Aug 2008
Phases of 4D Scalar-tensor black holes coupled to
Born-Infeld non-linear electrodynamics
Ivan Zh. Stefanov1∗, Stoytcho S. Yazadjiev1,2 †
1Dept. of Theoretical Physics, Faculty of Physics
St.Kliment Ohridski University of Sofia
5, James Bourchier Blvd., 1164 Sofia, Bulgaria
2Institut f¨ ur Theoretische Physik, Universit¨ at G¨ ottingen
Friedrich-Hund-Platz 1, D-37077 G¨ ottingen, Germany
Michail D. Todorov‡
Faculty of Applied Mathematics and Computer Science
Technical University of Sofia
8, Kliment Ohridski Blvd., 1000 Sofia, Bulgaria
Recent results show that when non-linear electrodynamics is considered the
no-scalar-hair theorems in the scalar-tensor theories (STT) of gravity, which are
valid for the cases of neutral black holes and charged black holes in the Maxwell
electrodynamics, can be circumvented [1, 2]. What is even more, in the present
work, we find new non-unique, numerical solutions describing charged black holes
coupled to non-linear electrodynamics in a special class of scalar-tensor theories.
One of the phases has a trivial scalar field and coincides with the corresponding
solution in General Relativity. The other phases that we find are characterized by
the value of the scalar field charge. The causal structure and some aspects of the
stability of the solutions have also been studied. For the scalar-tensor theories
considered, the black holes have a single, non-degenerate horizon, i.e., their causal
structure resembles that of the Schwarzschild black hole. The thermodynamic
analysis of the stability of the solutions indicates that a phase transition may
Among the most natural generalizations of General Relativity (GR) are the scalar-
tensor theories of gravity in which a single or multiple fundamental scalar fields are
present as a possible remnant of a fundamental unified theory like string theory or
higher dimensional gravity theories . Different modifications of scalar-tensor theories
are attracting much interest also in cosmology and astrophysics.
Possible deviations from GR, especially the existence of new effects, due to the
presence of the scalar field would be of considerable interest. According to the no-
scalar-hair conjecture, the black-hole solutions in the STT coincide with the solutions
from GR. No-scalar-hair theorems treating the cases of static, spherically symmetric,
asymptotically flat, electrically neutral black holes and charged black holes in the
Maxwell electrodynamics have been proved for a large class of scalar-tensor theories
[4, 5, 6]. The scalar field in these cases is constant, and thus trivial, if one demands
that the essential singularity at the center of symmetry is hidden in a regular event
There are no no-scalar-hair theorems, however, in the non-linear electrodynamics
(NLED). It was first introduced by Born and Infeld in 1934 to obtain finite energy
density model for the electron . For more information on recent interests in non-
linear lagrangians of electrodynamics please see , also – and references cited
therein for gravitational aspects of NLED.
In the case of NLED, the energy-momentum tensor of the electromagnetic field
has a non-vanishing trace which is non-trivially coupled to the scalar field. Hence,
the electro-magnetic field acts as a source of the scalar field and allows the existence
of asymptotically flat, hairy black holes. Such solutions have been found recently in
different non-linear electrodynamics [1, 2]. These solutions are hairy in a sense that
the scalar field is not trivial. That hair, however, is secondary since the solutions are
determined uniquely by the values of their magnetic charge and mass, and the value of
the scalar field at infinity. For other solutions describing asymptotically flat black holes
with scalar hair (in which, however, the potential of the scalar field is not positively
semi-definite) we refer the reader to .
Another interesting effect that occurs when NLED is considered is the presence of
multiple black-hole phases in a certain class of STT. Presence of non-unique solutions
in STT has been found by Damour and Esposito-Far` ese in their works on neutron
stars in the STT. One of the most interesting effects that occur in these solutions is
the spontaneous scalarization, a scalar field analogue of the spontaneous magnetization
of ferromagnets [32, 33, 34]. In the present work we consider the same STT and find
multiple-phase black-hole solutions numerically. Unlike the situation in [1, 2], they are
not determined in a unique way by the values of their magnetic charge and mass and
the value of the scalar field at infinity. The phase diagram of the present solutions is
reminiscent of the phase transition between caged black holes and black strings in higher
dimensions [35, 36]. The preliminary analysis gives us a reason to suppose that in the
system we study a phase transition might occur. The analysis of the thermodynamics
of the solutions, however, is not sufficient to make a conclusion about the stability of
the solutions and about the existence of phase transitions. The full examination of the
problem requires a perturbative analysis.
