The Kundt Family

The Kundt class consists of spacetimes admitting a null geodesic congruence with no expansion, shear, or twist. It contains pp-waves and many impulsive or type-D solutions, but the optical definition is broader than any one field equation or algebraic type. Degenerate Kundt metrics are especially important because distinct local geometries can share all polynomial scalar curvature invariants.

Required background. Frames, Coframes, and Anholonomy supplies null frames and screen projections; Affine and Variational Geodesics supplies geodesic congruences and optical scalars.

Helpful background. Einstein’s Equation supplies vacuum and matter restrictions; The Riemann Curvature Tensor supplies curvature operators; Exact Solutions of Einstein’s Equations supplies invariant classification; The Wave Equation and Characteristics supplies wavefront interpretation.

Optical definition

Let $k^a$ be null and geodesic, with an affine rescaling chosen when possible. Project $\nabla_bk_a$ onto the $(D-2)$-dimensional screen transverse to $k^a$ and an auxiliary null vector. The resulting optical matrix decomposes as

$$B_{ij}=\frac{\theta}{D-2}\delta_{ij} +\sigma_{ij}+\omega_{ij},$$

where $\theta$ is expansion, $\sigma_{ij}$ is symmetric trace-free shear, and $\omega_{ij}$ is antisymmetric twist. A Kundt congruence satisfies

$$\theta=0, \qquad \sigma_{ij}=0, \qquad \omega_{ij}=0.$$

These are properties of a specified null congruence. A spacetime belongs to the class if at least one such congruence exists; another null congruence in the same geometry may have nonzero optics.

Canonical coordinates

Coordinates $(u,v,x^i)$ adapted to the congruence can be chosen locally with $k=\partial_v$ and

$$ds^2=-2\,du\left[dv+H(u,v,x)\,du +W_i(u,v,x)\,dx^i\right] +g_{ij}(u,x)\,dx^idx^j.$$

The absence of $v$ from $g_{ij}$ is precisely the vanishing of the symmetric optical matrix: the transverse metric does not expand or shear along the rays. Since $k_a$ is proportional to $du$, hypersurface orthogonality gives zero twist. The functions $H$ and $W_i$ may retain $v$ dependence; setting them independent of $v$ is an extra restriction.

A pp-wave is the much narrower case with a covariantly constant $k^a$, flat transverse metric, removable transverse one-form in the standard local form, and $H=H(u,x)$. Conversely, a general Kundt metric need not describe radiation, need not be Ricci-flat, and need not be Petrov type N. Exact subfamilies include nonexpanding waves in flat or constant-curvature backgrounds, direct-product geometries, and aligned pure-radiation solutions. The four-dimensional classification is organized in Stephani et al. 2003, §§31.1–31.5.

Degenerate Kundt and invariant failure

In a null frame, tensor components carry boost weight under $k\mapsto e^\lambda k$ and $n\mapsto e^{-\lambda}n$. A degenerate Kundt spacetime admits an aligned Kundt frame in which all positive boost-weight components of the Riemann tensor and all of its covariant derivatives vanish. This persistence through every derivative is stronger than the zeroth-order optical definition.

The consequence is an invariant-classification exception. In four-dimensional Lorentzian geometry, metrics not characterized locally by their scalar polynomial curvature invariants lie in the degenerate Kundt class. Some members are VSI, with every such scalar zero; others are CSI, with every scalar constant. Scalar equality can then fail to imply local isometry. The exact scope and curvature-operator argument are given in Coley, Hervik, and Pelavas 2009, §§7–9, pp. 21–36, Open PDF.

This does not make invariants useless. They still constrain Ricci and Weyl eigenvalues, screen candidate families, and distinguish many Kundt metrics. They are simply incomplete in the degenerate branch; Cartan frame invariants or an equivalence algorithm must finish the comparison.

Contrasts and checks

The Robinson–Trautman class also admits a geodesic, shear-free, twist-free null congruence, but with nonzero expansion. Thus $\theta=0$ is the decisive optical distinction. A visual coordinate resemblance between the two canonical metrics is not enough.

For the canonical Kundt form, $k_a=-\nabla_a u$ gives $k_{[a}\nabla_bk_{c]}=0$, checking twist-free hypersurface orthogonality. The screen metric obeys $\mathcal L_k g_{ij}=\partial_vg_{ij}=0$, checking expansion and shear simultaneously. To establish the degenerate subclass, however, one must also compute aligned boost-weight components of curvature and its derivatives; the optical check alone does not suffice.

Exercise

Explain why a nonflat vacuum pp-wave and Minkowski spacetime defeat classification by scalar polynomial curvature invariants.

Solution

Both are VSI: every complete scalar contraction of the Riemann tensor and its derivatives vanishes. Minkowski has $R_{abcd}=0$, whereas a nonflat pp-wave has $R_{uiuj}=-\tfrac12\partial_i\partial_jH\neq0$ in an adapted frame. Their scalar lists agree, but their Cartan components do not, so they are not locally isometric.

References

• Coley, Alan, Sigbjørn Hervik, and Nicos Pelavas. “Spacetimes Characterized by Their Scalar Curvature Invariants.” Classical and Quantum Gravity 26 (2009): 025013. DOI; Open PDF.
• Stephani, Hans, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt. Exact Solutions of Einstein’s Field Equations. 2nd ed. Cambridge: Cambridge University Press, 2003. DOI.