Horizon definition free#

This last is because any topological density, Φ ^, satisfies δ 0 Φ ^ = d δ 0 Φ. In order to do that, there are only two options: the addition of boundary terms Φ or the addition of topological densities Φ ^. Essentially, the process of regularization corresponds to the addition of terms to the action principle that do not alter the EOM but satisfy the three aforementioned conditions. For asymptotically AdS spaces, this implies the need to implement a regularization process on the action principle.

In addition, the action must have proper conserved charges. This implies that Θ ( ϕ, δ 0 ϕ ) must vanish (on shell), provided the boundary conditions are satisfied. One aspect to consider, in addition, is that it is necessary to have a proper action principle, meaning that the action principle must be finitely evaluated on any solution that satisfies the boundary conditions and having an extreme subjected to those boundary conditions. See, about the correlation between the emission modes and temperature of the event horizon in Einstein Gauss Bonnnet gravity. This last idea is reinforced by the fact that it is direct to check that the Kerr–Newman–AdS thermodynamics behaves as the thermodynamics of a conformal theory, see, for instance.

On the other hand, the independence on l, or equivalently on the cosmological constant, of the conserved charges is usually considered a hint of a conformal pedigree of these charges, in spite of the gravitational theory is not conformal. On one hand, this seems to introduce to a sort regulator or a naïve regularization parameter into the computations defined as r / l with r, a radial coordinate. To our knowledge, in doing this, any method defines charges that are independent of AdS radius l (or the cosmological constant). However, the AdS asymptotia requires the introduction of a regularization process to define the Noether’s charges computed at infinity.

Horizon definition free#

For asymptotic flat spaces, the process of defining conserved charge can be cumbersome but it is usually free of divergences. It is worth noticing that Wald’s approach requires implementing certain considerations before the computations can be performed properly. Conversely, only changing the cosmological constant and not the rest of the coupling constants in the Lovelock Lagrangian accordingly, see below, would spoil the asymptotic behavior of the solutions. This can be considered one of the simplest generalizations of changing the cosmological constant in four dimensions. This maintains the asymptotia and, at the same time, allows us to study the changes of that scale. To avoid this, one can make each of the constant functionals dependent on a single scale, using the one provided by the AdS radii. Evidently, if the coupling constants vary independently, not only the values of λ i 2 would change, but also the degeneration of each of the ground states k and also the asymptotia of the non-ground state solutions. Lovelock gravity has different k-fold degenerated ground states, each being a manifold of constant curvature λ i 2 with λ i and k defined by the coupling constants in the Lagrangian see below.

For Lovelock gravity, this can be confirmed as follows. Unlike four dimensions moving in that enlarged space of parameters, we can modify the asymptotia of the solutions. In principle, for each of these new coupling constants, one could introduce an additional extra dimension in the space of the parameters mentioned above. This scenario changes in higher dimensions where additional terms can be incorporated in the Lagrangians, and with them, additional coupling constants arise. To begin with a discussion, one can notice that in four dimensions, in the absence of matter, there are only two coupling constants.