Metric and Units

Relativistic formulas are only meaningful after their conventions are fixed. We use the mostly-minus metric, keep $c$ and $\hbar$ visible in dimensional formulas, and treat a particle’s electric charge $q$ as a signed quantity. With these choices,

$$\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1), \qquad x^\mu=(ct,\mathbf x), \qquad p^\mu=(E/c,\mathbf p).$$

Natural units, $c=\hbar=1$, will be used only when explicitly announced. A formula from another source must first be translated into these conventions; changing a metric sign without changing all dependent definitions is not a valid translation.

This page assumes elementary special relativity and dimensional analysis.

Spacetime metric and index placement

Greek indices $\mu,\nu,\ldots$ run over $0,1,2,3$, while Latin spatial indices $i,j,k$ run over $1,2,3$. Repeated upper-lower index pairs are summed. The Minkowski inner product is

$$a\cdot b = \eta_{\mu\nu}a^\mu b^\nu = a^0b^0-\mathbf a\cdot\mathbf b.$$

The metric lowers an index and its inverse raises one:

$$a_\mu=\eta_{\mu\nu}a^\nu, \qquad a^\mu=\eta^{\mu\nu}a_\nu.$$

For $a^\mu=(a^0,\mathbf a)$ this gives

$$a_\mu=(a^0,-\mathbf a).$$

The minus sign on the spatial covariant components is easy to lose. It is also why $a_\mu b^\mu$ is not an ordinary Euclidean dot product of four displayed components.

The invariant interval is

$$ds^2 = \eta_{\mu\nu}\,dx^\mu dx^\nu = c^2dt^2-d\mathbf x^2.$$

Accordingly, a nonzero displacement is timelike, null, or spacelike when $ds^2$ is positive, zero, or negative. A source using the mostly-plus metric $\operatorname{diag}(-1,1,1,1)$ reverses these signs. The wave operator and Clifford algebra must then be written consistently with that choice.

Coordinates, derivatives, and the wave operator

Because $x^0=ct$, the covariant and contravariant derivatives are

$$\partial_\mu = \frac{\partial}{\partial x^\mu} = \left(\frac{1}{c}\frac{\partial}{\partial t},\nabla\right), \qquad \partial^\mu = \eta^{\mu\nu}\partial_\nu = \left(\frac{1}{c}\frac{\partial}{\partial t},-\nabla\right).$$

The d’Alembertian is therefore

$$\Box = \partial_\mu\partial^\mu = \frac{1}{c^2}\frac{\partial^2}{\partial t^2}-\nabla^2.$$

These definitions make the contraction $p_\mu x^\mu$ especially useful:

$$p\cdot x = Et-\mathbf p\cdot\mathbf x.$$

The default positive-energy plane wave is

$$e^{-ip\cdot x/\hbar} = \exp\!\left[-\frac{i}{\hbar} \left(Et-\mathbf p\cdot\mathbf x\right)\right].$$

Acting on this phase gives

$$i\hbar\partial_\mu e^{-ip\cdot x/\hbar} = p_\mu e^{-ip\cdot x/\hbar}.$$

This identity fixes the operator substitution used to pass from the energy-momentum relation to relativistic wave equations. Sources using $e^{+ip\cdot x}$ compensate with different Fourier or differential-operator signs.

For proper Lorentz transformations, $\det\Lambda=+1$, so the oriented four-volume element

$$d^4x=dx^0\,dx^1\,dx^2\,dx^3=c\,dt\,d^3x$$

is invariant. The equal-time spatial element $d^3x$ alone is not a Lorentz invariant four-volume; it belongs to a chosen spacelike slice.

Orientation and the Levi-Civita symbol

The orientation convention is

$$\epsilon^{0123}=+1.$$

Lowering all four indices with the mostly-minus metric gives

$$\epsilon_{0123}=-1.$$

The three-dimensional symbol obeys $\epsilon_{123}=+1$. The four-dimensional symbol is a totally antisymmetric tensor density in general coordinates; here we use it only in inertial Cartesian coordinates. These signs matter in $\gamma^5$, dual field strengths, and traces containing five gamma matrices.

