The Planck length (symbol ℓP) is the natural unit of length defined by the dimensional combination of the reduced Planck constant and two other universal constants, the Speed of light in vacuum c and the Gravitational constant G: ℓP = √(ħG/c³). With the 2022 CODATA recommended values curated by NIST, ℓP = 1.616255(18) × 10⁻³⁵ m, corresponding to a relative standard uncertainty of 1.1 × 10⁻⁵. In the same adjustment, c and h (and therefore ħ) are exact by SI definition, while G remains experimentally determined with significantly larger relative uncertainty than most other constants. According to the NIST constants database, G = 6.67430(15) × 10⁻¹¹ m³·kg⁻¹·s⁻² (2022 CODATA). These values fix the numerical value and uncertainty of ℓP through the defining formula. NIST: Planck length;
NIST: speed of light;
NIST: Planck constant;
NIST: Newtonian constant of gravitation;
NIST constants site/version history.
Origins and definition
- –The Planck system of natural units was introduced by Max Planck in 1899 by combining fundamental constants to define units of length, mass, time, and temperature independent of anthropogenic artifacts. Modern expositions trace the historical and conceptual basis of the “Planck scale” to the unique role of c, ħ, and G in demarcating where relativistic, quantum, and gravitational effects coincide.
Stanford Encyclopedia of Philosophy: Quantum Gravity;
Britannica: physical constant.
- –By definition, the Planck time tP satisfies tP = √(ħG/c⁵) and ℓP = c tP. Using CODATA 2022, tP = 5.391247(60) × 10⁻⁴⁴ s.
NIST: Planck time;
Numerical value and constants infrastructure
- –Since the 2019 SI redefinition, the Planck constant h and the speed of light c have exact values, which propagate into derived constants. NIST notes h = 6.62607015 × 10⁻³⁴ J·s (exact) and c = 299 792 458 m·s⁻¹ (exact), while ħ = h/2π inherits exactness. The residual uncertainty in ℓP thus arises entirely from the experimental uncertainty of G.
Rinaldi et al., 2022;
Role in black hole thermodynamics and quantum geometry
- –In semiclassical gravity, the Bekenstein–Hawking entropy of a Black hole is S = kB A / (4ℓP²), making the Planck area ℓP² the fundamental unit in which horizon entropy is counted. Authoritative reviews present this formula and its derivations from Hawking radiation and semiclassical action methods.
Living Reviews in Relativity: Wald (2001).
- –Candidate quantum-gravity frameworks often exhibit discreteness at the Planck scale. In Loop quantum gravity, area and volume operators have discrete spectra with eigenvalues proportional to ℓP² (up to the Immirzi parameter), indicating quantization of spatial geometry.
Conceptual significance in quantum gravity
- –The Planck scale is commonly identified as the regime where familiar notions of spacetime may fail, motivating the search for a theory of Quantum gravity. Philosophical and scientific surveys emphasize that this identification is largely dimensional and heuristic but has guided theory building for decades.
- –Several theoretical approaches incorporate a minimal length on the order of ℓP. Generalized uncertainty principle (GUP) models modify the Heisenberg relation to imply a nonzero minimum position uncertainty, often tied to ℓP. Representative treatments include analyses connecting deformed algebras with minimal length.
Maggiore, Phys. Rev. D 49, 5182 (1994); overview discussions in the literature frequently denote the minimal length with the Planck scale.
Generalized uncertainty principle (overview).
- –Doubly special relativity proposes two observer‑independent scales—a velocity c and an invariant length/energy—so that the Planck length or energy remains invariant under modified Lorentz transformations. Foundational expositions motivate this to preserve relativity principles in the presence of a fundamental scale.
Related Planck‑scale quantities
- –The Planck mass mP and energy mPc² set characteristic energy scales, mPc² ≈ 1.220890(14) × 10¹⁹ GeV in CODATA 2022. These, together with ℓP and tP, delineate the “Planck scale” frequently invoked in early‑universe cosmology and high‑energy gravity.
Measurement and observability
- –No experimental method currently probes distances comparable to ℓP, and SEP surveys note that empirical tests at this scale are beyond present capabilities; instead, the scale functions as a theoretical marker indicating where quantum and gravitational effects intertwine.
Terminology and system context
- –The Planck length belongs to the system of Planck units, in which c = ħ = G = kB = 1 by convention. Within SI, the exact definitions of h and c since 2019, combined with best‑available G, ensure that ℓP remains a derived constant with an uncertainty set by G.