Gravitational Spin-Orbit Interaction

• Gravitational spin-orbit interaction is the coupling between intrinsic spin and orbital angular momentum arising in advanced gravitational theories.
• Cubic torsion-curvature invariants in Poincaré gauge theory lead to nontrivial modifications of black hole spacetimes, revealing new dynamical features.
• Analytic studies in the slow rotation regime highlight potential observational signatures in astrophysical compact objects and gravitational wave phenomena.

Gravitational spin-orbit interaction refers to the dynamical coupling between a system's intrinsic (spin) and extrinsic (orbital) angular momentum in the context of gravitational theories, most notably in general relativity and its Poincaré gauge extensions. In atomic and nuclear physics, spin-orbit coupling underlies fine structure corrections; its gravitational analogue is not merely formal but arises naturally in geometric gauge theories where spacetime incorporates both curvature and torsion. Recent developments in Poincaré gauge theory have shown how cubic invariants in torsion-curvature couplings lead to concrete spin-orbit terms in analytic solutions, with implications for both the structure of black-hole spacetimes and the emergence of new physical phenomena not present in torsionless (pure GR) models (Bahamonde et al., 27 Aug 2025).

1. Poincaré Gauge Theory and Gravitational Spin-Orbit Interaction

Poincaré gauge theory generalizes Einstein's general relativity by promoting both local translations and Lorentz transformations to gauge symmetries. This extension naturally introduces torsion, with the total connection $\tilde{\Gamma}^\gamma_{\mu\nu}$ containing antisymmetric (torsional) parts beyond the Levi-Civita connection. The resulting torsion tensor,

$$T^\gamma_{\mu\nu} = 2 \tilde{\Gamma}^\gamma_{[\mu\nu]},$$

and Riemann–Cartan curvature tensor,

$$\tilde{R}^\gamma_{\rho\mu\nu} = \partial_\mu \tilde{\Gamma}^\gamma_{\rho\nu} - \partial_\nu \tilde{\Gamma}^\gamma_{\rho\mu} + \tilde{\Gamma}^\gamma_{\sigma\mu}\tilde{\Gamma}^\sigma_{\rho\nu} - \tilde{\Gamma}^\gamma_{\sigma\nu}\tilde{\Gamma}^\sigma_{\rho\mu},$$

are the fundamental field strengths.

A crucial feature is the freedom to introduce higher-order invariants: quadratic and especially cubic terms involving products of torsion and curvature. The explicit cubic action for the sector of interest is

$$S = \frac{1}{16\pi} \int d^4x \sqrt{-g} \Bigl\{ -R + \text{[quadratic invariants]} + \sum_i h_i \cdot (\text{cubic invariants of } \tilde{R} \text{ and } T) + \text{mass terms for torsion} \Bigr\}.$$

These couplings can generate interactions between intrinsic and extrinsic angular momenta (“spin–orbit interaction”), manifesting as additional terms in the field equations and Lagrangian density (Bahamonde et al., 27 Aug 2025).

2. Field Equations and Cubic Torsion-Gravity Couplings

Variation of the total action leads to two sets of field equations: one from the tetrad (coframe) and one from the spin connection. The latter, in compact notation,

$$E^{\lambda\mu\nu} = \frac{1}{2} (X^{\lambda\nu\mu} - X^{\lambda\mu\nu} + X^{\mu\lambda\nu} - X^{\mu\nu\lambda}) + 2[K^{\mu}_{\ \alpha\kappa} Y^{\lambda\alpha\nu\kappa} - K^{\lambda}_{\ \alpha\kappa} Y^{\mu\alpha\nu\kappa} - \nabla_\alpha Y^{\mu\lambda\nu\alpha}] = 0,$$

with $X$ and $Y$ defined via functional derivatives of the Lagrangian with respect to the connection and contortion tensors, encodes the complicated nonlinear couplings between torsion, curvature, and the spacetime metric. Mass terms, controlled by coefficients like $N_1$, $N_2$, etc., govern the propagation and possible “propagating” nature of the model's torsion modes.

The presence of cubic torsion-curvature terms fundamentally alters the analytic structure and admits richer solution spaces, including interactions that mix spin (axial torsion) and macroscopic rotation (extrinsic angular momentum) (Bahamonde et al., 27 Aug 2025).

3. Analytical Solutions: Slow Rotation and Degenerate Limit

Due to the highly nonlinear and coupled nature of the full field equations—with up to 27 unknown functions under stationary, axisymmetric ansatz—the analysis specializes to the slow rotation regime. Here, the rotation parameter $a$ (extrinsic angular momentum) is treated perturbatively, and torsion and metric variables are expanded:

$$q_i(r,\theta) = q_i^{(0)}(r,\theta) + \kappa_s\,\bar{q}_i(r,\theta), \quad f_i(r,\theta) = \kappa_s\,\bar{f}_i(r,\theta),$$

where $\kappa_s$ is the “spin charge” associated with the axial torsion mode.

