The mimumim length-scale: max GR tension smacking the quantum force scale!

In a previous post, we showed how, if the ultimate fate of our Universe is space empty of matter...but not quite....of energy (a de Sitter space), then this future cosmic event horizon (CEH) radius of our current, quantum, Universe set a natural maximum length-scale. Amazingly, the future CEH radius also defines the cosmological constant $\Lambda$.

  

For an observer at O inside the cosmic event horizon (CEH) with radius $l_{\Lambda}$, the universe can be divided into two sub-vacuums, $(A)$ inside the CEH, and $(B)$, outside. The horizon surface $\Sigma$ has entanglement entropy $S_{dS}$ and rest energy $E_H$

What about a Universal minimum length-scale? Quantum localisation plus gravitational collapse gives $\ell_{min}\gtrsim L_p$ (Planck length $L_p$) being a standard minimal length argument.  

Maximum Force  

 Barrow and Gibbons proposed, from pure classical GR, that the maximum tension sustainable by any physical system (e.g. consider a rope spanning a Schwarzschild horizon) is $F_{max}=c^4/4G$. 

Lets ask: What length does quantum mechanics associate with this force? The quantum force scale $F_{QG}$ associated with a relativistic excitation localised to $\ell$ is: $$F_{QG} \sim  \frac{\hbar c}{\ell_{\min}^2} \sim \frac{E_{UV}}{\ell_{\min}}$$ If we set  $F_{QG}=F_{max}$$$\ell_{min}=\ell_{UV}=2L_p$$ This shows $\ell_{min}$ as the UV cutoff where the quantum of action $\hbar c$ and the maximum gravitational tension  $c^4/4G$ meet.  

$$\boxed{F_{max}\ell^2_{min}=\hbar c}$$   

The "Minimal Mass" a UV Mass Gap

Now we have established the minimal length $\ell_{UV} = 2L_P$. What is the minimal mass? Although, as we shall see, "minimal mass" isn't really the correct term, there is no localised particle. In fact, it is the mass gap of space itself, or alternatively, modulus of rupture for a Clifford geometric algebra $Cl(\Re^3)$ spacetime geometry. That is: $$E_{UV}=\frac{1}{2}M_p c^2$$ where $M_p$ is the Planck mass. $$\boxed{m_{UV}=\frac{M_p}{2}}$$

A long time ago, Wesson conjectured that in a Universe with a positive cosmological constant $\Lambda$, there must be a quantum-scale rest mass, which we define as $m_s=m_{UV}$. Later, Boehmer and Harko proved that in classical GR, the presence of a positive $\Lambda$ sets a minimum local mass bound for a stable, gravitationally bound system of radius $r$: $$m(r) \geqslant \frac{\Lambda c^{2}}{12G} r^{3}.$$ We do not insert $r=2L_p$ (irrelevant). Instead, we evaluate the BH bound directly at the de Sitter horizon ($R_{IR} = \sqrt{3/\Lambda}$) then: $$m(r_{IR}) = \frac{c^2}{4G} r_{IR}.$$ This indicates when gravity saturates a casual horizon, mass-energy scales linearly with radius. We then propose a IR/UV duality: that the quantum minimum acts as a microscopic casual horizon mirroring the macroscopic casual horizon. Then we can apply this saturated linear scaling to the UV cutoff $r =\ell_{UV}= 2L_P$, so:

$$m_{UV} = \frac{c^2}{4G}(2L_P) = \frac{c^2 L_P}{2G}=\frac{M_P}{2}$$  $m_{UV}$ It is not a localised "particle" mass, but the mass gap of the spacetime geometry itself. This also shows that $L_p$ is the Schwarzschild radius of the mass gap, while $2L_p$ is the minimum operational resolution radius. 

Maximum Acceleration and the Gravitational Schwinger Effect

Now that we have the minimal mass $m_s$, we can determine the maximum local Universal acceleration $a_{UV}$ at this minimal scale. Just as the macroscopic horizon has an acceleration $H = c/R_{IR}$, the microscopic horizon has an acceleration bounded by $l_{UV}$: $$a_{UV} = \frac{c^2}{l_{UV}} = \frac{c^2}{2L_P} = \frac{a_{\rm Planck}}{2}.$$ This maximum acceleration has a direct physical interpretation. The Schwinger critical field in QED sets the threshold for spontaneous electron-positron pair production when the field does work $2m_e c^2$ over one Compton wavelength. However, we can't simply use liner work ($W = F \cdot d$). In gravity, a quantum fluctuation’s energy is its coupling charge, requiring a non-linear bootstrap mechanism rooted in the Equivalence Principle. 

