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Emergent Lorentzian dispersion relations from a Euclidean scalar-tensor theory

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Pith's one-line read A Euclidean scalar-tensor theory can admit massless tensor modes with Lorentzian dispersion relations, so the low-energy gravity sector would look exactly Lorentzian.

desk verdict A concrete tensor-sector existence result with an unproved mode-suppression argument; worth refereeing after cleanup. read the letter →

arxiv 2505.00112 v2 pith:CTJ7BRV5 submitted 2025-04-30 gr-qc

Justin C. Feng , Shinji Mukohyama , Sante Carloni This is my paper · ORCID
classification gr-qc
keywords emergentLorentzsignatureclockfieldEuclideangravityscalar-tensortheorytensorperturbationsdispersionrelationstachyonicmodesboundaryconditions
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 asks whether a theory formulated on a Euclidean (all-plus-signature) manifold can nevertheless produce degrees of freedom that obey the Lorentzian rules of space and time. It studies small perturbations of a renormalizable shift-symmetric scalar-tensor theory around a flat background in which a clock scalar field has a constant nonzero gradient. It finds a parameter regime where the tensor perturbations, the analogues of gravitational waves, obey the massless Lorentzian dispersion relation $\omega^2 = (\mu_2/\mu_3) k^2$ with a positive kinetic term. The remaining modes satisfy Euclidean dispersion relations or have large tachyonic masses, and the paper argues they can be suppressed by boundary conditions. The upshot is an explicit existence scenario in which time and light cones are emergent, not fundamental.

What carries the argument

The central object is the quadratic perturbation action for the theory, split into tensor, vector, and scalar sectors. The load-bearing identity is the low-momentum dispersion relation $\omega^2 = (\mu_2/\mu_3) k^2$ for tensor modes, with $\mu_3 = X_0(\beta_0+\gamma_0)+Z$ controlling the kinetic sign and $\mu_2 = X_0\gamma_0 - Z$ controlling the spatial-gradient term; their product must be positive for Lorentzian propagation. For the scalar sector, the machinery is the matrix method in which the equation-of-motion matrix $E = \omega^2 K + i\omega(M-M^T) + V$ has determinant proportional to the product of dispersion relations, and expanding $\det E$ in powers of $k^2$ yields the masses $m^2_a$ and effective gradient factors $G'_a$ that determine whether each degree of freedom is Lorentzian, Euclidean, or tachyonic.

What would settle it

On a bounded Euclidean region around the flat background, with the sample parameters $\lambda=\eta=\alpha_0=1$, $q=\gamma_0=3$, $l=2$, impose zero boundary values for the vector modes and for $\psi$, then solve the linearized equations; if any nonzero bulk mode survives, or if the nonlinear system develops such modes starting from generic data on one side of the signature-change surface, the suppression claim fails.

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Extended reading notes

Core claim

Expanding the Euclidean scalar-tensor action around the flat background with $\bar\varphi = \sqrt{X_0}\,t$, the quadratic tensor action is, in the long-distance limit $k \ll M_{\rm Pl}$, that of a massless field with dispersion relation $\omega^2 = (\mu_2/\mu_3) k^2$. The conditions $\mu_3 = X_0(\beta_0+\gamma_0)+Z > 0$ and $\mu_2\mu_3 = (\gamma_0 X_0 - Z)\mu_3 > 0$ ensure a positive kinetic term and a Lorentzian, rather than Euclidean, propagation. For the scalar sector, the paper computes the low-momentum gradient factors $G'_a$ from the determinant of the equation-of-motion matrix; with sample parameters such as $\lambda=\eta=\alpha_0=1$, $q=\gamma_0=3$, $l=2$, the massless scalar has $G'_1 = 1/5$ while the tensor mode has positive kinetic term. The paper concludes that a massless tensor degree of freedom can, with an appropriate choice of parameters, satisfy a Lorentzian dispersion relation, and it argues that the leftover Euclidean, tachyonic, and harmonic modes can be set to zero by boundary conditions.

Load-bearing premise

The result rests on the assumption that every non-Lorentzian degree of freedom (the tachyonic vector and scalar modes and the harmonic scalar $\psi$) can be set to zero by boundary conditions and that nonlinear effects or the signature-change surface do not regenerate them; the paper explicitly labels this an argument rather than a proof.

