Noether and Abbott-Deser-Tekin conserved quantities in scalar-tensor theory of gravity both in Jordan and Einstein frames

Abstract

We revisit the thermodynamic aspects of the scalar-tensor theory of gravity in the Jordan and in the Einstein frame. Examining the missing links of this theory carefully, we establish the thermodynamic descriptions from the conserved currents and potentials by following both the Noether and the Abbott-Deser-Tekin (ADT) formalism. With the help of conserved Noether current and potential, we define the thermodynamic quantities, which we show to be conformally invariant. Moreover, the defined quantities are shown to fit nicely in the laws of (the first and the second) black hole thermodynamics formulated by the Wald's method. We stretch the study of the conformal equivalence of the physical quantities in these two frames by following the ADT formalism. Our further study reveals that there is a connection between the ADT and the Noether conserved quantities, which signifies that the ADT approach provide the equivalent thermodynamic description in the two frames as obtained in Noether prescription. Our whole analysis is very general as the conserved Noether and ADT currents and potentials are formulated off-shell and the analysis is exempted from any prior assumption or boundary condition.PACS numbers:

Black hole mechanics and thermodynamics in the light of Weyl transformations

“…Finally, one has now to use Einstein field equations to introduceT ab inside Eq. (17). As already mentioned above, Einstein equations are not invariant under conformal transformations.…”

“…Therefore, only the second and last terms on the right-hand side will be kept in Eq. (17). Finally, one has now to use Einstein field equations to introduceT ab inside Eq.…”

“…Various authors have investigated the conformal behavior of Hawking temperature and Bekenstein-Hawking entropy, either (i ) based on the classical definitions of surface gravity and black hole horizons [10][11][12][13][14][15], or (ii ) [10,11,13,[15][16][17] based on the various methods devised in the literature for recovering those famous formulas [18][19][20][21] (see also the nice review in Ref. [22]).…”

Abbott–Deser–Tekin like conserved quantities in Lanczos–Lovelock gravity: beyond Killing diffeomorphisms

“…There is another elegant approach developed independently in General theory of Relativity (GR) for the formulation of the thermodynamic description from the conserved charges, which is popularly known as the Abbott-Deser-Tekin (abbreviated as ADT) approach [4][5][6][7]11]. This formulation works well not only for asymptotically flat black hole spacetimes in GR, but also for the anti-deSitter (AdS) spacetimes in GR [4] as well as higher order gravity theories [5][6][7] (For the thermodynamic description using the ADT approach, also see the recent works [12][13][14][15][16][17][18][19][20][21][22]). Recently, the ADT approach has been extended successfully by us in the scalar-tensor theory as well, where we have also shown that the two approaches (i.e.…”

Mass and angular momentum of charged rotating Gödel black holes in five-dimensional minimal supergravity

“…Consequently, in some sense, the offshell generalized ADT formalism could also be viewed as the cousin of both the BBC and phase space approaches. Till now, there have existed a series of developments and applications of the off-shell ADT formulation in a range of gravity theories regardless of the asymptotic behavior of the spacetime metrics [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46].…”