Planck Scales

Planck scales are combinations of $G$, $c$, $\hbar$, and $k_{\mathrm B}$ with dimensions set by gravity, relativity, and quantum mechanics. This record uses the values and status collected under Fundamental Constants and the dimensional conventions summarized in Units and Dimensional Conventions. The defining SI constants $c$, $h$, and $k_{\mathrm B}$ are exact; $\hbar=h/(2\pi)$ is therefore derived-exact, while every standard uncertainty below comes from the measured value of $G$ BIPM 2026, §2.2 and Table 1, printed pp. 123–124, PDF.

Planck combinations and values

The five commonly tabulated scales are

$$\begin{aligned} \ell_{\mathrm P}&=\sqrt{\frac{\hbar G}{c^3}}, &t_{\mathrm P}&=\frac{\ell_{\mathrm P}}{c} =\sqrt{\frac{\hbar G}{c^5}},\\ m_{\mathrm P}&=\sqrt{\frac{\hbar c}{G}}, &E_{\mathrm P}&=m_{\mathrm P}c^2 =\sqrt{\frac{\hbar c^5}{G}},\\ T_{\mathrm P}&=\frac{E_{\mathrm P}}{k_{\mathrm B}}. \end{aligned}$$

The 2022 CODATA adjustment gives Mohr et al. 2025, Table XXXIII, p. 025002-44, NIST-hosted PDF:

Quantity	CODATA value	Relative standard uncertainty
Planck length $\ell_{\mathrm P}$	$1.616\,255(18)\times10^{-35}\ \mathrm{m}$	$1.1\times10^{-5}$
Planck time $t_{\mathrm P}$	$5.391\,247(60)\times10^{-44}\ \mathrm{s}$	$1.1\times10^{-5}$
Planck mass $m_{\mathrm P}$	$2.176\,434(24)\times10^{-8}\ \mathrm{kg}$	$1.1\times10^{-5}$
Planck energy $E_{\mathrm P}$	$1.220\,890(14)\times10^{19}\ \mathrm{GeV}$	$1.1\times10^{-5}$
Planck temperature $T_{\mathrm P}$	$1.416\,784(16)\times10^{32}\ \mathrm{K}$	$1.1\times10^{-5}$

Useful powers and densities derived from the same constants are

$$\begin{aligned} A_{\mathrm P}&=\ell_{\mathrm P}^2=\frac{\hbar G}{c^3} \simeq2.6123\times10^{-70}\ \mathrm{m^2},\\ \mathcal R_{\mathrm P}&=\ell_{\mathrm P}^{-2}=\frac{c^3}{\hbar G} \simeq3.8281\times10^{69}\ \mathrm{m^{-2}},\\ \rho_{\mathrm P}&=\frac{m_{\mathrm P}}{\ell_{\mathrm P}^3} =\frac{c^5}{\hbar G^2} \simeq5.1548\times10^{96}\ \mathrm{kg\,m^{-3}},\\ \varepsilon_{\mathrm P}&=\frac{E_{\mathrm P}}{\ell_{\mathrm P}^3} =\rho_{\mathrm P}c^2 =\frac{c^7}{\hbar G^2} \simeq4.6329\times10^{113}\ \mathrm{J\,m^{-3}}. \end{aligned}$$

Thus $\rho_{\mathrm P}$ is a mass density, whereas $\varepsilon_{\mathrm P}$ is an energy density; they differ by $c^2$ in SI units. These definitions divide by the scale volume $\ell_{\mathrm P}^3$, not by the volume of a sphere. Factors such as $4\pi/3$ therefore describe a different normalization, not a different measured constant or a physical object of Planck density.

Uncertainty and Planck-mass conventions

For any derived quantity $Q\propto G^p$, first-order propagation gives

$$u_{\mathrm r}(Q)=\lvert p\rvert\,u_{\mathrm r}(G), \qquad u_{\mathrm r}(G)=2.2\times10^{-5}.$$

Consequently $\ell_{\mathrm P}$, $t_{\mathrm P}$, $m_{\mathrm P}$, $E_{\mathrm P}$, and $T_{\mathrm P}$ have relative standard uncertainty $u_{\mathrm r}(G)/2$; $A_{\mathrm P}$ and $\mathcal R_{\mathrm P}$ have $u_{\mathrm r}(G)$; and $\rho_{\mathrm P}$ and $\varepsilon_{\mathrm P}$ have $2u_{\mathrm r}(G)\simeq4.5\times10^{-5}$. This rule propagates uncertainty, not a possible future shift in the recommended central value of $G$.

The tabulated uncertainties are not independent error bars. Because the same measured $G$ supplies their only input uncertainty, quantities with equal-sign powers of $G$ are correlated and those with opposite-sign powers are anticorrelated at first order. Exact defining ratios and products must be evaluated with that shared dependence; combining their quoted uncertainties in quadrature would double-count the uncertainty in $G$.

