Gravitational Splittings in Physics

• Gravitational splittings are phenomena where gravity lifts degeneracies by separating frequencies or energy levels in systems ranging from celestial mechanics to quantum transitions.
• They manifest as differences in orbital epicyclic frequencies, quantum energy shifts in ultracold-neutron spectroscopy, and relativistic fine-structure effects observable via modified spin-orbit and Zeeman-like couplings.
• Observable effects depend on gravitational multipoles, coordinate choices, and external perturbations, providing diverse probes into both geometric and dynamical aspects of gravitational fields.

Searching arXiv for papers on gravitational splittings and closely related usages of the term. Gravitational splittings denote a family of phenomena in which gravity lifts degeneracies, separates characteristic frequencies, or otherwise resolves a single mode into distinct components. The term is used in several technically distinct senses. In celestial mechanics, it refers to the splitting of radial and vertical epicyclic frequencies in axisymmetric Newtonian potentials, yielding a Newtonian counterpart of the Shirokov effect (Idrissov et al., 22 Jun 2026). In relativistic quantum mechanics on curved spacetime, it refers to fine-structure-like level separations generated by gravitational Darwin and spin–orbit terms (Jentschura et al., 2015). In curved-spacetime spinor dynamics, it denotes Zeeman-like energy splitting induced by an axial-vector gravitational coupling (Mukhopadhyay et al., 2018). In quantum systems, it may also mean gravity-induced energy splittings of bound states, resonant transition frequencies, sideband formation, or relative phases, as in ultracold-neutron spectroscopy (Landry et al., 2016), gravitational Aharonov–Bohm sidebands (Chiao et al., 2023), Schrödinger–Newton dephasing (Großardt, 2023), and birefringent quantum electrodynamics (Grosse-Holz et al., 2017). The phrase also appears in more formal settings, such as Hilbert-space localization in perturbative gravity (Donnelly et al., 2018) and the splitting theory formulation of Regge–Teitelboim gravity (Grad et al., 2018). Across these usages, the common structure is the emergence of distinguishable gravitationally controlled sectors, frequencies, energies, or observables.

1. Newtonian frequency splitting in axisymmetric gravity

In the Newtonian epicyclic problem, a test particle of specific angular momentum $\ell$ moves in the effective potential

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$

and a circular reference orbit lies at a stationary point $(r_0,\theta_0)$ satisfying

$$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$

With scaled displacements $x=\delta r$ and $y=r_0\delta\theta$, the small oscillations obey

$\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$

where

$$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$

The normal-mode frequencies are

$$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$

with exact splitting

$\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$

(Idrissov et al., 22 Jun 2026).

For an axisymmetric Newtonian source expanded in multipoles,

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$0

the paper identifies a sharp selection rule. A quadrupole splits the radial and vertical epicyclic frequencies according to

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$1

positive for an oblate body, so $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$2 (Idrissov et al., 22 Jun 2026). This is presented as the Newtonian analogue of Shirokov’s splitting and is equivalent to the classical statement that an oblate body’s apsidal and nodal rates differ.

The dipole is exceptional. Since $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$3, it encodes only the choice of origin and is removable by re-centering at the center of mass. The paper therefore states

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$4

and shows that the apparent first-order coupling cancels at the true tilted equilibrium. The octupole, by contrast, produces genuine mixed modes with surviving $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$5, leading at linear order to

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$6

(Idrissov et al., 22 Jun 2026).

The resulting selection rule is not parity-based. Even moments split via $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$7 with $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$8 at the equator; odd moments with $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$9 split because a nonzero $(r_0,\theta_0)$0 survives at the tilted equilibrium; only the dipole does not split because it is pure gauge. The paper summarizes this as: every genuine multipole splits except the dipole (Idrissov et al., 22 Jun 2026).

2. Orbital observables, Shirokov drift, and solar-system scales

The same Newtonian analysis distinguishes two observables. Frequency splitting probes oblateness, while orbital-plane tilt probes center-of-mass offset. For a dipolar offset along the symmetry axis,

$(r_0,\theta_0)$1

so

$(r_0,\theta_0)$2

The paper explicitly describes this as a geometric observable, distinct from any frequency splitting (Idrissov et al., 22 Jun 2026).

