All symmetry groups of pp-waves in teleparallel gravity

A. Landry; D. D. McNutt

REVIEW 3 major objections 4 minor 77 references

All symmetry groups of pp-waves in teleparallel gravity

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper claims that every symmetry group of teleparallel pp-wave spacetimes is either a surviving 2D or 3D group from general relativity, a reduced higher-dimensional group, or one of two new GR solutions found here.

desk verdict Useful classification with a real completeness gap: the 'all' claim leans on a 1986 case list the paper itself shows to be incomplete. read the letter →

arxiv 2411.11420 v1 pith:P7NS6UGC submitted 2024-11-18 gr-qc

A. Landry , D. D. McNutt This is my paper · ORCID

classification gr-qc MSC 83D0583C35 PACS 04.20.-q04.30.-w04.50.Kd

keywords teleparallelgravitypp-wavesaffineframesymmetriesCartan-KarlhedealgorithmVSIspacetimestorsionTEGRKillingvectorfields

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper undertakes a complete symmetry census of pp-wave spacetimes in teleparallel gravity, asking which isometry groups survive when a spacetime symmetry must also preserve the torsion. It concludes that all two- and three-dimensional GR symmetry groups are realized, that the five-, six- and seven-dimensional groups are reduced (for instance, a GR G6 becomes a G4 or G3), and that two pp-wave solutions with symmetry groups G3 and G4 exist in both GR and teleparallel gravity but were missed in the standard GR classification. This matters because teleparallel gravity is a widely studied alternative to GR, and the resulting explicit TppW solutions give concrete gravitational-wave-like backgrounds whose stability and perturbations can now be studied.

What carries the argument

The machinery is the teleparallel Cartan-Karlhede algorithm combined with the VSI proper-frame ansatz. The frame has a null coframe built from H(u,x,y) and M0(u,x,y), and the torsion tensor is determined by those functions. The Cartan-Karlhede algorithm normalizes the torsion and its covariant derivatives into an invariantly defined frame, leaving a residual linear isotropy of dimension s and t_p functionally independent invariants; the affine frame symmetry dimension is then r = s + 4 - t_p. This counting formula is what turns the symmetry-group question into a computation of frame invariants.

What would settle it

Find a teleparallel frame that satisfies the TppW field equations and admits an affine frame symmetry group not listed in the Section VII table, for example a Case 10 solution with four independent affine frame symmetries instead of the claimed three, or show that a higher-dimensional GR pp-wave isometry group omitted from ref. [27] has a teleparallel analogue outside the table; either observation would break the claim that the list is complete.

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Extended reading notes

Core claim

The central claim is that the symmetry groups of the proper-frame teleparallel pp-wave metrics are exactly the ones collected in the table of Section VII. For every GR pp-wave isometry group of dimension 2 or 3 (Cases 1–8), the analogous teleparallel frames admit the same group, with explicit H and M0 functions; for the larger groups, the extra Killing vector fields of GR are generally not affine frame symmetries because they fail to annihilate the torsion, so Case 9 keeps G5, Case 10 drops from G5 to G3, Cases 11–14 drop from G6 to G4, Case 15 drops from G6 to G3, and Cases 16–17 drop from G7 to G5. The two new GR solutions are a G3 family with H = h1(u)x + h2(u,y) and M0 = M0(u,y), and a G4 family with H = h(u+C0 y) and M0 = m0(u+C0 y); these were missed in ref. [27] and are permitted both in GR and in teleparallel gravity.

Load-bearing premise

The paper's claim to have found all TppW symmetry groups rests on the assumption that the older GR classification it extends has not missed any higher-dimensional pp-wave symmetry groups, even though the paper itself shows that same classification is incomplete at lower dimension.

