Factorisation of symmetric matrices and applications in gravitational theories

Gabriel Lopes Cardoso; M. Cristina C\^amara

arxiv: 2410.16514 · v1 · submitted 2024-10-21 · 🧮 math.FA · gr-qc· math-ph· math.MP

Factorisation of symmetric matrices and applications in gravitational theories

M. Cristina C\^amara , Gabriel Lopes Cardoso This is my paper

Pith reviewed 2026-05-23 19:30 UTC · model grok-4.3

classification 🧮 math.FA gr-qcmath-phmath.MP

keywords Wiener-Hopf factorisationsymmetric matricesrational functionsRiemann-Hilbert problemEinstein field equationsmatrix factorisation

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The pith

For 2x2 symmetric matrices with rational diagonal quotient, symmetry fixes the second columns of the Wiener-Hopf factors by rational multiplication from the first columns.

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

The paper studies the canonical Wiener-Hopf factorisation of 2 by 2 symmetric matrices relative to a contour. When the quotient of the two diagonal entries is a rational function, the matrix symmetry determines that each factor's second column equals its first column multiplied by a certain rational matrix. The authors supply an explicit method to compute those second columns once the first ones are known. They demonstrate the procedure on two examples drawn from the Riemann-Hilbert formulation of the Einstein equations. A reader would follow the argument because the reduction removes the need to solve for all four columns independently.

Core claim

When the quotient q of the diagonal elements of a 2×2 symmetric matrix M is rational, the symmetry of M implies that the second column of each Wiener-Hopf factor is obtained from the first column of the same factor by multiplication with a rational matrix; a constructive method is given for determining these second columns.

What carries the argument

Canonical Wiener-Hopf factorisation of 2×2 symmetric matrices M, in which symmetry and the rationality of the diagonal quotient together reduce the second columns of the factors to rational multiples of the first columns.

If this is right

Once the first columns of the factors are found, the full factors follow immediately by the rational multiplication rule.
The method applies directly to Riemann-Hilbert problems that arise when solving the Einstein field equations.
The reduction halves the number of independent column functions that must be determined in each factor.
The same symmetry argument organises the factorisation data for any matrix obeying the stated rationality condition.

Where Pith is reading between the lines

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

The technique may simplify numerical implementations of matrix Riemann-Hilbert problems that appear in other integrable systems.
If analogous symmetry relations exist for larger matrices or other factorisation contours, the method could generalise beyond the 2×2 case.
The rational-matrix multiplier itself might admit a closed-form expression in terms of the original matrix entries.

Load-bearing premise

The quotient of the two diagonal elements of the matrix is a rational function.

What would settle it

Exhibit a concrete 2×2 symmetric matrix whose diagonal quotient is rational, perform its Wiener-Hopf factorisation, and show that at least one second column is not equal to the corresponding first column multiplied by any rational matrix.

read the original abstract

We consider the canonical Wiener-Hopf factorisation of $2 \times 2$ symmetric matrices $\mathcal M$ with respect to a contour $\Gamma$. For the case that the quotient $q$ of the two diagonal elements of $\mathcal M$ is a rational function, we show that due to the symmetric nature of the matrix $\mathcal M$, the second column in each of the two matrix factors that arise in the factorisation is determined in terms of the first column in each of these matrix factors, by multiplication by a rational matrix, and we give a method for determining the second columns of these factors. We illustrate our method with two examples in the context of a Riemann-Hilbert approach to obtaining solutions to the Einstein field equations.

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.

Desk Editor's Note private letter to a colleague

Symmetry of 2x2 matrices with rational diagonal ratio determines the second Wiener-Hopf factor columns from the first via a rational matrix, plus a method and gravity examples.

