Entropy and mass bounds of Kerr–de Sitter spacetimes

We consider Kerr–de Sitter spacetimes and evaluate their mass, angular momentum, and entropy according to the boundary counterterm prescription. We provide a physical interpretation for angular velocity and angular momentum at future/past infinity. We show that the entropy of the four-dimensional Kerr–de Sitter spacetimes is less than of pure de Sitter (dS) spacetime in agreement with the entropic $N$ bound. Moreover, we show that maximal mass conjecture which states “any asymptotically de Sitter spacetime with mass greater than de Sitter has a cosmological singularity” is respected by asymptotically de Sitter spacetimes with rotation. We furthermore consider the possibility of strengthening the conjecture to state that “any asymptotically dS spacetime will have mass greater than dS if and only if it has a cosmological singularity” and find that Kerr–de Sitter spacetimes do not respect this stronger statement. We investigate the behavior of the $c$ function for the Kerr–de Sitter spacetimes and show that it is no longer isotropic. However an average of the $c$ function over the angular variables yields a renormalization group flow in agreement with the expansion of spacetime at future infinity.

Figure 1

Entropy of Kerr-dS spacetime with respect to rescaled mass $m/\ell$ and rotational parameter $a/\ell$. The horizontal sheet denotes the $N$ bound of $\pi {\ell }^2$.

Figure 2

The $c$ function of Kerr-dS spacetime plotted as a function of $\tau$ for three different values of angular variable $\theta =0$ (lower solid curve), $\frac{\pi }{5}$ (dotted curve), $\frac{\pi }{3}$ (upper dashed curve). The cosmological constant $\Lambda =3$ and the mass and rotational parameters are, respectively, set to 2 and 3. The cosmological horizon is located at ${\tau }_c=0.6449$.

Figure 3

The $\overline{c}$ function of Kerr-dS spacetime plotted as a function of $\tau$. The cosmological constant $\Lambda =3$ and the mass and rotational parameters are, respectively, set to 2 and 3. The cosmological horizon is located at ${\tau }_c=0.6449$.