Editorial extensions

If this is right

If the enumeration is correct, the complete list of TppW symmetry groups is the Section VII table: every 2D and 3D GR case survives with the same orbit dimension, while every higher-dimensional case is reduced to dimension 5 or smaller.
The two newly found GR solutions, one with a G3 and one with a G4 symmetry group, should be added to the standard GR pp-wave classification.
Because explicit H and M0 are given for every class, each TppW solution can be perturbed and its stability examined in TEGR in the same style as perturbations of Minkowski space.
The systematic reduction of symmetry groups from GR to teleparallel gravity means wave-like backgrounds in TEGR typically have less symmetry than their GR counterparts, which may affect how plane gravitational waves are modeled in teleparallel theories.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The completeness of the higher-dimensional half of the table inherits the completeness of the GR classification in ref. [27]; an independent re-derivation of that classification would be needed before the word 'all' is fully unconditional.
If the Cartan-Karlhede method found omissions at low dimension, similar omissions may exist in other GR symmetry classifications of Kundt or VSI spacetimes, so teleparallel analogues could reveal further overlooked GR solutions.
The same machinery could be applied to constant-scalar-invariant teleparallel geometries to search for pp-wave-like gravitational waves propagating on (anti-)de Sitter backgrounds, a direction the paper itself mentions.
The explicit TppW solutions provide testbeds for whether the reduced symmetry changes observable gravitational-wave properties in TEGR, though this consequence is not tested in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short take: this is a solid extension of the authors' own program, and the two missed GR symmetry groups are a genuine find. But the headline claim of completeness is not supported as it stands.

What's new: The paper classifies the affine frame symmetry groups of teleparallel pp-wave spacetimes, building on the VSI frame form from their prior work (ref. [56]). The two overlooked GR solutions—a G3 from null-rotation isotropy (Remark V.6) and a G4 that is a special case of it (Proposition V.11)—are genuinely absent from Sippel and Goenner's 1986 list. That is a concrete, checkable contribution. The summary table showing how GR symmetry dimensions drop (e.g., G5→G3, G6→G4, G7→G5) is useful and well organized.

The soft spots are real, and one is load-bearing. The phrase "all possible symmetry groups" rests on the completeness of ref. [27] for the higher-dimensional cases (Cases 9–17). But the paper itself demonstrates that [27] is incomplete in the low-dimensional regime. The new G3 and G4 families are not checked for further specialization into larger symmetry groups; they could in principle admit additional affine frame symmetry generators that would appear among the higher-dimensional cases. Until the authors either prove the Sippel–Goenner enumeration is complete for the relevant branches or explicitly check that the new families don't extend, "all" is not justified. This isn't a small stylistic quibble; it's the main claim.

The other soft spot is the repeated reliance on "straightforward to solve" and "tedious exercise" for the differential equations. That is standard in this literature, but it means the classification cannot be verified from the text. A serious referee can ask for the intermediate equations and the explicit Killing verification for the new G3/G4. Both are addressable.

The self-citation point is minor. Ref. [56] shares an author and supplies the VSI frame input, but that is the natural foundation for this work, not a problem.

Who is this for? People working on teleparallel gravity and exact solution classification. It will not change the field's direction, but it is a correct and useful subfield contribution.

Recommendation: send it to review, but require the authors to close the completeness gap or soften the claim, and to supply the omitted DEs in an appendix or supplementary material. With that, this could be a solid paper.

Referee Report

3 major / 4 minor

Summary. The paper determines the symmetry groups of teleparallel pp-wave (TppW) spacetimes by combining the known VSI proper-frame ansatz with frame-based symmetry methods and the Cartan-Karlhede algorithm. It classifies the allowed affine frame symmetry groups in terms of the function H(u,x,y) and M0(u,x,y), presenting a table that compares the GR pp-wave symmetry groups of Sippel and Goenner with their teleparallel analogues. The authors find that most low-dimensional GR groups survive in TEGR, while the higher-dimensional groups are substantially reduced (e.g., G5 to G3 in Case 10, G6 to G4 in Cases 11-14, G7 to G5 in Cases 16-17). They also claim to have found two GR pp-wave symmetry groups missed in the Sippel-Goenner classification: a G3 (Remark V.6, eq. 56) and a G4 (Proposition V.11, eq. 63).

Significance. If the completeness claim holds, this is a useful and nontrivial contribution: it gives the first systematic enumeration of affine frame symmetry groups for teleparallel pp-wave spacetimes, provides explicit frame data for each case, and demonstrates that the CK algorithm can detect isometry groups missed by older GR classifications. The identification of two previously overlooked GR pp-wave symmetry groups is itself an interesting byproduct. The paper is not a black-box computation: the frame ansatz is grounded in the VSI teleparallel classification of ref. [56], and the Cartan-Karlhede framework is applied in a structured way. However, the central claim is conditional on the completeness of the Sippel-Goenner case list, and that dependency is not resolved by the paper's own evidence.

major comments (3)