Carried through to Shirokov’s original observable, the secular transverse drift after $(r_0,\theta_0)$3 orbits, the quadrupole splitting yields

$(r_0,\theta_0)$4

and, with $(r_0,\theta_0)$5,

$(r_0,\theta_0)$6

For the Sun with $(r_0,\theta_0)$7, $(r_0,\theta_0)$8, $(r_0,\theta_0)$9, and $$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$0, the paper reports $$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$1 at $$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$2 and $$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$3 at $$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$4 (Idrissov et al., 22 Jun 2026). These are stated to be comparable in scale to Shirokov’s original Schwarzschild estimate.

The same work gives representative solar-system estimates for the relative quadrupole and octupole splittings and for the solar center-of-mass offset dominated by Jupiter. At $$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$5,

$$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$6

while

$$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$7

For the barycentric offset estimate, the paper quotes $$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$8, giving

$$\partial_r V_{\rm eff}=\partial_\theta V_{\rm eff}=0.$$9

at $x=\delta r$0 (Idrissov et al., 22 Jun 2026).

A direct inversion formula is given for oblateness:

$x=\delta r$1

This establishes the splitting as a coordinate-independent dynamical probe of $x=\delta r$2, complementary to the geometric probe $x=\delta r$3 (Idrissov et al., 22 Jun 2026).

3. Quantum energy-level splittings and transition spectroscopy

In quantum mechanics, gravitational splittings can refer to discrete energy differences in a gravitational potential. A central example is the ultracold-neutron quantum bouncer. For a neutron of mass $x=\delta r$4 above a perfectly reflecting surface in Earth’s uniform field, the vertical Schrödinger equation is

$x=\delta r$5

with $x=\delta r$6. The solutions are Airy functions with characteristic length

$x=\delta r$7

and energies

$x=\delta r$8

The gravitational energy splittings are therefore

$x=\delta r$9

(Landry et al., 2016).

The paper reports $y=r_0\delta\theta$0 and, for the $y=r_0\delta\theta$1 transition,

$y=r_0\delta\theta$2

(Landry et al., 2016). Since the spectrum is non-linear, distinct level pairs have distinct resonant frequencies, enabling spectroscopy of gravitationally bound states by driving at $y=r_0\delta\theta$3.

The proposed perturbation comes from an oscillating nearby spherical mass $y=r_0\delta\theta$4 with $y=r_0\delta\theta$5, producing

$y=r_0\delta\theta$6

with

$y=r_0\delta\theta$7

At resonance, the transition probability grows as $y=r_0\delta\theta$8 in the coherent short-time regime. For the parameter choice $y=r_0\delta\theta$9, $\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$0, $\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$1, and the $\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$2 transition, the paper gives a prefactor $\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$3 and, with an optimal drive time $\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$4 for neutron lifetime $\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$5, obtains

$\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$6

(Landry et al., 2016). This makes the splitting experimentally addressable as a resonant quantum transition rather than as a static shift alone.

A different quantum usage appears in the Schrödinger–Newton analysis of a Stern–Gerlach interferometer. There, a self-gravitational interaction between the two spin-conditioned trajectories produces a relative phase,

$\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$7

for a homogeneous spherical particle of radius $\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$8 with sharply localized wave packets, separation $\ddot x=-Ax-Cy,\qquad \ddot y=-Cx-By,$9, and interaction time $$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$0 (Großardt, 2023). The paper explicitly interprets this as measurable gravitational splitting in the form of a relative phase between quasi-classical branches, with the dominant self-energy term scaling as $$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$1.

4. Relativistic fine structure, Zeeman-like splittings, and precision spectroscopy

In relativistic quantum mechanics on curved spacetime, “gravitational splittings” often designate fine-structure-like energy differences generated by the Foldy–Wouthuysen reduction of the Dirac equation in a Schwarzschild background. For a static isotropic metric,

$$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$2

the Hermitian Dirac–Schwarzschild Hamiltonian is

$$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$3

and its Foldy–Wouthuysen form includes

$$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$4

(Jentschura et al., 2015). These are identified respectively as the gravitational relativistic potential correction, the gravitational Darwin term, and the gravitational spin–orbit coupling.

The resulting bound-state spectrum is

$$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$5

with

$$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$6

The second term produces the gravitational fine-structure splittings and lifts the $$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$7-multiplet degeneracy (Jentschura et al., 2015). For the electron–proton system, the paper quotes

$$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$8

so the splittings are extraordinarily small.