2 Basic equations and qualitative investigation
The theory we consider is presented [1, 2] both in the Jordan and in the Einstein
frames. In the Einstein frame the action of the theory we consider is
d4x√−g?R − 2gµν∂µϕ∂νϕ − 4V (ϕ) + 4A4(ϕ)L(X,Y )?.
V (ϕ) is the potential of the scalar field, L(X) is the Lagrangian of the electromagnetic
field, A4(ϕ) is the function which determines the coupling between the electromagnetic
field and the scalar field. We also have that
FµνgµαgνβFαβ, Y =A−4(ϕ)
where “⋆” stands for the Hodge dual with respect to the Einstein frame metric gµν.
The type of the STT is determined by the explicit choice of the functions V (ϕ) and
A4(ϕ). The field equations obtained from action (1) take the following form
Rµν= 2∂µϕ∂νϕ + 2V (ϕ)gµν− 2∂XL(X,Y )
−2A4(ϕ)[L(X,Y ) − Y ∂YL(X,Y )]gµν,
∇µ[∂XL(X,Y )Fµν+ ∂YL(X,Y )(⋆F)µν] = 0,
∇µ∇µϕ =dV (ϕ)
− 4α(ϕ)A4(ϕ)[L(X,Y ) − X∂XL(X,Y ) − Y ∂YL(X,Y )],
where α(ϕ) =
In what follows the truncated1Born-Infeld electrodynamics described by the La-
LBI(X) = 2b
will be considered. And V (ϕ) will be equal to zero.
The anzats for metric of a static, spherically symmetric space-time can be taken in
the following form
ds2= gµνdxµdxν= −f(r)e−2δ(r)dt2+dr2
f(r)+ r2?dθ2+ sin2θdφ2?.
Since the Born-Infeld NLED is invariant under electric-magnetic duality rotations we
will study only the magnetically charged black holes for which the electromagnetic field
is given by
F = P sinθdθ ∧ dφ
and the magnetic charge is denoted by P.
1Here we consider the pure magnetic case for which Y = 0.
The field equations reduce to the following coupled system of ordinary differential
−4α(ϕ)A4(ϕ)[L(X) − X∂XL(X)] − rf
f = 1 −2m
and X reduces to:
We will be searching for solutions which have a regular horizon on which the scalar
field ϕ does not diverge. The regularity of the transition between the Einstein and the
Jordan conformal frames requires the following restrictions on the coupling function
0 < A(ϕ) < ∞ for r ≥ rH, where rHis the radius of the horizon, and we will consider
STT for which it is satisfied. The diversity and the properties of the solutions depend
strongly on the choice of the functions A(ϕ) (respectively on α(ϕ)). Even when the
functions A(ϕ) are chosen to satisfy the experimental constraints (see, for example,
) we are left with infinitely many possibilities. That is why some restrictions on the
functions A(ϕ) representing the STT should be imposed . In the present work we will
consider only theories for which ϕα(ϕ) is non-negative or non-positive for all values of
ϕ. As we prove below, theories for which ϕα(ϕ) ≥ 0 for all values of ϕ do not admit
(asymptotically flat) black hole solutions with non-trivial scalar field. In theories with
ϕα(ϕ) ≤ 0 black holes (if they exist) have a single non-degenerate event horizon when
α(ϕ) = 0 admits only the solution ϕ = 0.
Using the following equation
= 4r2e−δα(ϕ)A4(ϕ)[X∂XL(X) − L(X)],
which is another form of equation (8) and the fact that for the Born-Infeld Lagrangian
X∂XL(X) − L(X) > 0
we can draw some conclusions about the general properties of the solutions.
Let us multiply equation (10) by ϕ and then integrate it in the interval r ∈ [rH,∞)
where we denote the radius of the outer horizon (the event horizon) with rH
r2e−δϕα(ϕ)A4(ϕ)[X∂XL(X) − L(X)]dr,
and after integrating by parts we get
r2e−δϕα(ϕ)A4(ϕ)[X∂XL(X) − L(X)]dr. (13)
In the second line we take advantage of the fact that we are looking for asymptotically
flat, black hole solutions so f(rH) = 0, limr→∞f(r) = 1, limr→∞δ(r) = 0 and for the
asymptotic value of the scalar field we impose limr→∞ϕ(r) = ϕ∞, where ϕ∞= 0. The
constant D denotes the scalar charge which is defined as
D = − lim
Since ϕ is vanishing at infinity we finally get
r2e−δϕα(ϕ)A4(ϕ)[X∂XL(X) − L(X)]dr.
