Units and Dimensional Conventions

A gravity formula is dimensionally usable only after its unit system and mass normalization are known. The site normally suppresses $c$, $\hbar$, and $k_{\mathrm B}$ but keeps $G$ visible. This card records what that choice does, how a declared geometrized-unit calculation differs, and how to restore SI dimensions without confusing a physical mass with its gravitational length. The operational meaning of energy, temperature, and measurement scales is developed in Special Relativity and Physical Foundations.

The site’s default unit system

Unless a page declares otherwise,

$$c=\hbar=k_{\mathrm B}=1, \qquad G\ \text{explicit}.$$

Setting a constant to one identifies dimensions that the constant ordinarily converts; it does not make every quantity dimensionless. With $c=1$, time and length have the same dimension and mass and energy have the same dimension. With $\hbar=1$, energy has inverse-length dimension. With $k_{\mathrm B}=1$, a temperature denotes its associated energy scale. The underlying SI quantity, its numerical value, and its unit remain distinct, as emphasized by BIPM 2026, §§ 2.3.3–2.3.4, printed pp. 132–137, PDF.

In $d$ spacetime dimensions, the Einstein–Hilbert action and Einstein equation give the following length dimensions in this default system:

Quantity	Length dimension	Comment
$x$	$L$	coordinates measured as lengths
$\partial$, $M$, $E$, $T$, $\omega$	$L^{-1}$	$T$ is a thermal energy scale and $\omega$ is angular frequency
$S$, entropy	$L^0$	really $S/\hbar$ and entropy divided by $k_{\mathrm B}$
$G_d$	$L^{d-2}$	fixed by $G_dT_{ab}\sim R_{ab}$
$R_{abcd}$, $R_{ab}$, $R$, $\Lambda$	$L^{-2}$	curvature uses two derivatives of the metric
$T_{ab}$	$L^{-d}$	energy density in $d$-dimensional natural units

Thus, in four dimensions, $[G]=L^2$ and $[T_{ab}]=L^{-4}$ while the Einstein tensor has dimension $L^{-2}$. This is an immediate check on $G_{ab}+\Lambda g_{ab}=8\pi G T_{ab}$.

Some pages instead use geometrized units, $G=c=1$, for a sustained black-hole calculation. They must say so. In that system a physical mass is represented by a length, but the two must not share an undeclared symbol when dimensions are restored.

Electromagnetic rationalization is a separate convention

Setting constants to one is not the same operation as moving factors of $4\pi$ between an electromagnetic field, its sources, and its flux charge. The Physical Foundations Maxwell chapter uses rationalized Heaviside–Lorentz variables, while the exact Reissner–Nordström solution uses Gaussian variables. To describe the same physical solution, mark those variables by subscripts $\mathrm{R}$ and $\mathrm{G}$ and rescale the complete package:

$$\begin{aligned} A_{\rm R}&=\frac{A_{\rm G}}{\sqrt{4\pi}}, &F_{\rm R}&=\frac{F_{\rm G}}{\sqrt{4\pi}},\\ J_{\rm R}&=\sqrt{4\pi}\,J_{\rm G}, &e_{\rm R}&=\sqrt{4\pi}\,e_{\rm G}. \end{aligned}$$

Here $J$ is a source current and $e$ is a probe charge. The corresponding flux charges and horizon-to-infinity potentials obey

$$Q_{\rm G}=\frac{1}{4\pi}\int_{S^2}{*F_{\rm G}}, \qquad Q_{\rm R}=\int_{S^2}{*F_{\rm R}} =\sqrt{4\pi}\,Q_{\rm G}, \qquad \Phi_{\rm R}=\frac{\Phi_{\rm G}}{\sqrt{4\pi}}.$$

These are field redefinitions, not different black holes. The action density and stress tensor agree because

$$-\frac{1}{16\pi}F_{{\rm G}\,ab}F_{\rm G}^{ab} =-\frac14F_{{\rm R}\,ab}F_{\rm R}^{ab},$$ $$\frac{1}{4\pi} \left(F_{{\rm G}\,ac}F_{{\rm G}\,b}{}^c -\frac14g_{ab}F_{{\rm G}\,cd}F_{\rm G}^{cd}\right) =F_{{\rm R}\,ac}F_{{\rm R}\,b}{}^c -\frac14g_{ab}F_{{\rm R}\,cd}F_{\rm R}^{cd}.$$

