Noether charge and black hole entropy in teleparallel gravity

Abstract

The Noether charge associated to diffeomorphism invariance in teleparallel gravity is derived. It is shown that the latter yields the ADM mass of an asymptotically flat spacetime. The black hole entropy is then investigated based on Wald's prescription that relies on the Noether charge. It is shown that, like in general relativity, the surface gravity can be factored out from such a charge. Consequently, the similarity with the first law of thermodynamics implied by such an approach in general relativity does show up also in teleparallel gravity. It is found that, based on the expression of the first law of black hole mechanics, which is preserved in teleparallel gravity, entropy can thus be extracted from such a Noether charge. The resulting entropy can very naturally be expressed as a volume integral, though. As such, it is shown that the conformal issue that plagues the entropy-area law within general relativity does not arise in teleparallel gravity based on Wald's approach. The physics behind these features is discussed.

Energy and entropy in the Geometrical Trinity of gravity

“…Despite these details, the entropy formula in Ref. [166] agrees with the expression (170), and they also noted the benefit of teleparallel formulation and other 1 st order formulations [167], that one does not have to invoke a bifurcation surface and resort to a Killing vector that vanishes on that surface. The Killing vectors of the metric are parallel transported by the metric Levi-Civita connection.…”

“…and thus (172) is the straightforward generalisation of (174). An entropy formula for metric teleparallelism had been derived earlier by Hammad et al [166], and is the correct limit (170) when adapted to the metric teleparallel geometry. Hammad et al noted that the result can be rewritten as a volume integral…”

“…Though, we do not know what forbids using the Stokes law in GR exactly the same way. Also, we note technical issues with the computations in [166], as their energy formula would give a divergent result (the regularisations discussed in V A were not used explicitly), and their entropy formula would not either give the correct result (since an inertial frame was not identified), but the result was forced by implementing additional constraints on the Killing vectors which we will briefly discuss below. Despite these details, the entropy formula in Ref.…”

What can we learn from the conformal noninvariance of the Klein-Gordon equation?

“…Moreover, since the dynamics of the scalar ϕ(x) should not be tied up to the dynamics of any other vector apart from the spacetime itself, the natural vector species that comes to mind is the set of tetrad vectors of spacetime e a µ . The tetrad field does indeed transform as required [66], ẽa µ = Ωe a µ , because it is related to the metric of spacetime by g µν = η ab e a µ e b ν . To recover a curved spacetime vector ξ µ from the tetrad, we would then only need to project the tetrad along a given tangent-space direction ξ a according to ξ µ = ξ a e a µ .…”

“…Investigating the consequences of a Weyl conformal transformation on spacetime usually leads to very interesting insights about physical phenomena related to the dynamics of spacetime, ranging from the concept of quasi-local masses in general relativity [6][7][8] to the physics of wormholes [9][10][11] and black holes [10,[12][13][14][15][16][17][18][19]. Unfortunately, more often than not, the noninvariance of the KGE under a Weyl conformal transformation is mentioned in the literature as a pathology or, at best, simply as one of the arguments for introducing a non-minimal coupling of the Klein-Gordon scalar field to gravity [1,2,20].…”