[§IV, §VI, §VII] The central claim 'all possible symmetry groups for the TppW spacetimes' is not yet supported, because the enumeration inherits the completeness of the Sippel-Goenner list [27] while simultaneously showing that this list is incomplete. In §IV the paper explicitly uses ref. [27] as a framework for the small symmetry groups, and §VI takes Cases 9-17 from [27] as the complete set of higher-dimensional GR isometry groups. But Remark V.6 and Proposition V.11 exhibit a missed G3 and a missed G4 that are not in the [27] case list used in §IV. The paper never checks whether these new families, with H = h1(u)x + h2(u,y) (eq. 56) or H = h(u + C0 y) (eq. 63), can be specialized further to admit additional affine frame symmetry generators that would place them among the higher-dimensional cases. Until either an independent completeness argument for the TppW classification or a direct check of these new branches against the Cases 9-17 enumeration is supplied, the word 'all' in the abstract and in §VII is premature.
[§IV, §V, §VI] Several load-bearing classifications are asserted without derivations. The text states that 'the resulting DE coming from (5) are straightforward to solve' (§IV), 'It is straightforward to solve the resulting DEs' (Propositions V.7 and V.9), and 'A tedious exercise in solving these DEs yields...' (Proposition V.11). For a completeness claim, these omitted calculations are not merely expository: they are needed to verify that no additional branches arise. This is especially true for Case 10, where the reduction from a GR G5 to a TPG G3 depends on the unstated system (72)-(73) and an existence-uniqueness argument for a second-order ODE. The authors should provide the key differential equations, or a supplementary file containing the calculations, at least for the locally homogeneous cases and for Case 10.
[§VI.A, Case 10] The deduction that Case 10 has exactly a 3-dimensional symmetry group rests on an implicit linear-independence assumption for the two solutions of the second-order DE for f1 and f2. The text says that 'from the uniqueness and existence theorem for second order DEs, this will yield two solutions' and then concludes that the coefficients in eq. (73) must vanish. This requires that the two solutions are linearly independent over the relevant function space and that no degeneracy occurs for special choices of m1, m2, m4. The argument should be stated explicitly and, ideally, the two basis solutions exhibited, otherwise the table's entry 'Case 10: 5 -> 3' is not fully verifiable.

minor comments (4)

[§V, Proposition V.9] The proof begins 'Using the conditions in Proposition V.7', but the relevant conditions for spin isotropy appear to come from Proposition V.4, not Proposition V.7. Please correct the cross-reference.
[§V, Proposition V.9, eq. (62)] The parameter branch structure is unclear: in the first branch c0 is a constant with c1 = 0, while in the second branch c0 is set to -√2/(4u), a function of u. Please state the domain of u and clarify how c0 can be both a constant and a function in the two branches.
[§VI.A, eq. (67)] The coordinate transformation after eq. (66) appears to contain a typo: the expression for y' uses 'ρy − σy' where the context suggests 'ρy − σx'. Please check and correct this transformation, and specify which of the two solution families in Proposition V.7 is recovered.
[§VI.B, Case 15, eq. (90)-(91)] The function m(u) in eq. (90) is not constrained explicitly; eq. (91) involves division by m and m^2, so the condition m ≠ 0 and the allowed domain of u should be stated. The reality and sign conditions on m are also not specified.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the symmetry classification is a direct application of Lie-derivative equations and the CK algorithm to independently cited input classifications; the 'all' claim inherits a completeness caveat from [27], which is a correctness risk, not a circular reduction.

full rationale

The derivation chain is not circular. The paper takes two external inputs: the VSI teleparallel frame class from ref. [56] (eq. 6) and the GR pp-wave isometry group list from ref. [27]. It then solves the affine frame symmetry conditions (eq. 5) for the TppW coframe (28) and applies the modified Cartan-Karlhede algorithm (Sec. II.E). None of the output symmetry groups or the two new solutions (Remark V.6, eq. 56; Proposition V.11, eq. 63) is fitted to a subset of the output and then re-predicted; the new cases are obtained by imposing linear-isotropy conditions (e.g. eqs. 57-58) and solving the resulting DEs. Refs. [53] and [56] are self-citations through author McNutt, but they are published external classifications rather than self-asserted uniqueness theorems, and the paper independently proves the main exclusions (Propositions V.1-V.3) and independently re-derives the large-symmetry cases with the CK algorithm; therefore the self-citations are not load-bearing in a circular sense. The principal weakness is the completeness of the 'all' claim: Section IV explicitly uses ref. [27] as the framework for the small-symmetry enumeration, Section VI imports Cases 9-17 from [27], and Remark V.6 demonstrates that [27] missed a G3 (and Proposition V.11 a G4). If [27] also missed higher-dimensional GR isometry groups, those teleparallel analogues could be absent from the table. This is a real completeness/correctness gap, and the paper also leaves many DEs as 'straightforward' or 'tedious' rather than displaying them (e.g., Section VI), but it is not circularity: no listed result is equivalent by construction to its own input, and no fitted parameter is renamed as a prediction. The score of 2 reflects the mild self-citation involvement in the input frame and isotropy classifications, with the central derivation retaining independent computational content.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The classification rests on the VSI frame form from ref. [56] and the GR case list from ref. [27]. No parameters are fitted to data; constants in the solution families are integration constants. The paper introduces no new physical entities.

