Thermodynamics of a class of non-asymptotically flat black holes in ...
Thermodynamics of a class of non-asymptotically flat black holes in ...
Thermodynamics of a class of non-asymptotically flat black holes in ...
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The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong><br />
<strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> <strong>holes</strong> <strong>in</strong><br />
E<strong>in</strong>ste<strong>in</strong>-Maxwell-Dilaton theory<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
Universidade Federal do Pará<br />
Faculdade de Física<br />
II Amazonian Workshop on Black Holes and Analogue Models <strong>of</strong> Gravity<br />
June 6, 2013<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
Abstract<br />
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
We analyse <strong>in</strong> detail the thermodynamics <strong>in</strong> the ca<strong>non</strong>ical<br />
and grand ca<strong>non</strong>ical ensembles <strong>of</strong> a <strong>class</strong> <strong>of</strong><br />
<strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> <strong>holes</strong> <strong>of</strong> the E<strong>in</strong>ste<strong>in</strong>-(anti)<br />
Maxwell-(anti) Dilaton theory <strong>in</strong> 4D with spherical symmetry.<br />
We present the first law <strong>of</strong> thermodynamics, the thermodynamic<br />
analysis <strong>of</strong> the system through the geometrothermodynamics<br />
methods, We<strong>in</strong>hold, Ruppe<strong>in</strong>er, Liu-Lu-Luo-Shao and the most<br />
common, that made by the specific heat. We also analyse the<br />
local and global stability <strong>of</strong> the thermodynamic system<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
Outl<strong>in</strong>e<br />
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
1 The action and the specific model;<br />
2 <strong>Thermodynamics</strong> analysis;<br />
3 Conclusion.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The model<br />
The action <strong>of</strong> E<strong>in</strong>ste<strong>in</strong>-Maxwell-Dilaton theory is given by:<br />
∫<br />
S = dx 4√ ]<br />
−g<br />
[R − 2 η 1 g µν ∇ µ ϕ∇ ν ϕ + η 2 e 2λϕ F µν F µν , (1)<br />
where the first term is the usual E<strong>in</strong>ste<strong>in</strong>-Hilbert gravitational<br />
term, while the second and the third are respectively a k<strong>in</strong>etic<br />
term <strong>of</strong> the scalar field (dilaton or phantom) and a coupl<strong>in</strong>g term<br />
between the scalar and the Maxwell fields, with a coupl<strong>in</strong>g<br />
constant λ that we assume to be real. The coupl<strong>in</strong>g constant η 1<br />
can take either the value η 1 = 1 (dilaton) or η 1 = −1<br />
(anti-dilaton). The Maxwell-gravity coupl<strong>in</strong>g constant η 2 can<br />
take either the value η 2 = 1 (Maxwell) or η 2 = −1<br />
(anti-Maxwell).<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
In order to obta<strong>in</strong> the str<strong>in</strong>gs models, we choose a general<br />
conformal factor<br />
ĝ µν = e −2ωϕ g µν , (2)<br />
The action (1), which is <strong>in</strong> the E<strong>in</strong>ste<strong>in</strong> frame, can be <strong>class</strong>ified<br />
<strong>in</strong>to the three follow<strong>in</strong>g types:<br />
a) Type I Str<strong>in</strong>gs<br />
As we need the usual Str<strong>in</strong>gs models 1 , mak<strong>in</strong>g use <strong>of</strong> the<br />
