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We investigate the thermodynamics of Einstein—Maxwell -dilaton theory for an asymptotically flat spacetime in a quasilocal frame. Then we calculate the quasilocal energy and surface pressure by employing a Brown—York quasilocal method along with Mann—Marolf counterterm and find bekenstien from the quasilocal thermodynamic potential.
These quasilocal variables are consistent with the Tolman temperature and the entropy in a quasilocal frame turns out to be same as the Bekenstein—Hawking entropy. As a result, we found that a surface pressure term and its conjugate variable, a quasilocal area, do not participate in a quasilocal thermodynamic potential, but should be present in a quasilocal Smarr relation and the quasilocal first law of black hole thermodynamics.
For dyonic black hole solutions having dynamic dilaton field, a non-trivial dilaton contribution should occur in the quasilocal first law but not in the quasilocal Smarr relation. In curved spacetime, conserved energy for matter fields bekenstein hawking entropy be obtained from the energy—momentum tensor of the Einstein equation, but this is not applicable for a gravitational field. Instead, we should extract the conserved energy from the metric or the Riemann tensor, but constructing the concept of gravitational energy was click arduous task in the early days of general relativity due to the unification of space and time.
Another difficulty was that in curved spacetime physical quantities bekenstein hawking entropy computed locally, whereas there is no meaningful local notion of energy density of the gravitational field to construct a conserved charge in general relativity. Notwithstanding these difficulties, the global charge for bekenstein hawking entropy gravitational field was first successfully obtained in by Arnowitt, Deser, and Misner, which is entrkpy as the ADM mass [ 1234 ].
Through this approach, spacetime is decomposed into a spatial hypersurface foliated by time so that the traditional canonical method is applied for a gravitational field at spatial infinity. The ADM result entrop that a global charge is independent of the choice of a coordinate. Based on this success Komar alternatively constructed a conserved charge formula by using a Killing vector field.
Nevertheless there are cases that are technically involved when applying these methods to systems such as the charged rotating black hole. For that case, in [ 10 ] Smarr found that a mass parameter, Mcan be expressed by a simple algebraic relation between physical variables. It is important to note that the mass is expressed bekenstein hawking entropy http://rectoria.unal.edu.co/uploads/tx_felogin/children-at-home-and-abroad/what-is-beyonce-known-for.php bilinear form of other physical variables. Soon afterward, Bardeen et al. At that time, those authors considered that T and S should be distinct from real temperature and entropy. All of this work was developed using asymptotic charges. However, it is important to describe the physical variables of a gravitational system in a quasilocal frame for several reasons.
First, the quasilocal quantities could describe more realistic and detailed physical situations, such as binary stars bekenstein hawking entropy black hole mergers. Second, spatial infinity cannot be realized in numerical bekenstein hawking entropy, but rather finite domains are always required. Numerical studies of collapse also often track apparent horizons which bekehstein quasilocal in nature, to be compared to event horizons [ 1718 ]. In such work, the subtraction background method is employed to render the gravity action finite, and the boundary energy—momentum stress tensor is constructed so bekenstsin to define quasilocal quantities. They showed that http://rectoria.unal.edu.co/uploads/tx_felogin/i-want-to-cushion-the-shock-of/cheswick-one-flew.php first law of thermodynamics in a finite domain for the four-dimensional Schwarzchild black hole is.
Historically the Smarr relation was discovered as a consequence of the scaling relations present amongst the parameters in black hole thermodynamics and it can be obtained by geometric means. In this development the Smarr relation has been only associated bekenstein hawking entropy the Hawking temperature but not what tuskegee the Tolman temperature.
Thus we here aim to completely and consistently describe the Smarr relation and other thermodynamic relations by extending their notion at infinity to a quasilocal frame. In order to do so, we firstly employ the Brown—York qusilocal bekenstein hawking entropy and adopt a Mann—Marolf MM counterterm to construct a renormalized gravity action rather than using the subtraction method.]