A Hamiltonian Oscillator Extension of Wave Propagation in Schwarzschild Spacetime: Dispersion Index, Radial Wavenumber Dynamics, Covariant Derivation, and Observational Constraints on a Single Parameter
Abstract
v5 update (October 2026) — corrections. (1) Paper 2 is replaced by draft v0.5, which corrects computational errors in v0.4: Table 1 (impact-parameter columns), Section 7 and Eq. 20 (orbital frequency from the full dispersive Hamiltonian; correctly normalised Lyapunov exponent; retrograde ring added). The prograde correction is δΩ_ph/Ω_ph = +0.657 Λ̂E², not −0.219 Λ as stated in the v3 paragraph below, and eikonal corrections grow as l³. The photon-ring coefficients C_pro and C_ret, the chromatic E² scaling and the superradiant-threshold result are unchanged. (2) Paper 1: the quartic term of the scalar action (Eq. 7) is quartic in the field and does not produce the stated linear k⁴ dispersion (it drops out at linear order and has mass dimension 8, not 6). A linear dimension-6 candidate with the Laplacian of the preferred slices reproduces ω² = c²k²(1 + Λk²) and, in the eikonal limit, the Paper 2 Hamiltonian; a revised Paper 1 is not yet included. With the present Boyer–Lindquist/ZAMO foliation, the horizon boundary conditions for quasinormal modes are not fixed by the model, so the QNM results remain eikonal estimates. Details and check scripts: github.com/Kretski/Lambda-model (notes/: A0_LINEAR_ACTION, A1_KERR_CHECKS, A2a_SCHWARZSCHILD_BOUNDARY, EIKONAL_D_LAMBDA, THEORY_STATUS_GW). The CQG submission mentioned in the v2 paragraph (CQG-117140) was not accepted.
Complete covariant derivation of a one-parameter Hamiltonian extension of
massless wave propagation in Schwarzschild spacetime. The Hamiltonian
H = c k α(r) √(1 + Λk²) is derived from a scalar field action with quartic
local-frame correction. Includes closed-form analytical solutions, numerical
verification to machine precision, and EFT classification via dimension-6
SME operators.
v2 update (August 21, 2026): Added full manuscript PDF (v0.4, 11 pages, two
proved theorems), companion code files for Theorems 1–2 and einbein
falsification test, and key figures from the Kerr extension (photon ring,
chromatic shadow, Lyapunov analysis). Submitted to Classical and Quantum
Gravity (ref: CQG-117140).
v3 update (August 21, 2026): Added Paper 2 draft — Kerr spacetime extension
with four theoretical results: (1) Photon-ring correction: δb = C(a*, ring)
· Λ̂ · E², with C_pro = −0.6230 and C_ret = +11.495 (dimensionless, a*=0.9).
The E² scaling is proved analytically from the vanishing of g^μν k_μ k_ν at
the GR photon ring. (2) Chromatic shadow: the black hole shadow size acquires
a frequency-dependent shift, δθ ∝ Λ̂ · E² / D. The corresponding sensitivity
scale satisfies Λ_max ∝ ω⁻² (exact algebraic scaling, confirmed numerically
to slope −2.0000). (3) Superradiance: the kinematic threshold ω = mΩ_H is
unchanged by the quartic dispersion; proof via the ZAMO 4-velocity shows
n^μ k_μ → 0 at threshold, hence k⁴_loc → 0 identically. The amplification
factor away from threshold is not addressed here. (4) Eikonal QNM
continuation: the classical photon-orbit/QNM correspondence
ω_QNM ≈ Ω_ph(Λ)·l − i(n+½)·λ_Lyap(Λ) is used as an effective eikonal
approximation, with δΩ_ph/Ω_GR ≈ −0.219·Λ̂ in the Λ→0 expansion. The Λ→0
limit reproduces known Schwarzschild and Kerr photon-orbit benchmarks
(Bardeen 1973) to numerical precision (Schwarzschild error < 2×10⁻⁷).