Electromagnetic four-potential and signed charge

In SI units the four-potential is

$$A^\mu=\left(\frac{\Phi}{c},\mathbf A\right), \qquad A_\mu=\left(\frac{\Phi}{c},-\mathbf A\right).$$

For a particle with signed charge $q$, define

$$D_\mu = \partial_\mu+\frac{iq}{\hbar}A_\mu.$$

Then

$$i\hbar D_\mu=i\hbar\partial_\mu-qA_\mu,$$

and the spatial kinetic momentum is

$$\boldsymbol\pi = -i\hbar\nabla-q\mathbf A.$$

With a real gauge function $\chi(x)$, the compatible transformation is

$$A_\mu' = A_\mu-\partial_\mu\chi, \qquad \psi'=e^{iq\chi/\hbar}\psi, \qquad D_\mu'\psi'=e^{iq\chi/\hbar}D_\mu\psi.$$

In three-vector notation this is

$$\mathbf A'=\mathbf A+\nabla\chi, \qquad \Phi'=\Phi-\frac{\partial\chi}{\partial t}.$$

The electron has $q=-e$, where $e>0$ is the elementary charge magnitude. The symbols $q$ and $e$ are therefore not interchangeable. Keeping $q$ signed prevents the magnetic-moment sign from being inserted by memory later.

SI units and natural units

This bridge defaults to SI dimensions because the nonrelativistic limit and electromagnetic coupling are easiest to check when $c$ and $\hbar$ remain visible. Their exact SI values are

$$c=299\,792\,458\ \mathrm{m\,s^{-1}},$$

and

$$h=6.626\,070\,15\times10^{-34}\ \mathrm{J\,s}, \qquad \hbar=\frac{h}{2\pi}.$$

In natural units, $c=\hbar=1$, so mass, momentum, and energy share one unit, while length and time have inverse-energy dimension:

Quantity	SI dimension	Natural-unit dimension
energy $E$	$\mathrm{J}$	$E$
momentum $p$	$\mathrm{J}/c$	$E$
mass $m$	$\mathrm{J}/c^2$	$E$
inverse length $k$	$E/(\hbar c)$	$E$
inverse time $\omega$	$E/\hbar$	$E$
four-derivative $\partial_\mu$	$1/\mathrm m$	$E$

The last column records mass dimension, not that every quantity has become dimensionless. In $3+1$ dimensions a dimensionless action and canonically normalized kinetic terms give

$$[x^\mu]=-1, \qquad [\partial_\mu]=[m]=1, \qquad [\phi]=1, \qquad [\psi]=\frac32,$$

where square brackets now mean powers of an energy unit. The scalar and spinor entries are field-theory dimensions; they are included because they appear when this bridge reaches QFT, not because a one-particle wavefunction has acquired the same engineering dimension.

Setting constants equal to one is a choice of units, not an approximation. It does not mean that light travels infinitely fast or that quantum effects vanish.

Restoring constants safely

The safest procedure is to start from a dimensionally complete invariant. For example, the natural-unit mass shell

$$E^2=\mathbf p^2+m^2$$

becomes

$$E^2=\mathbf p^2c^2+m^2c^4,$$

because every term must have dimensions of energy squared. Likewise the natural-unit phase $p\cdot x$ becomes $p\cdot x/\hbar$, and the natural-unit reduced Compton wavelength $1/m$ becomes

$$\lambda_C=\frac{\hbar}{mc}.$$

Here $\lambda_C$ denotes the reduced Compton wavelength; the unreduced Compton wavelength is $h/(mc)=2\pi\lambda_C$.

The useful conversion follows from $\hbar c$:

$$1\ \mathrm{GeV}^{-1} \approx 0.1973269804\ \mathrm{fm}.$$

Dimensional analysis alone may not determine every power when several quantities carry the same dimensions. In that case, return to the covariant definition or derive the formula with constants retained; do not guess from a single term.

