Abstract
The gravitational shock waves have provided crucial insights into entanglement structures of black holes in the AdS/CFT correspondence. Recent progress on the soft hair physics suggests that these developments from holography may also be applicable to geometries beyond negatively curved spacetime. In this work, we derive a remarkably simple thermodynamic relation which relates the gravitational shock wave to a microscopic area deformation. Our treatment is based on the covariant phase space formalism and is applicable to any Killing horizon in generic static spacetime which is governed by arbitrary covariant theory of gravity. The central idea is to probe the gravitational shock wave, which shifts the horizon in the $u$ direction, by the Noether charge constructed from a vector field which shifts the horizon in the $v$ direction. As an application, we illustrate its use for the Gauss-Bonnet gravity. We also derive a simplified form of the gravitational scattering unitary matrix and show that its leading-order contribution is nothing but the exponential of the horizon area: $\mathcal{U}=\exp(i \text{Area})$.
Highlights
Recent developments at the interface between quantum gravity and quantum information theory in the AdS=CFT correspondence have provided useful tools to address conceptual puzzles concerning quantum aspects of black holes on less ambiguous settings [1,2,3,4,5,6,7,8,9]
Recent progress on the soft hair physics suggests that these developments from holography may be applicable to geometries beyond negatively curved spacetime
Our treatment is based on the covariant phase space formalism and is applicable to any Killing horizon in generic static spacetime which is governed by arbitrary covariant theory of gravity
Summary
Some general expressions of thermodynamic relations for soft charges have been derived for the Schwarzschild black hole and several other geometries [58] While these developments may provide a further insight into quantum aspects of the gravitational shock waves, it remains unclear how the soft charges may be measured in a. The implications of the soft hair physics on conceptual puzzles of quantum black holes still remain vague, in comparison with concrete developments within the framework of the AdS=CFT correspondence. We will extend the Noether charge method to include variations of the matter fields at arbitrary order This refinement enables us to obtain the microscopic thermodynamic relations [as in Eq (7)] which directly associate the area deformation to the energy-momentum tensor variation. The phase factor of the scattering matrix is proportional to the horizon area (or equivalently, the soft charge of an infalling matter measured by an outgoing matter)
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