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Complex degenerate metrics in general relativity: a covariant extension of the Moore-Penrose algorithm

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper constructs a covariant Moore-Penrose pseudoinverse that gives complex degenerate metrics unique, well-defined curvature tensors.

desk verdict The advertised covariant Moore-Penrose construction fails its own adjoint conditions for generic complex metrics, so the central uniqueness/tensoriality claims are unsupported; the geodesic completeness proof and the FLRW holomorphy observation are worth preserving. read the letter →

arxiv 2502.10053 v1 pith:HNN7NZ2K submitted 2025-02-14 gr-qc math-phmath.MP

Arthur Garnier , Emmanuele Battista This is my paper · ORCID
classification gr-qcmath-phmath.MP MSC 15A0953B2083C0583F05
keywords covariantMoore-PenrosepseudoinversedegeneratemetriccomplexmetricstorsiontensorcomplexifiedSchwarzschildspacetimeFLRWcosmologygeodesicsofgeneralrelativity
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 tries to establish that general relativity can be extended to metrics that are complex and nowhere invertible, provided one replaces the inverse metric by a pseudoinverse selected by coordinate-covariant conditions. The central object is a symmetric tensor $\tilde{g}^{ab}$ defined by four tensor identities that generalize the Moore-Penrose axioms, with the adjoint evaluated using a fixed background Riemannian metric $\zeta_{ab}$. The authors argue that this pseudoinverse is unique, transforms as a tensor, and yields a metric-compatible connection whose torsion measures the degeneracy. If correct, this gives well-defined curvature tensors, Einstein equations, and geodesic notions for non-invertible complex spacetimes, covering complexified black holes and FLRW cosmology.

What carries the argument

The machinery is a covariant dagger adjoint built from a fixed Riemannian metric $\zeta$: for tensor components, the adjoint uses $\zeta$ to raise and lower indices after complex conjugation, so the standard Moore-Penrose conditions become tensor equations. Existence and uniqueness are obtained by writing $\zeta = {}^t B B$ (the Cholesky decomposition), pulling the degenerate metric $g$ back to $g_0 = {}^t B^{-1} g B^{-1}$, taking its ordinary Moore-Penrose inverse, and pushing the result back. This object carries the argument because it makes the connection (55), the torsion (62), and the curvature invariants well-defined coordinate-invariant quantities.

What would settle it

Take a small constant-rank complex symmetric matrix $g$ that does not commute with a chosen positive-definite $\zeta$, construct the candidate $\tilde{g} = B^{-1}({}^t B^{-1} g B^{-1})^{+} {}^t B^{-1}$ from the Cholesky factor $B$ of $\zeta$, and test whether the component identities (52c) and (52d) hold exactly; a single failure would falsify the general existence theorem as stated.

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Extended reading notes

Core claim

For any symmetric complex degenerate metric of constant rank, the covariant Moore-Penrose conditions (51) have a unique solution $\tilde{g}$ with respect to any fixed Riemannian metric $\zeta$; the proof proceeds by Cholesky decomposition of $\zeta$ in local charts and gluing. This makes $\tilde{g}$ a true tensor, so the connection (55), Riemann tensor (63), Ricci tensor (67), and Einstein tensor (71) are genuine tensors whose coordinate transformation is unproblematic. The connection is $g$-compatible, but $\tilde{g}$ is generally not covariantly constant, and the antisymmetric part of the connection yields a torsion tensor (62) that vanishes only for maximal-rank or highly symmetric metrics. Applications show that complexified Schwarzschild and Reissner-Nordström metrics satisfy the vacuum or Einstein-Maxwell equations with the same Kretschmann scalar as their ordinary counterparts, while the FLRW model yields complex Friedmann equations and interprets holomorphy of the scale factor as the condition of vanishing torsion.

