Discreteness of the Laplace–Beltrami Spectrum in Einstein–Cartan Spacetime with a Weyssenhoff–Raabe Spin Fluid
Description
Abstract: We study the spectral properties of the Laplace–Beltrami operator in a class of Einstein–Cartan–Sciama–Kibble (ECSK) spacetimes sourced by a vortical Weyssenhoff–Raabe spin fluid. We work with a stationary 4-dimensional spacetime metric whose spatial part is a cylindrically symmetric warped product; separating variables reduces the problem to a one-parameter family of radial Sturm–Liouville operators on a compact angular fiber. Starting from the algebraic Cartan field equation, we derive a self-consistent nonlinear equation for the warping function of the spatial metric, f″(r) = K²f(r)³, and solve it in closed form, with regular initial data f(0) = f₀ > 0, f′(0) = 0, in terms of the Jacobi elliptic function cn: f(r) = f₀/cn(Kf₀r, 1/2). This solution is smooth and regular at the origin and exhibits a genuine finite-radius blow-up at r → R₀* = 𝕂(1/2)/(Kf₀), where 𝕂(1/2) is the complete elliptic integral of the first kind. Near this radius, f(r) behaves asymptotically as a simple pole, f(r) ~ C₁/(R₀*−r). Applying the Grigor'yan criterion, we show that the resulting Riemannian manifold is stochastically incomplete. We then invoke Harmer's discreteness criterion for the Sturm–Liouville operator on a half-open interval and prove, via the leading-order asymptotic behaviour of f near the blow-up radius, that the associated limit vanishes identically. This establishes our main result: the spectrum of the spatial Laplace–Beltrami operator on this class of vortical ECSK manifolds is strictly discrete. We discuss the relation of this result to known discreteness criteria in spectral geometry (Baider 1979; Kleine 1988, 1989; Brüning 1989; recent isoperimetric approaches to warped-product spectral gaps) and to the more general theory of stochastically incomplete manifolds. The result is established independently of any assumption on the value of the asymptotic scaling ratio between consecutive spectral levels, and is presented here purely as a statement in spectral geometry, without reference to any specific physical scale.
License: This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
Summary of terms: You are free to copy and redistribute the material in any medium or format, and to remix, transform, and build upon the material for non-commercial purposes only. You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. If you remix, transform, or build upon the material, you must distribute your contributions under the same license as the original.
Full license terms: Full license text available at https://creativecommons.org/licenses/by-nc-sa/4.0/
Other
This is a substantially revised and expanded version of the manuscript. While the core mathematical result — the strict discreteness of the Laplace–Beltrami spectrum in Einstein–Cartan spacetimes — remains unchanged, the derivation of the background geometry has been made fully rigorous and self-contained, and the mathematical context has been significantly clarified.
Key Revisions in Version 2:
- Rigorous Derivation of the Governing Equation from ECSK Theory:
Provided a complete, step-by-step derivation of the radial governing equation directly from the Einstein–Cartan field equations and the effective spin-density tensor, incorporating explicit references to Hehl et al. (1976, Eq. 5.18) and Popławski (2010, Eq. 8).
Corrected the tensorial structure of the algebraic Cartan field equation and unified conventions throughout Section 2.
- Kinematic & Thermodynamic Foundations:
Justified the cylindrical metric ansatz from the isometry group with explicit references to Stephani et al. and Wald.
Established the constitutive relation as a canonical consequence of local thermodynamic equilibrium in relativistic spin hydrodynamics (referencing Becattini et al. and Florkowski et al.).
- Clarification of Regularity Conditions:
Updated Section 2.4 to explicitly state the regularity condition at the origin, ensuring smooth geometry and absence of conical singularities on the symmetry axis without requiring artificial internal boundaries.
- Geometric Classification & Boundary Terminology:
Refined the description of the blow-up boundary in Section 4.1. Removed the imprecise term "finite-volume end" to eliminate ambiguity regarding radial volume integrals, strictly characterizing the geometry as a finite-radius blow-up regime ( with a diverging warping function).
- Refinement of Mathematical Discussions and Citations:
Added a dedicated mathematical discussion (Section 4) placing the result within the broader spectral geometry literature.
Updated references to operator stability theorems (Kleine, Brüning) to focus on the stability of the discrete spectrum under relatively compact perturbations on warped products.
Other
What's new in this version (v3):
-
Rigorous Mathematical Foundation for Self-Adjointness (Section 3.1.1): Added a rigorous proof of the essential self-adjointness of the radial Sturm-Liouville operator using Weyl's limit-point criterion. Explicitly verified the square-integrability of the basis solutions near the dynamically generated boundary r = R₀*, confirming that the spectrum is well-defined without the need to impose auxiliary boundary conditions.
