Planck Length Calculator
Compute the Planck length Lp=√(ħG/c³) and related Planck units (time, mass, energy, temperature). Lp=1.616×10⁻³⁵ m, the quantum-gravity scale.
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At a glance:Planck units (Max Planck, 1899) are natural units constructed from universal constants ħ (reduced Planck constant), G (Newton's gravitational constant), c (speed of light) and k_B (Boltzmann constant). The Planck length Lp=√(ħG/c³)=1.616255×10⁻³⁵ m is the scale where quantum mechanics and gravity are both relevant — a candidate for the minimal length (spacetime 'grain'). Physical meaning: Lp is where the Compton wavelength of a mass equals its Schwarzschild radius, so quantum and gravitational effects are comparable. Below Lp, classical spacetime may break down (a quantum-gravity regime). History: Planck introduced them in 1899 as 'natural units' independent of human artifacts; modern relevance grew with quantum gravity (1960s+). Classic values: Lp=1.616×10⁻³⁵ m (10⁻²⁰ of a proton), Tp=5.39×10⁻⁴⁴ s, Mp=2.18×10⁻⁸ kg (~22 μg, Planck-mass black hole radius = Lp), Ep=1.96×10⁹ J (~10¹⁹ GeV, the GUT/quantum-gravity energy), Tp=1.42×10³² K (upper temperature limit). Applications: (1) quantum gravity / string theory (string length ~Lp); (2) loop quantum gravity (area quantization in units of Lp²); (3) black-hole entropy S=kA/(4Lp²); (4) cosmology (Planck epoch, inflation); (5) dimensional analysis (setting ħ=G=c=1).
Formula
Planck length: Lp = √(ħG/c³) = 1.616×10⁻³⁵ m
Planck time: Tp = √(ħG/c⁵) = 5.391×10⁻⁴⁴ s
Planck mass: Mp = √(ħc/G) = 2.176×10⁻⁸ kg
Planck energy: Ep = Mp·c² = 1.956×10⁹ J (≈1.22×10¹⁹ GeV)
Planck temperature: Tp = ħc/(k_B·Lp) = 1.417×10³² K
$$L_P = \sqrt{\frac{\hbar G}{c^3}}, \quad t_P = \sqrt{\frac{\hbar G}{c^5}}, \quad m_P = \sqrt{\frac{\hbar c}{G}}, \quad E_P = m_P c^2, \quad T_P = \frac{\hbar c}{k_B L_P}$$How to Use
- The constants ħ, G, c are pre-filled with CODATA values.
- The tool computes Lp, Tp, Mp, Ep, Tp (temperature).
- Planck length Lp=1.616e-35 m is the quantum-gravity length scale.
Planck Units
| Unit | Formula | Value |
|---|---|---|
| Length Lp | √(ħG/c³) | 1.616×10⁻³⁵ m |
| Time Tp | √(ħG/c⁵) | 5.391×10⁻⁴⁴ s |
| Mass Mp | √(ħc/G) | 2.176×10⁻⁸ kg |
| Energy Ep | Mp·c² | 1.956×10⁹ J |
| Temperature Tp | ħc/(kB·Lp) | 1.417×10³² K |
Built from ħ, G, c (and kB). At the Planck scale quantum gravity dominates; below Lp classical spacetime likely breaks down. Planck mass is macroscopic (22 μg) but its black-hole radius is only Lp.
Case Studies
Black Hole Entropy and the Holographic Principle
Bekenstein-Hawking entropy S=k_B·A/(4Lp²), where A is the event-horizon area measured in Planck-area units (Lp²).
A solar-mass black hole has A/Lp² ~10⁷⁷ bits of information. This suggests information is stored on the surface, not the volume (holographic principle).
Planck length appears as the fundamental 'pixel' of spacetime area — a clue that geometry is emergent.
The Planck Epoch of the Early Universe
The first 10⁻⁴³ s (one Planck time) after the Big Bang, at temperature ~10³² K (Planck temperature), all forces were unified and quantum gravity dominated.
Our known physics (GR + Standard Model) breaks down before ~Tp; a quantum-gravity theory is needed to describe it.
Cosmological inflation began shortly after, diluting initial conditions — which is why we cannot directly observe the Planck epoch.
FAQ
What is so special about the Planck length?
Lp=√(ħG/c³) is the unique length scale formed from the three fundamental constants of quantum mechanics (ħ), gravity (G) and relativity (c). It is where the quantum Compton wavelength (ħ/Mc) of a mass equals its Schwarzschild radius (2GM/c²) — i.e. where quantum and gravitational effects are equally strong. It is a candidate minimal length: distances below Lp may be physically meaningless in a quantum theory of gravity. It is ~10⁻²⁰ of a proton radius.
Is the Planck length the smallest possible length?
Not proven, but widely expected. In many quantum-gravity approaches (string theory, loop quantum gravity, asymptotic safety) Lp acts as a minimal length or area quantum — spacetime may be discrete at that scale. However, it is not a 'hard wall' but a scale where classical geometry fails; some models allow sub-Planckian wavelengths in specific frames. It is a hint, not a theorem.
Why is the Planck mass macroscopic while length is tiny?
Mp=√(ħc/G)=2.18×10⁻⁸ kg (~22 micrograms) is large because gravity is extremely weak — you need a lot of mass for G to matter at quantum scales. Its Schwarzschild radius is 2GMp/c²=Lp, so a Planck-mass black hole has radius Lp. The mismatch (macroscopic mass, microscopic radius) reflects how weak gravity is compared to the other forces. This is why quantum gravity needs huge energies (Ep~10¹⁹ GeV) to probe Lp.
Can we measure the Planck length?
Directly, no — Lp is 10²⁰ times smaller than a proton, far beyond any accelerator (we reach ~10⁻²⁰ m, 15 orders short). Indirectly, we look for quantum-gravity signatures: Lorentz-invariance violation, cosmic-ray cutoff, gravitational-wave echoes, or CMB polarization. The Planck scale is the energy frontier of fundamental physics, motivating string theory and quantum-gravity research.
How does it appear in black-hole physics?
Black-hole entropy S=k_B·A/(4Lp²) counts horizon area in Planck-area units. A Planck-mass black hole has radius Lp and entropy 1 bit (natural unit). This ties information, area and Lp together — the holographic principle says the universe's information content is bounded by its surface area in Planck units. Lp is thus the 'pixel size' of spacetime.
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References
Content reviewed by the Calculatorism editorial team. Results are for reference only; please refer to the relevant authorities for the official figures.