Load-bearing premise

The construction's uniqueness and tensoriality rest on Appendix A's claim that pulling back through the Cholesky factor of the fixed metric $\zeta$ always produces a tensor satisfying the covariant adjoint conditions; if this does not hold for complex metrics, the pseudoinverse metric and every derived object lose their coordinate invariance.

Editorial extensions

If this is right

  • Complexified Schwarzschild and Reissner-Nordström metrics can be handled as genuine spacetimes with $\tilde{g}$, and their Kretschmann scalars coincide with the real-valued solutions, locating the singularity at $r=0$.
  • For the complex FLRW metric, the covariant pseudoinverse gives complex Friedmann equations; the conservation law $\nabla_a T^{ab}=0$ holds exactly when the scale factor is a holomorphic function of $\tau = T + i t$.
  • In general degenerate geometries the Einstein tensor no longer satisfies the ordinary conservation law; the modified identity (73) contains extra terms involving $\nabla_a \tilde{g}^{bc}$ and the torsion.
  • When the metric has maximal rank or high symmetry, such as Schwarzschild, Reissner-Nordström, and Reissner-Nordström-de Sitter, the torsion vanishes and the connection reduces to the pseudoinverse-mediated Christoffel form (77).
  • Extremal curves, called contravariant autoparallels, provide the degenerate-metric analogue of geodesics; in the Schwarzschild model they run entirely inside either the Euclidean or the Lorentzian section.

Reading between the lines

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

  • Editorial inference: A direct numerical check of the four covariant identities for arbitrary constant-rank complex symmetric matrices, using a non-commuting positive-definite $\zeta$, would settle the existence claim independently of the Appendix A proof.
  • Editorial inference: If the construction extends, every degenerate symmetric $(2,0)$ tensor field, not just metrics, could be inverted covariantly, giving a general calculus for degenerate tensors in any dimension.
  • Editorial inference: The dependence on the auxiliary metric $\zeta$ is a genuine freedom, so physical predictions such as torsion and conservation laws may shift under different choices of $\zeta$; selecting $\zeta$ naturally becomes the next question the framework raises.
  • Editorial inference: The proof that Euclidean Schwarzschild is geodesically complete while Lorentzian extremals can reach $r=0$ suggests that whether a complexified spacetime is singular may depend on which real section the geodesic is confined to.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper proposes a covariant extension of the Moore-Penrose pseudoinverse for constant-rank symmetric complex degenerate metrics, replacing the ordinary adjoint by a ζ-twisted adjoint defined via an auxiliary Riemannian metric ζ. The authors claim that the resulting pseudoinverse is unique, that the connection (55) and curvature tensors (63)–(71) are genuine tensors, and that metric degeneracy is encoded in a torsion tensor. These tools are applied to complexified Schwarzschild, Reissner–Nordström, Reissner–Nordström–de Sitter, and FLRW metrics, and a new notion of extremal curves is introduced. The paper also contains a self-contained proof of geodesic completeness of the Euclidean section of Schwarzschild (Sec. II A 2).

Significance. If the central existence/uniqueness theorem were correct, the framework would provide a principled way to handle nowhere-invertible complex metrics and could be relevant to quantum-gravity path integrals and signature-change models. The paper is transparent about the role of the auxiliary metric ζ and includes a nontrivial original proof of Euclidean Schwarzschild geodesic completeness. However, the central construction fails for generic complex metrics: the component equations (52) are not equivalent to the ζ-twisted adjoint conditions (51), and a simple counterexample in the flat case shows that the Cholesky-translated pseudoinverse does not satisfy (51c). Since the connection, torsion, and curvature tensors all rely on this construction, the main claim of the paper is not established. The ζ-dependence of the torsion—acknowledged by the authors themselves—further weakens the physical interpretation.

