Orientation, Levi-Civita Symbols, and Volume Forms

Orientation supplies a sign choice; the metric supplies a magnitude. Keeping those roles separate prevents the numerical alternating symbol, the metric volume form, and a positive integration density from being mistaken for the same object. This record fixes their meanings for the site convention $(-,+,+,+)$ and gives the sign tests needed when taking Hodge duals or orienting boundaries. The full geometric treatment is Differential Forms, Integration, and Orientation.

Symbol, oriented volume form, and positive density

In an oriented coordinate chart $(x^0,x^1,x^2,x^3)$, define the numerical alternating symbol by

$$\widetilde\epsilon_{0123}=+1,$$

with total antisymmetry and zero whenever two indices coincide. Its entries are numbers assigned by the chosen ordering of coordinate labels. They are not the components of an ordinary tensor under a general coordinate change; equivalently, the symbol can be organized as a tensor density once a density-weight convention is declared. An orientation-reversing chart changes which coordinate ordering is called positive.

Let $g=\det(g_{\mu\nu})$. In four-dimensional Lorentzian signature, $g\lt0$, and the chosen orientation determines the metric volume form

$$\operatorname{vol}_g =\frac{1}{4!}\epsilon_{\mu\nu\rho\sigma} \,dx^\mu\wedge dx^\nu\wedge dx^\rho\wedge dx^\sigma, \qquad \epsilon_{\mu\nu\rho\sigma} =\sqrt{-g}\,\widetilde\epsilon_{\mu\nu\rho\sigma}$$

in every positively oriented chart. Thus the site convention is exactly

$$\epsilon_{0123}=+\sqrt{-g}, \qquad \epsilon^{0123}=-\frac{1}{\sqrt{-g}}.$$

The minus sign in the raised component is not optional: raising four indices introduces the sign of the inverse-metric determinant. If an upper numerical symbol is also written with $\widetilde\epsilon^{0123}=+1$, it is a separately defined array, not the result of raising the lower symbol. These metric and volume-tensor conventions are consistent with Tong 2019, § 3.1, printed pp. 91–99.

A positive density instead discards orientation information:

$$d\mu_g=\sqrt{|g|}\,|dx^0\,dx^1\,dx^2\,dx^3|.$$

It is unchanged when coordinates reverse orientation and can therefore be integrated without first orienting the manifold. By contrast, $\operatorname{vol}_g$ is an alternating tensor and changes sign when the chosen orientation is reversed. The distinction between top forms, orientations, and densities is developed in Guillemin and Haine 2019, Chapter 1, §§ 1.4–1.9, and Chapter 4, printed pp. 111–153.

Hodge dual and its sign maps

On an oriented pseudo-Riemannian $n$-manifold, the Hodge operator is fixed by

$$\alpha\wedge\star\beta =\langle\alpha,\beta\rangle_g\operatorname{vol}_g, \qquad \langle\alpha,\beta\rangle_g =\frac{1}{p!}\alpha_{\mu_1\cdots\mu_p} \beta^{\mu_1\cdots\mu_p}$$

for $p$-forms $\alpha$ and $\beta$. In components,

$$(\star\beta)_{\mu_{p+1}\cdots\mu_n} =\frac{1}{p!}\beta^{\mu_1\cdots\mu_p} \epsilon_{\mu_1\cdots\mu_n}.$$

If the metric has $s$ negative eigenvalues, then on $p$-forms

$$\star^2=(-1)^{p(n-p)+s}.$$

For four-dimensional $(-,+,+,+)$ signature, $n=4$ and $s=1$: $\star^2=+1$ on one- and three-forms, while $\star^2=-1$ on zero-, two-, and four-forms. In particular, real Lorentzian two-forms do not split into real $\star$ eigenforms with eigenvalues $\pm1$.

Two transformations that are often conflated have different effects. Reversing orientation at fixed metric sends $\operatorname{vol}_g\mapsto-\operatorname{vol}_g$ and hence $\star\mapsto-\star$ in every degree. Replacing the metric by $g'=-g$ while holding the orientation fixed leaves $|\det g|$ unchanged in four dimensions, but each raised index contributes a minus sign. Therefore the degree-$p$ operators obey

$$\star'_p=(-1)^p\star_p.$$

The Hodge-square rule itself is unchanged: the number of negative directions changes from one to three, altering its exponent by two.

