Planck units are natural units of length, mass, time, and charge. They are the dimensional content of universal constants $c$, $\hbar$, $G$, $\varepsilon_0$, $\mu_0$, and $e$. For the physicist they function as the invariant reference scale of dimensional physics: the scale at which each universal constant equals unity, and the scale against which every physical quantity can be expressed as a dimensionless ratio. This page collects the definitions, clarifies what Planck units accomplish that setting $\hbar = c = 1$ does not, and presents the decomposition that organizes much of the content on this site.
The base Planck units
The four base Planck units are natural units of length, mass, time, and charge. Their values can be calculated from the measured universal constants: $l_{\mathrm{P}}$, $m_{\mathrm{P}}$, and $t_{\mathrm{P}}$ follow from $\hbar$, $c$, and $G$; $q_{\mathrm{P}}$ additionally requires the electromagnetic coupling ($\varepsilon_0$, or equivalently $\alpha$).
| Quantity | Symbol | Value formula | Value (SI) |
|---|---|---|---|
| Planck length | $l_{\mathrm{P}}$ | $\sqrt{\hbar G / c^3}$ | $1.616 \times 10^{-35}$ m |
| Planck mass | $m_{\mathrm{P}}$ | $\sqrt{\hbar c / G}$ | $2.176 \times 10^{-8}$ kg |
| Planck time | $t_{\mathrm{P}}$ | $\sqrt{\hbar G / c^5}$ | $5.391 \times 10^{-44}$ s |
| Planck charge | $q_{\mathrm{P}}$ | $\sqrt{4\pi\varepsilon_0 \hbar c}$ | $1.876 \times 10^{-18}$ C |
The three mechanical base Planck units are mutually consistent: $l_{\mathrm{P}} = c \, t_{\mathrm{P}}$ and $E_{\mathrm{P}} = m_{\mathrm{P}} c^2$. Given any one of $l_{\mathrm{P}}$, $m_{\mathrm{P}}$, or $t_{\mathrm{P}}$ together with $c$, the other two follow directly.
Derived Planck units (energy, momentum, force, voltage, current, impedance, and so on) follow from dimensional combinations of the base quantities. The Planck force $F_{\mathrm{P}} = c^4/G \approx 1.21 \times 10^{44}$ N is of particular note: it is the one derived Planck quantity whose value can be calculated from $c$ and $G$ alone, and it appears as the universal force scale in both Newtonian gravity and Coulomb’s law when these are written in Planck form. A complete listing of derived Planck units appears on the Planck Units Reference page.
What $\hbar = c = 1$ does — and does not — do
Setting $\hbar = c = 1$ is the convention most physicists mean by “natural units.” It is a computational convenience: the numerical values of $\hbar$ and $c$ are replaced by unity and formulas strip down to their dimensional essentials. But the convention does not escape arbitrary units — it trades one arbitrary unit system for another. Setting $c = 1$ effectively declares that meters equal seconds; setting $\hbar = 1$ with $c = 1$ in turn makes kilograms equal joules. These are unit identifications, not statements of physical content. What the system gains in compactness, it loses in the ability to say what a given factor in an equation physically represents: whether a symbol carries the role of a length, a mass, a time, or an energy becomes invisible once the compound constants are set to unity.
Decomposing each compound universal constant into its Planck-unit components is a different operation. Rather than suppressing dimensions, it identifies the Planck-scale quantities the compound constants contain. $\hbar$ can be expressed as $m_{\mathrm{P}} l_{\mathrm{P}} c$ — the Planck mass, Planck length, and speed of light combined; the constant of proportionality between a photon’s wavelength and its momentum is literally a Planck mass times a Planck length times the universal velocity. $G$ can be expressed as $l_{\mathrm{P}}^3 / (m_{\mathrm{P}} t_{\mathrm{P}}^2)$ — a compound scale built of Planck length, Planck mass, and Planck time. These decompositions preserve every dimension and expose the Planck-scale quantities each compound constant contains.
Unit invariance: the ratio is geometry
A feature of Planck units that rewards attention: the ratio of any physical quantity to its corresponding Planck unit is dimensionless and unit-system independent. The numerical value of $c$ depends on what a meter is. The numerical value of the ratio $l_{\mathrm{P}}/\lambda$ does not depend on any choice of metric convention — it returns the same number in SI, CGS, Planck units, or any other consistent system.
This invariance is the structural basis for treating such ratios as geometric. A dimensionless number tied to the Planck scale carries information that a dimensioned quantity in arbitrary units does not: its value is a property of the physics itself, not of the measurement convention. The fine-structure constant is the canonical example — it can be expressed as $\alpha = (e/q_{\mathrm{P}})^2$, the squared ratio of the elementary charge to the Planck charge, and its value $1/137.036$ is the same in every consistent unit system.
