On `The conformal metric structure of Geometrothermodynamics': Generalizations
aa r X i v : . [ m a t h - ph ] M a r On ‘The conformal metric structure of Geometrothermodynamics’: Generalizations
Mustapha Azreg-A¨ınouBas¸kent University, Department of Mathematics, Ba˘glıca Campus, Ankara, Turkey
Abstract
We show that the range of applicability of the change of representation formula derived in J.Math. Phys. , 033513 (2013) [arXiv:1302.6928] is very narrow and extend it to include all physicalapplications, particularly, applications to black hole thermodynamics, cosmology and fluid thermo-dynamics. We comment on a couple of equations derived in [1] and generalize them to apply to a wide range ofphysical problems pertaining to black hole thermodynamics, cosmology and fluid thermodynamics.To persuade the reader of the importance of the above-mentioned generalization, we provide twoexamples from black hole thermodynamics. Consider the thermodynamics of Kerr-Newman black holegoverned by the equations (2.3), (2.5), (2.6) to (2.9) of [2]: M = { S + [ J + ( Q ) / ] / ( S ) + Q / } / (1)d M = T d S + W d J + f d Q (2)where T = ¶ M / ¶ S , W = ¶ M / ¶ J , f = ¶ M / ¶ Q are given in Eqs. (2.6) to (2.8) of [2]. It is easy to checkthat M is not homogeneous in ( S , J , Q ) for it is not possible to find a real b such that M ( l S , l J , l Q ) = l b M ( S , J , Q ) . The way the right-hand side (r.h.s) of (1) has been arranged indicates that M is homoge-neous in ( S , J , Q ) of degree 1/2: M ( l S , l J , l Q ) = l / M ( S , J , Q ) . (3)By Euler’s Theorem we re-derive Eq. (2.9) of [2]:12 M = ¶ M ¶ S S + ¶ M ¶ J J + ¶ M ¶ ( Q ) Q (4) = T S + W J + f Q ¶ M / ¶ ( Q ) = ( ¶ M / ¶ Q )[ ¶ Q / ¶ ( Q )] = f / ( Q ) . It is worth mentioning that theextensive variables ( S , J , Q ) in terms of which the first law (2) is written are not the same extensivevariables ( S , J , Q ) in terms of which the Euler identity (4) is expressed. We denote the former variablesby E a and the latter variables by E ′ a (in this example: E = E ′ = S , E = E ′ = J , E = Q , E ′ = Q ).The variables E ′ a are power-law functions of E a : E ′ a = ( E a ) p a (no summation), where p a depends on a .Thus, if F is homogeneous in E ′ a of degree b : F ( l E ′ a ) = l b F ( E ′ a ) , then the Euler identity reads b F = E ′ a ¶ F¶ E ′ a ( S over a , a = , , . . . ) (6) = E a p a ¶ F¶ E a ( S over a ) (7)1here we have used ¶ E ′ a / ¶ E a = p a ( E a ) p a − (no summation), while the first law is given byd F = I a d E a ( S over a ) . (8)where I a = ¶ F / ¶ E a ( I a = d ab I b ). The authors of [1] considered the case where all p a ≡
1, which is a veryrestrictive constraint and rarely met in black hole thermodynamics, cosmology or fluid thermodynamics.Now, if we rewrite (1) as M = { ( √ S ) + [( √ J ) + ( Q ) / ] / [ ( √ S ) ] + ( Q ) / } / (9)and regard M as a function of ( √ S , √ J , Q ) then M is homogeneous in ( √ S , √ J , Q ) of degree 1: M ( l √ S , l √ J , l Q ) = l M ( √ S , √ J , Q ) (10)where E ′ = √ S , E ′ = √ J , E ′ = Q with p = p = / p =
1. By Euler’s Theorem we obtain(see (6), (7)): M = ¶ M ¶ √ S √ S + ¶ M ¶ √ J √ J + ¶ M ¶ Q Q (11) = T S + W J + f Q (12)where we have used ¶ M / ¶ √ S = √ S ( ¶ M / ¶ S ) and ¶ M / ¶ √ J = √ J ( ¶ M / ¶ J ) . But (12) is just (5). Thismeans that (1) one can always choose b = p a depend on b : p a ≡ p a ( b ) . If ¯ p a denotes the values of the powers for b =
1, then on dividing both sides of (7) by b one obtains¯ p a = b p a ( b ) . (13)We can rewrite (1) any way we want: If g >
0, we bring it to the form M = { ( S g ) / g + [( J g ) / g + ( Q g ) / g / ] / [ ( S g ) / g ] + ( Q g ) / g } / where M appears to be homogeneous in ( S g , J g , Q g ) of degree ( / ) / g .As a general rule: if f is a homogeneous function of ( x , y , . . . ) of degree b then it is a homogeneousfunction of ( x g , y g , . . . ) of degree b / g . In the special choice g = b , f is homogeneous in ( x b , y b , . . . ) ofdegree 1.In another more instructive example consider the thermodynamics of Reissner-Nordstr¨om black holesin d -dimensions governed by [3] M ( S , Q ) = S D + Q DS D (cid:0) D ≡ d − d − (cid:1) (14)where T = ¶ M ¶ S = DS D − − Q S D + , f = ¶ M ¶ Q = Q DS D . (15)It is straightforward to check that M is not homogeneous in ( S , Q ), that is the powers p a can’t all be 1.Assuming M ( l S p S , l Q p Q ) = l b M ( S p S , Q p Q ) we find p S ( b ) = D / b , p Q ( b ) = / b , (16)Whatever the value of b we choose, it is not possible to have p Q = p S . If we choose b = p S = D and ¯ p Q =