dilaton case <strong>in</strong> which ω = η 1,2 = 1, λ = 1 2<br />
, the action (1) reads:<br />
∫<br />
S I =<br />
√ { [<br />
d 4 x −ĝ e −2ϕ ̂R + 4ĝ µν ̂∇µ ϕ ̂∇<br />
]<br />
ν ϕ − e −ϕ 2 ̂F } , (3)<br />
1 We could formulate the Str<strong>in</strong>gs models with phantom contribution.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
In order to obta<strong>in</strong> the str<strong>in</strong>gs models, we choose a general<br />
conformal factor<br />
ĝ µν = e −2ωϕ g µν , (2)<br />
The action (1), which is <strong>in</strong> the E<strong>in</strong>ste<strong>in</strong> frame, can be <strong>class</strong>ified<br />
<strong>in</strong>to the three follow<strong>in</strong>g types:<br />
a) Type I Str<strong>in</strong>gs<br />
As we need the usual Str<strong>in</strong>gs models 1 , mak<strong>in</strong>g use <strong>of</strong> the<br />
dilaton case <strong>in</strong> which ω = η 1,2 = 1, λ = 1 2<br />
, the action (1) reads:<br />
∫<br />
S I =<br />
√ { [<br />
d 4 x −ĝ e −2ϕ ̂R + 4ĝ µν ̂∇µ ϕ ̂∇<br />
]<br />
ν ϕ − e −ϕ 2 ̂F } , (3)<br />
1 We could formulate the Str<strong>in</strong>gs models with phantom contribution.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
b) Type IIA Str<strong>in</strong>gs<br />
For the parameters ω = η 1,2 = 1, λ = 0, the action (1)<br />
becomes:<br />
∫ √ { [<br />
S IIA = d 4 x −ĝ e −2ϕ ̂R + 4ĝ µν ̂∇µ ϕ ̂∇<br />
]<br />
ν ϕ − ̂F 2} . (4)<br />
c) Heterotic Str<strong>in</strong>gs<br />
For the parameters ±ω = η 1,2 = ∓λ = 1, the action (1) behaves<br />
as:<br />
∫ √ {<br />
S H = d 4 x −ĝ e −2ϕ ̂R + 4ĝ µν ̂∇µ ϕ ̂∇ ν ϕ − ̂F 2} , (5)<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
b) Type IIA Str<strong>in</strong>gs<br />
For the parameters ω = η 1,2 = 1, λ = 0, the action (1)<br />
becomes:<br />
∫ √ { [<br />
S IIA = d 4 x −ĝ e −2ϕ ̂R + 4ĝ µν ̂∇µ ϕ ̂∇<br />
]<br />
ν ϕ − ̂F 2} . (4)<br />
c) Heterotic Str<strong>in</strong>gs<br />
For the parameters ±ω = η 1,2 = ∓λ = 1, the action (1) behaves<br />
as:<br />
∫ √ {<br />
S H = d 4 x −ĝ e −2ϕ ̂R + 4ĝ µν ̂∇µ ϕ ̂∇ ν ϕ − ̂F 2} , (5)<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
This action (1) leads to the follow<strong>in</strong>g field equations:<br />
∇ µ<br />
[e 2λϕ F µα] = 0 , (6)<br />
□ϕ = − 1 2 η 1η 2 λe 2λϕ F 2 , (7)<br />
( )<br />
1<br />
R µν = 2η 1 ∇ µ ϕ∇ ν ϕ + 2η 2 e 2λϕ 4 g µνF 2 − Fµ σ F νσ (8).<br />
A possible solution for spherical symmetry, is given by<br />
dS 2 = r γ (r − r + )<br />
r 1+γ dt 2 − r 1+γ<br />
0<br />
r γ (r − r + ) dr 2 − r 1+γ<br />
0<br />
r 1−γ dΩ 2 , (9)<br />
0<br />
√<br />
( )<br />
1 + γ 1<br />
r 1−γ<br />
F = ∓ dr ∧ dt , e 2λϕ = , (10)<br />
2η 2 r 0 r 0<br />
where the parameter γ = (1 − η 1 λ 2 )/(1 + η 1 λ 2 ).<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
This action (1) leads to the follow<strong>in</strong>g field equations:<br />
∇ µ<br />
[e 2λϕ F µα] = 0 , (6)<br />
□ϕ = − 1 2 η 1η 2 λe 2λϕ F 2 , (7)<br />
( )<br />
1<br />
R µν = 2η 1 ∇ µ ϕ∇ ν ϕ + 2η 2 e 2λϕ 4 g µνF 2 − Fµ σ F νσ (8).<br />
A possible solution for spherical symmetry, is given by<br />
dS 2 = r γ (r − r + )<br />
r 1+γ dt 2 − r 1+γ<br />
0<br />
r γ (r − r + ) dr 2 − r 1+γ<br />
0<br />
r 1−γ dΩ 2 , (9)<br />
0<br />
√<br />
( )<br />
1 + γ 1<br />
r 1−γ<br />
F = ∓ dr ∧ dt , e 2λϕ = , (10)<br />
2η 2 r 0 r 0<br />
where the parameter γ = (1 − η 1 λ 2 )/(1 + η 1 λ 2 ).<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
