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Spherically Symmetric Geometrodynamics in Jordan and Einstein frames

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Pith's one-line read Spherically symmetric scalar-tensor frames are canonically equivalent only after gauge fixing, and a singular conformal map takes the FJNW naked singularity to the BBMB black hole.

desk verdict The corrected Jordan-frame momenta and the FJNW-to-BBMB solution map are solid, but the central canonicality claim on the reduced phase space is asserted rather than demonstrated—and the missing Dirac bracket computation is the load-bearing issue. read the letter →

arxiv 2501.08364 v2 pith:FBMMM3LQ submitted 2025-01-14 gr-qc hep-th

Matteo Galaverni , Gabriele Gionti This is my paper · ORCID
classification gr-qchep-th MSC 83C0583C5783D05
keywords JordanframeEinsteinconformaltransformationHamiltoniancanonicalsphericallysymmetricgeometrodynamicsFJNWsolutionBBMBblackholeADMboundaryterms
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

This paper studies spherically symmetric scalar-tensor gravity in the Hamiltonian (ADM) formalism and argues that the Jordan and Einstein frames are connected by a Hamiltonian canonical transformation once the lapse and radial shift are gauge-fixed and imposed as secondary constraints. The same conformal transformation has a singular point: applied to the Fisher–Janis–Newman–Winicour (FJNW) static solution, which has a naked curvature singularity, it produces the Bocharova–Bronnikov–Melnikov–Bekenstein (BBMB) black hole in the Jordan frame, a spacetime with a different singularity structure. The central message is that frame changes in scalar-tensor gravity are not automatically physical equivalences, and a singular conformal factor can map physically inequivalent spacetimes onto each other. The authors also emphasize that boundary terms in the action are needed to obtain the correct canonical momenta in the Jordan frame.

What carries the argument

The carrying object is the conformal (Weyl) transformation between frames, combined with the reduced-phase-space canonical formalism. Gauge-fixing the lapse and shift and treating them as secondary constraints turns the primary first-class constraints into second-class ones, so Dirac brackets replace Poisson brackets and, after strongly imposing the constraints, the frame map is canonical on $\{\Lambda,R,\phi,\pi_\Lambda,\pi_R,\pi_\phi\}$. The companion machinery is the explicit ADM reduction with all boundary terms kept; the paper argues that integration-by-parts terms in the Jordan-frame action change the canonical momenta, and that only with those terms do the inter-frame momentum relations (57)–(59) hold. The singularity of the conformal factor at $\phi=-\sqrt6$ is what makes the map between the FJNW and BBMB solutions singular.

What would settle it

Substitute the stated static fields (75)–(79) into each Jordan-frame equation of motion (63)–(68) and check the cancellation; a single non-cancelling term would break the solution-mapping claim. Independently, compute the second-class constraint matrix $\{\chi_i,\chi_j\}$ for the lapse and shift gauge fixings near the conformal singularity $\rho=b/2$; if the matrix is not invertible there, the frame map is not canonical at the radial coordinate on which the FJNW-to-BBMB mapping depends.

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Extended reading notes

Core claim

The paper claims two things. First, on the phase space reduced by gauge-fixing $N$ and $N^r$ and imposing them as secondary constraints, the conformal transformation $$\tilde g_{\mu\nu}=\left(1-\frac{\$phi^{2}$}{6}\right)g_{\mu\nu},\qquad \tilde\phi=\sqrt6\,\$tanh^{{-1}}$\!\left(\frac{\phi}{\sqrt6}\right)$$ is a Hamiltonian canonical transformation, with the momenta related by (57)–(59); without gauge fixing, the Poisson brackets of the lapse and shift with $\pi_\phi$ do not vanish, so the map is not canonical on the full phase space. Second, for the special value $\gamma=1/2$, the FJNW solution of the Einstein frame is carried by this transformation into the BBMB black hole of the Jordan frame, whose lapse is $N=1-b/(4\rho)$ in the radial coordinate $\rho$. The conformal factor $1-\phi^2/6$ vanishes at $\rho=b/2$ (equivalently $r=b$), which is why the naked-singularity spacetime and the black-hole spacetime differ: the transformation is singular exactly where the FJNW curvature singularity sits, so the FJNW manifold $r>b$ covers only the region $\rho>b/2$ of the BBMB manifold.

Load-bearing premise

The load-bearing premise is that the gauge-fixing construction from the authors' earlier work makes the frame transformation Hamiltonian canonical for this conformally coupled spherically symmetric system, together with the asserted but not displayed verification that the listed FJNW and BBMB fields solve the corresponding Hamiltonian equations.