The total sign of the left-hand side of (15) is negative for black hole solutions with
nontrivial ϕ. The sign of the integral on the right-hand side depends on the sign of
ϕα(ϕ). If ϕα(ϕ) ≥ 0 a contradiction is reached so the assumption for the existence
of asymptotically flat black holes with non-trivial scalar field in this theory is wrong.
Asymptotically flat black-hole solutions with nontrivial scalar field exist in scalar-tensor
theories for which ϕα(ϕ) ≤ 0. Through a similar examination we prove that these
black holes have a single, non-degenerate horizon. Let us admit that more than one
horizon exists. Then we multiply equation (10) by ϕ again and integrate it by parts
in the interval r ∈ [r−,r+] where we denote the first inner horizon and the outermost,
non-degenerate horizon with r−and r+, respectively
r2e−δϕα(ϕ)A4(ϕ)[X∂XL(X) − L(X)]dr < 0.
Having in mind that f(r−) = 0 = f(r+) and that f < 0 in the interval r ∈ [r−,r+] we
reach a contradiction, which means that the admission is incorrect.
Now, we only have to prove the non-existence of extremal black holes (black holes
with degenerate event horizon). For the scalar-tensor theories we consider α(ϕ) turns
to zero only when ϕ = 0. Let us admit that an extremal black hole with non-trivial
scalar field exists. In this case, the left-hand side of (10) is equal to zero. The right-
hand side is equal to zero only when α(ϕH) = 0, where ϕHis the value of the scalar
field on the horizon, which means that ϕH= 0. We also require that ϕ∞= 0, where
ϕ∞is the value of the scalar field at the spacial infinity. In this situation, the only
possibility to have a solution with non-trivial scalar field is the scalar field to have at
least one extremum. So let us integrate equation (10) in the interval r ∈ [rH,re], where
reis the point of the leftmost (the one which is nearest to the event horizon) extremum
of ϕ which is on the right of the event horizon
r2e−δα(ϕ)A4(ϕ)[X∂XL(X) − L(X)]dr.
Since ϕ ?= 0 in the interval (rH,re], the sign of α(ϕ) also does not change in this
interval. Hence, the integral on the right-hand side of (17) is non-zero and has a fixed
sign which depends on the sign of α(ϕ). The contradiction we reach means that no
extremal solutions with non-trivial scalar field can exist.
To sum up, we can say that if a black hole exists it will have a single horizon, i.e.,
its causal structure will be Schwarzschild-like2. In both conformal frames, inside the
event horizon a space-like singularity is hidden.
Finishing this section it is worth noting that the differential equations system (6–
8) is invariant under the rigid rescaling r → λr, m → λm, P → λP and b → λ−2b
where 0 < λ < ∞. Therefore, given a solution to (6–8) with one set of physical pa-
rameters (rh,M,P,b,D,T), the rigid rescaling produces new solutions with parameters
(λrh,λM,λP,λ−2b,λD,λ−1T). Here T denotes the temperature of the horizon.
2Another class of scalar-tensor theories which admit black holes of the Schwarzschild type are those
with negative function β(ϕ) =dα(ϕ)
for all values of ϕ and α(ϕ∞)=0. This can be shown by a method
similar to that presented above. One can also show that scalar-tensor theories with β(ϕ) > 0 for all
values of ϕ do not admit asymptotically flat black holes with nontrivial scalar field.
The nonlinear system (6)-(8) is inextricably coupled and the event horizon rHis a priori
unknown boundary. In order to be solved numerically, it is recast as a equivalent first
order system of ordinary differential equations. Following the physical assumptions of
the matter under consideration the asymptotic boundary conditions are set, i.e.,
r→∞m(r) = M
(M is the mass of the black hole in the Einstein frame),
r→∞δ(r) = lim
r→∞ϕ(r) = 0.
At the horizon both the relationship
f(rH) = 0
and the regularization condition
=?4α(ϕ)A4(ϕ)[X∂XL(X) − L(X)]???
concerning the spectral quantity rH must be held. For the treating the above posed
boundary-value problem (BVP) the Continuous Analog of Newton Method (see, for
example ,,) is used. After an appropriate linearization the original BVP is
rendered to solving a vector two-point BVP. On a discrete level sparse (almost diagonal)
linear algebraic systems with regard to increments of sought functions δ(r), m(r), and
ϕ(r) have to be inverted.
