Thermodynamic structure of a generic null surface and the zeroth law in scalar-tensor theory

Abstract

We show that the equation of motion of scalar-tensor theory, along the likes of GR and its higher order generalization i.e Lanczos-Lovelock gravity, acquires thermodynamic identity when projected on a generic null surface. The relevant projection is given by E ab l a k b , whereab represents the equation motion for gravitational field in presence of external matter, l a is the generator of the null surface and k a is the corresponding auxiliary null vector. Our analysis is done completely in a covariant way. Therefore all the thermodynamic quantities are in covariant form and hence can be used for any specific form of metric adapted to a null surface. We show this both in Einstein and Jordan frames and find that these two frames provide equivalent thermodynamic quantities. This is consistent with the previous findings for a Killing horizon. Also, a concrete proof of the zeroth law in scalar-tensor theory is provided when the null surface is defined by a Killing vector. ContentsC Proof the zeroth law for static spacetimes 28 D Proof of the relation (4.31) 29 E Proof of the relation (4.56) 29

Extended phase space thermodynamics of black holes: A study in Einstein's gravity and beyond

“…These theories can be described in either Jordan frame or in Einstein frame, which are conformally connected to each other. The physical equivalence of two conformal frames is a longstanding debate in literature [19,28,[33][34][35][36]. However, in our earlier work [19,[33][34][35][36], we have shown that the thermodynamic parameters, like entropy, temperature, internal energy (which is given by the mass of the black hole), angular momentum etc.…”

“…The above identity (38) will help us to obtain off-shell Noether current and potential due to the diffeomorphism invariance. Firstly, we want to obtain the change of the Lagrangian due to the diffeomorphism invariance x a −→ x a + ξ a , which can be obtained by replacing arbitrary variation δ in (36) by Lie derivative (£ ξ ). Thus, (36) yields…”

“…As we have mentioned, both f (R) and BD gravity can be studied as a special case of scalar-tensor gravity and can be described in two conformally connected frames i.e., Jordan and Einsten frame. In earlier works the equivalence of thermodynamic parameters, in the two conformally connected frames, has been established by us and other groups [19,[33][34][35][36]. But, those analysis were done in the non-extended phase space approach as the variation of the cosmological constant was not accounted.…”

“…Later we move on to discuss extended phase space thermodynamics in the conformally connected frames. Although there are debates regarding the physical equivalence of the two conformally connected frames [19,28,[33][34][35][36], it has been found that the thermodynamic parameters (of non-extended phase space) are equivalent in the two conformally connected frames [19,[33][34][35][36]. Therefore, following the extended phase space approach, here we study the connection of thermodynamic parameters in the two frames (especially pressure and volume).…”

Thermal nature of a generic null surface

“…Incidentally, one can provide a thermodynamical interpretation of a gravitational field equation by suitably projecting it on a generic null surface also [16][17][18][19][20][21]. For instance, Einstein's equation, contracted with null generator and corresponding auxiliary vector, yields a thermodynamical identity of the form T δ λ S = δ λ E + P δ λ V [16][17][18][19], where symbols have their usual meanings and the variation can be interpreted as the change due to virtual displacement of the null surface along an auxiliary null vector, parametrized by the parameter λ (generalization to Lanczos-Lovelock gravity [22] and scalartensor theory [23] has been done as well). Therefore, it has been realized that null surfaces which act as one-way membranes to a certain class of observers also possess thermodynamical attributions.…”