Editorial extensions

If this is right

  • If the frame transformation is canonical only after gauge fixing, then the full phase spaces of the Jordan and Einstein frames are not equivalent, and quantization or reduced-phase-space methods must specify which frame they use.
  • The FJNW naked singularity and the BBMB black hole, physically inequivalent in their curvature invariants, are still mapped into each other by the singular conformal transformation, so conformal maps can serve as generators of new solutions.
  • Correct Jordan-frame equations of motion require the boundary contributions to the action; dropping them breaks the momentum relations that make the canonical map work.
  • Conformal-frame equivalence should not be assumed in scalar-tensor gravity, since the singularity structure and horizon structure of a solution can change under a frame change.

Reading between the lines

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

  • Beyond the paper: the same gauge-fixing mechanism should make the frame map canonical for a broader class of scalar-tensor theories, since the argument uses only the lapse/shift gauge fixing and the momentum relations, not the specific conformal coupling.
  • A testable extension is to compute the Dirac-bracket matrix $\{\chi_i,\chi_j\}$ explicitly for this system; if it degenerates as the conformal factor vanishes, the claims of canonicality and of the FJNW-to-BBMB mapping would fail exactly at $\rho=b/2$.
  • The paper's result suggests a frame-invariant criterion for physical equivalence: two frames connected by a conformal factor that vanishes on a hypersurface should not be expected to share causal or singularity properties; checking this against other known conformal pairs would sharpen that criterion.
  • One could also derive the Jordan-frame equations from a covariant phase-space method to confirm that the boundary terms, not a choice of convention, are responsible for the momentum relations.
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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 / 4 minor

Summary. The paper constructs the ADM Hamiltonian formulation for spherically symmetric scalar-tensor gravity in the Einstein frame (general relativity minimally coupled to a massless scalar) and in the Jordan frame (a conformally coupled scalar), with a careful treatment of boundary terms for noncompact foliations. It derives the canonical momenta, the Hamiltonian and momentum constraints, and Hamilton's equations in both frames. The paper then claims that, on the reduced phase space obtained by gauge-fixing the lapse and the shift and implementing them as secondary constraints, the conformal frame transformation (55)-(56) is a Hamiltonian canonical transformation, even though it is not canonical on the extended phase space. The paper applies this result to map the FJNW naked-singularity solution in the Einstein frame to the BBMB black hole in the Jordan frame, and it explains the difference in singularity structure through the vanishing of the conformal factor at a particular radius.

Significance. The paper's strengths are its systematic treatment of boundary terms and its explicit, benchmarked formulas for the spherically symmetric Hamiltonian systems; the reductions to the Schwarzschild limit and to known FJNW and BBMB results are correctly reproduced, and the paper carefully checks the GR limit against the existing literature. If the canonicality claim is established, the paper would provide a concrete and nontrivial example of frame transformations being canonical only after gauge fixing, and of singular conformal transformations connecting physically inequivalent solutions. However, the central canonicality claim is deferred to prior work and is not verified for the present system, which limits the current significance of the paper until the missing computation is supplied.

major comments (3)
  1. [Section 4, after Eq. (62)] The load-bearing claim that the transformation (56)-(59) is Hamiltonian canonical on the reduced phase space is not demonstrated in this manuscript. The gauge conditions χ_i are not given and the second-class constraint matrix C_{αβ} is not displayed, so the Dirac brackets that would replace the Poisson brackets are unknown. A direct Poisson-bracket check on the six-dimensional phase space shows that the map is not canonical: while {eΛ,eπφ}=0 and {eR,eπφ}=0, one finds {eπΛ,eπφ}=φπΛ/(6Ω^{1/2}) and {eπR,eπφ}=φπR/(6Ω^{1/2}) with Ω=1-φ²/6. The canonicality claim requires these extra terms to be cancelled by the Dirac-bracket corrections; whether they cancel depends on the explicit choice of gauge conditions, which is not stated. Please display the gauge conditions, the C matrix, and the resulting Dirac brackets among {Λ,R,φ,πΛ,πR,πφ}, and show explicitly that (56)-(59) preserve them.
  2. [Section 5.1 and 5.2, Eqs. (72)-(79)] The solution-mapping claim rests on the assertions that (72)-(74) satisfy the EF equations (37)-(42) and that (75)-(79) satisfy the JF equations (63)-(68) with H=0. The text says 'It is easy to verify' and 'It is easy to check' without displaying the verification. Given the length of the Jordan-frame Hamiltonian (53) and the number of terms in (63)-(68), a single algebraic slip would invalidate the mapping. Please include the verification, either as an explicit substitution or by stating that the substitutions were checked with a computer algebra system and providing the corresponding code or output.
  3. [Section 5.3, Eqs. (91)-(97)] The interpretation of the singularity structure relies on the conformal factor (1-φ²/6) vanishing at ρ=b/2. Please clarify the domain of the coordinate transformation (84) more carefully: the relation (90) is double-valued for ρ in (b/4,b/2) versus ρ>b/2, and the paper works only with the branch ρ>b/2. Stating this branch restriction explicitly would prevent an apparent ambiguity in the mapping between r>b and the BBMB manifold.
minor comments (4)
  1. [Section 4, Eq. (61)] The bracket {eN^r,eπφ} is claimed to be -N^r Ω^{1/2} φ/3, but from (56)-(59), eN^r=N^r and eπφ has no dependence on N^r or its conjugate momentum, so the bracket should vanish. If the computation is done in an extended phase space where N^r is treated as a canonical coordinate, the definition of eπφ and the phase-space variables must be stated explicitly; as written, Eq. (61) is inconsistent with the displayed transformation.
  2. [Abstract and Introduction] There are several typos and inconsistencies in the front matter: 'We discussed' in the abstract should be 'We discuss'; 'Jf' appears in the Introduction where 'JF' is meant; and the PACS line contains 'discribing' instead of 'describing'.
  3. [Section 5.3, Eqs. (91)-(93)] The notation in the limits labeled 'γ=1 −−−→ 0' is nonstandard and visually confusing; please use the standard notation lim_{γ→1} and place the limiting value on the right-hand side.
  4. [Section 5.2, after Eq. (75)] The phrase 'the other canonical variables in the JF are obtained using Eqs. (56)' is imprecise: N, Λ, and R are metric coefficients, not canonical variables. Consider saying 'the remaining metric coefficients in the JF are obtained...'.

