Hamiltonian dynamics in the space of asymptotically Kerr–de Sitter spacetimes

We determine all spacetime flows which lead to a Hamiltonian dynamics in the space of general-relativistic initial data sets with asymptotically Kerr–de Sitter ends. The corresponding Hamiltonians are calculated. Some implications for black-hole thermodynamics are pointed out.

Figure 1

The tangent space at $\partial V$. The vector field $m$ is tangent to $V$ and normal to $\partial V$, while $n$ is normal to $V$. The vector field $N$ is parallel to $X$ and normal to $\partial V$, and $M$ is orthogonal to both $N$ and $T\partial V$.

Figure 2

A projection diagram for the Kerr-Newman–de Sitter metric with four distinct zeros of ${\mathrm{\Delta }}_r$, from [22]. The shaded region corresponds to causality violations and the ring singularity, and the Minkowskian causality does not represent correctly the causality in spacetime there. The pattern continues indefinitely in all directions.