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The Horizon Quantum of Area

  Several independent semiclassical arguments converge on $$\boxed{\Delta A=8\pi l_{\mathrm{Pl}}^2}, \qquad l_{\mathrm{Pl}}^2=\frac{G\hbar}{c^3}$$ I got interested in this reading the 2018 paper  Bekenstein, I, and the quantum of black-hole surface area . Even though I disagree with Hod, its a great paper.  Honourable mentions also go to:   The Physical Interpretation of the Spectrum of Black Hole Quasinormal Modes , 2008  The Off-Shell Black Hole , 1994     On Black Hole Spectroscopy via Adiabatic Invariance , 2012 An Analytical Computation of Asymptotic Schwarzschild Quasinormal Frequencies , 2003  Asymptotic Black Hole Quasinormal Frequencies , 2003  1. Horizon Action Near any nonextremal horizon, imaginary time turns the normal geometry into a plane: $$ds_E^2\simeq d\rho^2+\rho^2d\Theta^2+\cdots.$$ Smoothness requires one horizon cycle: $$\oint d\Theta=2\pi.$$ The gravitational action identifies the momentum conjugate to $\The...
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Minimum and Maximum Mass from the Cosmic Horizon

 

A positive cosmological constant sets the future cosmic event-horizon radius of an expanding FLRW universe:

$$l_\Lambda=\sqrt{\frac{3}{\Lambda}}.$$

Associate two lengths with a positive mass $m$:

$$\bar\lambda_C=\frac{\hbar}{mc} \qquad r_g=\frac{2Gm}{c^2}$$

where $\bar\lambda_C$ is the reduced Compton wavelength and $r_g$ is the gravitational radius.

Requiring both lengths not to exceed the cosmic horizon,

$$\bar\lambda_C\le l_\Lambda \qquad r_g\le l_\Lambda,$$

gives

$$\boxed{ m_{\min}= m_s=\frac{\hbar}{cl_\Lambda} =\frac{\hbar}{c}\sqrt{\frac{\Lambda}{3}} }$$

$m_s$ is the minimum mass first pointed out by Wesson, 2003. This was also adopted by Bohmer and Harko, 2005 to derive a minimum density for static bodies  using the cosmological Buchdahl inequality.

and

$$\boxed{ m_{\max}=\frac{c^2l_\Lambda}{2G} =\frac{c^2}{2G}\sqrt{\frac{3}{\Lambda}} \equiv m_{\rm CEH}. }$$

Wesson also conjectured a maximum mass, but was off by a factor of 2, as well as incorrectly assigning it as the mass of the observable universe, rather than the cosmic event horizon. In an FLRW universe, matter and radiation dilute with time, while the apparent and event horizons approach $l_{\Lambda}$. The mass inside the limiting causal region therefore approaches $m_{CEH}$. 

Barrow, 2014 also re-derived the minimum and maximum masses to be $m_{min}=m_s$ and $m_{max}=m_{CEH}$, keep in mind there is a factor-of-two error in their published final result (Eqn. 26) for $m_{max}$.  

Thus

$$\boxed{ \frac{\hbar}{c}\sqrt{\frac{\Lambda}{3}} \le m\le \frac{c^2}{2G}\sqrt{\frac{3}{\Lambda}} }.$$

The upper endpoint is the vacuum mass enclosed by the cosmic horizon. The vacuum mass density is $\rho_\Lambda=\Lambda c^2/8\pi G$, so

$$\frac{4\pi}{3}\rho_\Lambda l_\Lambda^3 = \frac{\Lambda c^2l_\Lambda^3}{6G} = \frac{c^2l_\Lambda}{2G} = m_{\rm CEH}.$$

$m_{CEH}$ is a horizon-scale quasi-local mass.

The two endpoint masses satisfy

$$m_{\min}m_{\rm CEH} = \frac{\hbar c}{2G} = \frac{m_{\rm Pl}^2}{2}, \qquad m_{\rm Pl}=\sqrt{\frac{\hbar c}{G}}.$$

Therefore,

$$\boxed{ \bar\lambda_C(m_{\min}) = l_\Lambda = r_g(m_{\rm CEH}) \qquad 2 \ m_{s}m_{\rm CEH} = m_{\rm Pl}^2 }$$

The same cosmic horizon therefore defines a minimum mass whose reduced Compton wavelength fits inside it and a maximum enclosed mass whose gravitational radius reaches it. 

Of course, particles less than the minimum mass can exist (indeed even massless particles). The maximum is the mass associated with the cosmic horizon (the Misner–Sharp quasi-local mass of the static patch evaluated at its horizon) not the maximum mass of a black hole inside de Sitter space.

Furthermore, the  Brown-York energy associated with the length of the CEH is 

$$E_{BY}=\frac{c^4l_\Lambda}{G} = 2m_{CEH}c^2$$ i.e. the Brown-York mass is twice the CEH mass. 

So can we get a Minimum length? 

$$ \frac{\hbar}{c\,l_\Lambda} \le m \le \frac{c^2 l_\Lambda}{2G}$$ That mass $m$ requires $$ \frac{\hbar}{c\,l_\Lambda} \le \frac{c^2 l_\Lambda}{2G}$$ which means $$ l_\Lambda^2 \ge 2\,l_{\mathrm{Pl}}^2$$ We might call this a "minimum length", where the reduced Compton wavelength equals the Schwarzschild radius. 

Well, not so fast, because the minimum position uncertainty $\Delta x$ of a relativistic probe having to contend with both quantum mechanics and gravity is actually $\Delta x=2L_p$.  $\Delta x \gtrsim \frac{\hbar}{\Delta p} + \frac{G\,\Delta p}{c^3}$ so $\Delta x_{\min} = 2\ell_{\mathrm{Pl}}$. This also is a "minimum length". Actually Mead who in 1964 first made this argument, multiplied the uncertainties, but I think it is more physical to add them.  

All in one horizon!

$\boxed{ m_s m_{\mathrm{CEH}}=\frac{m_{\mathrm{Pl}}^2}{2}, \qquad \frac{m_{\mathrm{CEH}}}{m_s}=\frac{S_\Lambda}{2\pi k_B}, \qquad m_{\mathrm{CEH}}c^2=T_\Lambda S_\Lambda }$

The same horizon ties together UV localisation, IR gravitation, horizon entropy, and vacuum thermodynamics!


 

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