>[!warning] >This content has not been peer reviewed. # The Planck scale and the substrate's minimal resolution Below a certain scale, the classical description of space, time, and gravity is expected to break down — **quantum gravity** regime. The natural scales are the **Planck units**, built from $\hbar$, $c$, and $G$. In RST the substrate ([[Format]], A1) has **minimal resolution** and a **finite bit density**; the Planck scale is where that minimal resolution is expressed in geometric terms. So the *math* of the quantum gravity regime (Planck length, time, mass; Bekenstein–Hawking entropy per area; minimal length proposals) folds into RST as the **substrate's minimal resolution and finite information density**. --- ## What it is (the math) **Planck units** (dimensional analysis from $\hbar$, $c$, $G$): - **Planck length:** $\ell_P = \sqrt{\hbar G/c^3} \approx 1.616\times 10^{-35}$ m - **Planck time:** $t_P = \ell_P/c = \sqrt{\hbar G/c^5} \approx 5.391\times 10^{-44}$ s - **Planck mass:** $M_P = \sqrt{\hbar c/G} \approx 2.176\times 10^{-8}$ kg - **Planck energy:** $E_P = M_P c^2 = \sqrt{\hbar c^5/G} \approx 1.956\times 10^9$ J **Bekenstein–Hawking entropy:** A horizon of area $A$ has entropy $S = A/(4 \ell_P^2)$ (in natural units $\hbar=c=k_B=1$ with $G$ in $\ell_P$). So **one bit per 4 Planck areas** — the maximum information density on a boundary is $1/(4\ell_P^2)$ bits per unit area. **Minimal length:** Many quantum-gravity approaches (loop quantum gravity, string theory, causal sets) imply a **minimal length** or **discrete area spectrum** of order $\ell_P$ or $\ell_P^2$. The continuum is an approximation above that scale. **Lloyd's bound:** A region of mass $M$ and size $R$ can perform at most $\sim 2 M c^2 R/(\pi \hbar c)$ logical operations per second — so the substrate has a finite **rate of computation** per unit mass and size, set by $\hbar$, $c$, and the geometry. --- ## How RST folds it in | Quantum gravity / Planck | RST reading | |:---|:---| | **$\ell_P$** | Natural **minimal resolution** of the Format (A1): the substrate cannot resolve structure below $\ell_P$; the continuum description is valid only for scales $\gg \ell_P$. So $\ell_P$ is the **cell size** or **lattice spacing** of the relational network in geometric units. | | **$t_P$** | Minimal **refresh interval** $\tau$ (A4, [[Proper Time]]): the substrate's tick cannot be shorter than $\sim t_P$ for processes at the Planck energy scale. So $t_P$ is the **fastest tick** of the format when the relevant energy is $E_P$. | | **$S = A/(4\ell_P^2)$** | **Maximum information in a region** (Bekenstein bound) in natural units: the substrate has at most **1 bit per 4 Planck areas** on a boundary. So the Format's **bit density** is bounded; the Resource Triangle's finite budget $W$ is consistent with this ceiling. See **[[expanded theory/knowledge/Information in a region cannot exceed its boundary]]**. | | **Minimal length / discrete area** | The substrate has **discrete degrees of freedom** per region (finite bit count); the "continuum" is the limit when the observer's resolution is much coarser than $\ell_P$. So RST's finite resolution (A1, A2) is the **relational** version of minimal length. | | **$\ell \to \ell_P$ and $\mu \to 0$** | As the effective scale of a process approaches $\ell_P$, the substrate can no longer resolve it — **fidelity** $\mu \to 0$ ([[Fidelity]]). So singularities (e.g. Navier–Stokes blow-up at $r\to 0$) are regulated: the substrate **coarse-grains** below $\ell_P$. See **[[Navier-Stokes Smoothness (RST)]]**. | | **Lloyd bound** | **Maximum computation in a region** ([[Concrete properties of the substrate]]): the substrate's rate of logical operations is bounded by mass, size, $\hbar$, and $c$. So the Format not only has a minimal resolution and a max bit density, but a **max ops per second** per region. | --- ## Summary equations (RST use) - **Minimal resolution:** $\Delta x \gtrsim \ell_P$, $\Delta t \gtrsim t_P$ for the substrate's native scale. - **Max bits on a boundary:** $S \leq A/(4\ell_P^2)$. - **At scale $\ell \sim \ell_P$:** continuum fails; $\mu \to 0$ for processes that would probe below that scale. - **Planck mass/energy:** $M_P$, $E_P$ set the scale at which the substrate's "one cell" has energy $\sim E_P$; above that, semiclassical gravity applies; at/below, the substrate's discrete structure dominates. So the **math of the quantum gravity regime** (Planck units, area law, minimal length, Lloyd) is not separate from RST — it **is** the substrate's minimal resolution, finite bit density, and finite computation rate expressed in geometric and thermodynamic units. RST does not assume a particular theory of quantum gravity (LQG, strings, etc.); it assumes a **finite-resolution, finite-capacity** Format, and the Planck scale is where that finiteness appears in the usual dimensional analysis with $G$, $\hbar$, $c$. --- **Script:** `rst_planck_scale_math.py` in this folder computes $\ell_P$, $t_P$, $M_P$, $E_P$ and the Bekenstein–Hawking bits-per-area $1/(4\ell_P^2)$ in SI, so the math is explicit. Run from workspace root: `python "expanded theory/knowledge/rst_planck_scale_math.py"`. --- ## Links | Concept | Note | |:---|:---| | Substrate definition | **[[Format]]** | | All concrete properties | **[[Concrete properties of the substrate]]** | | Max information in region | **[[expanded theory/knowledge/Information in a region cannot exceed its boundary]]** | | Refresh, proper time | **[[Proper Time]]** | | Regulator at small scale | **[[Navier-Stokes Smoothness (RST)]]** (in further applications) | --- ## References - Bekenstein, J. D. (1973). *Black holes and entropy.* Phys. Rev. D **7**, 2333–2346. - Hawking, S. W. (1975). *Particle creation by black holes.* Commun. Math. Phys. **43**, 199–220. - Lloyd, S. (2000). *Ultimate physical limits to computation.* Nature **406**, 1047–1054. [DOI](https://doi.org/10.1038/35023282) - CODATA 2018: Planck length, time, mass (Tiesinga et al., Rev. Mod. Phys. **93**, 025010, 2021).