Hypersurface Normals and Tangential Projectors

A hypersurface is a codimension-one submanifold. Locally it can be described as a regular level set, and its conormal determines which ambient directions are tangent. A non-null conormal can be normalized to construct an orthogonal projector and a nondegenerate induced metric. A null hypersurface is qualitatively different because its normal is also tangent.

These distinctions are local but consequential. Projecting an ambient tensor onto a spacelike slice, a timelike wall, or a null boundary requires three different signature checks. A correct calculation begins by testing the normal’s norm and regularity; only then can it select the non-null induced metric or the null screen data. That order avoids formulas whose apparent simplicity hides a singular normalization.

Required background. Affine Connections and Covariant Derivatives supplies covariant tensor differentiation; this page constructs the tangential projection from the metric and a regular normal. The Riemann Curvature Tensor supplies the ambient curvature later decomposed by Gauss–Codazzi; Pseudo-Riemannian Metrics and Signature supplies causal character and orthogonality. Helpful background. Calculus and Linear Algebra for Relativity supplies regular level sets and projectors; Exterior Algebra and Differential Forms supplies conormals and orientation algebra.

Regular level sets and coorientation

Let $F$ be a smooth real function on an open set of spacetime. If $dF$ is nonzero at every point of

$$\Sigma=\{p:F(p)=0\},$$

then the regular-level-set theorem makes $\Sigma$ a smooth hypersurface. A vector $X^\mu$ is tangent precisely when

$$X^\mu\nabla_\mu F=0.$$

The one-form $\nabla_\mu F$ spans the conormal line. Replacing $F$ by a function with the same regular zero set may rescale that conormal, so the hypersurface alone does not choose its magnitude. Choosing one of the two continuous directions in the conormal line is a coorientation. For a timelike normal one commonly chooses future direction; for a boundary one commonly chooses outward direction.

A global defining function is convenient but not required. Local conormals can define a global coorientation even when no single preferred $F$ exists. Conversely, a hypersurface can fail to be coorientable, in which case no continuous unit normal exists globally even though local normals do.

Non-null unit normals

Suppose the normal is nowhere null. Normalize it so that

$$n^\mu n_\mu=\epsilon, \qquad \epsilon\in\{-1,+1\}.$$

The tangential projector and induced metric are

$$h^\mu{}_\nu=\delta^\mu{}_\nu-\epsilon n^\mu n_\nu, \qquad h_{\mu\nu}=g_{\mu\nu}-\epsilon n_\mu n_\nu.$$

They satisfy

$$h^\mu{}_\nu n^\nu=0, \qquad h^\mu{}_\alpha h^\alpha{}_\nu=h^\mu{}_\nu.$$

If $\epsilon=-1$, the normal is timelike and the hypersurface is spacelike; the induced metric is positive definite. If $\epsilon=+1$, the normal is spacelike and the hypersurface is timelike; the induced metric has one negative direction. Reversing $n^\mu$ does not change $h_{\mu\nu}$, although it will reverse every quantity linear in the normal.

For any vector,

$$V^\mu=h^\mu{}_\nu V^\nu+\epsilon(n_\nu V^\nu)n^\mu.$$

Applying this identity twice checks both the coefficient and the sign. The same projection is applied to every index of a tensor when its fully tangential part is required. Gourgoulhon develops normals in Gourgoulhon 2007, §2.3.2, pp. 21–22, PDF and the orthogonal projector for a spacelike hypersurface in Gourgoulhon 2007, §2.4.1, pp. 29–31, PDF.

The null case is separate

If $\nabla_\mu F$ is null on $\Sigma$, its raised vector $k^\mu$ is tangent as well as normal. The pullback of $g_{\mu\nu}$ is degenerate, with $k^\mu$ in its kernel. There is no unit normalization and no value $\epsilon=0$ that turns the non-null formula into a projector: setting $\epsilon$ to zero would simply return the identity.

Choose an auxiliary null vector $\ell^\mu$ satisfying $k\cdot\ell=-1$. The codimension-two screen projector is

$$q_{\mu\nu} =g_{\mu\nu}+k_\mu\ell_\nu+\ell_\mu k_\nu.$$

It annihilates both null directions and represents the positive-definite quotient of the null tangent space by its generator. The paired rescaling $k\mapsto e^f k$, $\ell\mapsto e^{-f}\ell$ leaves $q_{\mu\nu}$ unchanged. Any null extrinsic data must track this normalization freedom explicitly.

Examples and checks

For $t=\text{constant}$ in Minkowski spacetime, $n=\partial_t$ is future timelike, $\epsilon=-1$, and $h$ is the Euclidean spatial metric. For the timelike plane $x=\text{constant}$, $n=\partial_x$, $\epsilon=+1$, and the induced metric has signature $(-,+,+)$. These examples test that the causal character of a hypersurface is opposite to that of its normal.

The cone $t-r=0$ is null away from its vertex. Its normal one-form $d(t-r)$ has zero norm and its raised vector lies along the cone’s generators. The vertex is not a smooth hypersurface point: the generators meet there with different tangent directions, so no coordinate change can supply a unique tangent hyperplane. The ambient Minkowski spacetime and individual geodesics remain regular in Cartesian coordinates, but the null-hypersurface construction applies only away from the apex.

Common pitfalls

Normal covectors and vectors are related by the metric. A level set first supplies $dF$. Its causal character is computed with the inverse metric before normalization.

Coorientation is extra structure. The induced metric survives $n\mapsto-n$, but extrinsic curvature and boundary orientation change sign.

Null hypersurfaces are not a continuous sign substitution. Their tangent metric is degenerate and their generator normalization is free. Use a null screen or quotient construction from the start.

Exercises

Let $\Sigma$ be the hyperboloid $t^2-r^2=\tau_0^2$ with $\tau_0\gt0$ and $t\gt0$ in Minkowski spacetime. Show that it is spacelike and find a future-directed unit normal.

Solution

Take $F=t^2-r^2-\tau_0^2$. Raising $dF$ gives a vector proportional to $-t\partial_t-r\partial_r$, which is past directed. Its squared norm on $\Sigma$ is $-4\tau_0^2$. Therefore the future-directed unit normal is

$$n=\frac{t\partial_t+r\partial_r}{\tau_0}.$$

It has $n^2=-1$, so $\epsilon=-1$ and the hypersurface is spacelike. The projector is $h^\mu{}_\nu=\delta^\mu{}_\nu+n^\mu n_\nu$.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, 2007. arXiv:gr-qc/0703035.