Hydro & Thermo Dynamics at Causal Boundaries, Examples in 3d Gravity

Abstract

We study 3-dimensional gravity on a spacetime bounded by a generic 2-dimensional causal surface. We review the solution phase space specified by 4 generic functions over the causal boundary, construct the symplectic form over the solution space and the 4 boundary charges and their algebra. The boundary charges label boundary degrees of freedom. Three of these charges extend and generalize the Brown-York charges to the generic causal boundary, are canonical conjugates of boundary metric components and naturally give rise to a fluid description at the causal boundary. Moreover, we show that the boundary charges besides the causal boundary hydrodynamic description, also admit a thermodynamic description with a natural (geometric) causal boundary temperature and angular velocity. When the causal boundary is the asymptotic boundary of the 3d AdS or flat space, the hydrodynamic description respectively recovers an extension of the known conformal or conformal-Carrollian asymptotic hydrodynamics. When the causal boundary is a generic null surface, we recover the null surface thermodynamics of [1] which is an extension of the usual black hole thermodynamics description.

Figures (5)

• Figure 1: Portion of AdS$_3$ bounded by a generic timelike boundary ${\mathcal{C} }_r$. We formulate physics in the shaded region. The wiggles on ${\mathcal{C} }_r$ are to highlight the boundary degrees of freedom, where the Brown-York-type charges ${\cal T}^{ab}$ are canonical conjugates of boundary metric components $\gamma_{ab}$.
• Figure 2: A generic time-like boundary ${\mathcal{C} }_r$ in flat space. The wiggles on ${\mathcal{C} }_r$ depict the boundary degrees of freedom which are associated with the boundary metric and its geometry.
• Figure 3: A generic null boundary in $AdS_3$. The null boundary ${\cal N}$ may be viewed as future (Left figure) or past (Right figure) Poincaré horizon on AdS. We formulate physics in the shaded region and focus on the boundary degrees of freedom residing at ${\cal N}$.
• Figure 4: Null boundaries ${\cal N}$ in a flat $3d$ space. The left and right figures respectively show a future or past null boundary. We are interested in formulating physics in the shaded regions bounded by ${\cal N}$.
• Figure 5: Asymptotic boundary of AdS$_3$ (Left figure) and $3d$ flat space (Right figure). These asymptotic boundaries may be viewed as asymptotic limits of respective timelike boundaries.