A closely related analysis of the nonrelativistic limit of the Dirac–Schwarzschild Hamiltonian gives the effective spin–orbit term

$$A=\left.\frac{\partial^2 V_{\rm eff}}{\partial r^2}\right|_{r_0,\theta_0},\quad B=\left.\frac{1}{r_0^2}\frac{\partial^2 V_{\rm eff}}{\partial \theta^2}\right|_{r_0,\theta_0},\quad C=\left.\frac{1}{r_0}\frac{\partial^2 V_{\rm eff}}{\partial r\,\partial \theta}\right|_{r_0,\theta_0}.$$9

together with a gravitational Darwin term proportional to $$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$0 (Jentschura et al., 2013). The paper stresses that no direct $$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$1 coupling appears, so parity is preserved. It also shows that the corrected electromagnetic transition current acquires $$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$2 gravitational terms, modifying amplitudes but not selection rules (Jentschura et al., 2013).

In curved-spacetime spinor dynamics, a different relativistic splitting arises from the effective axial-vector coupling

$$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$3

In a stationary weak-gravity regime, the effective Hamiltonian contains $$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$4 and related spin-momentum terms, producing the “Gravitational Zeeman Effect” (Mukhopadhyay et al., 2018). The paper identifies the leading spin splitting with the term $$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$5 and gives, for neutrino and antineutrino branches,

$$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$6

in the nonrelativistic or weak-gravity limit, and

$$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$7

for ultra-relativistic neutrinos (Mukhopadhyay et al., 2018). In Schwarzschild spacetime $$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$8, so no such Zeeman-like splitting occurs; in Kerr and anisotropic cosmologies, $$\omega_\pm^2=\frac{A+B}{2}\pm\sqrt{\left(\frac{A-B}{2}\right)^2+C^2},$$9 and the effect is present.

A further spectroscopic setting appears in high-precision atomic and molecular spectroscopy in weak gravity. The generalized weak-field Dirac analysis shows that atomic transitions remain equivalence-principle compliant to leading order, because the universal $\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$0 scaling cancels in proper time. Genuine splittings require gradients, tidal terms, or spin–curvature couplings (Jentschura, 2018). The Fokker precession term,

$\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$1

produces true $\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$2-dependent splittings but is numerically tiny on Earth (Jentschura, 2018). By contrast, in diatomic molecules the first-order gradient term need not vanish and yields orientation-dependent shifts,

$\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$3

or, for ionization-related bond-length changes,

$\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$4

The paper reports Earth-surface shifts of approximately $\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$5 for HF, $\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$6 for $\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$7, and $\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$8 for $\omega_+^2-\omega_-^2=\sqrt{(A-B)^2+4C^2}$9, and characterizes these as surprisingly large compared with atomic tidal and spin–curvature effects (Jentschura, 2018).

5. Spectral sidebands, atomic anisotropy, and position-dependent local QED

Another class of gravitational splittings arises from time-dependent gravitational phases rather than static curvature corrections. In the gravitational Aharonov–Bohm setup, a quantum system in free fall on a slightly elliptical orbit experiences no local gravitational force on its internal degrees of freedom, yet its internal Hamiltonian acquires a time-dependent scalar potential energy

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$00

where $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$01 is the semi-major axis, $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$02 the eccentricity, and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$03 the orbital frequency (Chiao et al., 2023). The phase modulation depth is

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$04

and the time evolution factor acquires the Jacobi–Anger expansion

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$05

As a result, a transition near $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$06 develops sidebands at $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$07 with amplitudes proportional to $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$08 (Chiao et al., 2023). The paper explicitly identifies these sidebands as the signature of the gravitational Aharonov–Bohm effect.

For low Earth orbit with $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$09 and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$10, the paper gives $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$11 for an electron and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$12 for a nucleon, corresponding to characteristic offsets of about $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$13 and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$14, respectively (Chiao et al., 2023). In this sense, “gravitational splittings” are a frequency-domain comb produced by periodic gravitational phase modulation.

A distinct mechanism is proposed in the unified gravity extension of the Standard Model. There, the gravity gauge field rescales hydrogenic Dirac energies by a coefficient

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$15

so that

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$16

and the transition redshift becomes

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$17

(Partanen et al., 27 Jun 2025). More unusually, the gravitational potential gradient modifies the nuclear Coulomb potential anisotropically through the gravity-modified Maxwell equation. The perturbation takes the form

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$18

with

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$19

The diagonal first-order shift vanishes,

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$20

but off-diagonal couplings with $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$21 and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$22 split degenerate manifolds after diagonalization (Partanen et al., 27 Jun 2025). The paper estimates that even near neutron stars the splittings are only a few hertz to a few hundred hertz for high-$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$23 ions, so direct observation remains remote.