This is an exact solution <strong>of</strong> a spherically symmetric, <strong>non</strong><br />
<strong>asymptotically</strong> <strong>flat</strong> (NAF), static and electrically charged <strong>black</strong><br />
hole, with event horizon r + (r + ≥ 0). The parameters r + and r 0<br />
(r 0 ≥ 0) are related to the physical mass 2 and the electric<br />
charge by:<br />
√<br />
(1 − γ)<br />
(1 + γ)<br />
M = r + , q = ±r 0 . (11)<br />
4<br />
2η 2<br />
Consider<strong>in</strong>g the normal and phantom cases, one gets the<br />
follow<strong>in</strong>g <strong>in</strong>terval γ ∈ (−∞, +1).<br />
2 We use the quasi-local mass.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
Figure: Penrose diagrams for NAF <strong>black</strong> <strong>holes</strong>.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The thermodynamics variables<br />
The Hawk<strong>in</strong>g temperature, the entropy and electric potential <strong>of</strong><br />
the <strong>black</strong> hole are give by<br />
[<br />
g ′ ]<br />
00<br />
T =<br />
4π √ = r γ +<br />
−g 00 g 11 r = r + 4πr 1+γ , (12)<br />
0<br />
S = 1 4 A = 1 ∫<br />
√g22 g 33 dθdφ∣ = πr 1+γ<br />
4<br />
0<br />
r 1−γ<br />
+ , (13)<br />
r=r+<br />
√<br />
A 0 = − r + 1 + γ<br />
, (14)<br />
2r 0 2η 2<br />
which satisfies the first law <strong>of</strong> <strong>black</strong> hole thermodynamics,<br />
dM = TdS + η 2 A 0 dq . (15)<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
Also, we obta<strong>in</strong> the follow<strong>in</strong>g equations <strong>of</strong> state<br />
( ) ( )<br />
∂M<br />
∂M<br />
= T , = η 2 A 0 . (16)<br />
∂S<br />
∂q<br />
q<br />
For study<strong>in</strong>g the thermodynamics <strong>of</strong> a geometric form, it is<br />
useful to write the temperature, entropy and the electric<br />
potential <strong>in</strong> functions <strong>of</strong> the mass and the electric charge. To do<br />
this, from (11) we get<br />
r 0 = q<br />
√<br />
S<br />
2η 2<br />
1 + γ , r + = 4M<br />
1 − γ , (17)<br />
which, from (12), (13) and (14) provides the temperature, the<br />
entropy and the electric potential <strong>in</strong> functions <strong>of</strong> the mass and<br />
the electric charge<br />
T = T 1 M γ q −(1+γ) , T 1 = 2 3γ−5<br />
2<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
π<br />
[η 2 (1 + γ)] 1+γ<br />
2<br />
(1 − γ) γ , (18)<br />
(19)<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
Also, we obta<strong>in</strong> the follow<strong>in</strong>g equations <strong>of</strong> state<br />
( ) ( )<br />
∂M<br />
∂M<br />
= T , = η 2 A 0 . (16)<br />
∂S<br />
∂q<br />
q<br />
For study<strong>in</strong>g the thermodynamics <strong>of</strong> a geometric form, it is<br />
useful to write the temperature, entropy and the electric<br />
potential <strong>in</strong> functions <strong>of</strong> the mass and the electric charge. To do<br />
this, from (11) we get<br />
r 0 = q<br />
√<br />
S<br />
2η 2<br />
1 + γ , r + = 4M<br />
1 − γ , (17)<br />
which, from (12), (13) and (14) provides the temperature, the<br />
entropy and the electric potential <strong>in</strong> functions <strong>of</strong> the mass and<br />
the electric charge<br />
T = T 1 M γ q −(1+γ) , T 1 = 2 3γ−5<br />
2<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
π<br />
[η 2 (1 + γ)] 1+γ<br />
2<br />
(1 − γ) γ , (18)<br />
(19)<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
S = S 1 M 1−γ q 1+γ , S 1 =<br />
π2 5−3γ<br />
2<br />
(1 − γ) γ−1 , (20)<br />