Editorial extensions

If this is right

  • Long-wavelength tensor perturbations of the Euclidean theory would be indistinguishable from massless gravitational waves on a Lorentzian spacetime, so a low-energy observer could not tell that the underlying geometry has Euclidean signature.
  • The sector is not fully Lorentz invariant unless the scalar and vector modes are actually removed: the massless scalar is either a ghost with a Lorentzian dispersion relation or a harmonic function, and the paper argues the harmonic option is preferable because its boundary value can be set to zero.
  • Euclidean tachyonic modes with large masses are acceptable because they obey elliptic equations and are suppressed in the bulk once set to zero at the boundary, so they never propagate.
  • If the mechanism extends to curved backgrounds with curvature scales below the momenta considered here, the same Lorentzian tensor dispersion would hold for black-hole-like geometries, a direction the paper identifies for future work.
  • Adding matter coupled to the effective metric would force the tensor speed and matter speed to match if Lorentz invariance is to be recovered, which would impose additional constraints on the parameters.

Reading between the lines

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

  • We infer that the suppression of the harmonic scalar $\psi$ is the fragile step: in a finite region with generic boundary data, the boundary value of $\psi$ cannot necessarily be tuned to zero independently of the data sourcing the tensor modes, so a bounded-domain analysis may reveal residual Euclidean modes.
  • We infer a concrete observable consequence: if matter couples to the effective metric of the theory, the ratio of gravitational-wave speed to photon speed would generically equal $\sqrt{\mu_2/\mu_3}$, so a precise measurement of that ratio would bound the parameter combination.
  • We infer that the full suppression argument could be tested by studying the coupled linearized system on a compact Euclidean region with the signature-change surface included; existence and uniqueness of solutions with zero boundary data would settle whether the Euclidean modes stay absent.
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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 studies a shift-symmetric scalar-tensor theory on a Euclidean-signature manifold, with a clock field phi, and asks whether the theory can, in the long-distance limit, produce degrees of freedom obeying Lorentzian dispersion relations. Around the flat background (8) with constant phi-gradient, the authors expand the action to quadratic order, separate the tensor, vector, and scalar sectors, and derive dispersion relations. The main positive result is in Sec. III B: under conditions (14)-(15), the tensor modes h_+ and h_x acquire a massless Lorentzian dispersion relation omega^2 = (mu2/mu3) k^2 with a positive kinetic term. The vector and scalar sectors are argued to be Euclidean or tachyonic and therefore suppressible by boundary conditions, leaving the tensor mode as the only long-distance Lorentzian degree of freedom. The paper presents two explicit parameter sets as existence demonstrations and concludes that a massless tensor degree of freedom can, with an appropriate choice of parameters, satisfy a Lorentzian dispersion relation.

Significance. If the central claim is sound, the paper provides an explicit proof-of-principle that a Euclidean, renormalizable gravity theory can yield a Lorentzian tensor dispersion relation at low momenta. The tensor-sector calculation is explicit and self-contained, and the conditions (14)-(15) are transparent; the paper is honest that this is an existence demonstration rather than a prediction. The significance is reduced, however, by two problems: the scalar-sector algebra contains an apparent numerical inconsistency and undefined symbols, and the claim that all non-Lorentzian modes can be suppressed by boundary conditions is asserted rather than demonstrated. Because the abstract and Sec. V present this suppression as part of the scenario, the paper's broader claim is not yet established even though the tensor-sector result appears sound.