-
Extended Perturbation Analysis (Section 3.4): Strengthened the analytical treatment for non-zero angular (m ≠ 0) and longitudinal (k_z ≠ 0) modes. Utilized Kato's and Reed–Simon's perturbation theory to rigorously demonstrate that the effective potential term V_m(r) constitutes a relatively compact perturbation, preserving the strict discreteness of the essential spectrum across all modes.
-
Bibliographic and Editorial Refinement: Updated the bibliography with foundational mathematical physics literature regarding operator theory and linear operators ([25, 26]), and refined the Abstract to explicitly incorporate essential self-adjointness alongside stochastic incompleteness and Harmer's discreteness criterion.
Files
article_A_en5_Zen.pdf
Additional details
Dates
- Created
-
2026-06-27
- Updated
-
2026-09-30
- Updated
-
2026-10-31
References
- [1] A. Baider (1979), Noncompact Riemannian manifolds with discrete spectra, J. Differ. Geom. 14, 41–57. [2] R. Kleine (1988), Discreteness conditions for the Laplacian on complete noncompact Riemannian manifolds, Math. Z. 198, 127–141. [3] R. Kleine (1989), Warped products with discrete spectra, Results Math. 15, 81–103. [4] J. Brüning (1989), On Schrödinger operators with discrete spectrum, J. Funct. Anal. 85, 117–150. [5] H. Donnelly and P. Li (1979), Pure point spectrum and negative curvature for noncompact manifolds, Duke Math. J. 46, 497–503. [6] V. Kondratiev and M. Shubin (1999), Discreteness of spectrum for the Schrödinger operators on manifolds of bounded geometry, in Operator Theory: Advances and Applications, vol. 110 (Birkhäuser, Basel), pp. 185–226. [7] V. A. Kondratiev, V. Maz'ja, and M. A. Shubin (2004), Discreteness of spectrum and strict positivity criteria for magnetic Schrödinger operators, Comm. Partial Differential Equations 29, 489–521. [8] V. Maz'ya and M. Shubin (2005), Discreteness of spectrum and positivity criteria for Schrödinger operators, Ann. of Math. 162, 919–942. [9] G. P. Bessa and J. F. Montenegro (2003), Eigenvalue estimates for submanifolds with locally bounded mean curvature, Ann. Global Anal. Geom. 24, 279–290. [10] M. Harmer (2009), Discreteness of the spectrum of the Laplacian and stochastic incompleteness, J. Geom. Anal. 19(2), 358–372, https://doi.org/10.1007/s12220-008-9055-6. [11] J. Weyssenhoff and A. Raabe (1947), Relativistic dynamics of spin-fluids and spin-particles, Acta Phys. Pol. 9, 7. [12] F. W. Hehl, P. von der Heyde, G. D. Kerlick, and J. M. Nester (1976), General relativity with spin and torsion: Foundations and prospects, Rev. Mod. Phys. 48, 393. [13] T. W. B. Kibble (1961), Lorentz invariance and the gravitational field, J. Math. Phys. 2, 212. [14] D. W. Sciama (1962), On the analogy between charge and spin in general relativity, in Recent Developments in General Relativity (Pergamon Press). [15] N. J. Popławski (2010), Cosmology with torsion: An alternative to cosmic inflation, Phys. Lett. B 694, 181. [16] F. Becattini, V. Chandra, L. Del Zanna, and E. Grossi (2013), Relativistic distribution function for particles with spin at local thermodynamical equilibrium, Ann. Phys. 338, 32. [17] A. Grigor'yan (1999), Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Am. Math. Soc. 36(2), 135. [18] A. Banerjee (2026), Isoperimetric inequalities and spectral consequences in warped product manifolds, arXiv:2603.22069 [math.DG] (preprint). [19] A. Grigor'yan (2009), Heat Kernel and Analysis on Manifolds, AMS/IP Studies in Advanced Mathematics, vol. 47 (American Mathematical Society, Providence / International Press, Boston). [20] NIST Digital Library of Mathematical Functions (2025), Chapter 22 (Jacobian Elliptic Functions), https://dlmf.nist.gov/22, Release 1.2.4 of 2025-01-15. [21] W. Florkowski, B. Friman, A. Jaiswal, and E. Speranza (2018), Relativistic fluid dynamics with spin, Phys. Rev. C 97, 041901. [22] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt (2003), Exact Solutions of Einstein's Field Equations, 2nd ed. (Cambridge University Press, Cambridge), Chaps. 19–20. [23] R. M. Wald (1984), General Relativity (University of Chicago Press, Chicago), Chap. 3 and Appendix C. [24] K. A. Bronnikov, N. O. Santos, and A. Wang (2020), Cylindrically symmetric spacetimes and gravitation, Phys. Rep. 880, 1–65. [25] M. Reed and B. Simon (1978), Methods of Modern Mathematical Physics, Vol. IV: Analysis of Operators. [26] T. Kato (1980), Perturbation Theory for Linear Operators, 2nd ed. (Springer, Berlin).