major comments (3)
  1. [Appendix A; Sec. III A, Eqs. (51)-(52)] The component equations (52) do not reproduce the covariant adjoint conditions (51) for complex metrics. The right and left dagger operations (47) and (48) include complex conjugation of the tensor components, but the printed component identities (52c) and (52d) contain no overbars. For a complex metric, (52c) is therefore not equivalent to (g˜g)† = g˜g, and the claim in Sec. III A that (52) is 'equivalently' (51) is incorrect. The uniqueness proof in Appendix A is carried out for the wrong set of equations in the complex case.
  2. [Appendix A; Sec. III A] The Cholesky-based construction in Appendix A does not yield a tensor satisfying the ζ-twisted adjoint conditions. With ζ = B^T B, the paper sets g_0 = B^{-T} g B^{-1} and tilde g = B^{-1} tilde g_0 B^{-T}, where tilde g_0 is the standard Moore-Penrose inverse. But the standard conditions only imply that g_0 tilde g_0 is Hermitian, not that g tilde g is ζ-self-adjoint. For a concrete failing example, take g = [[1,i],[i,-1]] and ζ = diag(2,1), so B = diag(sqrt(2),1). The construction gives tilde g = [[1/9,-2i/9],[-2i/9,-4/9]], and direct computation yields g tilde g = [[1/3,-2i/3],[i/3,2/3]], whereas the ζ-twisted adjoint of g tilde g evaluates to [[1/3,-i/6],[4i/3,2/3]], which is not equal to g tilde g. Hence condition (51c) fails even locally in flat space, so the existence and uniqueness claim of Sec. III A is false as stated; consequently the tensorial character of the connection (55), the torsion (62), and the curvature tensors (63)–(71) rests on an invalid premise.
  3. [Sec. III B, Eq. (62); Appendix B; Sec. V] The torsion tensor (62) and the connection (55) depend on the arbitrary auxiliary metric ζ_ab. Condition (B1c) in Appendix B is introduced as a normalization without a geometric or physical justification, and the paper itself states in Sec. V that no natural choice of ζ is evident and that the dependence of physical predictions on ζ deserves separate study. As a result, the claim that metric degeneracy is 'geometrically represented' by a torsion tensor is not gauge-invariant: different choices of ζ can change the torsion even for the same degenerate metric, so the physical content of the framework is not fully determined by the metric alone.
minor comments (3)
  1. [Sec. III A] The metric is described as a '(2,0)-tensor field' but also as 'a global section of T*M⊗T*M'; a (2,0) tensor is a section of TM⊗TM, and later examples write gab with lower indices. Please make the index convention consistent throughout.
  2. [Sec. II A 2] The proof of geodesic completeness refers to Eq. (27) before that equation is derived; moving the derivation of the radial equation ahead of the initial-condition discussion would improve readability.
  3. [Sec. III A, Eqs. (47)-(48)] The index placement in the dagger operations (47) and (48) is unclear: for a field X^a_b, the right-hand side ζ^{ac} X^d_c ζ_{db} has the same index type as X, so the transposition of X is not displayed. Writing the complex transpose explicitly, for example ζ^{ac} \overline{X^c_d} ζ_{db}, would remove the ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the covariant Moore-Penrose construction is derived from stated assumptions and standard external theorems, with admitted auxiliary-metric dependence rather than circular reasoning.

full rationale

The paper's central claim is the existence and uniqueness of a covariant pseudoinverse metric satisfying the dagger conditions (51). This is presented as a theorem whose proof (Appendix A) uses the standard, externally established Moore-Penrose inverse, the Cholesky decomposition, and a gluing argument; it does not assume the conclusion it is proving. The auxiliary metric ζ_ab is an arbitrary input, and the paper explicitly acknowledges in Sec. V that 'the extent to which physical predictions depend on the choice of ζ_ab is an important topic that deserves consideration in a separate paper.' Dependence on an auxiliary structure is a limitation, not circularity. The torsion tensor (62) is a derived consequence of the explicitly constructed connection (55), and the statement that degenerate metrics can be represented geometrically by torsion is an interpretation of that construction, not an empirical prediction fitted to data. Self-citations (Refs. [29], [32], [56], [78], etc.) are used for background results and technical details, not as the load-bearing justification of the central existence/uniqueness claim. Whether Appendix A's Cholesky-based construction actually satisfies (51c,d) for complex metrics is a mathematical correctness question, not a circularity question: no step in the paper equates its output with its input by definition, nor does any prediction reduce to a fit or to a self-citation chain.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The central claim rests on a new auxiliary background metric ζ_ab, chosen by hand, plus standard smoothness results for Moore-Penrose inverses and Cholesky decompositions. The connection is singled out by ad hoc criteria (B1). No data or fitting is involved, but the dependence on ζ is a free input that determines all physical outputs.