read the letter

The paper's core result is that for 2x2 symmetric M with rational q (ratio of diagonal entries), the symmetry forces the second column of each Wiener-Hopf factor to be the first column multiplied on the right by a rational matrix, and they give an explicit way to compute those columns. They demonstrate it on two examples drawn from Riemann-Hilbert constructions of Einstein solutions. That algebraic shortcut is the actual new piece here; it reduces the independent data you have to find when doing the factorization. The examples are useful because they show the method producing factors that can be plugged into the gravitational context without extra fitting. The restriction to rational q is stated up front, so the claim does not overreach. The work stays within functional analysis applied to a specific physical setting and does not pretend to handle general q or larger matrices. One soft spot is that the abstract and stress-test give no detail on how the method guarantees the factors meet the exact Wiener-Hopf analyticity and invertibility conditions on the contour; the paper presumably checks this in the examples, but a referee would want to see the verification steps written out clearly. Another minor point is that the scope is narrow by design, which is fine but means the result will mainly interest people already doing these Riemann-Hilbert problems in GR. Overall the argument looks internally consistent and the symmetry implication follows from the setup they describe. This is the kind of targeted technical note that belongs in a journal like Journal of Mathematical Physics or a factorization-focused venue. It deserves peer review because the reduction is concrete and reproducible from the stated assumptions, even if the audience is specialized.

Referee Report

0 major / 3 minor

Summary. The manuscript examines the canonical Wiener-Hopf factorization of 2×2 symmetric matrices M with respect to a contour Γ. For the case in which the quotient q of the two diagonal entries of M is a rational function, it claims that symmetry of M implies that the second column of each of the two factors is obtained from the first column by right-multiplication by a rational matrix, supplies an explicit constructive method for determining those columns, and illustrates the procedure with two examples drawn from a Riemann-Hilbert formulation of the Einstein field equations.

Significance. If the central derivation holds, the result supplies a structural simplification that reduces the number of independent functions to be determined in the factorization of symmetric matrices; the constructive method together with the two concrete gravitational examples constitutes a verifiable and immediately applicable contribution within the Riemann-Hilbert approach to exact solutions of the Einstein equations.

minor comments (3)

[Abstract] Abstract: the statement that 'the second column … is determined … by multiplication by a rational matrix' would be clearer if the abstract indicated the explicit form of that rational matrix (or at least its degree) rather than leaving the reader to infer it from the later sections.
§2 (or wherever the factorization is first written): the notation M = M₊ M₋ and the precise function spaces (e.g., whether the factors belong to the Wiener algebra or to a Hardy-space setting) should be stated once at the outset so that the subsequent column-wise construction can be read without ambiguity.
Examples section: the two gravitational illustrations would benefit from a short table or explicit listing of the rational functions q that arise, so that the reader can immediately check that the constructed rational multiplier indeed reproduces the second column.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of its content, and recommendation for minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

Derivation is self-contained from symmetry and rationality of q

full rationale

The central result states that when the quotient q of the diagonal entries of the symmetric matrix M is rational, symmetry implies the second columns of the Wiener-Hopf factors are obtained from the first columns by right-multiplication by a rational matrix, with an explicit constructive method supplied. This follows directly from the matrix symmetry and the rationality assumption without any parameter fitting, self-citation load-bearing steps, or reduction of the claimed property to its own inputs by construction. The restriction to rational q is stated explicitly as the setting of the theorem rather than an unexamined premise. No load-bearing self-citations or ansatz smuggling are indicated in the abstract or reader's summary.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are mentioned in the abstract; the result is presented as following from the symmetry of M and the rationality of q.

pith-pipeline@v0.9.0 · 5658 in / 1187 out tokens · 34852 ms · 2026-05-23T19:30:55.984574+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages · 2 internal anchors

[1]

Aniceto, M

P. Aniceto, M. C. Cˆ amara, G. L. Cardoso, and M. Rossell´ o, Weyl metrics and Wiener-Hopf factorisation , JHEP 05 (2020) 124, 1910.10632

work page arXiv 2020
[2]

Bastos, Y

M. Bastos, Y. Karlovich, and A. dos Santos, Oscillatory Riemann-Hilbert problems and the corona theorem , Journal of Functional Analysis 197 (2003) 347

work page 2003
[3]

Breitenlohner and D

P. Breitenlohner and D. Maison, On the Geroch Group , Ann. Inst. H. Poincare Phys. Theor. 46 (1987) 215

work page 1987
[4]

Cˆ amara and G

M.C. Cˆ amara and G. L. Cardoso, Riemann-Hilbert problems, Toeplitz operators and ergosurfaces, JHEP 06 (2024) 027 [ 2404.03373]

work page arXiv 2024
[5]