Boundary orientation and time-slab signs

For a non-null boundary, choose the outward-pointing unit normal $n_{\mathrm{out}}$. The induced orientation is outward-normal-first: a boundary basis $(e_1,\ldots,e_{n-1})$ is positive precisely when $(n_{\mathrm{out}},e_1,\ldots,e_{n-1})$ is positive in the bulk. Equivalently, its oriented volume form is

$$\eta_{\partial M}=\iota_{n_{\mathrm{out}}}\operatorname{vol}_g.$$

This rule applies both to timelike boundaries, whose unit normals are spacelike, and to spacelike boundaries, whose unit normals are timelike. Time orientation is additional structure: it labels a timelike normal as future- or past-directed, but it does not replace the outwardness test. The same convention is the one entering the boundary term in Stokes’ theorem; see Wendl 2021–2022, §§ 9–13, printed pp. 69–118.

For a Minkowski time slab $t_1\le t\le t_2$ with $\operatorname{vol}_g=dt\wedge dx\wedge dy\wedge dz$, the final slice has outward normal $n_+=+\partial_t$ and

$$\eta_+=dx\wedge dy\wedge dz.$$

The initial slice has outward normal $n_-=-\partial_t$ and

$$\eta_-=-dx\wedge dy\wedge dz.$$

Consequently, the temporal contribution to a Stokes integral is the final-slice integral minus the initial-slice integral. A timelike outer wall is handled by the same outward-normal-first rule, with no extra sign inserted merely because the wall is timelike.

Null boundaries require normalization data

A null hypersurface has no unit normal: raising its null normal covector produces a null generator tangent to the hypersurface. The non-null formula must therefore never be continued by setting a symbol such as $\varepsilon=n^2$ to zero. Instead, choose a generator $k$ together with an auxiliary transverse null vector $\ell$, or a more general transverse rigging, and fix their relative normalization—for example,

$$k\cdot\ell=-1.$$

The pullback of $\iota_\ell\operatorname{vol}_g$ supplies a null hypersurface measure once its sign and the generator parameter have been declared; the screen-space area form can likewise be obtained from the double contraction with $k$ and $\ell$. Under $k\mapsto e^f k$ and $\ell\mapsto e^{-f}\ell$, the measure and generator parameterization rescale, so a null-boundary formula must keep that normalization dependence explicit. Directed null surface elements, joints, and boundary actions require precisely this extra bookkeeping; see Lehner, Myers, Poisson, and Sorkin 2016, §§ II.B–II.C and IV.

Diagnostic checks and limitations

• Raise every volume-tensor index. Starting from $\epsilon_{0123}=+\sqrt{-g}$ must give $\epsilon^{0123}=-1/\sqrt{-g}$.
• Dualize twice. A two-form in four-dimensional Lorentzian signature must return $-1$ times itself; a one-form must return itself.
• Reverse orientation. The positive density stays fixed, whereas $\operatorname{vol}_g$ and every Hodge dual change sign.
• Test a time slab. The future boundary carries the positive spatial form and the past boundary the negative one.
• Rescale a null generator. Any null measure or boundary integrand must transform consistently with the chosen auxiliary transverse vector and generator parameter.

These statements assume a smooth, nondegenerate metric. Degenerate metrics do not define the inverse metric or Hodge operator used here, and nonorientable manifolds have no global volume form, though their positive metric density remains available. For fuller treatments, see Orientation, Densities, and Volume Elements, Generalized Stokes Theorem, Hodge Dual in Lorentzian Signature, and Hypersurface Normals and Tangential Projectors.

References

• Guillemin, Victor, and Peter J. Haine. Differential Forms. Hackensack, NJ: World Scientific, 2019. Open PDF.
• Lehner, Luis, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin. “Gravitational Action with Null Boundaries.” Physical Review D 94 (2016): 084046. DOI. Open arXiv version.
• Tong, David. General Relativity. Cambridge Part III lecture notes, 2019. Official PDF.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022. Official PDF.