For our numerical solutions we have considered the STT studied by Damour and
Esposito-Far` ese in their works on neutron stars in the STT of gravity. In this particular
theory, the coupling function has the following form
A(ϕ) = e
where β is a negative constant. Observational data from binary-pulsar and solar-
system experiments restricts the admissible values of β. When α(ϕ = 0) = 0, as it is
in our case, the coupling constant should be β > −5 (see  for more details). We
have studied the solutions for values of the parameters which are in agreement with
the current observations but also for such values that are out the admissible interval
since the later have qualitatively different behavior from the former which makes them
interesting for the theory.
For this coupling function the field equations possess the discrete symmetry ϕ →
−ϕ. Let us also note that every general relativistic solution is a solution to the scalar
tensor theory under consideration with ϕ = 0.
A thorough study of the phase space would be difficult due to the large number of
parameters (β,b,P,M) in the problem. So in order to illustrate the general behavior of
the obtained solutions we give several representative figures considering several values
of β,b and P and varying the mass M of the black hole. For the cases presented here
b = 0.01; 0.2.
3.1 General description of the phase space
Even if the values of all four parameters β,b,P,M and the boundary conditions are
fixed the solutions of (6-8) are not uniquely determined, i.e. the solutions are not
unique. An additional parameter should be used for labeling of the different solutions.
One natural choice3would be to label the different solution by the value of the scalar
charge D, which here unlike the cases in [1, 2] is independent.
The global structure of the phase space changes with the variation of β. For β >
βcrit, where βcrit≈ −14.9 when b = 0.01 and P = 1.0, the M−D phase diagram consists
of three branches, while for β < βcritthe number of branches is five. The change of the
qualitative structure of that phase diagram with the variation of β is given in Figure
(1). The different cases have been studied in more details in the succeeding subsections.
Figure 1: The M − D relation for several different values of β, b = 0.01 and P = 1.0.
For β > βcritthe Middle branch disappears.
3.2Cases with three solutions, β = −4.0
In the left panel of Figure (2) the dependence D(M) of the scalar charge on the mass
for β = −4.0 is presented. There are two special points to be considered, namely A
and E. For P = 1.0 the points A and E lie at masses MA≈ 0.794 and ME≈ 0.46. For
values of the black hole mass in the interval M ∈ (ME,MA) three solutions co-exist.
We call them Outer and Trivial. The equations posses a discrete symmetry ϕ → −ϕ
as a result of which the Outer solution has a mirror image with respect to the abscissa,
whose scalar field charge D has an opposite sign. The symmetric solutions we will
denote as Outer±where the indices + and − refer to the sign of the scalar field charge
of the solutions. The radius and the temperature of the event horizon, are the same
for both solutions in the couple so when we comment on them will usually omit the +
and − indices.
3An alternative choice would be the value of the scalar field on the horizon
Figure 2: The M − D and the M − rHrelations, for β = −4.0, b = 0.01 and P = 1.0.
?? ?? ???
Figure 3: The M −T−1relation for the same values of the magnetic charge as in Figure
The Trivial brunch has scalar field ϕ ≡ 0 and represents the Einstein-Born-Infeld
solution in GR. For masses lower than MEonly the solutions which we call Outer exist
and for M > MAonly the Trivial solution remains.
The radii of the black holes are shown in the right panel of Figure (2). An object
with zero radius of the event horizon (a naked singularity) is reached for a finite value
of M for all three solutions - the Outer±and the Trivial.
The inverse temperature T−1of the solutions is presented in Figure (3).
3.3Cases with five solutions, β = −50.0
Solutions with Born-Infeld parameter b=0.013.3.1
Figure 4: The value of scalar field charge D as a function of the mass M of the black
hole, for two different values of the magnetic charge P = 1.5 and P = 3.0.
Figure 5: The M −rHrelation for the same values of the magnetic charge as in Figure
(4). A magnification in the encircled region is presented in Figure (6).
For β = −50.0 and b = 0.01 the D(M) dependence is given in Figure (4). Here, the
special points to be considered are four, A, B±and E. For P = 1.5 the points A and
B±lie at masses MA≈ 2.023 and MB±= MB≈ 2.37 while in the case of P = 3.0 we
Figure 6: A magnification of the encircled region in Figure (5).
