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Questions

  1. What does a negative surface tension of a de Sitter cosmological event horizon actually mean? Scotch in water, or something else?
  2. What does a zero ‘external’ vacuum pressure to the de Sitter event horizon mean?

Background to questions

A de Sitter space time with a positive cosmological constant $\Lambda$ has an event horizon around any observer with a radius $l_{\Lambda}$. This basically splits the universe into two sub-vacuums, within and without the event horizon.

The effective pressure of the bulk inside the event horizon is $P_{\Lambda} = -\frac{\Lambda c^4}{8 \pi G}$, i.e. negative.

This answer shows there is a significant history in considering an event horizon as a kind of fluid membrane, with an associated surface tension $\sigma$.

This 2017 paper (page 124 in the pdf) actually derives the surface tension of a de Sitter event horizon, giving (with units): \begin{equation} \tag{1} \sigma_{\Lambda} = - \frac{3c^4}{16\pi G l_{\Lambda}} \end{equation}

Assuming the authors got it right, this is a negative surface tension, which is puzzling (generally, surface tension is always positive), so along the lines of this answer (1) would seem to imply that the two ‘fluids’ (vacuums) will actually mix until evenly distributed, like scotch in water? The fact that the Gibbs free energy is negative (Eqn 33 in the 2017 paper), also implies the vacuums will mix (despite the event horizon)!? Perhaps, non-withstanding the event horizon, vacuum is vacuum, already 'mixed'? After all, the universe doesn't 'expand' into anything.

Unless (1) and negative Gibbs free energy has something to do with de Sitter vacuum being unstable under certain conditions / swampland, (yes, I know this 1986 paper claimed dS is stable)...???

Anyhow, assuming (1) as correct, if we apply a standard droplet analysis, (assuming this is legit, and, that the event horizon has only one surface, not two) with $P_{in}=P_{\Lambda}$ we get:

\begin{equation} \notag P_{in}-P_{out} = \frac{2\sigma}{l_{\Lambda}} \end{equation}

\begin{equation} \notag P_{out}= -\frac{\Lambda c^4}{8 \pi G} + \frac{\Lambda c^4}{8 \pi G} \end{equation}

\begin{equation} \tag{2} P_{out} = 0 \end{equation} This is also odd, so what does it mean? Is it that nothing ‘outside’ the de Sitter event horizon affects the inside? In a standard droplet analysis, because of the curvature of the bubble, pressure inside a bubble is higher than the pressure outside. Now, negative temperature is 'warmer' than absolute zero, so maybe, negative pressure of the 'in' vacuum might be considered ’higher’ than zero pressure of the 'out' vacuum? Yes, I know a pressure gradient doesn’t affect the expansion of the Universe.

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