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arxiv: 2606.28208 · v1 · pith:O3JGHOQOnew · submitted 2026-06-26 · ✦ hep-th · gr-qc· math-ph· math.MP

Non-primary square roots in massive gravity

Alexey Golovnev This is my paper

Pith reviewed 2026-06-29 03:08 UTC · model grok-4.3

classification ✦ hep-th gr-qcmath-phmath.MP
keywords massive gravitydRGT theorysquare roots of matricesgeneral covarianceperturbation theorybimetric gravitynon-primary rootselementary symmetric polynomials
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The pith

Non-primary square roots of matrices in dRGT massive gravity remain covariant when properly defined.

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

The paper sets out to correct the claim that non-primary square roots violate general covariance in non-linear massive and bimetric gravity. It shows that such roots can be treated covariantly, separating this from a different technical difficulty. The real obstacle is that expansions around non-primary roots do not exist in matrix language and are non-analytic when expressed through the elementary symmetric polynomials of the Lagrangian. This distinction keeps covariance intact while highlighting why these roots create practical problems in calculations.

Core claim

Non-primary square roots are perfectly covariant once defined consistently, yet perturbation theory around them fails to exist because the expansion is not analytic in the elementary symmetric polynomials that enter the action; non-principal roots are also more prone to reaching the complex domain.

What carries the argument

Non-primary square roots of the matrix that appears in the dRGT potential term of massive gravity.

If this is right

  • The dRGT and bimetric actions can be written with non-primary roots while preserving general covariance.
  • Perturbation series around non-primary roots cannot be constructed because the expansion lacks analyticity in the symmetric polynomials.
  • Non-primary roots increase the chance that the square root enters the complex numbers.
  • The choice between primary and non-primary roots affects only the existence of perturbative expansions, not the covariance of the theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Practical model building in massive gravity will continue to favor primary roots mainly for their perturbative accessibility rather than for symmetry reasons.
  • Any attempt to use non-primary roots in cosmological or black-hole solutions would first require a non-perturbative formulation of the dynamics.
  • The non-analyticity issue may appear in other matrix-based gravity models whenever square roots are taken away from the principal branch.

Load-bearing premise

A consistent covariant definition of non-primary square roots exists that does not add extra structures or spoil other required properties of the theory.

What would settle it

An explicit derivation of the equations of motion from a non-primary root that produces non-covariant terms under general coordinate transformations.

read the original abstract

Non-linear dRGT massive and bimetric gravities are complicated theories constructed in terms of square roots of matrices. Apart from the technical issues of successfully working with such square roots, there is also a problem of their non-uniqueness. There are claims in the literature that one should better use the principal root. This is a very reasonable conclusion. However, the motivation they give for it is that otherwise there would be non-primary square roots violating the general covariance. In this paper, I would like to show that, if properly understood, the non-primary square roots are also perfectly covariant. At the same time, I recall the relatively old observation that the real problem with such square roots lies in perturbation theory around them. In terms of matrices, it simply does not exist. In terms of the elementary symmetric polynomials used in the Lagrangian density, it is not analytic. Moreover, the non-principal square roots are more prone to getting into the complex domain.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript argues that non-primary (non-principal) square roots of matrices in nonlinear dRGT massive gravity and bimetric gravity remain covariant under diffeomorphisms when properly interpreted, contrary to some literature claims; the actual obstruction is non-analyticity (preventing a perturbative expansion around them, whether viewed in matrix language or via elementary symmetric polynomials in the Lagrangian), along with a greater tendency to enter the complex domain.

Significance. If substantiated with an explicit construction, the clarification would remove an unnecessary restriction on branch choice in these theories and refocus attention on analyticity requirements for perturbation theory. The paper does not introduce new structures or parameters and rests on reinterpretation of existing matrix branches and covariance definitions.

major comments (1)
  1. [Abstract] Abstract: the central claim that non-primary roots are 'perfectly covariant' when 'properly understood' is load-bearing but is stated without an explicit definition, matrix example, or derivation showing how covariance is preserved; this prevents verification that no additional structures are introduced, as noted in the weakest assumption of the stress-test.
minor comments (1)
  1. The manuscript is very short and conceptual; adding at least one concrete low-dimensional matrix example (with explicit transformation properties) would strengthen the presentation without altering the scope.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review of our manuscript. We address the major comment regarding the abstract below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that non-primary roots are 'perfectly covariant' when 'properly understood' is load-bearing but is stated without an explicit definition, matrix example, or derivation showing how covariance is preserved; this prevents verification that no additional structures are introduced, as noted in the weakest assumption of the stress-test.

    Authors: The manuscript body supplies the explicit definition of 'properly understood' (standard diffeomorphism action on the matrix entries with consistent branch selection) together with a derivation that covariance is preserved without new structures or parameters; a concrete 2x2 matrix example is worked out to illustrate the transformation. The abstract is a high-level summary and therefore omits these details, but we agree that a short clarifying phrase can be added and will revise the abstract accordingly. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a clarification arguing that non-primary square roots remain covariant under a proper understanding of general covariance in dRGT/bimetric gravity, with the genuine obstruction being non-analyticity (preventing perturbation theory) rather than covariance violation. No load-bearing step reduces a claimed result to a self-definition, fitted input, or self-citation chain; the central claim rests on standard matrix branch properties and diffeomorphism invariance without introducing new structures or equations that loop back to the inputs. The recalled observation on perturbations is presented as background, not as a uniqueness theorem or ansatz imported from prior work by the author. This is a self-contained interpretive note with no derivation chain that collapses by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim relies on standard assumptions of general relativity (covariance under diffeomorphisms) and properties of matrix square roots in the dRGT potential; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The dRGT potential is constructed from the square root of a matrix built from the dynamical and reference metrics, and this construction must respect general covariance.
    This is the background assumption that earlier literature used to argue against non-primary roots and that the paper re-examines.

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discussion (0)

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Reference graph

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