The mass $m_{\mathrm P}=\sqrt{\hbar c/G}$ used above is the unreduced Planck mass. Field-theory and gravity references often instead use

$$\bar M_{\mathrm P}=\sqrt{\frac{\hbar c}{8\pi G}} =\frac{m_{\mathrm P}}{\sqrt{8\pi}}.$$

The factor $\sqrt{8\pi}$ is conventional, not an uncertainty; a formula must state which mass it uses. In $c=\hbar=1$ units, $G=m_{\mathrm P}^{-2}$ signals the unreduced convention, while

$$\frac{1}{16\pi G}=\frac{\bar M_{\mathrm P}^2}{2}$$

is the common reduced-mass normalization of the Einstein–Hilbert coefficient. Neither convention changes $G$. It only moves factors of $8\pi$ between the mass scale and gravitational couplings, so convention conversion must precede any numerical comparison.

Dimensional and natural-unit checks

With $[G]=L^3M^{-1}T^{-2}$, $[\hbar]=ML^2T^{-1}$, $[c]=LT^{-1}$, and $[k_{\mathrm B}]=ML^2T^{-2}\Theta^{-1}$, the definitions above have dimensions $L$, $T$, $M$, $ML^2T^{-2}$, and $\Theta$, respectively. The area, inverse-area, mass-density, and energy-density formulas then have dimensions $L^2$, $L^{-2}$, $ML^{-3}$, and $ML^{-1}T^{-2}$.

Several exact identities provide quick transcription and unit checks:

$$\frac{\ell_{\mathrm P}}{t_{\mathrm P}}=c, \qquad \frac{E_{\mathrm P}}{m_{\mathrm P}}=c^2, \qquad \frac{E_{\mathrm P}}{T_{\mathrm P}}=k_{\mathrm B}, \qquad A_{\mathrm P}\mathcal R_{\mathrm P}=1, \qquad \frac{\varepsilon_{\mathrm P}}{\rho_{\mathrm P}}=c^2.$$

A table or code path that violates one of these relations has mixed conventions, units, or rounded inputs.

In the site default $c=\hbar=k_{\mathrm B}=1$ with $G$ explicit, the same definitions reduce to

$$\ell_{\mathrm P}=t_{\mathrm P}=\sqrt G, \quad m_{\mathrm P}=E_{\mathrm P}=T_{\mathrm P}=G^{-1/2}, \quad A_{\mathrm P}=G, \quad \mathcal R_{\mathrm P}=G^{-1}, \quad \rho_{\mathrm P}=\varepsilon_{\mathrm P}=G^{-2}.$$

Equivalently, $G=m_{\mathrm P}^{-2}=(8\pi\bar M_{\mathrm P}^2)^{-1}$. This is the standard dimensional simplification supplied by natural units, not a change in physics Tong 2006, §0.1, Eqs. (0.1)–(0.2), PDF.

Curvature and interpretation boundary

$\mathcal R_{\mathrm P}=\ell_{\mathrm P}^{-2}$ is an inverse-area scale, not the value of a particular curvature scalar. A physical comparison must name an invariant. For Schwarzschild, the Compactness and Curvature record gives

$$K=R_{abcd}R^{abcd}=\frac{48G^2M^2}{c^4r^6}, \qquad \left.\ell_{\mathrm P}^2\sqrt K\right|_{r=2GM/c^2} =\frac{\sqrt3}{2}\left(\frac{m_{\mathrm P}}{M}\right)^2.$$

The dimensionless horizon curvature therefore becomes small for $M\gg m_{\mathrm P}$ even though the horizon compactness remains of order unity. Outside cases where $K$ is a suitable diagnostic, use a named invariant or frame-defined tidal observable with its own absolute-value or norm convention; no single scalar diagnoses every geometry. Even a dimensionless curvature ratio near unity is only a warning that a curvature expansion may need re-examination. The conclusion also depends on the operators retained, their coefficients, derivative scales, quantum state, and observable.

Finally, a dimensionful Planck value by itself establishes neither a discrete spacetime nor a minimum observable length. It also does not prove singularity resolution, identify new degrees of freedom, or specify quantum-gravity dynamics. Such a claim requires a dimensionless control parameter, a stated observable and regime. The surrounding Quantum-Gravity Scales, Regimes, and Evidence chapter supplies that framework; its discussion of Planck Units and Invariant Expansion Parameters develops the dynamical boundary in detail.

References

• Bureau International des Poids et Mesures. The International System of Units (SI). 9th ed., version 4.01. Sèvres: BIPM, 2026. DOI. Official PDF.
• Mohr, Peter J., David B. Newell, Barry N. Taylor, and Eite Tiesinga. “CODATA Recommended Values of the Fundamental Physical Constants: 2022.” Reviews of Modern Physics 97 (2025): 025002. DOI. NIST-hosted PDF.
• Tong, David. Quantum Field Theory. Cambridge Part III Mathematical Tripos lecture notes, 2006. PDF.