Every formula in physics can be written so that it reads the same way: a Planck-scale quantity, multiplied by one or more dimensionless ratios that scale it to the system being measured. This page introduces that two-part structure — the organizing idea behind the original research on this site — and works one example all the way to a number you can check.
The two-part pattern
The universal constants $\hbar$, $G$, and $c$ carry Planck-scale quantities inside their unit dimensions: $\hbar = l_{\mathrm{P}}\, m_{\mathrm{P}}\, c$, $G = (l_{\mathrm{P}}/m_{\mathrm{P}})\,c^2$, and $c = l_{\mathrm{P}}/t_{\mathrm{P}}$. (See What are Planck units? for the full account.) When the constants in any physical formula are written this way, the formula reorganizes itself into a consistent shape:
The Planck-scale quantity — a Planck momentum, a Planck energy, a Planck acceleration — sets the natural scale of the phenomenon. Each dimensionless ratio then does two things at once: it converts a measured value from arbitrary human units into an invariant number, the same in every unit system; and it represents a physical property of the system — a wavelength, a mass, a velocity, a distance — compared against nature’s own scale.
Three formulas, side by side
The pattern is easiest to see across domains. Here are a quantum formula, a classical formula, and a gravitational formula in conventional and universal form:
| Quantity | Conventional form | Universal form | Planck scale |
|---|---|---|---|
| Momentum | $p = \dfrac{\hbar}{\bar{\lambda}}$ | $p = \dfrac{l_{\mathrm{P}}}{\bar{\lambda}} \times m_{\mathrm{P}}c$ | $m_{\mathrm{P}}c = 6.52$ kg·m/s |
| Kinetic energy | $E_K = \tfrac{1}{2}m_0v^2$ | $E_K = \tfrac{1}{2} \times \dfrac{l_{\mathrm{P}}}{\bar{\lambda}_{dB}} \times \dfrac{v}{c} \times E_{\mathrm{P}}$ | $E_{\mathrm{P}} = 1.96 \times 10^{9}$ J |
| Gravitational acceleration | $g = \dfrac{GM}{r^2}$ | $g = \dfrac{M}{m_{\mathrm{P}}} \times \left(\dfrac{l_{\mathrm{P}}}{r}\right)^{2} \times a_{\mathrm{P}}$ | $a_{\mathrm{P}} = 5.56 \times 10^{51}$ m/s² |
Three different domains — quantum mechanics, classical mechanics, gravitation — and one shape. In each case the conventional constants have dissolved into a Planck-scale quantity and ratios of measured properties to Planck units. The ratios that appear — $l_{\mathrm{P}}/\bar{\lambda}$ for a wavelength, $M/m_{\mathrm{P}}$ for a mass, $v/c$ for a velocity, $l_{\mathrm{P}}/r$ for a distance — recur across formulas with the same physical meaning every time. These are exact algebraic rearrangements: no approximation is made, and every prediction is identical to the standard form.
A worked example: Earth’s surface gravity
To see that the universal form computes real numbers, take the gravitational acceleration at the Earth’s surface. Two ratios are needed: the Earth’s mass against the Planck mass, and the Planck length against the Earth’s radius.
The familiar $9.8$ m/s² emerges from the Planck acceleration — a quantity fifty-one orders of magnitude larger — cut down by the Earth’s mass ratio and the square of its distance ratio. The same two kinds of ratio, with different values, give the surface gravity of the Moon, the Sun, or a neutron star. The formula is not a special trick for the Earth; it is Newton’s law, read in nature’s own units.
Why the same ratios keep appearing
The deeper observation is that the same dimensionless ratios appear across formulas that conventional notation treats as unrelated. The wavelength ratio $l_{\mathrm{P}}/\bar{\lambda}$ that sets a photon’s momentum also sets its energy and its mass-equivalent. The mass ratio $M/m_{\mathrm{P}}$ that sets the Earth’s surface gravity also sets its Schwarzschild radius. A single measured property of a system, expressed once as an invariant ratio, feeds many formulas — which is precisely what it means for physical quantities to be correlated rather than independent.
For the cleanest demonstration, see the cesium-133 photon — a single physical event whose wavelength, period, energy, and mass collapse to one invariant number and its reciprocal. And for the framework applied at full scale, the Stefan-Boltzmann decomposition computes the Sun’s luminosity from factors spanning 160 orders of magnitude.
Where to go next
- What are Planck units? — the invariant reference scale, and what decomposition does, that setting ħ = c = 1 does not.
- Many Formulas, One Structure — fifteen universal constants decomposed, with every cancellation shown.
- What is momentum? · What is energy? · Why is gravity so weak? — the explainer articles that put the structure to work.
- Published Papers — the peer-reviewed development of the framework.