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Aspects of Geometrodynamics in the Jordan and Einstein Frames

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Gauge-fixing the lapse and shift turns the conformal Jordan-to-Einstein frame change into a Hamiltonian canonical transformation, making the FJNW naked singularity map onto the BBMB black hole.

desk verdict A useful review-style consolidation of the author's Hamiltonian frame-equivalence work, but the central gauge-fixing claim is outsourced to prior papers and the ω=-3/2 example is applied at a point where the transformation is singular. read the letter →

arxiv 2505.02504 v1 pith:LRTPI4PB submitted 2025-05-05 gr-qc hep-th

Gabriele Gionti , S.J. This is my paper · ORCID
classification gr-qchep-th
keywords JordanframeEinsteinBrans-DicketheoryHamiltoniancanonicaltransformationDiracsecond-classconstraintsgaugefixinglapseandshiftsphericalsymmetryFJNWBBMBsolutions
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 establishes that the conformal transformation from the Jordan frame to the Einstein frame in scalar-tensor gravity is a Hamiltonian canonical transformation only after gauge-fixing the lapse and shift functions and reducing the phase space by the corresponding second-class constraints. On the full phase space the transformation is not canonical, because the Poisson brackets of the transformed lapse and shift with the scalar-field momentum do not vanish. By imposing $N = c(x)$ and $N_i = c_i(x)$ as secondary constraints, building the Dirac bracket, and imposing the constraints strongly, the paper obtains a reduced phase space on which the frame map obeys the canonical bracket relations. It applies this result in spherical symmetry, with boundary terms handled by background subtraction, to map the FJNW naked-singularity solution of the Einstein frame into the BBMB black hole solution of the Jordan frame. If the claim is correct, conformal frame changes become a reliable solution-generating tool, while the physical equivalence of observables between frames remains open.

What carries the argument

The load-bearing device is Dirac's second-class constraint reduction applied to the gauge-fixing conditions $N - c(x) \approx 0$ and $N_i - c_i(x) \approx 0$. Together with the primary constraints $\pi_N \approx 0$ and $\pi_i \approx 0$, these conditions form second-class pairs; the induced Dirac bracket projects out the lapse and shift directions, and imposing the constraints strongly yields the reduced phase space on which the frame map is canonical. The frame map itself is the conformal relation $\tilde h_{ij} = (1 - \phi^2/6) h_{ij}$ between the spatial metrics, with the induced rescalings of the momenta, lapse, and shift. In the spherically symmetric sector the same reduction is carried out with the ADM metric written in terms of $\Lambda(r)$ and $R(r)$, and the boundary terms needed for a well-posed variational principle are treated by subtracting a background geometry.

What would settle it

Take a non-static, spherically symmetric solution in the Jordan frame, choose a concrete gauge such as $N=1$ and $N_r=0$, integrate the Dirac-bracket equations of motion to obtain a Hamiltonian trajectory, map that trajectory to the Einstein frame with the conformal transformation, and check whether the image satisfies the Einstein-frame equations of motion in the same gauge with the same Dirac brackets; a mismatch in any component would falsify the claim.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Jordan-to-Einstein transformation is not canonical on the extended phase space, because the Poisson brackets of the transformed lapse and shift with the scalar momentum are nonzero, but it becomes canonical on the reduced phase space obtained by fixing $N = c(x)$ and $N_i = c_i(x)$ and treating these gauge conditions as secondary second-class constraints. The Dirac-bracket construction removes the gauge degrees of freedom, and after imposing the constraints strongly the remaining canonical variables of the two frames satisfy the required bracket relations, so solutions of the equations of motion map from one frame to the other. For Brans-Dicke theory with $\omega \neq -3/2$, the first-class constraint algebras of the two frames match; for $\omega = -3/2$ an extra conformal constraint appears and the frame transformation is singular, so that case is exceptional. The solution-mapping property is demonstrated concretely: the static FJNW solution in the Einstein frame is transformed into the static BBMB black hole solution in the Jordan frame, with the conformal factor vanishing at the horizon and turning the naked singularity into a regular null surface.