[η 2 (1 + γ)] 1+γ<br />
2<br />
( )<br />
M<br />
1 + γ<br />
A 0 = −Ā0<br />
q , Ā0 = η 2 . (21)<br />
1 − γ<br />
One can <strong>in</strong>vert (20) for writ<strong>in</strong>g the mass <strong>in</strong> terms <strong>of</strong> the entropy<br />
and the electric charge<br />
( ) 1<br />
M(S, q) = q − 1+γ<br />
1−γ<br />
S 1−γ<br />
. (22)<br />
S 1<br />
Then, one can express the specific heat as<br />
( ) ( )<br />
∂M ∂M<br />
/ ( ∂ 2 M<br />
C q = =<br />
=<br />
∂T ∂S<br />
q<br />
q<br />
∂S 2 )q<br />
(1 − γ)<br />
S . (23)<br />
γ<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The usual thermodynamics analysis<br />
Let us start our study <strong>of</strong> the thermodynamics <strong>of</strong> the <strong>class</strong> <strong>of</strong><br />
NAF <strong>black</strong> hole solutions <strong>of</strong> EMD theory, us<strong>in</strong>g the analysis <strong>of</strong><br />
the specific heat. The specific heat calculated by the use <strong>of</strong> the<br />
mass is directly proportional to the entropy, as shown <strong>in</strong> (23).<br />
Then, as we have a s<strong>in</strong>gle horizon r + , there are no extreme<br />
case nor phase transition. So, the geometric methods that<br />
present acceptable results must reproduce the result <strong>of</strong> the<br />
specific heat. In general, the thermodynamics system presents<br />
thermodynamic <strong>in</strong>teraction, s<strong>in</strong>ce both the specific heat and the<br />
temperature <strong>of</strong> the <strong>black</strong> hole are <strong>non</strong> null, but do not possess<br />
the extreme case, nor phase transition.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The We<strong>in</strong>hold method<br />
One <strong>of</strong> the first to formulate a geometric analysis for a<br />
thermodynamic system was We<strong>in</strong>hold. His method, known as<br />
the We<strong>in</strong>hold method , is to def<strong>in</strong>e a metric <strong>of</strong> the<br />
thermodynamic space <strong>of</strong> equilibrium states, through the mass<br />
as thermodynamic potential. This metric is used for the<br />
calculation <strong>of</strong> the curvature scalar R W , which determ<strong>in</strong>e<br />
whether the system possesses phase transitions. The<br />
We<strong>in</strong>hold metric is given by<br />
dl 2 W (M)<br />
= ∂2 M<br />
∂S 2 dS2 + 2 ∂2 M<br />
∂S∂q dSdq + ∂2 M<br />
∂q 2 dq2<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
=<br />
γq − 1+γ<br />
1−γ<br />
S 2 (1 − γ) 2 ( S<br />
S 1<br />
) 1<br />
1−γ<br />
dS 2 − 2<br />
2(1 + γ)<br />
−<br />
(1 − γ) 2 q 3−γ<br />
1−γ<br />
2<br />
(1 + γ)q− 1−γ<br />
S(1 − γ) 2 ( ) 1<br />
S 1−γ<br />
dSdq<br />
S 1<br />
( ) 1<br />
S 1−γ<br />
dq 2 .<br />
S 1<br />
(24)<br />
The curvature scalar <strong>of</strong> this metric is identically null and this<br />
shows that the system can not be analysed as hav<strong>in</strong>g or not<br />
phase transition.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The Ruppe<strong>in</strong>er method<br />
Ruppe<strong>in</strong>er also <strong>in</strong>troduced a metric for def<strong>in</strong><strong>in</strong>g the geometry <strong>of</strong><br />
the thermodynamic space <strong>of</strong> the equilibrium states, but <strong>in</strong> this<br />
case, with the entropy as a thermodynamic potential. Aga<strong>in</strong>,<br />
this metric provides a curvature scalar R R , which shows if the<br />