1. On applying (7) we obtain M = T S / D + f Q or DM = T S + D f Q , which is2ndependent of the choice of b . It is straightforward to verify the validity of this latter equation uponsubstituting into its r.h.s the expressions of T and f given in (15).Our starting point is the set of Eqs. (31) to (38) of [1] which we intend to generalize. Those equationswere derived constraining F to obey the very special Euler identity b F = I a E a (Eq. (34) of [1]). Eqs.(31) to (33) of [1] remain valid in the general case (7). Eqs. (35) and (36) of [1] have extra misprinted ormissing factors. Considering the general case (7) with p a =
1, the equations generalizing Eqs. (35) and(36) of [1] are derived on substituting (7), (8), and Eqs. (32) and (33) of [1] into Eq. (31) of [1]. Theyread respectively g E ( i ) = b (cid:20) x ( i )( i ) E ( i ) p ( i ) + (cid:229) j = i (cid:18) x ( i )( i ) p ( i ) − x jj b (cid:19) I j E j I ( i ) (cid:21) × h − L ( i ) c ( i )( i ) I ( i ) d E ( i ) ⊗ d I ( i ) − L ( i ) c ( i )( i ) (cid:229) j = i I j I ( i ) d E j ⊗ d I ( i ) − (cid:229) j = i L j c jj I ( i ) d E j ⊗ d I j + (cid:229) j = i L j c jj I j I ( i ) d E j ⊗ d I ( i ) i , (17) g E ( i ) = b (cid:20) x ( i )( i ) E ( i ) p ( i ) + (cid:229) j = i (cid:18) x ( i )( i ) p ( i ) − x jj b (cid:19) I j E j I ( i ) (cid:21) × h − (cid:229) k L k c kc I ( i ) d I k ⊗ d E c + (cid:229) j = i (cid:16) L j c jj − L ( i ) c ( i )( i ) (cid:17) I j I ( i ) d E j ⊗ d I ( i ) i . (18)where we have assumed, as in [1], c kc = d kc or c kc = h kc = diag [ − , , . . . , ] . Notice the absence of theleftmost factor ‘ − / I ( i ) ’ in both expressions and the presence of the same factor in front of the first S signin (18). The constraints (37) of [1] remain unchanged L ( i ) = L j c jj / c ( i )( i ) , ∀ j = i ( no S over j ) . (19)We see that the changes appear in the first factor of each expression only. The final expression of theinduced metric, generalizing Eq. (38) of [1], reads g E ( i ) = − b I ( i ) (cid:20) x ( i )( i ) E ( i ) p ( i ) + (cid:229) j = i (cid:18) x ( i )( i ) p ( i ) − x jj b (cid:19) I j E j I ( i ) (cid:21) [ x ab I a E b ] − g F . (20)As we have seen earlier, we can always choose b =
1, and this choice is not an extra constraintobeyed by some physical systems only as one may infer that to be the case from [1]. So, assume that F is homogeneous in some set of thermodynamic variables, find the set of variables with respect to which F is homogeneous of degree 1 ( b = g F is chosen such that theconstraints (19) are satisfied, and if x ( i )( i ) = x jj =
1, then the induced metric reduces to g E ( i ) = − (cid:20) E ( i ) ¯ p ( i ) I ( i ) + (cid:18) p ( i ) − (cid:19) (cid:229) j = i I j E j I ( i ) (cid:21) ( I a E a ) − g F (21) = − F − (cid:229) j = i I j E j + (cid:229) j = i ( ¯ p − ( i ) − ¯ p − j ) I j E j I ( i ) ( I a E a ) g F (22)3eneralizing Eq. (53) of [1]. Here we have used (7) with b = F = I ( i ) E ( i ) / ¯ p ( i ) + (cid:229) j = i I j E j / ¯ p j .In black hole thermodynamics, if the mass depends only on two extensive variable: M ( S , Z ) where Z = Q or Z = J . In this case, setting Z = E j , Y ≡ ¶ M / ¶ Z = I j , E ( i ) = S , I ( i ) = T and M = T S / ¯ p S + Y Z / ¯ p Z in (22) we obtain g S = − T h p S − Y ZT S + Y Z i g M (23)provided g M is chosen such that the constraints (19) are satisfied. In the case of Reissner-Nordstr¨om blackholes in d -dimensions, Z = Q , Y = f and ¯ p S = D .Our next point is to generalize Eqs. (51) and (52) of [1]. These two equations have been derived fromEq. (50) of [1] on substituting F by the special form I a E a / b . Their generalizations are straightforwardlyderived on substituting in Eqs. (51) and (52) of [1] I a E a / b by I a E a / ( b p a ) = I a E a / ¯ p a [Eq. (7)]. References [1] A. Bravetti, C.S. Lopez-Monsalvo, F. Nettel, and H. Quevedo, J. Math. Phys. , 033513 (2013).arXiv:1302.6928[2] P.C.W. Davies, Proc. R. Soc. Lond. A , 499 (1977)[3] J.E. ˚Aman and N. Pidokrajt, Phys. Rev. D73