The Virial Horizon

 

In general relativity, defining "energy" is notoriously difficult. A specific point of confusion often arises in de Sitter space: Why is the quasi-local energy of the horizon exactly double the effective gravitational mass?

By imposing a fundamental consistency condition— Horizon Uncertainty—we can re-derive the thermodynamics of the de Sitter horizon. We find that the horizon behaves exactly like a quantum-stretched membrane, where the "factor of two" is simply the result of the Virial Theorem applied to spacetime itself.

Horizon Position Uncertainty

Whether through holographic arguments, diffusion models, or entanglement entropy, a causal horizon is never sharp. Its position fluctuates. The variance of the horizon position $\Delta x^2$ scales with the geometric mean of the Planck length $L_p$​ and the horizon radius $l_{\Lambda}$​: 

\begin{equation}
    \Delta x^2 = L_P \, l_\Lambda, \qquad L_P = \sqrt{\frac{G\hbar}{c^3}}
\end{equation}  

What mass? $m_{CEH}$ vs $m_0$

In de Sitter space, we must distinguish between two distinct energy definitions evaluated at the cosmological horizon ($r=l_{\Lambda}$).

The Misner–Sharp Mass ($m_{CEH}$​) This is the "active" gravitational mass contained within the horizon. It represents the energy that sources the curvature:

\begin{equation}
    m_{CEH} = \frac{c^2 l_\Lambda}{2G}
\end{equation} The Brown–York Mass ($m_0$​) This is the quasilocal energy defined by the work required to assemble the boundary surface (the horizon) against the vacuum stiffness. It is the total energy of the region: 

\begin{equation}
    m_0 = \frac{c^2 l_\Lambda}{G}
\end{equation} Immediately, we see the geometric identity:

\begin{equation}
    \boxed{ m_0 = 2 m_{CEH} }
\end{equation}

Deriving the Surface Tension

The Smarr energy relation, which is just the first law of black hole dynamics (Hawking et al), gives the horizon effective rest mass-energy (Misner-Sharp) $E_H$:

\begin{equation}
E_{H}=-4 \ \sigma_{\Lambda}A_{H}
\end{equation} So the effective surface tension $\sigma_{\Lambda}$ of the CEH is:

\begin{equation}
\label{eq:7}
\sigma_{\Lambda} = - \frac{c^4}{32\pi G l_{\Lambda}}
\end{equation} The enthalpy (total energy) of the dS space-time is zero: \begin{equation}
\textsf H = E_{H} + PV = 0. 
\end{equation} The Gibbs free energy is:
\begin{equation}
\notag
G = U - TS + PV
\end{equation} With $U_{H}=E_{H}$, $E_H=T_{dS}S_{dS}$ and $PV=-E_{H}$ from the Smarr relation, the Gibbs free energy is: 
\begin{equation}
G = (U_{H}) - (E_H) + (-E_H) = -E_{H}
\end{equation}

Vacuum spring constant 

For a system of size $l_{\Lambda}$, the effective vacuum spring constant $k_{\Lambda}$ scales as Force (the inverse of the Einstein gravitational constant $\kappa$) divided by Distance, or equivalently, the Young's modulus times the Area (where, as we discussed previously, Area is the inverse of the cosmological constant $\Lambda$) divided by the length scale.   

\begin{equation}
    k_{\Lambda} =  \frac{1}{\kappa l_{\Lambda}}= Y_\Lambda \frac {1} {l_\Lambda \Lambda} = \frac{c^4}{8\pi G \, l_\Lambda}
\end{equation} We can see the cosmological (effective) Young's modulus associated with the cosmological constant is: \begin{equation}
    Y_\Lambda = \frac{\Lambda c^4}{8\pi G}\end{equation} 

Equation of State and the Virial Theorem

Consider the Virial Theorem for a Harmonic Oscillator. For a spring with potential energy $U= \frac{1}{2}kx^2$, the total energy of the system $H$ (assuming equipartition where kinetic energy equals potential energy) is: \begin{equation}
    H = 2 \langle U \rangle = k x^2
\end{equation}

By identifying the effective spring constant of the vacuum as $k_{\Lambda}=2 \sigma$: \begin{equation}
 E_0 = k_{\Lambda}  \ 16 \pi \ x^2 = \frac{c^4 l_\Lambda}{G} = H
\end{equation} We resolve the factor of two ambiguity 

1. $m_{0}$ (Brown-York) is the Total Hamiltonian Energy of the oscillating horizon membrane.
2. $m_{CEH}$ (Misner-Sharp) is the Potential Energy component that sources gravity.