First law of black ring thermodynamics in higher dimensional dilaton gravity withp+1strength forms

Abstract

04.50.+h, 98.80.Cq.We derive the first law of black rings thermodynamics in n-dimensional Einstein dilaton gravity with additional (p + 1)-form field strength being the simplest generalization of five-dimensional theory containing a stationary black ring solution with dipole charge. It was done by means of choosing any cross section of the event horizon to the future of the bifurcation surface. I. INTRODUCTIONRecently there has been a great resurgence of interests in n-dimensional generalization of black holes motivated by various attempts of constructing the unified theories. As far as the static n-dimensional black holes is concerned the uniqueness theorem for them is quite well established [1]. But for stationary axisymmetric ones the problem is much more complicated. In Ref.[2] it was shown that even in five-dimensional spacetime there is the so-called black ring solution possessing S 2 × S 1 topology of the event horizon and having the same mass and angular momentum as spherical five-dimensional stationary axisymmetric black hole. On the contrary, when one assumes the spherical topology of the horizon S 3 the uniqueness proof can be conducted (see Ref.[3] for the vacuum case and Ref.[4] for the stationary axisymmetric self-gravitating σ-model). As was remarked in [5] due to the vast number of the ways of forming of these objects, one can take the view that the spirit of no-hair theorem is preserved.The black ring solution was vastly treated in literature, i.e., the solution possessing the electric and magnetic dipole charge was found [6,7], static black ring solution in five-dimensional Einstein-Maxwell-dilaton gravity was presented in [8] and systematically derived in [9] both in asymptotically flat and non-asymptotically flat case. Recently, the rotating non-asymptotically flat black ring solution in charged dilaton gravity was given [10]. Also the supersymmetric black ring solutions were discovered [11].As far as the black hole thermodynamics is concerned, Wald [12] referred two versions of the black hole thermodynamics. The first one, the so-called equilibrium state version was treated for the first time in the seminal paper of Bardeen, Carter and Hawking [13]. In this attitude the linear perturbations of a stationary electrovac black hole to another stationary black hole were taken into account. In Ref.[14] arbitrary asymptotically flat perturbations of a stationary black hole were considered. The first law of black hole thermodynamics valid for an arbitrary diffeomorphism invariant Lagrangian with metric and matter fields possessing stationary and axisymmetric black hole solutions was given in Refs.[15]- [18], while the higher curvature terms and higher derivative terms in the metric were considered in [19]. The case when the Lagrangian is an arbitrary function of metric, Ricci tensor and a scalar field

The first law of heterotic stringy black hole mechanics at zeroth order in α′

“…The proof of the first law demands that we use the so-called Page charge, which in this context is conserved, localized and on-shell gauge invariant. Only when we use this charge definition for the 1-forms, the closed 1-form associated to the KR potentials Φ i over the bifurcation sphere appears [6,32] and the term Φ i δQ i of the first law associated to the "dipole charges" [6,32,[37][38][39][40] can be identified.…”

DC conductivities and Stokes flows in Dirac semimetals influenced by hidden sector

“…For instance studying five-dimensional spacetime, we have to consider the torus topology of the event horizon for the so-called black ring, being a stationary axisymmetric black object solution. The closed form assumption enables us to define potential and charges, i.e., calculating the Noether charges over the even horizon we obtain a constant value (potential) multiplied by the adequate charge [63][64][65][66]. Namely, using the Hodge theorem, the closed p-form, in ndimensions, can be rewritten on the event horizon as a sum of an exact and harmonic forms.…”

Conserved charge of a gravity theory with p -form gauge fields and its property under Kaluza-Klein reduction

“…In this section, our main goal is to extend the general derivations in the previous section to systematically investigate the conserved charges of D-dimensional covariant gravity theories consisting of a gravitational field g µν , m scalar fields φ (k) (k = 1, 2, · · ·, m) and a number of p-form potentials A (p) , where 1 ≤ p ≤ n − 1 and 2 ≤ n ≤ D. In [31,36,37], the conserved charges of the gravity theories with p-form potentials were also investigated in other methods. The Lagrangian describing these theories is assumed to take the generic…”

“…For example, when Eq. (3.17) is applied to compute the off-shell ADT potential for the gravity theory with a single p-form potential given in the works [36] and [37], where potentials for this theory were derived through the LIW approach and the BBC method respectively, one finds that all the potentials agree with each other. On the other hand, as a consequence of the generality for the Lagrangian L M , the potential (3.17) can be widely used to compute the conserved charges of gravity theories in the presence of scalar fields and p-form gauge fields.…”