An entropy current and the second law in higher derivative theories of gravity

Abstract

We construct a proof of the second law of thermodynamics in an arbitrary diffeomorphism invariant theory of gravity working within the approximation of linearized dynamical fluctuations around stationary black holes. We achieve this by establishing the existence of an entropy current defined on the horizon of the dynamically perturbed black hole in such theories. By construction, this entropy current has non-negative divergence, suggestive of a mechanism for the dynamical black hole to approach a final equilibrium configuration via entropy production as well as the spatial flow of it on the null horizon. This enables us to argue for the second law in its strongest possible form, which has a manifest locality at each space-time point. We explicitly check that the form of the entropy current that we construct in this paper exactly matches with previously reported expressions computed considering specific four derivative theories of higher curvature gravity. Using the same set up we also provide an alternative proof of the physical process version of the first law applicable to arbitrary higher derivative theories of gravity.

Snowmass white paper: Quantum information in quantum field theory and quantum gravity

“…In contrast, the generalization of the extremal surface has non-vanishing extrinsic curvature and the resulting expression includes such contributions. This higher curvature corrected entropy formula also has the correct structure to give quasilocal entropy of dynamical horizons respecting the second law [152][153][154].…”

The zeroth law of black hole thermodynamics in arbitrary higher derivative theories of gravity

“…It should be convincing that this transformation should leave the metric invariant, since the metric functions depend on the coordinates (r, v) only through their product. This is called the boost transformation and due to this the stationary black hole configurations are said to enjoy a boost symmetry, see [17], [18] for details. Alternatively, we can also explain the boost symmetry, that we described above, in the following way.…”

“…Through the discussions in the previous paragraphs, we are actually trying to emphasize the following point. It was, therefore, indeed essential for the analysis in [18] to have the stationary metric written in the form given in eq.(3.1). In this section, we will argue that if the zeroth law is satisfied one can perform a coordinate transformation that changes the space-time metric from eq.…”

“…There have been various attempts at designing a proof for the second law that will be valid for arbitrary higher derivative theories of gravity [6][7][8][9][10][11][12][13][14] 2 . Also, recently the construction of an entropy current in such theories was studied [17][18][19], following the work of [12].…”

“…For example, in [12], a proof of linearized second law for arbitrary higher derivative theories of gravity was developed using this symmetry. Also, in [17] and [18], it was crucial to determine the structure of the equations of motion to construct an entropy current with non-negative divergence. In our present paper, assuming that the zeroth law is being satisfied at the order, O(α m ) of the α-expansion, this boost-symmetry enables us to constrain the off-shell structure of the equations of motion at the next order O(α m+1 ) as the desired one.…”