= Euclidean de Sitter action and entropy
{c}
{title2=$I_E=-3\pi/\Lambda,\quad S=3\pi/\Lambda$}
In units $G=\hbar=c=k_B=1$, a smooth <Wick rotation> of the four-dimensional <static patch of de Sitter spacetime> has Euclidean time period $2\pi\sqrt{3/\Lambda}$. Its compact geometry is a round four-sphere of radius $\sqrt{3/\Lambda}$. The <Einstein field equations> give $R=4\Lambda$, and direct integration gives volume $24\pi^2/\Lambda^2$, hence gravitational <Euclidean action> $I_E=-3\pi/\Lambda$. With zero assigned internal energy, $I_E=\beta E-S$ gives <Bekenstein-Hawking entropy> $S=3\pi/\Lambda=A/4$.
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