Comparing common conventions

Feature	Convention here	Common alternative	What must change together
metric	$(+---)$	$(-+++)$	scalar products, $\Box$, Clifford algebra, lowered components
time coordinate	$x^0=ct$	$x^0=t$	components of $p^\mu$, $A^\mu$, and derivatives
plane wave	$e^{-ip\cdot x/\hbar}$	$e^{+ip\cdot x/\hbar}$	Fourier and momentum-operator signs
charge	signed $q$	electron magnitude $e>0$	substitute $q=-e$ everywhere, not selectively
units	SI by default	$c=\hbar=1$	dimensions and all restored factors
orientation	$\epsilon^{0123}=+1$	opposite orientation	$\gamma^5$, duals, and epsilon traces

Two convention sets can describe the same physics. A hybrid expression made from pieces of both usually cannot.

Common pitfalls

Lowering a spatial index without changing its sign. With the mostly-minus metric, $a_i=-a^i$. Write one explicit lowering operation before manipulating a long contraction.

Using $\partial^\mu=(\partial_t/c,\nabla)$. Raising the derivative index changes the spatial signs, so $\partial^\mu=(\partial_t/c,-\nabla)$.

Treating $e$ as the electron’s signed charge. Here $e$ is positive and an electron has $q=-e$. Magnetic terms must inherit this substitution.

Restoring only $c$. A natural-unit expression may also hide $\hbar$ in a phase, derivative, wavelength, or spin term. Check dimensions and at least one known limit.

Exercises

1. Lowered four-momentum

Starting from $p^\mu=(E/c,\mathbf p)$, find $p_\mu$ and verify the mass-shell scalar $p_\mu p^\mu$.

Solution

Lowering with $\eta_{\mu\nu}=\operatorname{diag}(1,-1,-1,-1)$ gives

$$p_\mu=\left(\frac{E}{c},-\mathbf p\right).$$

Therefore

$$p_\mu p^\mu = \frac{E^2}{c^2}-\mathbf p^2.$$

For a particle of invariant mass $m$, this scalar equals $m^2c^2$.

2. Plane-wave derivative

Show directly that $i\hbar\partial_\mu e^{-ip\cdot x/\hbar}=p_\mu e^{-ip\cdot x/\hbar}$.

Solution

For constant $p_\mu$,

$$\partial_\mu e^{-ip_\nu x^\nu/\hbar} = -\frac{i}{\hbar}p_\mu e^{-ip_\nu x^\nu/\hbar}.$$

Multiplication by $i\hbar$ gives the stated result.

3. Gauge covariance

Verify $D_\mu'\psi'=e^{iq\chi/\hbar}D_\mu\psi$ using the conventions on this page.

Solution

Substitute $A_\mu'=A_\mu-\partial_\mu\chi$ and $\psi'=e^{iq\chi/\hbar}\psi$:

$$\begin{aligned} D_\mu'\psi' &= \left(\partial_\mu+\frac{iq}{\hbar}A_\mu' \right)e^{iq\chi/\hbar}\psi \\ &= e^{iq\chi/\hbar} \left[ \partial_\mu+\frac{iq}{\hbar}\partial_\mu\chi +\frac{iq}{\hbar}A_\mu -\frac{iq}{\hbar}\partial_\mu\chi \right]\psi \\ &= e^{iq\chi/\hbar}D_\mu\psi. \end{aligned}$$

4. Natural-unit Compton scale

Restore $c$ and $\hbar$ in the natural-unit statement that the inverse mass sets a particle’s reduced Compton length.

Solution

A length has SI dimensions obtained from inverse energy by multiplication with $\hbar c$. The rest energy is $mc^2$, so

$$\lambda_C = \frac{\hbar c}{mc^2} = \frac{\hbar}{mc}.$$

References

• Bureau International des Poids et Mesures, The International System of Units (SI Brochure), 9th ed., version 4.01, 2026, doi:10.59161/AUEZ1291.
• C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, 1980.
• P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, “CODATA Recommended Values of the Fundamental Physical Constants: 2022,” Journal of Physical and Chemical Reference Data 54, 033105, 2025, doi:10.1063/5.0279860.
• F. Takahashi et al. (Particle Data Group), “Review of Particle Physics,” International Journal of Modern Physics A 41, 2630011, 2026, constants and units.
• S. Weinberg, The Quantum Theory of Fields, Volume I: Foundations, Cambridge University Press, 1995.