Circularity Check

1 steps flagged · score 4.0 of 10

The canonicality of the Jordan/Einstein transformation on the reduced phase space is asserted by reference to the authors' own prior work [16] without an in-paper Dirac-bracket computation, making the central Hamiltonian-equivalence claim self-citation load-bearing; the FJNW-to-BBMB solution mapping itself is not circular.

  1. self citation load bearing [Section 4, paragraph following Eq. (62); restated in Section 6]
    "We can avoid this problem considering a reduced phase space where we gauge fix the lapse function N and the shift function N r, implementing them as secondary constraints χi as discussed in [16]. ... Afterward - strongly imposing the second class constraints - we reduce the dynamical variables to:{Λ, R, ϕ, πΛ, πR, πϕ}. On this reduced phase space, defined by gauge fixing the lapse and the shift function, the transformation from EF and JF is Hamiltonian canonical."

    The paper's central Hamiltonian-equivalence claim is not derived for the present spherically symmetric conformally coupled system: the second-class constraint matrix Cαβ is never displayed and no Dirac-bracket computation is shown. The only support offered is '[16]', a prior paper by the same authors, so the claim is imported rather than established from the paper's own equations. This is load-bearing because the advertised result and the interpretation of the FJNW-to-BBMB map as a 'Hamiltonian canonical' transformation depend on it. The paper itself derives only the negative statement, Eqs.

full rationale

The paper's negative result on the full phase space is derived in-paper: Eqs. (60)-(61) explicitly show nonzero Poisson brackets, so that part is self-contained and honest. The problematic step is the positive assertion after Eq. (62): after gauge-fixing N and N^r and implementing them as secondary constraints 'as discussed in [16]', the transformation is declared Hamiltonian canonical on the reduced phase space. No constraint matrix, Dirac brackets, or reduced bracket computation is provided for the conformally coupled system, so the claim is not checked against the paper's own equations; it is inherited from [16], whose authors overlap with the present paper. The FJNW-to-BBMB mapping is not circular: it is obtained by applying the conformal transformation (55)-(56) to a known solution and then checking the equations of motion, with no fitted parameter or target result used as input. The 'It is easy to verify/check' passages are omitted proofs rather than circular steps. Overall, the paper has substantial independent content in its ADM derivation and solution mapping, but its central canonicality claim is load-bearing self-citation, so the circularity score is moderate rather than zero.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper fits no data and introduces no new physical entities. The only numerical parameters are the integration constants b and gamma of the known FJNW solution family (gamma=1/2 is chosen by hand in Section 5.2 only for the explicit BBMB display). The derivation leans on standard ADM machinery, the standard conformal frame map of Dicke/Bekenstein, the authors' own gauge-fixed canonicality framework from [16] (asserted, not re-derived here), and two unshown 'easy to check' solution verifications. This is a moderate axiom burden, concentrated in the inherited canonicality claim and the asserted verifications.