free parameters (1)
  • Auxiliary Riemannian metric ζ_ab = Flat Euclidean metric in the Schwarzschild, Reissner-Nordström, and FLRW examples
    Introduced in Eq. (80) to define the covariant adjoint. The pseudoinverse, connection, torsion, and curvature all depend on ζ, and the paper admits no natural choice exists (Sec. V).
assumptions (5)
  • domain assumption The metric g has constant rank on the spacetime manifold M.
    Stated in Sec. III: 'any symmetric degenerate tensor having constant (but non-maximal) rank.' Excludes metrics that are degenerate only on hypersurfaces, which are the typical signature-changing spacetimes.
  • domain assumption The auxiliary metric ζ_ab is Riemannian (positive definite).
    Needed for the Cholesky decomposition and its smoothness in Appendix A. A Lorentzian ζ would not admit such a decomposition.
  • standard math The standard Moore-Penrose inverse of a constant-rank matrix is smooth and unique (Penrose's theorem).
    Used repeatedly in Appendix A to establish smoothness of the local pseudoinverse, including the derivative formula (A8).
  • standard math The Cholesky factor map ζ ↦ B(ζ) is smooth.
    Invoked in Appendix A via Refs. [99,100] and Eq. (A9) to ensure the glued pseudoinverse is smooth.
  • ad hoc to paper The connection Γ defined by (55) is the unique connection satisfying the criteria (B1a)-(B1c).
    The criteria are chosen to single out this connection; they are not derived from physical principles, making the entire connection scheme contingent on this choice.
invented entities (1)
  • Background Riemannian metric ζ_ab
    purpose: Provides a coordinate-independent definition of the adjoint (dagger) operation and thereby the covariant Moore-Penrose inverse.
    It is a fixed, non-dynamical structure with no physical origin or predicted observable; its arbitrariness is acknowledged in Sec. V.

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Pith. "Pith review of Complex degenerate metrics in general relativity: a covariant extension of the Moore-Penrose algorithm." pith.science (2026). https://pith.science/paper/HNN7NZ2K

@misc{pith2026250210053,
  author       = {Pith},
  title        = {Pith review of: Complex degenerate metrics in general relativity: a covariant extension of the Moore-Penrose algorithm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HNN7NZ2K}},
  note         = {Machine review of arXiv:2502.10053}
}
read the original abstract

The Moore-Penrose algorithm provides a generalized notion of an inverse, applicable to degenerate matrices. In this paper, we introduce a covariant extension of the Moore-Penrose method that permits to deal with general relativity involving complex non-invertible metrics. Unlike the standard technique, this approach guarantees the uniqueness of the pseudoinverse metric through the fulfillment of a set of covariant relations, and it allows for the proper definition of a covariant derivative operator and curvature-related tensors. Remarkably, the degenerate nature of the metric can be given a geometrical representation in terms of a torsion tensor, which vanishes only in special cases. Applications of the new scheme to complex black hole geometries and cosmological models are also investigated, and a generalized concept of geodesics that exploits the notion of autoparallel and extremal curves is presented. Relevance of our findings to quantum gravity and quantum cosmology is finally discussed.

Discussion (0). Continue with ORCID to comment.

Forward citations

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