major comments (3)
  1. [Sec. IV B, Eq. (34), Eq. (36), Eq. (43)] For the sample parameters lambda=eta=alpha0=1, q=gamma0=3, l=2 used in Eq. (36), the formula for G'_1 in Eq. (34) gives G'_1 = -2 X0 mu2 nu1/(mu3 mu7) = 2/5, while the reduced Lagrangian in Eq. (43) gives G'_1 = - X0 nu1 mu2/(mu3 mu7) = 1/5 under the same dispersion convention as Eq. (30). The value tabulated in Eq. (36) is 1/5, so the two derivations are inconsistent by a factor of two. This is not a typographical nuance: the determinant method of Sec. IV B and the explicit integration-out of Sec. IV C must agree. Please locate the source of the discrepancy and recompute the scalar-sector sample values before the scalar-sector conclusions can be accepted.
  2. [Sec. IV B, Eq. (34), Table I] The scalar-sector formulas are not reproducible as printed. Equation (34) uses the symbol mu9, which is not defined in Table I or anywhere in the text, and Table I itself contains two entries labeled mu6 with different expressions. Because G'_3 and G'_4 are presented numerically in Eqs. (36)-(37) and used to support the claim that all remaining modes are Euclidean or tachyonic, these missing or duplicate definitions block verification of a load-bearing part of the analysis. Please define every symbol and re-derive the scalar-sector matrix algebra, checking the duplicate entries.
  3. [Sec. IV B, Sec. IV C, Sec. V] The suppression of the non-Lorentzian modes is asserted, not proved. The perturbation calculation is performed on the infinite, constant-gradient background (8), where there is no boundary and no signature-change hypersurface; the 'suitable boundary conditions' invoked for the vector modes after Eq. (16), for the scalar psi after Eq. (44), and in Sec. V are therefore not part of the solved problem. No explicit boundary-value construction is given, and the argument based on the quadratic action alone does not control nonlinear effects: cubic and higher-order terms in S' will couple h to B, psi, and the massive scalars, so a nonzero tensor perturbation can act as a source for the elliptic equations of the unwanted modes. This is load-bearing because the abstract and Sec. V present suppression as the mechanism by which the effective low-energy sector is Lorentzian. Please either construct the boundary-value problem, including the matching conditions across the degenerate effective-metric surface, or explicitly weaken the claim to the existence of a Lorentzian tensor sector within the unrestricted perturbation space.
minor comments (5)
  1. [Eq. (36)] The equation lists G'_3 twice; the second entry should read G'_4.
  2. [Table I] The table contains two entries labeled mu6; please give the second one a distinct subscript so that formulas in the text referring to mu6 are unambiguous.
  3. [Sec. III C] The text refers to vector degrees of freedom 'Bx and By', but the metric perturbation (9) contains By and Bz; the labels should be corrected.
  4. [Eq. (34)] The expression for G'_2 is ambiguous as typeset; please use parentheses to show that the numerator is -3Z + X0(2 beta0 + gamma0) and the denominator is 3 mu3.
  5. [Sec. IV C, text after Eq. (43)] The phrase 'the massless dispersion relation in Eq. (34)' is ambiguous because Eq. (34) contains several dispersion-relation formulas; please refer explicitly to the formula for G'_1.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the dispersion-relation calculation is a self-contained perturbation analysis of a given Euclidean scalar-tensor action.

full rationale

The paper explicitly takes the shift-symmetric Euclidean scalar-tensor action from Mukohyama's prior work as its starting point and then performs an independent, explicit perturbative expansion around the flat background (8). The central result is conditional: given the action and the parameter inequalities (14) and (15), the tensor mode h has positive kinetic term and satisfies a massless Lorentzian dispersion relation in the long-distance limit. The parameter choices in Eqs. (35)-(37) are existence demonstrations, not fits to data, and the dispersion relation is read off from the computed quadratic action, not imposed by construction. The self-citations to refs. [2,6,13,14] provide the framework and the scalar-sector reduction technique, but the tensor sector calculation is carried out in the paper itself and does not reduce to those citations. The main weakness—the assertion that Euclidean tachyonic and harmonic modes can be suppressed by boundary conditions (Secs. IV B, IV C, and V)—is a gap in justification rather than a circular step: the paper labels it as an argument, and no explicit boundary-value construction is given. That concerns correctness and completeness, not circularity, because the suppression claim is not equivalent to the input action or to the derived dispersion relation. Therefore the paper's derivation chain is self-contained with respect to its stated goal of showing that the chosen action can, for suitable parameters, yield a massless Lorentzian tensor degree of freedom.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

Everything in the paper hangs on the shift-symmetric Euclidean action of Eq. (6) and on the assumptions of a flat background with a constant clock gradient, a long-distance truncation that drops higher-derivative terms, and the boundary-condition suppression of non-Lorentzian sectors. The first three are conventional model assumptions; the boundary-condition suppression is the fragile one because it is asserted, not demonstrated. There are no data fits: the many couplings are free parameters of the model, and the examples are existence demonstrations.