Load-bearing premise

The claim rests on treating the lapse and shift as pure gauge, so that fixing them to arbitrary functions and imposing those choices as second-class constraints removes no physical dynamics; if this gauge reduction changes the Hamiltonian dynamics, the canonical equivalence between the frames collapses.

Editorial extensions

If this is right

  • For $\omega \neq -3/2$, the Jordan and Einstein frames have identical first-class constraint algebras, so the reduced Hamiltonian dynamics of the two frames is structurally the same once lapse and shift are fixed.
  • Static spherically symmetric solutions map across frames: the Einstein-frame FJNW solution is carried into the Jordan-frame BBMB black hole solution by the canonical frame transformation.
  • The FJNW naked singularity becomes a regular horizon in the Jordan frame because the conformal factor vanishes at the value the scalar field reaches at that surface, showing that singularity structure is not frame-invariant.
  • The canonical equivalence is a mathematical equivalence of the Hamiltonian dynamics; which frame's observables correspond to measurements remains open, so the two frames are not automatically physically equivalent.
  • In the $\omega = -3/2$ case, the extra conformal constraint makes the frame transformation singular and the constraint algebras in the two frames differ, marking that case as exceptional.

Reading between the lines

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

  • Inference: if the reduction works for generic non-minimal couplings $f(\phi)R$, the same gauge-fixing procedure should turn the frame transformation into a canonical map for any scalar-tensor theory, not only Brans-Dicke; repeating the Dirac analysis for generic $f(\phi)$ would test this extension.
  • Inference: the singularity-to-horizon mapping implies that geodesic completeness and singularity theorems stated in one conformal frame need not carry over to the other, so numerical studies of collapse should specify which frame's metric is used for causal structure.
  • Inference: a testable consequence is that gauge-invariant quantities, such as the BBMB horizon area and the FJNW scalar charge, should be related by the frame transformation independently of the gauge choice; computing these Dirac observables explicitly would check the consistency of the mapping.
  • Inference: the same canonical reduction could be used to set initial data in numerical relativity, where conformal maps between frames are common, since the gauge-fixed Hamiltonian equations provide a consistency condition for choosing lapse and shift.
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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 / 5 minor

Summary. The paper argues that the transformation between the Jordan and Einstein frames in Brans-Dicke theory becomes a Hamiltonian canonical transformation once the lapse and radial shift functions are gauge-fixed and implemented as second-class constraints. It summarizes earlier work by the author for both ω ≠ −3/2 and ω = −3/2, then develops the ADM formulation for spherically symmetric scalar-tensor gravity, including boundary terms, and presents an explicit mapping from the Fisher–Janis–Newman–Winicour (FJNW) solution in the Einstein frame to the Bocharova–Bronnikov–Melnikov–Bekenstein (BBMB) solution in the Jordan frame for γ = 1/2. The paper concludes that, after gauge reduction, the frame transformation is canonical and can be used to generate solutions.

Significance. If the central claim is correct, the paper provides a Hamiltonian justification for the long-debated Jordan/Einstein frame equivalence and offers a solution-generating technique with an explicit nontrivial example. The FJNW-to-BBMB mapping is a well-known result and adds an external anchor to the discussion. The paper also gives a useful summary of the author's previously published constraint analyses. However, the key derivation—the Dirac-bracket calculation on the reduced phase space—is not reproduced here, and the formulation is not fully self-contained. The claim as stated covers ω = −3/2, a sector where the paper's own tables show differing constraint algebras and where the transformation is singular, so the general result as written is not established.

