Boundaries, Corners, and Orientation Conventions

Integrating a geometric identity over a spacetime region turns orientations into signs. Smooth boundary pieces inherit an orientation from the ambient manifold and an outward normal, while corners inherit an orientation by applying the boundary rule again. Timelike, spacelike, and null pieces require different normalizations, so a mixed-signature boundary cannot be handled by one unsigned “surface element.”

Required background. Affine Connections and Covariant Derivatives supplies covariant differentiation of normal fields; The Riemann Curvature Tensor supplies curvature tensors that may be integrated; Pseudo-Riemannian Metrics and Signature supplies causal character and volume geometry, and Metric Volume, Divergence, and Integration supplies the metric divergence theorem. Helpful background. Calculus and Linear Algebra for Relativity supplies normal decompositions; Exterior Algebra and Differential Forms supplies the form algebra, while the generalized Stokes theorem supplies induced boundary orientation and the bulk–boundary identity.

Outward-normal-first orientation

Let $(M,\varepsilon_M)$ be an oriented $n$-manifold with boundary. On a non-null boundary component choose the outward-pointing unit normal $n^\mu$, with $n^2=\epsilon=\pm1$. An ordered tangential basis $(e_1,\ldots,e_{n-1})$ is positively oriented on the boundary when

$$(n,e_1,\ldots,e_{n-1})$$

is positively oriented in $M$. This is the outward-normal-first convention. In differential-form language, the induced signed boundary form is represented by

$$\varepsilon_{\partial M}=\iota_n\varepsilon_M$$

when evaluated on positively oriented tangent vectors. The metric measure $d\Sigma=\sqrt{|h|}\,d^{n-1}y$ is nonnegative; orientation resides in the directed surface element or in the form, not in the square root.

With these definitions the normal component of a vector is $V_\perp=\epsilon(V\mathbin{\cdot}n)n$. Consequently, the pullback of $\iota_V\varepsilon_M$ to the boundary is $\epsilon(V\mathbin{\cdot}n)\varepsilon_{\partial M}$, and the divergence theorem is

$$\int_M \nabla_\mu V^\mu\,dV_g =\int_{\partial M}\epsilon V^\mu n_\mu\,d\Sigma$$

piece by piece, with $n^\mu$ outward on every non-null face and $\epsilon=n^2$. The factor $\epsilon$ is invisible on a timelike boundary, whose normal is spacelike, but essential on a spacelike cap, whose normal is timelike. An equivalent convention can absorb this sign into a directed surface element; one must not use both conventions at once. Future and past spacelike caps have opposite outward directions even though both have timelike normals.

A rectangular sign check

Take an oriented rectangle $[t_1,t_2]\times[x_1,x_2]$ in two-dimensional Minkowski spacetime with orientation $dt\wedge dx$. The outward normals are future directed on the top cap, past directed on the bottom cap, right pointing on the right wall, and left pointing on the left wall.

Contracting the volume form gives

$$\iota_{\partial_t}(dt\wedge dx)=dx, \qquad \iota_{-\partial_t}(dt\wedge dx)=-dx,$$

for the top and bottom. On the right and left walls,

$$\iota_{\partial_x}(dt\wedge dx)=-dt, \qquad \iota_{-\partial_x}(dt\wedge dx)=dt.$$

Thus opposite faces inherit opposite orientations. Applying the boundary operator once more gives opposite endpoint signs on each face; summing all four oriented face boundaries cancels every corner. This explicit calculation verifies $\partial(\partial M)=0$ and prevents a future-normal convention from being used accidentally on both temporal caps.

The divergence theorem supplies a second sign check. For $V=f(t)\partial_t$, the bulk integral is

$$\int_M\nabla_\mu V^\mu\,dt\,dx =\bigl(f(t_2)-f(t_1)\bigr)(x_2-x_1).$$

Both caps have $\epsilon=-1$. On the top cap $V\mathbin{\cdot}n=-f(t_2)$, while on the bottom cap $V\mathbin{\cdot}n=+f(t_1)$. Thus $\epsilon V\mathbin{\cdot}n$ produces exactly the bulk result after the two cap integrals are added; omitting $\epsilon$ would reverse it.

Corners and joints

When two smooth boundary faces meet transversely, their intersection is a codimension-two corner. Its orientation can be obtained as the boundary of either face. The two results are opposite where they represent the same corner in the total boundary-of-boundary sum.

Geometric data at a non-null joint include the two outward normals and their relative angle or rapidity. The signature of their span, rather than either normal’s causal character alone, determines the appropriate parameter. In a positive-definite normal two-plane one uses an ordinary angle. In a Lorentzian normal two-plane one uses a boost parameter, with the relevant causal branches specified. Even two spacelike unit normals can span a Lorentzian plane: if $c=n_1\cdot n_2$, their Gram determinant is $1-c^2$, which is negative when $|c|\gt1$. Reversing a normal or changing which region is retained changes the signed joint data.

This page fixes orientation and local geometric data only. The coefficient and functional form of a joint contribution depend on the variational problem or action being studied and are not inferred from orientation alone.

Null faces and normalization

A null boundary normal $k_\mu$ is also tangent and cannot be normalized to $\pm1$. A second null direction or a transverse rigging is needed to define codimension-two measures and compare joints. Under

$$k^\mu\mapsto e^f k^\mu,$$

null extrinsic data and any nonaffinity generally transform. Boundary conditions or endpoint choices must fix this freedom, or the final expression must be invariant under it.

Null and mixed-signature joints therefore require additional conventions beyond outward-normal-first. Lehner and collaborators give a systematic account of null normals, joints, and reparametrization issues in Lehner et al. 2016, §§II.B–II.H, pp. 7–25. Those action-specific applications motivate the warning, but gravitational boundary actions themselves are outside this page.

A boundary whose causal character changes cannot be covered by a single smooth unit normal through the transition. Divide it into regular pieces, use non-null or null data on each, and account for their joints. Treating the null location as an $\epsilon=0$ substitution loses both normalization and the degenerate tangent direction.

Common pitfalls

Outward and future are not synonyms. The past cap of a spacetime slab has a past-directed outward normal. State both causal and geometric direction where needed.

Measures and orientations are different. $\sqrt{|h|}\,d^{n-1}y$ is positive; a sign enters through an oriented form or directed normal.

Corners are not extra smooth faces. Their codimension is two, and their induced orientation depends on an ordered sequence of normal choices.

Null normalization is free. A formula containing a null normal must either state how it is normalized or demonstrate the relevant rescaling invariance.

Exercises

For the oriented Euclidean square $0\le x,y\le1$ with area form $dx\wedge dy$, list the induced one-forms on the bottom, right, top, and left edges in counterclockwise traversal order.

Solution

The outward normals are $-\partial_y$, $\partial_x$, $\partial_y$, and $-\partial_x$. Contracting them into $dx\wedge dy$ gives

$$dx,\qquad dy,\qquad -dx,\qquad -dy,$$

respectively. These forms orient the bottom from left to right, the right edge upward, the top from right to left, and the left edge downward. Each vertex occurs twice with opposite endpoint signs, so the oriented boundary of the boundary vanishes.

References

• Lehner, Luis, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin. “Gravitational Action with Null Boundaries.” Physical Review D 94, 084046 (2016). arXiv:1609.00207.