Universal constants as Planck composites
Each universal constant can be expressed systematically as a product of Planck-unit quantities and dimensionless numbers. The following table collects the standard cases:
| Constant | Planck decomposition | Dimensionless factor |
|---|---|---|
| Speed of light | $c = l_{\mathrm{P}} / t_{\mathrm{P}}$ | $1$ |
| Reduced Planck constant | $\hbar = m_{\mathrm{P}} l_{\mathrm{P}}^2 / t_{\mathrm{P}} = m_{\mathrm{P}} l_{\mathrm{P}} c$ | $1$ |
| Gravitational constant | $G = l_{\mathrm{P}}^3 / (m_{\mathrm{P}} t_{\mathrm{P}}^2) = (l_{\mathrm{P}}/m_{\mathrm{P}}) c^2$ | $1$ |
| Permittivity | $\varepsilon_0 = t_{\mathrm{P}}^2 q_{\mathrm{P}}^2 / (4\pi \, l_{\mathrm{P}}^3 m_{\mathrm{P}})$ | $1/(4\pi)$ |
| Permeability | $\mu_0 = 4\pi \, m_{\mathrm{P}} l_{\mathrm{P}} / q_{\mathrm{P}}^2$ | $4\pi$ |
| Elementary charge | $e = \sqrt{\alpha} \cdot q_{\mathrm{P}}$ | $\sqrt{\alpha}$ |
The structural observation here is that the universal constants of physics can be expressed in terms of four dimensional scales — the base Planck units of length, mass, time, and charge — together with a single dimensionless number, the fine-structure constant $\alpha$. The compound constants $c$, $\hbar$, and $G$ each carry a coefficient of exactly one: each is a pure product of Planck-scale quantities, with nothing else along for the ride. The electromagnetic constants $\varepsilon_0$ and $\mu_0$ carry factors of $4\pi$ — a geometric convention fixed by the choice of rationalized units — together with $q_{\mathrm{P}}^2$. The elementary charge $e$ is the sole fundamental quantity whose Planck decomposition introduces a number, $\sqrt{\alpha}$, not determined by the system itself.
The decomposition of physical formulas
Replacing each constant in a physical formula by its Planck composite has a systematic effect: the formula separates into a Planck-scale quantity and a set of dimensionless ratios. The Planck-scale quantity is the value the formula would yield at the Planck reference; the dimensionless ratios scale that value to the regime where the measurement actually takes place.
Consider photon energy. The conventional expression is
Substituting $h = 2\pi \, m_{\mathrm{P}} l_{\mathrm{P}} c$ and $c = l_{\mathrm{P}}/t_{\mathrm{P}}$, and collecting the Planck factors:
The formula separates cleanly: $E_{\mathrm{P}}$ is the energy a photon would carry if its wavelength equalled the Planck length; $l_{\mathrm{P}}/\lambda$ is the dimensionless ratio that scales this Planck-scale energy down to the photon’s actual wavelength. The same structural pattern appears in every formula of physics: a Planck-scale quantity is multiplied by one or more dimensionless ratios that position the phenomenon relative to the Planck reference.
For a more thorough treatment — including gravitational, electromagnetic, and quantum formulas expressed in this form — see The Structure of Natural Formulas. A number of worked numerical examples illustrate the pattern concretely, beginning with the cesium hyperfine photon as a single physical event that produces four consistent Planck-scale magnitudes at once.
Relation to the published literature
The invariance and decomposition properties described above are developed at length in the peer-reviewed papers underlying this site. The foundational treatment appears in “Understanding the natural units and their hidden role in the laws of physics” (European Journal of Physics 45, 055802, 2024), which presents the Planck-unit formulation of standard equations and the invariant product $\lambda \cdot m = l_{\mathrm{P}} \cdot m_{\mathrm{P}}$ (verified to eleven significant digits across the three charged leptons). “The implicit structure of Planck’s constant” (European Journal of Applied Physics, 2022) gives the dedicated treatment of $\hbar$ as a Planck composite. “Natural Planck units and the structure of matter and radiation” extends the framework to momentum, kinetic energy, and the matter–radiation correspondence.
The predictions of the decomposed formulas are identical to those of the standard formalism. The decomposition is an organizing choice, not a revised physics. Its value, when there is value, lies in what it makes visible: the common Planck-scale structure shared by formulas that the conventional notation keeps apart, and the dimensionless quantities that recur across domains of physics with the same interpretation.
- Planck Units — Complete Reference: the full table of base and derived Planck quantities, with values, formulas, and cross-calculation relations.
- The Structure of Natural Formulas: the decomposition approach applied across mechanics, gravity, electromagnetism, and quantum formulas.
- Published papers: peer-reviewed development of the framework in European Journal of Physics, European Journal of Applied Physics, and International Journal of Quantum Foundations.