The maximum localized vacuum fluctuation is $\Delta E_{\max} = M_p c^2 / 2$. This zero-point energy inherently acts as an inertial "virtual mass" resisting acceleration: $$m_{\text{virt}} = \frac{M_P}{2}$$ A causal (Rindler) horizon evaluated at the minimal operational length $\ell_{UV} = 2L_P$ dictates the maximum kinematic acceleration $$a_{UV} = \frac{c^2}{2L_P}$$ The instantaneous force exerted on the spacetime vacuum by its own maximal fluctuation under maximal acceleration exactly saturates the Barrow-Gibbons limit $$F = m_{\text{virt}} a_{UV} = \left(\frac{M_P}{2}\right)\left(\frac{c^2}{2L_P}\right) = \frac{M_P c^2}{4L_P}$$ Substituting $M_P c^2 = \hbar c / L_P$ and $L_P^2 = \hbar G / c^3$$$\boxed{F = \frac{\hbar c}{4L_P^2} = \frac{c^4}{4G} = F_{\max}}$$
The Gravitational Schwinger Effect is not a field pulling a particle pair apart; it is a structural limit of spacetime. When the vacuum fluctuation ($M_p/2$) is subjected to maximal acceleration ($a_{UV}$), its inertial resistance matches the absolute maximum tension spacetime can sustain ($c^4/4G$). At this threshold, the vacuum ruptures. Instead of yielding fermions, it nucleates microscopic black holes (spacetime foam), acting as an absolute UV cutoff. Of course, all this is a semi-classical heuristic.  

Maximum Force from the GUP

With our minimum length, we can also recover the maximum force bound from the Generalised Uncertainty Principal (GUP). While in standard QM $\Delta x$ can be made as small as you like by increasing momentum $\Delta p$, in a gravitational framework, concentrating momentum (i.e. energy) into a small region curves spacetime, creating a localised uncertainty that bounds resolving power. 

1. The GUP is: $$\Delta x \Delta p \ge \frac{\hbar}{2} (1 + \beta (\Delta p)^2)$$ 2. Then, the minimal position uncertainty is  $$\Delta x_{\min} = \hbar \sqrt{\beta}$$ 3. Use our minimal length $\ell_{UV}$: $$\Delta x_{\min} = 2 L_P \implies \beta = 4 G / (\hbar c^3)$$ 4. Saturating momentum becomes: $$\Delta p_{\rm sat} = 1/\sqrt{\beta} = M_P . c / 2$$ 5. The maximum localised energy fluctuation becomes: $$\Delta E_{\max} = c \Delta p_{\rm sat} = M_P c^2 / 2$$ 6. And the maximum force is: $F_{\max} = \Delta E_{\max} / \Delta x_{\min} = c^4 / 4 G$  

De Sitter

For a long time, it has been observed that the evolution of our Universe can be considered as two asymptotic de Sitter epochs connected by a transition phase parameterised by the brief moment of matter-radiation equality.  We are presently living in the second epoch - our accelerating Universe is quasi-de Sitter. 

Perhaps then, the most natural state of our Universe is dS space, as Sean Carroll posited. The past dS state is during and up to the end of inflation. Although then, by logical extension (i.e. we exist!) it also implies that dS space is highly unstable, giving common ground with Swampland, i.e. in string theory it is basically impossible to construct a meta-stable de Sitter theory. 

This implies that the Universe before inflation would be the actual initial past boundary condition, with the dS-like start of inflation being a later phase change. A key takeaway here is that during and up during and up to the end of inflation, the proper distance $l_{\Lambda}$ to the CEH stays constant (i.e. a dS-like state), while the proper distance between two points increases exponentially.  I know, this is a little difficult to get your head around. What it means is that inflation is not superluminal expansion.

As Lineweaver explains, during inflation (a period of around 60 e-folds) all the energy is in the inflaton which has very few degrees of freedom and low entropy. Inflation ends with a period of reheating, during which the inflation's energy is transferred into a relativistic fluid. This is also known as the Hot Big Bang. After reheating (in $\Lambda$CDM) during radiation domination, the CEH is approximately constant (the CEH proper radius increases as $l_{\Lambda} \propto a$ ),  and in the DE dominated future, the CEH is a constant proper radius. Here $a$ the cosmic scale factor (refer Figure below, also from Lineweaver).  

Alright then, but what about the minimal length-scale? Well, there is an argument (derived from the holographic principle), that the entire universe must be contained within the past horizon of a so-called eternal observer.

Now, there are multiple lines of evidence that at Planckian scales spacetime behaves effectively two-dimensional, so the natural "cell" for information is a 2-sphere of radius $L_p$ (linear size $l_{UV} = 2L_p$, i.e., its diameter). The unit of information is then the area of that sphere in Planck units:
$$I_c = \frac{A(L_P)}{L_P^2} = \frac{4\pi L_P^2}{L_P^2} = 4\pi.$$ This means the accessible cosmic information $I_c = 4\pi$ is the properly normalised number of modes that cross the Hubble radius during inflation.

Topological entropy

Using the minimum length, we can equate the semi-classical Bekenstein-Hawking entropy to the Gass-Bonnet theorum.  $$\frac{S_{\text{BH}}(2L_P)}{k_B} = \int_{S^2} K \, dA = 4\pi$$. Although, instead of viewing this as a smooth geometrical sphere, it is better interpreted through Wald entropy in higher-derivative gravity. If the most fundamental, indivisible unit of localised 3D space requires the topology of $S^2$ ($\chi=2$), then $4\pi$ represents the absolute minimum topological entropy.