Abstract
We present a new set of boundary conditions for General Relativity on AdS3, where the dynamics of the boundary degrees of freedom are described by two independent left and right members of the Gardner hierarchy of integrable equations, also known as the “mixed KdV-mKdV” hierarchy. This integrable system has the very special property that simultaneously combines both, the Korteweg-de Vries (KdV) and modified Korteweg-de Vries (mKdV) hierarchies in a single integrable structure. This relationship between gravitation in three-dimensional spacetimes and two-dimensional integrable systems is based on an extension of the recently introduced “soft hairy boundary conditions” on AdS3, where the chemical potentials are now allowed to depend locally on the dynamical fields and their spatial derivatives. The complete integrable structure of the Gardner system, i.e., the phase space, the Poisson brackets and the infinite number of commuting conserved charges, are directly obtained from the asymptotic analysis and the conserved surface integrals in the gravitational theory. These boundary conditions have the particular property that they can also be interpreted as being defined in the near horizon region of spacetimes with event horizons. Black hole solutions are then naturally accommodated within our boundary conditions, and are described by static configurations associated to the corresponding member of the Gardner hierarchy. The thermodynamic properties of the black holes in the ensembles defined by our boundary conditions are also discussed. Finally, we show that our results can be naturally extended to the case of a vanishing cosmological constant, and the integrable system turns out to be precisely the same as in the case of AdS3.
Highlights
Asymptotic region of spacetime in the gravitational theory
We present a new set of boundary conditions for General Relativity on AdS3, where the dynamics of the boundary degrees of freedom are described by two independent left and right members of the Gardner hierarchy of integrable equations, known as the “mixed Korteweg-de Vries (KdV)-modified Korteweg-de Vries (mKdV)” hierarchy
This integrable system has the very special property that simultaneously combines both, the Korteweg-de Vries (KdV) and modified Korteweg-de Vries hierarchies in a single integrable structure. This relationship between gravitation in three-dimensional spacetimes and two-dimensional integrable systems is based on an extension of the recently introduced “soft hairy boundary conditions” on AdS3, where the chemical potentials are allowed to depend locally on the dynamical fields and their spatial derivatives
Summary
We describe the asymptotic behavior (fall-off) of the gravitational field without specifying yet what is fixed at the boundary of spacetime, i.e., without imposing at this step of the analysis a precise boundary condition. For the purpose of simplicity and clarity in the presentation, we mostly work in the Chern-Simons formulation of threedimensional Einstein gravity with a negative cosmological constant [29, 30]. The action for General Relativity on AdS3 can be written as a Chern-Simons action for the gauge group SL(2, R) × SL(2, R). The gauge connections A± are 1-forms valued on the sl(2, R) algebra, and are related to the vielbein e and spin connection ω through A± = ω±el−1. The level in (2.1) is given by κ = l/4G, where l is the AdS radius and G is the Newton constant, while the bilinear form. The sl(2, R) generators Ln with n = 0, ±1, obey the commutation relations [Ln, Lm] = (n − m) Ln+m, with non-vanishing components of the bilinear form given by L1L−1 = −1, and L20 = 1/2
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