Figure 7: The M −T−1relation for the same values of the magnetic charge as in Figure
have MA≈ 3.60 and MB±= MB≈ 4.53. As it can be seen, for values of the black hole
mass in the interval M ∈ (MA,MB) five solutions co-exist. They can be separated in
three groups which we name Outer, Middle and Trivial. The symmetric solutions we
will denote as Outer±and Middle±with the same convention for the + and − indices
as in the previous case. Both points B+and B−are projected on the same point on
the M − rHand M − T−1diagrams, which we denote simply as B.
The Trivial has scalar field ϕ ≡ 0 and represents the Einstein-Born-Infeld solution
in GR. For masses lower than ME≈ 1.00 and ME≈ 2.42 for P = 1.5 and P = 3.0,
respectively, only the solutions which we call Outer exist and for M > MB only the
Trivial solution remains.
The radii of the black holes are shown in Figures (5) and (6). Again, for three of
the solutions - the Outer and the Trivial a naked singularity is reached for a finite
value of M. The Outer solutions have a larger radius than the Trivial for low masses,
but as it can be seen in Figure (6), which is a magnification of the encircled region in
Figure (5), with the increase of the mass, in point C, the situation changes and the
black hole with zero scalar charge becomes larger. The approximate position of point
C is shown also on Figure (4) with a dotted vertical line. The Middle solution black
holes are smaller than the other three for all values of M.
The inverse temperature T−1of the solutions is presented in Figure (7). With the
decrease of the mass M the inverse temperature of the Trivial solution passes through
a local maximum which gets sharper with the increase of the magnetic charge P and
leads to numerical calculation difficulties. In the limit of vanishing radii of the black
holes their inverse temperature decreases steeply.
3.3.2Solutions with Born-Infeld parameter b=0.2
Figure 8: The M − D and the M − rHrelations, for b = 0.2 and P = 1.0.
Here we present an example of solutions for values of the parameters for which the
Trivial solution reaches an extremal black hole instead of a naked singularity with the
decrease of the mass M. In Figure (8) the dependence D(M) of the scalar charge on
the mass and the radii of the black holes are presented. The points A and B±lie at
masses MA≈ 1.13 and MB±= MB≈ 1.54. Again, for values of the black hole mass
Figure 9: A magnification in the encircled region of the M − rHrelation from Figure
(8) and the M − T−1relation, for b = 0.2 and P = 1.0. .
in the interval M ∈ (MA,MB) five solutions co-exist. The Trivial reaches an extremal
black hole at ME≈ 0.93 and this can be seen on the M − rHdiagram.
In Figure (9) a magnification of the encircled region from Figure (8) and the inverse
temperature are shown. The point C is once again indicated with a vertical line.
With the approaching of the extremal black hole the inverse temperature T−1rises
Black holes have long been known to be thermodynamical systems . The First Law
(FL) of black hole thermodynamics in the presence of a scalar field has the following
δM = TδS + ΨHδP + Dδϕ∞,
where T, S and P are the temperature, the entropy, and the magnetic charge of the
black hole, respectively, and ϕ∞is the value of the scalar field at spacial infinity.
The quantity Ψ conjugate to the magnetic charge is the potential of the magnetic
field which is given by the following definition
On the other hand the magnetic field is defined as
Hµ= − ⋆ Gµνξν,
Gµν= −2∂ (A4(ϕ)L)
∂tis the Killing vector generating time translations and “⋆” is the Hodge star
Since in our case the asymptotic value of scalar field is fixed, the term which contains
its variation vanishes and the FL reduces to the form it has in GR.
In the situation when the solutions are not unique a natural question is which of
them are stable. Certainly, the answer of this question requires linear perturbative
analysis of our system of coupled differential equations and solving the corresponding
eigenvalue problem. In certain cases, however, some information on the stability can be
inferred by using only the equilibrium thermodynamical characteristics of the solutions
via the so-called “Poincare” or “turning point” method. For a nice discussion of the
method we refer the reader to  and references therein.
method is its remarkable simplicity. The “turning point” method, however, may hide
many uncertainties and should be applied with caution. The method consists in the
following. Consider a system with thermodynamical parameters µi. The equilibrium
states (stable or not) are extrema of an appropriate Massieu function S. At equilibrium
state we can define the conjugate variables βi(µj) =
change of stability can only occur at turning points4or bifurcations in the equilibrium
sequence on the conjugate diagram βi(µi). In the absence of bifurcations the stability
character changes only when the equilibrium curve meets a turning point and if one
single point of an equilibrium sequence is shown to be fully stable, then all equilibria
in the sequence are fully stable up to the first turning point. At the turning point the
branch with negative slope is always unstable while the branch with a positive slope is
more stable than the one with negative slope.