major comments (3)
  1. [Abstract and Section 3 (Tables 1 and 2)] The abstract and Section 3 claim canonical equivalence for both ω ≠ −3/2 and ω = −3/2, but the manuscript itself states that for ω = −3/2 the transformation is singular and that the constraint algebras in the two frames differ (Table 2). A canonical transformation must preserve the Dirac bracket algebra of the remaining constraints; the differing brackets in Table 2, for example {C_ϕ, H_i} ≠ 0 in the Jordan frame versus 0 in the Einstein frame, contradict canonicity in that sector. The claim should be explicitly restricted to ω ≠ −3/2, or a separate proof for ω = −3/2 must be supplied.
  2. [Section 3, Eqs. (15)–(19)] The central claim is asserted rather than demonstrated. After defining the gauge-fixing conditions as secondary constraints and writing the Dirac bracket formula, the paper jumps to the statement that on the reduced phase space the transformation is canonical, citing [8] and [15] without showing the computation. Since this is the main result, the manuscript should reproduce the Dirac-bracket calculation for the transformed canonical variables and verify that the gauge-fixed constraint algebra is preserved, or provide a detailed appendix. As written, the claim is unverifiable from the manuscript alone.
  3. [Section 5, Eqs. (42)–(48)] The FJNW-to-BBMB example applies the conformal transformation at a point where the transformation is singular. For γ = 1/2, the FJNW solution has φ̃ → −∞ as r → b, so the Jordan-frame scalar field approaches ϕ = −√6, making Ω = 1 − ϕ²/6 vanish and the field redefinition φ̃ = √6 tanh⁻¹(ϕ/√6) diverge. The Jordan-frame metric is regular at the corresponding ρ = b/2, but the canonical transformation itself is not defined at that point. The manuscript acknowledges that the conformal factor becomes null but does not explain how the canonical equivalence extends there. A limiting argument or an explicit restriction to the open domain Ω > 0 is required; as written, the example does not establish solution generation within the transformation's domain.
minor comments (5)
  1. [Section 3, Eq. (16)] The second gauge-fixing condition in the Einstein frame is incomplete; the expression eNi − ci(x)(16πGϕ)^{1/2} should be followed by '≈ 0' as in the Jordan frame.
  2. [Section 5, Eq. (42)] Equation (42) is garbled and unreadable as printed; the intended coordinate transformation between r and ρ should be written explicitly, for example (1 − b/r)^{1/2} = 1 − b/(2ρ) or the equivalent relation used to derive Eq. (48).
  3. [Section 5, text after Eq. (37)] The statement 'the FJNW metric is valid only for r < b' appears to be opposite to the usual domain r > b for γ ∈ (0,1); please verify the domain of validity with reference [38] and correct if necessary.
  4. [Section 4 and Section 5] The paper does not explicitly identify which results are new (e.g., the spherical-symmetry boundary-term analysis) and which are reproduced from the author's previous papers [8], [11]–[15]. A clearer statement of novelty would help the reader assess the contribution.
  5. [Throughout] There are several typographical errors, such as 'trasndormation' after Eq. (33), and the notation around the conformal factor Ω = 1 − ϕ²/6 is used inconsistently (sometimes as Ω, sometimes spelled out). A careful proofreading pass is needed.

Circularity Check

2 steps flagged · score 4.0 of 10

Central canonicity claim is self-cited rather than derived in this manuscript; the FJNW/BBMB example imports external content, so circularity is partial, not total.

  1. self citation load bearing [Section 3, after Eq. (19)]
    "On this reduced phase space, the transformation from the Jordan to the Einstein frames is a Hamiltonian canonical transformation [8]."

    This sentence states the paper's central result, repeated in the abstract and conclusion and used as the justification for mapping solutions in Section 5. The decisive step is not demonstrated in the present manuscript: no Dirac-bracket algebra on the reduced phase space is computed, no symplectic-form preservation is checked, and no independent theorem is invoked. The only support offered is the author's own previous article [8], with related claims from the same group ([11]-[15]). A self-citation is normal in a summary paper, but here the load-bearing premise of the whole derivation is imported from that self-citation rather than established in this text.

  2. self citation load bearing [Section 5, paragraph after Eq. (41)]
    "Applying the transformation from the Jordan to the Einstein frame on the FJNW solution, we can do it since it is a Hamiltonian canonical trasformation according to [8], of the EF case, see Eqs. (39)-(41), we obtain a solution in the JF."