system possesses thermodynamic <strong>in</strong>teraction and phase<br />
transitions. The Ruppe<strong>in</strong>er metric is given by<br />
dl 2 R(S)<br />
= − ∂2 S<br />
∂M 2 dM2 − 2 ∂2 S<br />
∂M∂q dMdq − ∂2 S<br />
∂q 2 dq2<br />
= S 2 1 γ(1 − γ)M−1−γ q 1+γ dM 2 − 2S 1 (1 + γ)(1 − γ)M −γ q γ ×<br />
×dMdq − S 1 γ(1 + γ)M 1−γ q −1+γ dq 2 . (25)<br />
The curvature scalar <strong>of</strong> this metric is identically null. This<br />
means that there is no thermodynamic <strong>in</strong>teraction, which does<br />
not agree with the result obta<strong>in</strong>ed by the specific heat.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The geometrothermodynamics method<br />
Hernando Quevedo has improved the follow<strong>in</strong>g metric for the<br />
thermodynamic space <strong>of</strong> the equilibrium states<br />
(<br />
dlG(Φ) 2 = E c ∂Φ ) (<br />
∂E c η ad δ di ∂ 2 )<br />
Φ<br />
∂E i E b dE a dE b , (26)<br />
where Φ is the thermodynamics potential and E a are the<br />
extensive thermodynamics variables. For the thermodynamic<br />
potential be<strong>in</strong>g the entropy (20), the metric (26) has to be given<br />
by<br />
(<br />
dlG(S) 2 = M ∂S<br />
∂M + q ∂S ) (<br />
)<br />
− ∂2 S<br />
∂q ∂M 2 dM2 + ∂2 S<br />
∂q 2 dq2 (27)<br />
= 2S 2 1 γ(1 − γ)M−2γ q 2(1+γ) dM 2<br />
+ 2S 2 1 γ(1 + γ)M2(1−γ) q 2γ dq 2 . (28)<br />
The curvature scalar <strong>of</strong> this metric is identically zero<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
Mak<strong>in</strong>g the calculus for the metric (26), us<strong>in</strong>g the mass (22) as<br />
thermodynamic potential, on gets<br />
dlG(M) 2 =<br />
(1+γ<br />
γ2 −2 ( ) 2<br />
q<br />
(1−γ) S 1−γ<br />
dS 2 2γ(1 + γ)<br />
S 2 (1 − γ) 3 − S 1 (1 − γ) 3 q− 1−γ ×<br />
( ) 2<br />
S 1−γ<br />
× dq 2 .<br />
S 1<br />
(29)<br />
The curvature scalar <strong>of</strong> this metric is also identically zero. Once<br />
aga<strong>in</strong> the <strong>in</strong>compatibility with the analysis <strong>of</strong> GTD with that <strong>of</strong><br />
specific heat has been obta<strong>in</strong>ed. This was first shown <strong>in</strong><br />
pathological solutions <strong>of</strong> phantom <strong>black</strong> <strong>holes</strong> 3 and next <strong>in</strong><br />
(anti) Reissner-Nordstrom-AdS <strong>black</strong> <strong>holes</strong> 4 .<br />
3 Manuel E. Rodrigues and Zui A.A. Oporto, Phys.Rev. D85 (2012) 104022.<br />
4 Deborah F. Jardim, Manuel E. Rodrigues and Stephane J. M. Houndjo,<br />
Eur.Phys.J.Plus 127 (2012) 123.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The Liu-Lu-Luo-Shao method<br />
Recently, Liu-Luo-Lu-Shao, <strong>in</strong> the same order as We<strong>in</strong>hold, has<br />
formulated a metric <strong>of</strong> the thermodynamic space <strong>of</strong> the<br />
equilibrium states, which provides a curvature scalar that<br />
determ<strong>in</strong>es whether the system possesses phase transitions.<br />
This metric is a new thermodynamic metric based on the<br />
Hessian matrix <strong>of</strong> several free energy, which <strong>in</strong> the case <strong>of</strong> the<br />
Helmholtz free energy, is given by<br />
dlLLLS(F 2 )<br />
= −dTdS + dA 0 dq = − γT ( ) γ<br />
(1+γ)<br />
1<br />
q− (1−γ) S 1−γ<br />
dS<br />
2<br />
S 1<br />
−<br />
S(1 − γ)<br />
q −2 ( ) 1<br />
1−γ S 1−γ [Ā0 − (1 + γ)T 1 S ] dSdq<br />
S(1 − γ) S 1<br />
+ 2Ā0 (3−γ)<br />
q− (1−γ)<br />
(1 − γ)<br />
( S<br />