Cˆ amara and A.F

M.C. Cˆ amara and A.F. dos Santos, Wiener-Hopf factorization for a class of oscillatory symbols , Integral Equations and Operator Theory 36 (2000) 409

work page 2000
[6]

Cˆ amara, A.F

M.C. Cˆ amara, A.F. dos Santos, and M.P. Carpentier, Explicit Wiener-Hopf factorization and nonlinear Riemann-Hilbert problems , Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132 (2002) 45

work page 2002
[7]

Y. I. Karlovich and G. S. Litvinchuk, On generalization of the factorization of measurable matrix-functions and its applications to Fredh olm theory for certain classes of convolution type equations , Demonstratio Mathematica 26 (1993) 597

work page 1993
[8]

H. Lu, M. J. Perry, and C. N. Pope, Inﬁnite-dimensional symmetries of two-dimensional coset models coupled to gravity , Nucl. Phys. B 806 (2009) 656–683, 0712.0615

work page internal anchor Pith review Pith/arXiv arXiv 2009
[9]

Mikhlin and S

S. Mikhlin and S. Pr¨ ossdorf, Singular integral operators, Springer-Verlag, Berlin, 1986. Translated from German by Albrecht B¨ ottcher and Reinhard Lehmann

work page 1986
[10]

Weyl metrics and the generating conjecture

L. Richterek and J. Horsky, Weyl metrics and the generating conjecture , Czech. J. Phys. 54 (2004) 1451 [ gr-qc/0408049]. 20

work page internal anchor Pith review Pith/arXiv arXiv 2004

[1] [1]

Aniceto, M

P. Aniceto, M. C. Cˆ amara, G. L. Cardoso, and M. Rossell´ o, Weyl metrics and Wiener-Hopf factorisation , JHEP 05 (2020) 124, 1910.10632

work page arXiv 2020

[2] [2]

Bastos, Y

M. Bastos, Y. Karlovich, and A. dos Santos, Oscillatory Riemann-Hilbert problems and the corona theorem , Journal of Functional Analysis 197 (2003) 347

work page 2003

[3] [3]

Breitenlohner and D

P. Breitenlohner and D. Maison, On the Geroch Group , Ann. Inst. H. Poincare Phys. Theor. 46 (1987) 215

work page 1987

[4] [4]

Cˆ amara and G

M.C. Cˆ amara and G. L. Cardoso, Riemann-Hilbert problems, Toeplitz operators and ergosurfaces, JHEP 06 (2024) 027 [ 2404.03373]

work page arXiv 2024

[5] [5]

Cˆ amara and A.F

M.C. Cˆ amara and A.F. dos Santos, Wiener-Hopf factorization for a class of oscillatory symbols , Integral Equations and Operator Theory 36 (2000) 409

work page 2000

[6] [6]

Cˆ amara, A.F

M.C. Cˆ amara, A.F. dos Santos, and M.P. Carpentier, Explicit Wiener-Hopf factorization and nonlinear Riemann-Hilbert problems , Proceedings of the Royal Society of Edinburgh: Section A Mathematics 132 (2002) 45

work page 2002

[7] [7]

Y. I. Karlovich and G. S. Litvinchuk, On generalization of the factorization of measurable matrix-functions and its applications to Fredh olm theory for certain classes of convolution type equations , Demonstratio Mathematica 26 (1993) 597

work page 1993

[8] [8]

H. Lu, M. J. Perry, and C. N. Pope, Inﬁnite-dimensional symmetries of two-dimensional coset models coupled to gravity , Nucl. Phys. B 806 (2009) 656–683, 0712.0615

work page internal anchor Pith review Pith/arXiv arXiv 2009

[9] [9]

Mikhlin and S

S. Mikhlin and S. Pr¨ ossdorf, Singular integral operators, Springer-Verlag, Berlin, 1986. Translated from German by Albrecht B¨ ottcher and Reinhard Lehmann

work page 1986

[10] [10]

Weyl metrics and the generating conjecture

L. Richterek and J. Horsky, Weyl metrics and the generating conjecture , Czech. J. Phys. 54 (2004) 1451 [ gr-qc/0408049]. 20

work page internal anchor Pith review Pith/arXiv arXiv 2004