Traditional and universal forms of the core equations of mechanics, electromagnetism, and quantum physics, shown side by side.
In SI units, the law of gravitation reads $F = G\, m_1 m_2 / r^2$. The constant $G$ carries the dimensions and the numerical scale; the product $m_1 m_2 / r^2$ is a combination of particle properties. In universal form, the same law reads $F = F_{\mathrm{P}}\, (m_1/m_{\mathrm{P}})(m_2/m_{\mathrm{P}})(l_{\mathrm{P}}/r)^2$. The Planck force $F_{\mathrm{P}}$ sets the scale; three dimensionless ratios determine the actual force.
Both expressions give the same numerical answer. The universal form makes the scale dependence transparent: the gravitational force on two Planck masses separated by one Planck length is exactly $F_{\mathrm{P}}$. At any other scale, the force is reduced by the product of three ratios, each of which has a clear geometric meaning.
The tables below collect the traditional and universal forms of the standard formulas across mechanics, electromagnetism, and quantum physics. Planck unit values: $l_{\mathrm{P}} = 1.6163 \times 10^{-35}$ m, $m_{\mathrm{P}} = 2.1764 \times 10^{-8}$ kg, $t_{\mathrm{P}} = 5.3912 \times 10^{-44}$ s, $E_{\mathrm{P}} = 1.9561 \times 10^{9}$ J, $F_{\mathrm{P}} = 1.2103 \times 10^{44}$ N, $q_{\mathrm{P}} = 1.8755 \times 10^{-18}$ C.
Mechanics
| Quantity | Traditional | Universal form |
|---|---|---|
| Newton’s second law | $F = m a$ | $\dfrac{F}{F_{\mathrm{P}}} = \dfrac{m}{m_{\mathrm{P}}} \cdot \dfrac{a}{a_{\mathrm{P}}}$ |
| Momentum | $p = m v$ | $\dfrac{p}{m_{\mathrm{P}} c} = \dfrac{m}{m_{\mathrm{P}}} \cdot \dfrac{v}{c}$ |
| Kinetic energy (matter) | $E_K = \tfrac{1}{2} m v^2$ | $\dfrac{E_K}{E_{\mathrm{P}}} = \tfrac{1}{2}\,\dfrac{m}{m_{\mathrm{P}}} \cdot \left(\dfrac{v}{c}\right)^{\!2}$ |
| Rest energy | $E = m c^2$ | $\dfrac{E}{E_{\mathrm{P}}} = \dfrac{m}{m_{\mathrm{P}}}$ |
| Newton’s gravitation | $F = \dfrac{G m_1 m_2}{r^2}$ | $\dfrac{F}{F_{\mathrm{P}}} = \dfrac{m_1}{m_{\mathrm{P}}} \cdot \dfrac{m_2}{m_{\mathrm{P}}} \cdot \left(\dfrac{l_{\mathrm{P}}}{r}\right)^{\!2}$ |
| Gravitational potential | $U = -\dfrac{G m_1 m_2}{r}$ | $\dfrac{U}{E_{\mathrm{P}}} = -\dfrac{m_1}{m_{\mathrm{P}}} \cdot \dfrac{m_2}{m_{\mathrm{P}}} \cdot \dfrac{l_{\mathrm{P}}}{r}$ |
| Angular momentum | $L = m v r$ | $\dfrac{L}{\hbar} = \dfrac{m}{m_{\mathrm{P}}} \cdot \dfrac{v}{c} \cdot \dfrac{r}{l_{\mathrm{P}}}$ |
Electromagnetism
| Quantity | Traditional | Universal form |
|---|---|---|
| Coulomb’s law | $F = \dfrac{1}{4\pi\varepsilon_0} \dfrac{q_1 q_2}{r^2}$ | $\dfrac{F}{F_{\mathrm{P}}} = \dfrac{q_1}{q_{\mathrm{P}}} \cdot \dfrac{q_2}{q_{\mathrm{P}}} \cdot \left(\dfrac{l_{\mathrm{P}}}{r}\right)^{\!2}$ |
| Coulomb potential | $V = \dfrac{q}{4\pi\varepsilon_0 r}$ | $\dfrac{V}{V_{\mathrm{P}}} = \dfrac{q}{q_{\mathrm{P}}} \cdot \dfrac{l_{\mathrm{P}}}{r}$ |
| Electrostatic energy | $U = \dfrac{q_1 q_2}{4\pi\varepsilon_0 r}$ | $\dfrac{U}{E_{\mathrm{P}}} = \dfrac{q_1}{q_{\mathrm{P}}} \cdot \dfrac{q_2}{q_{\mathrm{P}}} \cdot \dfrac{l_{\mathrm{P}}}{r}$ |
| Ohm’s law | $V = I R$ | $\dfrac{V}{V_{\mathrm{P}}} = \dfrac{I}{I_{\mathrm{P}}} \cdot \dfrac{R}{Z_{\mathrm{P}}}$ |
| Lorentz force | $\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$ | $\dfrac{F}{F_{\mathrm{P}}} = \dfrac{q}{q_{\mathrm{P}}}\left(\dfrac{E}{E^{\mathrm{field}}_{\mathrm{P}}} + \dfrac{v}{c} \cdot \dfrac{B}{B_{\mathrm{P}}}\right)$ |
Quantum physics
| Quantity | Traditional | Universal form |
|---|---|---|
| Photon energy | $E = h\nu = \dfrac{hc}{\lambda}$ | $\dfrac{E}{E_{\mathrm{P}}} = \dfrac{l_{\mathrm{P}}}{\bar{\lambda}}$ |
| de Broglie wavelength | $\lambda = \dfrac{h}{p}$ | $\dfrac{\bar{\lambda}}{l_{\mathrm{P}}} = \dfrac{m_{\mathrm{P}} c}{p}$ |
| Compton wavelength | $\bar{\lambda}_{\mathrm{C}} = \dfrac{\hbar}{m c}$ | $\dfrac{\bar{\lambda}_{\mathrm{C}}}{l_{\mathrm{P}}} = \dfrac{m_{\mathrm{P}}}{m}$ |
| Uncertainty principle | $\Delta x \cdot \Delta p \ge \dfrac{\hbar}{2}$ | $\dfrac{\Delta x}{l_{\mathrm{P}}} \cdot \dfrac{\Delta p}{m_{\mathrm{P}} c} \ge \tfrac{1}{2}$ |
| Hydrogen energy levels | $E_n = -\dfrac{\alpha^2 m_e c^2}{2 n^2}$ | $\dfrac{E_n}{E_{\mathrm{P}}} = -\dfrac{\alpha^2}{2 n^2} \cdot \dfrac{m_e}{m_{\mathrm{P}}}$ |
| Black-body peak (Wien) | $h\nu_{\mathrm{peak}} \approx 2.82\, k_{\mathrm{B}} T$ | $\dfrac{h\nu_{\mathrm{peak}}}{E_{\mathrm{P}}} \approx 2.82 \cdot \dfrac{k_{\mathrm{B}} T}{E_{\mathrm{P}}}$ |
| Stefan–Boltzmann | $j = \sigma T^4$ | $\dfrac{j}{P_{\mathrm{P}}/l_{\mathrm{P}}^2} = \dfrac{\pi^2}{60} \left(\dfrac{k_{\mathrm{B}} T}{E_{\mathrm{P}}}\right)^{\!4}$ |
Recurring structural patterns
Several patterns recur across the tables above.
The side-by-side comparison makes the two observations compatible with each other. The traditional form is efficient for calculation in a chosen unit system; the universal form makes the structural content — which ratios scale which way — visible at a glance.