free parameters (2)
  • gamma (FJNW scalar-charge parameter) = gamma = 1/2 (chosen by hand for the explicit BBMB example)
    Section 5.2 picks gamma=1/2 to exhibit the BBMB form explicitly (Eqs. (80)-(89)); the qualitative claims (conformal factor vanishing at r=b) hold for the general 0<gamma<1 family, so this choice does not carry the argument.
  • b (FJNW and BBMB scale parameter) = Arbitrary; set by the physical mass scale (b = 2 sqrt(m^2 + q^2/2))
    b is the integration constant of the known FJNW/BBMB solution family (Eq. (71)); it labels the solution rather than being fitted or tuned for the derivation.
assumptions (5)
  • standard math ADM 3+1 decomposition with full boundary terms (Eq. (1)) gives a well-defined action for asymptotically flat, non-compact foliations after background subtraction (Eqs. (4)-(5)).
    Invoked in Sections 2.1-2.2 and used to define momenta; standard framework cited to Hawking-Hunter [30] and Poisson [31].
  • domain assumption The conformal map g_tilde = (1 - phi^2/6) g, phi_tilde = sqrt(6) atanh(phi/sqrt(6)) (Eq. (55)) sends the JF action (43) to the EF action (25).
    Used in Section 4 to connect frames and in Section 5 to generate the BBMB solution; standard Dicke/Bekenstein result, cited to [29,47].
  • domain assumption Gauge-fixing N and N^r as secondary constraints and using Dirac brackets yields a reduced phase space on which the frame transformation is Hamiltonian canonical.
    Section 4, after Eqs. (60)-(62), imports this from the authors' [16]; the constraint matrix and Dirac brackets are not displayed for this system.
  • ad hoc to paper The FJNW fields (72)-(74) satisfy the EF Hamiltonian equations (37)-(42), and the conformally transformed fields (75)-(79) satisfy the JF equations (63)-(68) (with H=0).
    Asserted in Sections 5.1-5.2 ('It is easy to verify/check') with no substitution shown; these verifications are load-bearing for the solution-mapping claim.
  • domain assumption Fall-off conditions (28)-(30) suffice for the background subtraction defining the physical action.
    Section 3, Eqs. (28)-(30); standard asymptotically flat ADM assumptions cited to Kuchar [17] and Loll-Pires [40].

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Pith. "Pith review of Spherically Symmetric Geometrodynamics in Jordan and Einstein frames." pith.science (2026). https://pith.science/paper/FBMMM3LQ

@misc{pith2026250108364,
  author       = {Pith},
  title        = {Pith review of: Spherically Symmetric Geometrodynamics in Jordan and Einstein frames},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FBMMM3LQ}},
  note         = {Machine review of arXiv:2501.08364}
}
read the original abstract

Spherically symmetric geometrodynamics is studied for scalar-tensor theory and Einstein General Relativity minimally coupled to a scalar field. We discussed the importance of boundary terms and derived the equations of motion in the Hamiltonian canonical formalism both in the Jordan and Einstein frames. These two frames are connected through an Hamiltonian canonical transformation on the reduced phase space obtained gauge-fixing the lapse and the radial shift functions. We discussed the effects of the singularity of the Hamiltonian canonical transformation connecting Jordan and Einstein frames for two static solutions (Fisher, Janis, Newman and Winicour solution in the Einstein frame and Bocharova-Bronnikov-Melnikov-Bekenstein black hole solution in the Jordan frame).

Figures

Figures reproduced from arXiv: 2501.08364 by the authors.

Figure 1
Figure 1. The 4-dimension manifold (M, gµν) can be foliated in 3-dimensions space-like sub-manifolds at a given time (Σt, hij ) with normal vector n µ . The time-like boundaries correspond to a 3-dimension time-like foliation (B, γab) with normal vec￾tor u µ . Note that in general the space-like and the time-like foliations are not orthogonal (n µuµ ̸= 0). The boundary of M is nothing but ∂M = Σt0 ∪Σt1 ∪B. The sub-manifolds Σ… view at source ↗
Figure 2
Figure 2. Coordinate r as a function of the new radial coordinate ρ defined in Eq. (84). The conformal transformation (55)-(56) between JF and EF is well posed for ρ > b/2, therefore the FJNW manifold r > b is mapped to the part of BBMB mani￾fold with ρ > b/2 (red-dashed line). 5.3 Peculiar aspects of the transformation between JF and EF We study here the properties of the static solutions pre￾sented earlier and evaluate the … view at source ↗
Figure 3
Figure 3. , and it is not possible to connect the two frames with the conformal transformation (55). Moreover, the scalar field ϕe in the EF diverges, see Eqs. (70). In other words, the conformal transformation between JF and EF (56) is well posed when (1 − ϕ 2 6 ) > 0, or ρ > b/2 (see [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

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Forward citations

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