Natural Units and Planck Units: When c = ℏ = 1

Why Set Constants to 1?

The equations of physics contain several constants that appear everywhere: the speed of light c, the reduced Planck constant ℏ (h-bar), the gravitational constant G, the Boltzmann constant k_B, and the elementary charge e. In SI units, these have specific numerical values with units attached:

• c = 299,792,458 m/s
• ℏ = 1.054571817 × 10⁻³⁴ J·s
• G = 6.674 × 10⁻¹¹ N·m²/kg²

In many theoretical physics calculations, carrying these constants through every equation adds bookkeeping without adding insight. The equations become harder to read, and the physical relationships they express are obscured by numerical clutter.

Natural unit systems solve this by choosing units in which selected constants have the value exactly 1. When c = 1, velocities are expressed as fractions of the speed of light, and the factor c² in E = mc² becomes simply E = m — energy and mass are the same quantity measured in the same units. When ℏ = 1, angular momenta are dimensionless numbers.

The constants do not disappear from physics — they are absorbed into the unit scale. Converting a result in natural units back to SI requires multiplying by the appropriate powers of the constants. The choice of which constants to set to 1 defines which system of natural units you are using.

Planck Units (Length, Time, Mass, Temperature, Charge)

Planck units were first proposed by Max Planck in 1899, the year before quantum mechanics formally began. Planck observed that the constants G, ℏ, and c could be combined to form unique combinations with the dimensions of length, time, mass, and temperature — without any human-chosen reference.

The five Planck base units, derived from c, ℏ, G, and k_B:

In Planck units, all five of these constants equal 1: G = c = ℏ = k_B = 1 (and 4πε₀ = 1 in the rationalized version). Any physical quantity in Planck units is expressed as a pure dimensionless number.

Physical significance: The Planck length (≈ 1.6 × 10⁻³⁵ m) is approximately 20 orders of magnitude smaller than a proton (≈ 0.84 × 10⁻¹⁵ m). Most theories of quantum gravity predict that spacetime itself has a granular structure at or near the Planck scale — below which the classical notion of a smooth manifold breaks down. The Planck time (≈ 5.4 × 10⁻⁴⁴ s) is the time it takes light to cross one Planck length. The Planck temperature (≈ 1.4 × 10³² K) is believed to be the maximum temperature at which current physics remains valid.

The Planck mass (≈ 2.18 × 10⁻⁸ kg ≈ 21.8 µg) is the only Planck unit at a humanly imaginable scale — it is roughly the mass of a flea egg.

Particle Physics Natural Units (c = ℏ = 1)

Particle physics uses a natural unit system defined by setting c = ℏ = 1. In this system, all quantities are expressed in terms of a single energy unit, typically the electron-volt (eV) or gigaelectron-volt (GeV).

With c = 1: energy E = mc² becomes E = m, so mass is measured in the same units as energy. The proton mass is 938.272 MeV/c² in SI; in particle physics natural units it is simply 938.272 MeV.

With ℏ = 1: length has units of inverse energy (ℏc / E has units of length in SI, and with ℏ = c = 1, length = 1/E). The proton's charge radius (~0.84 fm) in particle physics units is ~0.84 fm × (197.3 MeV·fm) = ~166 MeV⁻¹.

The factor ℏc = 197.3269804 MeV·fm is the standard conversion factor between the SI and particle physics unit systems. It appears constantly in cross-section calculations:

1 femtometer = 1 fm = 10⁻¹⁵ m = 1/(197.3 MeV) in natural units

Cross-sections (areas) in particle physics are measured in barns: 1 barn = 10⁻²⁸ m². In natural units: 1 barn = 1/(ℏc)² × 10⁻²⁸ m² = 2568 GeV⁻².

The elementary charge in particle physics natural units is characterized by the fine-structure constant: α = e²/(4πε₀ℏc) ≈ 1/137.036 — a dimensionless number that controls the strength of electromagnetic interactions.

Geometrized Units in General Relativity (c = G = 1)

General relativity (GR) uses a different natural unit convention: set c = 1 and G = 1. This is called the geometrized unit system (or geometric units). With these choices, mass, length, and time all have the same dimension.

The conversion factors are: - 1 meter of length = 1 / (c²/G) kg of mass = 1 m × c²/G ≈ 1.347 × 10²⁷ kg/m (inverted) - Equivalently: 1 kg of mass = G/c² meters ≈ 7.425 × 10⁻²⁸ m

The Schwarzschild radius of a black hole is r_s = 2GM/c². In geometrized units (G = c = 1): r_s = 2M. The mass of the Sun is M_☉ = 1.989 × 10³⁰ kg. In geometrized units: M_☉ = G × 1.989 × 10³⁰ / c² ≈ 1,477 m (the Sun's geometrized mass is 1.477 km). The Sun's Schwarzschild radius is therefore 2 × 1,477 m ≈ 2.954 km — the radius at which the Sun would become a black hole.

GR equations take their simplest form in geometrized units. The Einstein field equations: G_µν = 8π T_µν (with G = c = 1) rather than G_µν = 8πG/c⁴ × T_µν in SI.

Stoney Units vs Planck Units

George Johnstone Stoney proposed a natural unit system in 1874, predating Planck's by 25 years, based on the constants c, G, and the elementary charge e (rather than ℏ):

Stoney units differ from Planck units by a factor of √α (the square root of the fine-structure constant, ≈ 1/11.7). In Stoney units, the elementary charge e = 1; in Planck units, e = √α ≈ 0.0854.

In modern usage, Planck units are more common in quantum gravity because they incorporate ℏ (the quantum of action) rather than e (the quantum of charge). Stoney units appear occasionally in historical discussions and in alternative theories where the electron charge plays a more fundamental role.

Converting Between Natural and SI Units

To convert a result from Planck units to SI, multiply by the appropriate combination of Planck scales:

Example: The Hawking temperature of a black hole with Planck mass M (in Planck units) is T_H = 1/(8πM). For M = 10¹⁰ m_P:

T_H (Planck) = 1/(8π × 10¹⁰) ≈ 3.98 × 10⁻¹²

T_H (SI) = 3.98 × 10⁻¹² × T_P = 3.98 × 10⁻¹² × 1.417 × 10³² K ≈ 5.64 × 10²⁰ K

For particle physics: to go from natural units (GeV) to SI (meters), multiply by ℏc / (1 GeV) = (6.582 × 10⁻²⁵ GeV·s × 2.998 × 10⁸ m/s) / (1 GeV) = 1.973 × 10⁻¹⁶ m.

Why It Matters: Simplifying Physics Equations

Beyond aesthetics, natural units have a functional benefit: they expose when quantities that look different in SI are actually the same physical thing.

In special relativity: energy and mass are unified (E = mc² → E = m). Time and distance are unified (a light-year in time is numerically the same as a light-year in distance). Momentum and energy have the same units.

In quantum mechanics: energy and frequency are unified (E = ℏω → E = ω when ℏ = 1). Inverse length and momentum have the same units (de Broglie: p = ℏ/λ → p = 1/λ).

In thermodynamics: temperature and energy are unified (E ~ k_B T → E ~ T when k_B = 1). Entropy becomes dimensionless.

These identifications are not just notational shortcuts — they reflect genuine physical unifications. Setting c = 1 makes the Lorentz transformation a rotation in a mathematical sense; setting ℏ = 1 makes phase space volume naturally dimensionless; setting k_B = 1 makes entropy count the number of microstates on an absolute scale.

For everyday unit conversions involving length, time, and weight, SI units remain the practical standard. Natural units are the domain of theoretical work where the fundamental relationships between physical quantities are the focus.