On a Quasi-Local Energy Outside the Cosmological Horizon

We investigate a quasi-local energy naturally introduced by Kodama’s prescription for a spherically symmetric space-time with a positive cosmological constant $\mathrm{\Lambda }$. We find that this quasi-local energy is well behaved inside a cosmological horizon. However, when there is a scalar field with a long enough Compton wavelength, the quasi-local energy diverges in the course of its time evolution outside the cosmological horizon. This means that the quasi-local energy has a meaning only inside the cosmological horizon.