Three short invariants catch a mixed convention:

$$q^2=GQ_{\rm G}^2=\frac{GQ_{\rm R}^2}{4\pi}, \qquad J_{\rm G}\mathbin{\cdot}A_{\rm G} =J_{\rm R}\mathbin{\cdot}A_{\rm R}, \qquad e_{\rm G}\Phi_{\rm G}=e_{\rm R}\Phi_{\rm R},$$

and the mechanical work term is unchanged:

$$\Phi_{\rm G}\,\delta Q_{\rm G} =\Phi_{\rm R}\,\delta Q_{\rm R}.$$

Thus a Gaussian metric term $GQ_{\rm G}^2/r^2$ and a rationalized term $GQ_{\rm R}^2/(4\pi r^2)$ are identical only when the potential, flux definition, source or probe charge, and Maxwell action are translated with them. The rationalized mechanics convention is used in the stationary Einstein–Maxwell first law.

Restoring a black-hole scale

Let $M_{\mathrm{SI}}$ be a physical mass in kilograms. Its natural length and time scales are

$$\ell_M=\frac{G M_{\mathrm{SI}}}{c^2}, \qquad t_M=\frac{G M_{\mathrm{SI}}}{c^3}.$$

If a dimensionless calculation uses $\hat r=r/\ell_M$, $\hat t=t/t_M$, and $\hat\omega=\omega_{\mathrm{SI}}t_M$, then

$$r_{\mathrm{SI}}=\ell_M\hat r, \qquad t_{\mathrm{SI}}=t_M\hat t, \qquad \omega_{\mathrm{SI}} =\frac{c^3}{G M_{\mathrm{SI}}}\hat\omega.$$

Here $\omega$ is angular frequency. A cyclic frequency is $f=\omega/(2\pi)$, so replacing one by the other without the factor $2\pi$ breaks the round trip. The same scale restores curvature invariants:

$$R_{\mathrm{SI}}=\frac{\hat R}{\ell_M^2}, \qquad K_{\mathrm{SI}}=\frac{\hat K}{\ell_M^4},$$

where $K=R_{abcd}R^{abcd}$. The units are inverse square metres and inverse fourth-power metres, respectively; neither expression converts a coordinate component into an invariant.

For the four-dimensional Schwarzschild solution, the horizon radius is

$$r_h=2\ell_M=\frac{2GM_{\mathrm{SI}}}{c^2}.$$

The factor $2$ is dynamical information, not something dimensional analysis can recover; the solution and its domain are developed in Deriving the Schwarzschild Metric and documented in Reall 2013, § 12, printed pp. 131–148, PDF.

Restoring quantum and thermal constants

For an angular-frequency scale and its corresponding energy and temperature,

$$E=\hbar\omega, \qquad T=\frac{E}{k_{\mathrm B}}.$$

This is a conversion between scales, not a claim that every physical system is in thermal equilibrium. A formula derived with $\hbar=k_{\mathrm B}=1$ is restored by asking which quantity carries energy dimension and then inserting the constants required by the target units. Tong gives the natural-unit dimension assignments in Tong 2006, § 0.1, Eqs. (0.1)–(0.2), PDF.

The two common Planck-mass conventions are

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

Consequently, when $c=\hbar=1$,

$$G=m_{\mathrm P}^{-2} =\frac{1}{8\pi\overline M_{\mathrm P}^{,2}}.$$

The bar, the word “reduced,” or an explicit defining equation is essential: dropping it silently moves a factor $8\pi$ between the gravitational coupling and every power-counting formula. The longer derivation and invariant expansion parameters are collected in Planck Units and Invariant Expansion Parameters.

Conversion checks and limitations

A reliable conversion passes three checks.

1. Dimensions close. Every term in a sum has the same target unit, and the result of an exponent or logarithm is dimensionless.
2. The map round-trips. Converting $\hat\omega$ to SI and multiplying by $t_M$ returns exactly $\hat\omega$; the same test applies to length and curvature.
3. Dimensionless physics agrees. Ratios such as $r/\ell_M$, the norm of a four-velocity, or $G M^2$ are unchanged after both representations are translated consistently.

Dimensional analysis cannot determine a sign, a factor $2$, $2\pi$, or $8\pi$, nor can it decide whether a source uses angular or cyclic frequency. It also cannot supply a current numerical value for $G$. Instrument-facing work should use SI and name the standards or data release; computational files should carry a machine-readable unit and convention contract. The applied workflow is given in Units, Conventions, and Data Contracts.

References

• Bureau International des Poids et Mesures. The International System of Units (SI). 9th ed., version 4.01, 2026. DOI; Official PDF.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes, 2013. Open PDF.
• Tong, David. Quantum Field Theory. Cambridge Part III lecture notes, 2006. Open PDF.