v4 update (August 2026): Revised Paper 2 draft correcting several
presentation issues identified in internal review prior to journal
submission: (i) corrected the ZAMO 4-velocity / angular-velocity formula
used in the superradiance proof; (ii) introduced explicit dimensionless
normalization Λ̂ = Λ/(GM/c²)² for all quoted coefficients, removing the
earlier ambiguity between geometric-unit coefficients and SI-unit Λ;
(iii) reworded the abstract and Table 2 to distinguish theoretical
sensitivity estimates from observational constraints — no value of Λ has
been measured or fitted in this work; (iv) separated the A3.3 fixed-
tolerance scaling result (Λ_max ∝ ω⁻², the primary theoretical result)
from the A3.4 resolution-scaled toy-model illustration (slope ≈ −2.495,
not a physical prediction); (v) softened the eikonal-QNM correspondence
language to reflect that it is used as an effective approximation, with
a first-principles dispersive wave-equation derivation deferred to a
planned Paper 3; (vi) disambiguated "validated against GR" throughout to
mean specifically that the Λ→0 limit reproduces known benchmarks, not
that the Λ≠0 predictions have been independently confirmed. A new
companion feasibility test (A3.3 TeV analysis) is also included, showing
that the theoretical E² chromatic-shift scaling is confirmed numerically
(measured slope −2.000000 against expected −2.000000) while clarifying
that high photon energy alone does not provide spatial resolution: a
direct observational constraint on Λ requires the energy-dependent
photon-ring displacement to be spatially resolved, which current TeV
instruments do not provide model-independently.
All Λ→0 results are validated against known closed-form Schwarzschild and
Kerr benchmarks. The Λ≠0 sector consists of theoretical predictions and
sensitivity estimates; no observational bound on Λ is claimed in this
version. Submitted to Classical and Quantum Gravity (Paper 1 ref:
CQG-117140).
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BibTeX
@misc{kretski2026hamiltonian,
title = {{A Hamiltonian Oscillator Extension of Wave Propagation in Schwarzschild Spacetime: Dispersion Index, Radial Wavenumber Dynamics, Covariant Derivation, and Observational Constraints on a Single Parameter}},
author = {Dimitar Kretski},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23459797},
url = {https://zenodo.org/records/23459797},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Hamiltonian Oscillator Extension of Wave Propagation in Schwarzschild Spacetime: Dispersion Index, Radial Wavenumber Dynamics, Covariant Derivation, and Observational Constraints on a Single Parameter
AU - Dimitar Kretski
PY - 2026
DA - 2026/10/18
PB - Zenodo
DO - 10.5281/zenodo.23459797
UR - https://zenodo.org/records/23459797
AB - v5 update (October 2026) — corrections. (1) Paper 2 is replaced by draft v0.5, which corrects computational errors in v0.4: Table 1 (impact-parameter columns), Section 7 and Eq. 20 (orbital frequency from the full dispersive Hamiltonian; correctly normalised Lyapunov exponent; retrograde ring added). The prograde correction is δΩ_ph/Ω_ph = +0.657 Λ̂E², not −0.219 Λ as stated in the v3 paragraph below, and eikonal corrections grow as l³. The photon-ring coefficients C_pro and C_ret, the chromatic E² scaling and the superradiant-threshold result are unchanged. (2) Paper 1: the quartic term of the scalar action (Eq. 7) is quartic in the field and does not produce the stated linear k⁴ dispersion (it drops out at linear order and has mass dimension 8, not 6). A linear dimension-6 candidate with the Laplacian of the preferred slices reproduces ω² = c²k²(1 + Λk²) and, in the eikonal limit, the Paper 2 Hamiltonian; a revised Paper 1 is not yet included. With the present Boyer–Lindquist/ZAMO foliation, the horizon boundary conditions for quasinormal modes are not fixed by the model, so the QNM results remain eikonal estimates. Details and check scripts: github.com/Kretski/Lambda-model (notes/: A0_LINEAR_ACTION, A1_KERR_CHECKS, A2a_SCHWARZSCHILD_BOUNDARY, EIKONAL_D_LAMBDA, THEORY_STATUS_GW). The CQG submission mentioned in the v2 paragraph (CQG-117140) was not accepted. Complete covariant derivation of a one-parameter Hamiltonian extension of massless wave propagation in Schwarzschild spacetime. The Hamiltonian H = c k α(r) √(1 + Λk²) is derived from a scalar field action with quartic local-frame correction. Includes closed-form analytical solutions, numerical verification to machine precision, and EFT classification via dimension-6 SME operators. v2 update (August 21, 2026): Added full manuscript PDF (v0.4, 11 pages, two proved theorems), companion code files for Theorems 1–2 and einbein falsification test, and key figures from the Kerr extension (photon ring, chromatic shadow, Lyapunov analysis). Submitted to Classical and Quantum Gravity (ref: CQG-117140). v3 update (August 21, 2026): Added Paper 2 draft — Kerr spacetime extension with four theoretical results: (1) Photon-ring correction: δb = C(a*, ring) · Λ̂ · E², with C_pro = −0.6230 and C_ret = +11.495 (dimensionless, a*=0.9). The E² scaling is proved analytically from the vanishing of g^μν k_μ k_ν at the GR photon ring. (2) Chromatic shadow: the black hole shadow size acquires a frequency-dependent shift, δθ ∝ Λ̂ · E² / D. The corresponding sensitivity scale satisfies Λ_max ∝ ω⁻² (exact algebraic scaling, confirmed numerically to slope −2.0000). (3) Superradiance: the kinematic threshold ω = mΩ_H is unchanged by the quartic dispersion; proof via the ZAMO 4-velocity shows n^μ k_μ → 0 at threshold, hence k⁴_loc → 0 identically. The amplification factor away from threshold is not addressed here. (4) Eikonal QNM continuation: the classical photon-orbit/QNM correspondence ω_QNM ≈ Ω_ph(Λ)·l − i(n+½)·λ_Lyap(Λ) is used as an effective eikonal approximation, with δΩ_ph/Ω_GR ≈ −0.219·Λ̂ in the Λ→0 expansion. The Λ→0 limit reproduces known Schwarzschild and Kerr photon-orbit benchmarks (Bardeen 1973) to numerical precision (Schwarzschild error < 2×10⁻⁷). v4 update (August 2026): Revised Paper 2 draft correcting several presentation issues identified in internal review prior to journal submission: (i) corrected the ZAMO 4-velocity / angular-velocity formula used in the superradiance proof; (ii) introduced explicit dimensionless normalization Λ̂ = Λ/(GM/c²)² for all quoted coefficients, removing the earlier ambiguity between geometric-unit coefficients and SI-unit Λ; (iii) reworded the abstract and Table 2 to distinguish theoretical sensitivity estimates from observational constraints — no value of Λ has been measured or fitted in this work; (iv) separated the A3.3 fixed- tolerance scaling result (Λ_max ∝ ω⁻², the primary theoretical result) from the A3.4 resolution-scaled toy-model illustration (slope ≈ −2.495, not a physical prediction); (v) softened the eikonal-QNM correspondence language to reflect that it is used as an effective approximation, with a first-principles dispersive wave-equation derivation deferred to a planned Paper 3; (vi) disambiguated "validated against GR" throughout to mean specifically that the Λ→0 limit reproduces known benchmarks, not that the Λ≠0 predictions have been independently confirmed. A new companion feasibility test (A3.3 TeV analysis) is also included, showing that the theoretical E² chromatic-shift scaling is confirmed numerically (measured slope −2.000000 against expected −2.000000) while clarifying that high photon energy alone does not provide spatial resolution: a direct observational constraint on Λ requires the energy-dependent photon-ring displacement to be spatially resolved, which current TeV instruments do not provide model-independently. All Λ→0 results are validated against known closed-form Schwarzschild and Kerr benchmarks. The Λ≠0 sector consists of theoretical predictions and sensitivity estimates; no observational bound on Λ is claimed in this version. Submitted to Classical and Quantum Gravity (Paper 1 ref: CQG-117140).
ER -