S 1<br />
) 1<br />
1−γ<br />
dq 2 . (30)<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The curvature scalar <strong>of</strong> this metric is identically null. As we had<br />
seen, this result also is not <strong>in</strong> accordance with that <strong>of</strong> the<br />
specific heat. An <strong>in</strong>consistency with respect to the specific heat<br />
was shown for the topological case <strong>of</strong> <strong>black</strong> hole <strong>in</strong><br />
Horava-Lifshitz gravity 5 .<br />
5 Qiao-Jun Cao, Yi-X<strong>in</strong> Chen and Kai-Nan Shao, Phys.Rev.D 83: 064015<br />
(2011).<br />
Manuel E. Rodrigues and Glauber T. Marques <strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The local and global stability<br />
We now have to study the stability <strong>of</strong> this thermodynamic<br />
system. With respect to this, we may split <strong>in</strong>to two parts, the<br />
local stability and the global one.<br />
The local stability is easily analysed by the specific heat (23)<br />
C q =<br />
(1 − γ)<br />
S . (31)<br />
γ<br />
Here, we see that for the normal case <strong>of</strong> EMD (η 1 = η 2 = 1),<br />
the system is always locally stable for 0 < γ < 1, i.e. C q > 0,<br />
and locally unstable for −1 < γ < 0. But for the phantom case,<br />
the system is always locally unstable, s<strong>in</strong>ce γ ∈ (−∞, −1).<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
The local and global stability<br />
We now have to study the stability <strong>of</strong> this thermodynamic<br />
system. With respect to this, we may split <strong>in</strong>to two parts, the<br />
local stability and the global one.<br />
The local stability is easily analysed by the specific heat (23)<br />
C q =<br />
(1 − γ)<br />
S . (31)<br />
γ<br />
Here, we see that for the normal case <strong>of</strong> EMD (η 1 = η 2 = 1),<br />
the system is always locally stable for 0 < γ < 1, i.e. C q > 0,<br />
and locally unstable for −1 < γ < 0. But for the phantom case,<br />
the system is always locally unstable, s<strong>in</strong>ce γ ∈ (−∞, −1).<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
Now we can analyse the global stability <strong>of</strong> the thermodynamic<br />
system. First, we def<strong>in</strong>e the Gibbs potential <strong>in</strong> the grand<br />
ca<strong>non</strong>ical ensemble<br />
G = M − TS − η 2 A 0 q = M(1 − T 1 S 1 + η 2 Ā 0 ) =<br />
M<br />
1 − γ . (32)<br />
So, we see that there is no change <strong>of</strong> the Gibbs potential,<br />
unless for the variation <strong>of</strong> the parameter γ. For the normal<br />
case, γ ∈ (−1, 1), the potential is always positive and then, the<br />
system is globally unstable. But the phantom case,<br />
γ ∈ (−∞, −1), appears as the opposite <strong>of</strong> the previous one,<br />
and the system is globally stable. This is not consistent with our<br />
previous analysis, so that the Gibbs potential, for the grand<br />
ca<strong>non</strong>ical ensemble, does not seem suitable to do this analysis.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
On the other hand, the analysis made by the Helmholtz free<br />
energy <strong>in</strong> the ca<strong>non</strong>ical ensemble, shows a good agreement<br />
with the results already obta<strong>in</strong>ed <strong>in</strong> this work. We def<strong>in</strong>e the<br />
Helmholtz free energy as<br />
F = M − TS = M(1 − T 1 S 1 ) = − γM<br />
1 − γ . (33)<br />