free parameters (10)
  • lambda = 1 in the examples; free Lagrangian coupling
    Coupling of the Weyl-squared term in Eq. (6); appears in the tensor, vector, and scalar actions. Chosen by hand in the two example parameter sets in Sec. IV B.
  • eta = 1 or -1 in the examples; free
    Coupling for the chi^2 - chi R term in Eq. (6); its sign differs between the two branches studied in Sec. IV B.
  • alpha0 = 1 in the examples; free
    Coupling for the (box phi)^2 term in Eq. (6); set to 1 in the sample parameter points.
  • beta0 = fixed by the parameterization (35); free
    Coupling for phi_ab phi^ab in Eq. (6); in the examples it is determined by q, l, gamma0, and alpha0 through Eq. (35).
  • gamma0 = 3 (example 1), 1 (example 2); free
    Coupling for G_ab phi^a phi^b in Eq. (6); enters the key combinations mu2 and mu3 that control the tensor dispersion.
  • Z = of order M_Pl^2, not numerically fixed
    Coefficient of the Ricci scalar in Eq. (6); sets the Planck scale and is assumed positive in the examples.
  • X0 = q Z / gamma0, free
    Background clock-field gradient squared; must be positive; sets the emergent time direction and appears in every sector.
  • q = 3 (example 1), 4 (example 2); free
    Parameterization X0 = q Z / gamma0 in Eq. (35); q > 1 is required for mu2 > 0 (Lorentzian tensor dispersion).
  • l = 2 (example 1), 1/3 (example 2); free
    Parameterization of beta0 in Eq. (35); l > 1 gives mu7 < 0 (Lorentzian scalar ghost), while l < 1 gives mu7 > 0 (harmonic scalar).
  • P0 = 0
    Set to zero in Sec. II B to admit the flat background with X = X0; it is a tuning condition for the background solution.
assumptions (6)
  • domain assumption Shift symmetry phi -> phi + const and Z2 symmetry phi -> -phi are imposed on the clock field action.
    This is the defining property of the ELST framework from refs [2,6] and is stated in Sec. I; it forbids a potential for phi and forces the gradient to play the role of an emergent time direction.
  • domain assumption The flat Euclidean metric with phi = sqrt(X0) t is a valid background solution.
    Assumed in Sec. III A with P0 = 0. Shift symmetry makes a constant gradient trivially satisfy the scalar equation, and the potential terms vanish at X = X0; the metric equations are not explicitly checked.
  • domain assumption Long-distance limit: momenta and time derivatives are much smaller than M_Pl, so the O(1/lambda) higher-derivative terms can be dropped from the dispersion analysis.
    Stated in Sec. III B as k much smaller than M_Pl and time derivatives much smaller than M_Pl; the dispersion relations are extracted from the leading terms proportional to X0 and Z, which is an effective-field-theory truncation.
  • domain assumption The determinant det(E) factorizes into four dispersion relations with analytic k^2 dependence near k^2 = 0.
    Assumed in Sec. IV B, Eq. (30). Only even powers of k appear, so analyticity is plausible, but the reality and separation of the poles is not proven.
  • ad hoc to paper Non-Lorentzian sectors (Euclidean tachyonic vectors and scalars, and the harmonic scalar psi) can be suppressed by suitable boundary conditions.
    Asserted in Sec. IV B-C and V with phrases like 'can be easily controlled' and 'we argue'; no explicit boundary-value problem or global solution is constructed, and the interaction with the Lorentzian tensor mode is not analyzed.
  • domain assumption The theory is renormalizable and bounded below, and evades Ostrogradsky instabilities at the Euclidean level.
    The paper cites refs [6,7,22] for these properties and does not re-derive them; they underpin the claim that the higher-derivative Euclidean action is a legitimate starting point.

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Cite this review

Pith. "Pith review of Emergent Lorentzian dispersion relations from a Euclidean scalar-tensor theory." pith.science (2026). https://pith.science/paper/CTJ7BRV5

@misc{pith2026250500112,
  author       = {Pith},
  title        = {Pith review of: Emergent Lorentzian dispersion relations from a Euclidean scalar-tensor theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CTJ7BRV5}},
  note         = {Machine review of arXiv:2505.00112}
}
read the original abstract

Can one be fooled into thinking that space and time are fundamentally described by a Lorentzian manifold? In this article, we describe a scenario in which a theory constructed on a (Euclidean signature) Riemannian manifold can lead to degrees of freedom with Lorentzian dispersion relations, due to a nontrivial configuration of a scalar field. In particular, we perform a perturbative analysis of a renormalizable shift-symmetric scalar-tensor theory and find that it can, in principle, admit a massless tensor degree of freedom with a Lorentzian dispersion relation. While the remaining degrees of freedom in the gravity sector will, in general, satisfy Euclidean dispersion relations, we argue that they can be brought under control by elliptic equations with an appropriate choice of boundary conditions.

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