# What is the Komar mass of the de-Sitter spacetime?

The Komar mass of some spacetime is defined as an integral (volume or surface, depending on its formulation): https://en.wikipedia.org/wiki/Komar_mass

The de-Sitter metric in static coordinates is ($$\Lambda > 0$$ here): $$\begin{equation}\tag{1} ds^2 = (1 - \frac{\Lambda}{3} \, r^2) \, dt^2 - \frac{1}{1 - \frac{\Lambda}{3} \, r^2} \, dr^2 - r^2 \, d\Omega^2. \end{equation}$$ This metric has a timelike Killing vector for $$r < \ell$$, where $$\ell = \sqrt{3/\Lambda}$$: $$\begin{equation}\tag{2} \xi^{\mu} = (1, 0, 0, 0). \end{equation}$$ Thus $$\xi^{\mu} \, \xi_{\mu} = g_{00}$$. It is spacelike for $$r > \ell$$.

AFAIK, the Komar mass is defined only for asymptotically flat spacetimes, which the de-Sitter spacetime isn't. So I believe that the Komar mass cannot be defined for the de-Sitter spacetime. Is this true?

If the Komar mass was defined for the de-Sitter spacetime, it should already be well known today, but I can't find it anywhere. So I guess that the answer to my question is affirmative, but I need a clear confirmation of it.

If an observer (say at $$r = 0$$) has only access to the region inside the horizon located at $$r = \ell$$, could we define a Komar mass (or an energy of some sort) for that observer?