Cartan-Karlhede Equivalence

Invariant frame fixing overcomes the limits of scalar invariants by retaining the full curvature tensor and its covariant derivatives while systematically removing Lorentz-frame freedom. On regular connected regions, the stabilized Cartan–Karlhede data decide local metric equivalence, count essential coordinates, and recover the dimension of the local isometry group—even for VSI spacetimes whose scalar polynomial invariants all vanish.

Required background. The Riemann Curvature Tensor supplies curvature and covariant derivatives; Killing Fields and Isometry Groups supplies the isometry interpretation of residual isotropy; The Einstein Equation supplies solution data when metric-field equivalence, rather than metric equivalence alone, is intended.

Helpful background. Frames, Coframes, and Anholonomy supplies local Lorentz transformations and curvature components in a frame.

The local metric-equivalence problem

Let $g$ and $\widetilde g$ be $C^{q+2}$ Lorentzian metrics of the same dimension and signature on connected open sets $U$ and $\widetilde U$. Restrict to regular strata on which the ranks of the curvature-component maps and the dimensions of their stabilizers are constant. The question is whether a local diffeomorphism $\psi:\widetilde U\to U$ satisfies $\widetilde g=\psi^*g$. Field equations are not hypotheses of the metric-equivalence theorem; if fields are part of the physical solution, their normalized components and internal gauge freedom must be compared separately.

This is a local smooth-equivalence statement. It does not decide global identifications, completeness, boundary conditions, or whether two disconnected strata join smoothly.

The normalization sequence

Choose an orthonormal or null frame $e_a$ with constant inner products $\eta_{ab}$. At order zero, compute the frame components $R_{abcd}$ and use Lorentz transformations to put them into a canonical form. Record

• $H_0$, the residual Lorentz subgroup preserving that form; and
• $t_0$, the number of functionally independent normalized components.

At derivative order $q$, canonicalize the joint set

$$\mathcal R^{(q)} =\{R,\nabla R,\ldots,\nabla^{(q)}R\}_{\text{frame}},$$

then record its residual isotropy $H_q\subseteq H_{q-1}$ and the number $t_q$ of independent functions. Continue until both quantities stabilize at a new order:

$$H_q=H_{q-1}, \qquad t_q=t_{q-1}.$$

Frame choices must be made consistently, including discrete sign, permutation, and branch choices. A component that looks constant in an arbitrary frame is not an invariant datum; the count becomes meaningful only after normalization. Mergulhão and Batista give the algorithm, stopping rule, and worked low-dimensional examples in Mergulhão and Batista 2021, §§II–III, printed pp. 3–9, Open PDF, followed by a four-dimensional Schwarzschild identification in Mergulhão and Batista 2021, §IV, printed pp. 9–12, Open PDF.

Why the stabilized data are complete

Canonical curvature components that are functionally independent can serve as invariant coordinates. Equality of the complete functional relations between the two stabilized sets supplies the point correspondence; equality of the structure encoded by their derivatives supplies the local metric correspondence. Residual $H_q$ records frame rotations not distinguished by the geometry. This is the operational content of Cartan’s local equivalence method, specialized and bounded by Karlhede for relativity.

On a regular $n$-dimensional stratum, the local isometry-group dimension is

$$\dim\operatorname{Isom}_{\mathrm{loc}} =n-t_q+\dim H_q.$$

Here $n-t_q$ counts directions along which all invariants are unchanged, while $H_q$ counts isotropy at a point.

The familiar finite upper bounds are worst-case statements; most exact spacetimes stabilize at low order. At points where eigenvalues merge or gradients vanish, the canonicalization branch can change, so one must stratify the domain rather than force one frame across an exceptional surface.

Schwarzschild and the VSI test

For Schwarzschild with $m\ne0$, the Weyl tensor is type D. Its repeated principal null directions fix most of the null frame, and the essential Weyl component varies as $m/r^3$ up to the declared Newman–Penrose convention. A derivative fixes the remaining boost freedom. Stabilization gives one essential coordinate, $t_q=1$, and one-dimensional spatial rotational isotropy at a generic point, $\dim H_q=1$. Hence

$$\dim\operatorname{Isom}_{\mathrm{loc}}=4-1+1=4,$$

matching time translations plus $SO(3)$ rotations. The nontrivial functional data also recover the relation between an unfamiliar form and Schwarzschild; two apparent parameters in the example of Mergulhão and Batista are frame-coordinate artifacts, while the mass relation survives.

Now compare Minkowski spacetime with a nonflat vacuum pp-wave. All SPIs vanish in both, but Minkowski has $R_{abcd}=0$ in every frame, whereas the pp-wave has nonzero negative-boost-weight frame components. The algorithm separates them at curvature order zero. More subtly, it can separate distinct VSI waves through normalized derivatives. This is exactly the missing information exposed by the degenerate Kundt exception Coley, Hervik, and Pelavas 2009, §7, printed pp. 21–24, Open PDF.

Exceptional homogeneous branch

Constant-curvature spacetime is not a failure of the method. All normalized curvature components are constant, so $t_q=0$, while the full Lorentz isotropy survives with dimension $n(n-1)/2$. The formula gives $n+n(n-1)/2=n(n+1)/2$, the maximal Killing count. Large isotropy is information, not ambiguity.

Check your understanding

A four-dimensional regular metric stabilizes with $t_q=2$ and a two-dimensional residual isotropy group. How many local Killing generators does it have, and what does the count not establish?

Solution

The formula gives

$$4-2+2=4$$

local Killing generators. The count does not determine their Lie algebra, causal character, global completeness, or whether they descend through a quotient. Those require the actual normalized data and global domain. It also does not establish equivalence to another metric unless the complete stabilized functional relations agree.

References

• Coley, Alan, Sigbjørn Hervik, and Nicos Pelavas. “Spacetimes Characterized by Their Scalar Curvature Invariants.” Classical and Quantum Gravity 26 (2009), 025013. doi:10.1088/0264-9381/26/2/025013.
• Karlhede, Anders. “A Review of the Geometrical Equivalence of Metrics in General Relativity.” General Relativity and Gravitation 12 (1980), 693–707. doi:10.1007/BF00771861.
• Mergulhão, Thiago M., and Carlos Batista. “On the Equivalence of Spacetimes, the Cartan–Karlhede Algorithm.” General Relativity and Gravitation 53 (2021), 28. doi:10.1007/s10714-021-02820-4.