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Spacetime Has a Clock: Horizon Thermodynamics and the de Sitter Seesaw

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De Sitter space as a thermodynamic equilibrium

During both inflation and late-time cosmic acceleration, the Universe is well-approximated by a de Sitter (dS) spacetime with nearly constant curvature radius ($\ell_\Lambda $). Our present Universe may therefore be regarded as a quasi–de Sitter state, possessing a cosmic event horizon (CEH) associated with its vacuum energy density (cosmological constant).

A defining feature of de Sitter space is that the cosmological horizon is not merely a causal boundary but a thermodynamic object, endowed with temperature, entropy, and energy. In this context, the total bare (rest) energy associated with the horizon, defined via the Brown–York quasilocal energy, can be written as:
\begin{equation}
E_0 = 2\, k_B T_{dS} S_{dS}
= 2\, m_{CEH} c^2
= 2E_H 
\end{equation} This relation is a horizon version of the entanglement first law, from which the Einstein equations themselves can be derived. Here:

  • $S_{dS}$ is the de Sitter entropy (dimensionless, in nats), $k_B T_{dS}$ has dimensions of energy,
  • $m_{CEH}$ is the Misner–Sharp effective gravitational mass inside the cosmological horizon.

In de Sitter space, we have previously established the Misner–Sharp mass is exactly half the Brown–York mass $m_0$ enclosed by the horizon:
\begin{equation}
m_0 = 2 m_{CEH}
\end{equation} This factor of two is not arbitrary; it reflects the distinction between bulk gravitating energy (Misner–Sharp) and the full quasilocal energy including boundary contributions (Brown–York). 

The local energy quantum and the Gibbons–Hawking temperature

An inertial observer at the centre of the de Sitter static patch measures a thermal bath at the Gibbons–Hawking temperature
\begin{equation}
T_{dS} = \frac{\hbar H_\Lambda}{2\pi k_B}
\end{equation} This motivates the definition of a local rest energy quantum
\begin{equation}
E_s = m_s c^2 = \hbar H_\Lambda = 2\pi k_B T_{dS}.
\end{equation} In natural units ($c=\hbar=1$),
\begin{equation}
m_s = H_\Lambda
\end{equation} This quantity is not a particle mass. Instead, it is a kinematic mass scale:

  • Arises purely from spacetime geometry.
  •  Reflects the Euclidean time periodicity of de Sitter space.
  • Corresponds to a minimum resolvable energy or frequency, the fundamental ``tick rate'' of spacetime itself. $H_{\Lambda}$ acts as a natural (aka `bare') vacuum frequency.
There is no field with a dispersion relation $E^2 = p^2 + m_s^2$, no associated propagating degree of freedom, and no new contribution to the stress–energy tensor. Calling $m_s$ a ``mass'' is dimensional bookkeeping: in natural units, energy, mass, and frequency coincide. In this sense, $m_s$ plays the same conceptual role as the Unruh or Hawking temperature: it is observer and horizon dependent, not dynamical.

Planck-locked mass duality

The total Brown–York mass $m_0$ associated with the cosmological horizon and the local kinematic mass $m_s$ are not independent. The total (bare) mass $m_0$ and the mass of its fundamental quantum $m_s$ satisfy a precise seesaw relation locked to the Planck scale. 
\begin{equation}
M_p^2 = m_0\, m_s
\end{equation} where $M_p = \sqrt{\hbar c/G}$ is the Planck mass.

  • $m_s$ is UV-like: local, kinematic, and frequency-based.
  • $m_0$ is IR-like: global, holographic, and boundary-defined.
  • The Planck mass mediates between them as a geometric mean.

This structure mirrors the UV/IR duality familiar from holography and appears in entropy bounds, horizon complementarity, and vacuum energy arguments.