In the degenerate case, crucial kinetic terms for the axial torsion mode vanish (choice $N_2 = -N_1$), considerably simplifying the equations and constraining the metric to remain of slowly rotating Kerr form,

$$ds^2 = [\Psi(r) + O(a)] dt^2 - [\Psi(r)^{-1} + O(a)] dr^2 - r^2 d\theta^2 - r^2 \sin^2\theta d\phi^2 + O(a), \quad \Psi(r) = 1 - \frac{2m}{r}.$$

Despite this, the torsion field—while not back-reacting on the metric—induces nontrivial interactions and modifies the gravitational Lagrangian.

4. Emergence and Structure of the Gravitational Spin-Orbit Term

The gravitational Lagrangian density for these solutions splits as

$$\mathcal{L} = \frac{d_1 N_1^2 \kappa_s^2}{8\pi r^4} + \mathcal{L}_{\rm SOI},$$

where the spin-orbit interaction (SOI) term is

$$\mathcal{L}_{\rm SOI} = \frac{a d_1 N_1 \kappa_s}{2\pi r^4} \Big\{ r\partial_r F(r,\theta) - F(r,\theta) + \frac{r\Psi'(r) + 2\Psi(r) - 4\Psi^2(r)}{2r\Psi^2(r)} \cos\theta \Big\},$$

with $F(r,\theta)$ encoding the torsion axial field structure. For suitable $F(r,\theta)$ (for instance, a Thomas-precession–like term), this component is directly analogous to spin-orbit coupling in atomic physics, i.e., a deterministic interaction of the form

$$\mathcal{L}_{\rm SOI} = \frac{a d_1 N_1 \kappa_s G(r,\theta)}{2\pi},$$

where $G(r,\theta)$ is an arbitrary function related to the precise choice of the torsion field (Bahamonde et al., 27 Aug 2025).

This term quantifies a genuine coupling between the intrinsic angular momentum (via $\kappa_s$) and the black hole’s extrinsic rotation parameter $a$, even for solutions where the spacetime geometry remains formally Kerr.

5. Nonlinearity, Degeneracy, and Generalizations Beyond Kerr

The general (nondegenerate) field equations involve nonlinearities up to quartic in the spin charge $\kappa_s$ and include 27 coupled PDEs for the stationary, axisymmetric case. Only for degenerate parameter choices are analytic solutions tractable, resulting in the metric being “frozen” to Kerr and torsion confined to the action/Lagrangian.

In nondegenerate models (i.e., with $N_2 \ne -N_1$), the expectation is the emergence of broader classes of solutions in which intrinsic spin modifies the geometry, potentially shifting horizon structures, ergospheres, or causal properties compared to standard Kerr black holes. The constructed analytic solution thus serves as a “proof of concept” for the presence and structure of gravitational spin-orbit coupling at the level of field-theoretic actions rather than as an end in itself.

6. Key Formulas and Interpretation

The central formulas governing the gravitational spin-orbit interaction in this context are:

• Torsion and curvature:

$$T^\gamma_{\mu\nu} = 2 \tilde{\Gamma}^\gamma_{[\mu\nu]}, \qquad \tilde{R}^\gamma_{\rho\mu\nu} = \partial_\mu \tilde{\Gamma}^\gamma_{\rho\nu} - \partial_\nu \tilde{\Gamma}^\gamma_{\rho\mu} + \tilde{\Gamma}^\gamma_{\sigma\mu}\tilde{\Gamma}^\sigma_{\rho\nu} - \tilde{\Gamma}^\gamma_{\sigma\nu}\tilde{\Gamma}^\sigma_{\rho\mu} .$$

• Cubic Poincaré gauge action (schematic):

$$S = \frac{1}{16\pi} \int d^4x \sqrt{-g} \Bigl\{ -R + \text{quadratic} + \sum_i h_i(\text{cubic torsion-curvature}) + \text{masses} \Bigr\}.$$

• Gravitational spin-orbit Lagrangian term:

$$\mathcal{L}_{\rm SOI} = \frac{a d_1 N_1 \kappa_s G(r,\theta)}{2\pi}$$

where $a$ is the extrinsic rotation, $\kappa_s$ is the intrinsic spin charge (axial torsion), and $G(r,\theta)$ encodes detailed torsion field structure.

7. Physical Implications and Prospects

The existence of a gravitational spin–orbit term in the Poincaré gauge action parallels the physics of atomic and nuclear fine structure but arises at the geometric/gravitational level through torsion-curvature couplings. While in the degenerate limit the metric remains unaltered, this mechanism demonstrates that true gravitational spin–orbit interactions can:

• Exist independently of stress-energy sources (arising purely from spacetime structure).
• Modify the action and energy content of rotating black-hole spacetimes, yielding new invariants.
• In more general (nondegenerate) models, potentially deform the black-hole geometry, offering a route to spacetimes beyond Kerr, with associated signatures in horizon structure, gravitational-wave emission, or stability.

A plausible implication is that gravitational spin–orbit interactions in nondegenerate, cubic-Poincaré gauge theories will furnish both new classical solutions and testable predictions for astrophysical compact objects, particularly in regimes where intrinsic spins (e.g., fermionic fields or torsional matter) are present and dynamically relevant (Bahamonde et al., 27 Aug 2025).

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References

• "A gravitational spin-orbit interaction in Poincaré gauge theory" (Bahamonde et al., 27 Aug 2025)