Concerning the application of the “turning point” method to our case we shall
consider scalar-tensor black holes in the micro-canonical ensemble. In this case the
corresponding Massieu function is the entropy S(M)5. We will comment on the case
with five branches first. The conjugate variable is βM = T−1. For β < βcrit the
conjugate diagram M −T−1is shown on Figures (7) and (9). The uncertainties in our
case come from that fact that we do not know the full diagram, i.e., whether there are
other branches and bifurcation points different from point A. Assuming however that
there are no other bifurcation points and taking into account that there is only one
turning point B we may conclude that the Middle branch is unstable while the Outer
branch is more stable. The cusp which appears in point B on the M − rH diagram
in Figures (6) and (9) also indicates a change in the stability of the solutions. Since
the Outer branch is the unique solution (up to the discrete degeneracy Outer±) for
sufficiently small masses one might accept that the Outer branch is probably stable
there. Then according to the “turning point” method the Outer branch should be fully
stable up to the turning point B. The Trivial branch is unique solution for sufficiently
large masses and as a GR solution is known to be stable6there [42, 43]. So, assuming
that Trivial branch is stable for large masses, the “turning point” method asserts that
the Trivial branch should be stable up to the bifurcation point A. Since the entropy of
a black hole is proportional to the area of its event horizon, among the three black-hole
The advantage of this
∂µi. According to the method the
4The turning points are points where two equilibrium branches merge with a vertical tangent. A
bifurcation point is point where branching of equilibrium sequences occurs.
5We keep the magnetic charge fixed.
6In general, the stability of the trivial solution within the framework of GR does not guarantee its
stability within the “larger” scalar-tensor theory.
phases we consider, the one with the biggest radius would have the maximal entropy
(see Figures (5), (6), (8) and (9)). So for M < MC the Outer solutions would be
thermodynamically favorable and for M > MC - the Trivial. The Middle solutions
are thermodynamically unstable for all values of the mass since their radius is smaller
than the radii of the other three solutions. The point C is a candidate for a point of
a first order phase transition. Let us stress again that above analysis based on the
“turning point” method is only suggestive and cannot serve for a definitive solution
of the stability of the solutions. Reliable analysis of the risen questions will be given
elsewhere together with the solution of corresponding eigenvalue problem.
For β > βcritthe M −T−1diagram is given in Figure (3). No turning points can be
seen on it. As in the previously discussed case, since Outer±are the only solutions for
sufficiently small masses one might accept that they are probably stable there. Then
according to the “turning point” method the Outer branch should be fully stable up to
the bifurcation point A. Again, assuming that Trivial branch is stable for large enough
masses one can expect that to the left it should be stable at least up to the bifurcation
point A. From the right panel of Figure (2) we can see that the Outer solutions have
larger radius of the event horizon so they would be thermodynamically favorable.
The thermodynamical stability considerations were made in the Einstein frame. In
order to transfer the conclusions to the physical, Jordan frame properly, we need to
clarify the connection between the thermodynamic properties in the two conformal
frames. The temperature of the event horizon is invariant under conformal transfor-
mations of the metric that are unity at infinity . The properly defined entropy is
also the same in both conformal frames. It has been proved that in the Jordan frame
the entropy of the black hole is not simply one fourth of the horizon area [45, 46] as
in the Einstein frame and needs to be generalized. The entropy in the Jordan frame is
Passing to the Einstein frame we get
−(2)g = SE= S.
In the last two equations(2)˜ g and(2)g are the determinants of the induced metrics on
the horizon in the Jordan and in the Einstein frame, respectively.
The term in the FL (19) connected with the magnetic charge is also preserved under
the conformal transformations.
In order for the FL of thermodynamics to be satisfied in the Jordan frame the mass
should be properly chosen since the Arnowitt-Deser-Misner (ADM) masses in both
frames are not equivalent. It can be easily shown that in the STT considered the ADM
mass in the Jordan frame MJis equal ADM mass in the Einstein frame M
For the Jordan frame, the proper mass in the FL of thermodynamics is the ADM mass
in the Einstein frame M. Similarly, for boson and fermion stars the proper measure for
the energy of the system is again the ADM mass in the Einstein frame M. For more
details on the subject we would refer the reader to the works [47, 48, 49, 50].