Mass bounds in the presence of a cosmological constant

We have talked about these mass-scales before. Also, Barrow and Gibbons (2014) showed that in classical GR with a positive cosmological constant $\Lambda$, any physical mass must lie between strict lower and upper bounds:
\begin{equation}
M_{\text{lower}} = m_s > \frac{\hbar}{c}\sqrt{\frac{\Lambda}{3}},
\end{equation}
\begin{equation}
M_{\text{upper}} = m_0 < \frac{c^2}{G}\sqrt{\frac{3}{\Lambda}}.
\end{equation} De Sitter space saturates both bounds simultaneously. It is therefore an extremal spacetime. This further reinforces the interpretation of $m_s$ and $m_0$ as geometrically enforced scales rather than model-dependent inputs.

The vacuum energy seesaw

The observed vacuum energy scale,
\begin{equation}
M_\Lambda \approx 2.35 \times 10^{-12}\ \text{GeV}
\end{equation} can be written in a seesaw form closely related to the previous relations:
\begin{equation}
M_\Lambda
= \left(\frac{3}{8\pi}\right)^{1/4}
\sqrt{M_p\, m_s}
\end{equation} Equivalently, in terms of the vacuum energy density,
\begin{equation}
\rho_\Lambda = M_\Lambda^4
= \frac{3}{8\pi}\frac{M_p^2}{\ell_\Lambda^2}
= 3\, M_{Pl}^2\, L^{-2}
\end{equation} where $M_{Pl} = M_p/\sqrt{8\pi}$ is the reduced Planck mass and $L = \ell_\Lambda$. This expression is exactly consistent with the Friedmann equation
\begin{equation}
H_\Lambda^2 = \frac{\rho_\Lambda}{3 M_{Pl}^2}
\end{equation} The same seesaw structure,
\begin{equation}
M_\Lambda \sim \sqrt{M_p H_\Lambda}
\end{equation} has appeared repeatedly in the literature:

  • running vacuum models,
  • holographic dark energy,
  • string landscape arguments.

Its persistence is not coincidental. It is enforced by dimensional analysis plus horizon thermodynamics.

The picture that emerges is coherent and economical:

  • De Sitter space is a thermodynamic equilibrium state.
  • The relation $E_0 = 2\, k_B T_{dS} S_{dS}$ is a horizon entanglement identity.
  • $m_s = H_\Lambda$ is a kinematic mass scale, a horizon clock rate, not a particle.
  • $m_0$ is a boundary energy, not a local gravitating mass.
  • $M_\Lambda$ is emergent and collective, arising from the geometric mean of UV (Planck) and IR (horizon) cutoffs.

The only caveat is interpretational: $m_s$ should not be treated as a propagating degree of freedom unless embedded in a concrete effective field theory. Its role is geometric, not dynamical.

 The cosmological constant does not introduce a new fundamental mass scale by hand. Instead, it reflects a deep interplay between horizon thermodynamics, spacetime kinematics, and holographic duality. In this sense, spacetime itself has a clock, and its tick rate is set by the Hubble parameter of de Sitter space. What this means is that dark energy can be exactly constant in value while being emergent in origin. 

$\Lambda$ is is thermodynamic, not dynamical. 

However! 

 Our universe is not exactly de Sitter

  • inflation: slow-roll quasi–dS
  • today: matter + dark energy mixture

In that case:\begin{equation}
H(t) = H_\Lambda + \delta H(t)
\end{equation}

  •  horizon temperature slowly evolves,
  •  horizon entropy slowly evolves,
  •  Brown–York energy slowly evolves.

This can be described as:

  • running vacuum energy,
  • entropic exchange with matter,
  • backreaction of horizon degrees of freedom.

But:
\begin{equation}
w \approx -1, \qquad |\dot{\rho}_\Lambda| \ll \rho_\Lambda H
\end{equation} So any “variation” is:

  • higher order
  • suppressed by slow-roll parameters or ( $\Omega_m$ )
  • observationally tiny

This is why:

  • running vacuum models are allowed, but tightly constrained.


 

 

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