    The FJNW-to-BBMB example is presented as a consequence of the canonicity result, but that canonicity result is exactly the self-cited claim from the previous step. In this manuscript the solution-mapping property is not re-derived from first principles; it is borrowed from [8]. The external status of FJNW and BBMB as known solutions provides independent content, which prevents a fully definitional loop, but the inference 'solution in the Einstein frame maps to a solution in the Jordan frame' is carried by the same self-citation chain.

full rationale

Most of the formal material in the paper is standard and externally anchored: the conformal transformation (28), the FJNW metric (36)-(41), and the BBMB form (46)-(47) are all known results, and the Einstein-frame action follows from the Jordan-frame action by direct substitution rather than by fitting. No fitted parameter is renamed as a prediction, and no equation is defined in terms of the result it is supposed to prove. The main circularity concern is the central claim that, after gauge fixing lapse and shift, the Jordan-Einstein transformation is Hamiltonian canonical on the reduced phase space; this is asserted with a citation to the author's own prior work [8] and sibling papers, and it is not re-derived here. The FJNW-BBMB example depends on that self-cited claim, although the solutions themselves are known independently. The paper also states that for omega = -3/2 the transformation is singular and the Dirac constraint algebras differ (Table 2), which is a scope limitation on the abstract's unqualified claim but is a correctness concern rather than a circularity step. On balance, the self-citation is load-bearing but not definitionally circular, so the score is moderate.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new fitted constants or invented entities. It relies on standard Hamiltonian mechanics plus domain assumptions about the ADM action, the gauge-fixing reduction, and the conformal solution map. The gauge-fixing assumption is the most fragile item because the central canonical-equivalence claim depends on it.

assumptions (4)
  • domain assumption The ADM 3+1 decomposition with the boundary terms (21) correctly represents the gravitational action for non-compact spherically symmetric geometries.
    Used in Section 4 to define the Hamiltonian and momenta in the Jordan frame.
  • domain assumption The lapse and shift functions can be gauge-fixed as N=c(x), N_i=c_i(x) and promoted to second-class constraints without changing the physical content of the theory.
    This is the load-bearing premise that turns the noncanonical transformation into a canonical one on the reduced phase space (Section 3, Eqs. (15)-(19)).
  • domain assumption The conformal transformation g_tilde = (1 - phi^2/6) g is a valid solution-map between the Jordan and Einstein frames, including where the conformal factor vanishes.
    Used in Section 5 to map FJNW to BBMB; the conformal factor vanishes at rho=b/4, so invertibility there is assumed rather than proven.
  • standard math Dirac's algorithm for second-class constraints and Dirac brackets is valid.
    Invoked in Eqs. (17)-(19) for the gauge-fixed lapse and shift.

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Pith. "Pith review of Aspects of Geometrodynamics in the Jordan and Einstein Frames." pith.science (2026). https://pith.science/paper/LRTPI4PB

@misc{pith2026250502504,
  author       = {Pith},
  title        = {Pith review of: Aspects of Geometrodynamics in the Jordan and Einstein Frames},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LRTPI4PB}},
  note         = {Machine review of arXiv:2505.02504}
}
read the original abstract

We will summarize recent results on the Hamiltonian equivalence between the Jordan and Einstein frames based on the analysis of Brans-Dicke theory for both cases \omega\neq -\frac{3}{2} and \omega =-\frac{3}{2}. We will introduce and perform ADM analysis for spherically symmetric solutions of gravity. We will discuss with particular care the problem of the boundary terms to be introduced in the general case of spherical symmetry. These two frames are connected through a Hamiltonian canonical transformation on the reduced phase space obtained by gauge fixing the lapse and the radial shift functions. We introduce and discuss 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: 2505.02504 by the authors.

Figure 1
Figure 1. renormalization of the theory [28]. The action, in the Jordan frame, following from (22), is similar to (21) extending the considerations in [3] as explained in [15]. In general, this action is defined for compact geometries. For non-compact geometries, we need to define a physical action S PHY S that is the action above mentioned minus a suitable background geometry g0 [31] [15] in such a way the action is not dive… view at source ↗

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