assumptions (5)

domain assumption The explicit proper frames for the teleparallel analogue of the VSI class, including the Kundt frame ansatz (eq. 28), are complete for all VSI teleparallel geometries.
Invoked in Section I and III, relying on ref. [56]. If this frame form is not exhaustive for TppW spacetimes, the 'all' in the main result fails.
domain assumption The Sippel-Goenner classification [27] enumerates all Killing vector fields of pp-wave spacetimes in GR for each dimension considered.
Used in Section IV as 'a framework' and in Section VI to enumerate candidate KVF sets (Cases 9-17). The paper itself shows [27] missed a G3 and a G4, so this assumption is fragile but load-bearing.
standard math The modified Cartan-Karlhede algorithm for teleparallel geometries terminates and uniquely characterizes the local geometry, so that r = s + n - tp determines the affine frame symmetry group dimension.
Used throughout Section V; based on ref. [53]. Standard in the field.
standard math A KVF X is an affine frame symmetry if and only if L_X T^a_bc = 0 and L_X ∇^p T^a_bc = 0 for all p (eq. 5).
Definition/theorem from frame-based symmetry analysis, used to determine which GR KVFs survive in TPG.
domain assumption TppW spacetimes are VSI, so the torsion scalar T vanishes and the TEGR field equations apply.
Section I and II.C; follows from ref. [56] and the definition of VSI. It restricts the theory from F(T) to TEGR.

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Cite this review

Pith. "Pith review of All symmetry groups of pp-waves in teleparallel gravity." pith.science (2026). https://pith.science/paper/P7NS6UGC

@misc{pith2026241111420,
  author       = {Pith},
  title        = {Pith review of: All symmetry groups of pp-waves in teleparallel gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P7NS6UGC}},
  note         = {Machine review of arXiv:2411.11420}
}

read the original abstract

We elaborate on and further develop an approach to determining the teleparallel analogue of spacetimes in General Relativity (GR) by studying the Teleparallel analogue of pp-Wave (TppW) spacetimes. This relies on using the fact that these solutions belong to the Vanishing Scalar Invariant (VSI) subclass for which the explicit forms of the frame and spin-connection are known. By identifying the pp-wave (ppW) metric within this class, we are able to use frame based symmetry methods and the Cartan-Karlhede (CK) algorithm to determine the necessary form for the frame. Through this analysis we find two overlooked solutions that are permitted in teleparallel gravity (TPG) and in GR.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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Set the order of diﬀerentiation q to 0

work page

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Einstein and N

A. Einstein and N. Rosen, J. Franklin Inst. 223, 43 (1937)

work page 1937

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Determine the canonical form of the q-th covariant derivative of the torsion tensor

work page

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The dimension of Hq is the dimension of the remaining vertical freedom of the frame bundle

Fix the frame as much as possible, using this canonical form, and r ecord the remaining frame transformations that preserve this canonical form (the group of allowed frame tra nsformations is the linear isotropy group Hq). The dimension of Hq is the dimension of the remaining vertical freedom of the frame bundle

work page

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This tells us the remaining horizontal freedom

Find the number tq of independent functions of spacetime position in T q in the canonical form. This tells us the remaining horizontal freedom

work page

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The resulting non-zero components of T p constitute the Cartan invariants and we will denote them as T ≡ T p+1 so that: T = {Tabc, Tabc|d1,

If the dimension of Hq and number of independent functions are the same as in the previou s step, let p + 1 = q, and the algorithm terminates; if they diﬀer (or if q = 0), increase q by 1 and go to step 2. The resulting non-zero components of T p constitute the Cartan invariants and we will denote them as T ≡ T p+1 so that: T = {Tabc, Tabc|d1, . . . Tabc|...

work page