Now the situation seems to have been reversed. In the normal<br />
case, we have F < 0 for 0 < γ < 1, which leads to a globally<br />
stable system, such as the local stability made by the specific<br />
heat. However, the phantom case is always globally unstable.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
a) We calculated the extensive and <strong>in</strong>tensives variable for this<br />
thermodynamic system. The first law <strong>of</strong> thermodynamics is<br />
also established <strong>in</strong> (15).<br />
b) We analysed the thermodynamic system by the geometric<br />
methods <strong>of</strong> the geometrothermodynamics, We<strong>in</strong>hold,<br />
Ruppe<strong>in</strong>er and that <strong>of</strong> Liu-Lu-Luo-Shao. All analysis by the<br />
geometric methods provided an identically null curvature scalar<br />
for the thermodynamic space <strong>of</strong> the equilibrium states. This is<br />
<strong>in</strong>compatible with the analysis made by the usual specific heat,<br />
which provides an <strong>in</strong>teract<strong>in</strong>g system without extreme case or<br />
phase transition.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
a) We calculated the extensive and <strong>in</strong>tensives variable for this<br />
thermodynamic system. The first law <strong>of</strong> thermodynamics is<br />
also established <strong>in</strong> (15).<br />
b) We analysed the thermodynamic system by the geometric<br />
methods <strong>of</strong> the geometrothermodynamics, We<strong>in</strong>hold,<br />
Ruppe<strong>in</strong>er and that <strong>of</strong> Liu-Lu-Luo-Shao. All analysis by the<br />
geometric methods provided an identically null curvature scalar<br />
for the thermodynamic space <strong>of</strong> the equilibrium states. This is<br />
<strong>in</strong>compatible with the analysis made by the usual specific heat,<br />
which provides an <strong>in</strong>teract<strong>in</strong>g system without extreme case or<br />
phase transition.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
c) The local stability analysis done by the specific heat and the<br />
global stability made by the Helmholtz free energy <strong>in</strong> the<br />
ca<strong>non</strong>ical ensemble, shows that the normal case is locally and<br />
globally stable for −1 < λ < 1 (0 < γ < 1). On the other hand,<br />
for the other <strong>in</strong>tervals <strong>of</strong> γ, <strong>in</strong>clud<strong>in</strong>g all phantom cases, the<br />
system becomes unstable.<br />
Thank you very much.<br />
Muito Obrigado.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h
The action and the specific model<br />
Thermodyamics analysis<br />
Conclusion<br />
c) The local stability analysis done by the specific heat and the<br />
global stability made by the Helmholtz free energy <strong>in</strong> the<br />
ca<strong>non</strong>ical ensemble, shows that the normal case is locally and<br />
globally stable for −1 < λ < 1 (0 < γ < 1). On the other hand,<br />
for the other <strong>in</strong>tervals <strong>of</strong> γ, <strong>in</strong>clud<strong>in</strong>g all phantom cases, the<br />
system becomes unstable.<br />
Thank you very much.<br />
Muito Obrigado.<br />
Manuel E. Rodrigues and Glauber T. Marques<br />
<strong>Thermodynamics</strong> <strong>of</strong> a <strong>class</strong> <strong>of</strong> <strong>non</strong>-<strong>asymptotically</strong> <strong>flat</strong> <strong>black</strong> h