An additional spectroscopic framework appears in birefringent quantum electrodynamics on area-metric backgrounds. There, local QED observables depend ultralocally on perturbations $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$24 and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$25, and the hydrogen hyperfine line acquires a position-dependent stretch and, in general backgrounds, an anisotropic triplet splitting (Grosse-Holz et al., 2017). In a spherically symmetric weak field around a point mass, the effect reduces to a uniform stretch

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$26

with $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$27, while more general backgrounds can produce actual triplet splitting through the tracefree tensor $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$28 (Grosse-Holz et al., 2017).

6. Collective modes, formal Hilbert-space splittings, and other specialized usages

In fractional quantum Hall physics, the phrase “gravitational splitting” is used for the splitting of the Girvin–MacDonald–Platzman graviton mode. The long-wavelength projected density operator $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$29 produces a neutral spin-2 excitation at $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$30, the GMP graviton. The paper shows that in the composite-fermion hierarchy $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$31, this mode is undivided for the primary Jain sequence $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$32, but splits into two chiral gravitons for the secondary Jain sequence $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$33 with $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$34 (Balram et al., 2024). The dynamical structure factor is generalized from the single-mode approximation

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$35

to a composite-fermion exciton ansatz

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$36

and for $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$37 and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$38 the $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$39 sector shows two dominant peaks corresponding to a primary CF graviton and a parton graviton (Balram et al., 2024). Although the setting is condensed matter, the terminology of “graviton splitting” is explicit and tied to geometric response.

In perturbative quantum gravity, “gravitational splitting” denotes a statement about localization of quantum information rather than spectral lines. Expanding around Minkowski space with

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$40

one finds that diffeomorphism-invariant operators require dressings extending to infinity. Nevertheless, to leading order, one can define Hilbert subspaces $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$41 such that exterior operators are insensitive to interior microstate details except through total Poincaré charges (Donnelly et al., 2018). The characteristic factorization is

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$42

for states within the same charge sector (Donnelly et al., 2018). Here the “split” is a charge-labeled decomposition of the Hilbert space rather than a dynamical frequency separation.

A further specialized use appears in the splitting theory formulation of Regge–Teitelboim gravity, where spacetime is represented as a surface embedded in a higher-dimensional flat bulk and described by scalar fields $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$43 (Grad et al., 2018). In that context, “splitting theory” concerns the foliation of the bulk into nonintersecting $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$44-dimensional surfaces. The paper studies corresponding definitions of energy and finds, for an isolated Einsteinian solution, $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$45 and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$46 (Grad et al., 2018). This is terminologically related but conceptually distinct from spectral or dynamical splittings.

Loop-corrected effective field theory of gravity supplies yet another meaning. In the Barnes–Rivers decomposition of the graviton propagator, loop effects reweight the spin-$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$47 and off-shell scalar sectors through form factors $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$48 and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$49, encoded in a mixing matrix

$$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$50

The paper interprets the unequal tensor and scalar channel renormalizations as “gravitational splittings,” and argues that this obstructs the definition of a universal running Newton constant (Latosh, 2020). This suggests a broader abstract pattern: gravitational splitting can mean the emergence of distinct effective sectors under gravitational dressing, projection, or response.

7. Unifying themes and recurring distinctions

Despite their heterogeneity, the surveyed usages share several recurring distinctions.

First, many papers separate uniform gravitational shifts from genuine splittings. In the Newtonian Shirokov problem, a center-of-mass offset tilts the orbital plane but does not split frequencies, whereas the quadrupole and octupole do (Idrissov et al., 22 Jun 2026). In atomic gravity problems, a uniform redshift rescales all levels coherently, but gradients, curvature terms, or anisotropies are required to break degeneracy (Jentschura, 2018, Partanen et al., 27 Jun 2025). In the gravitational Aharonov–Bohm effect, the constant part of $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$51 is absorbed into a base energy, while the periodic part creates sidebands (Chiao et al., 2023).

Second, gauge or coordinate issues recur. The Newtonian dipole is removable by re-centering and therefore cannot appear in any coordinate-independent frequency (Idrissov et al., 22 Jun 2026). In precision spectroscopy and $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$52-factor analyses, apparent position dependence can disappear when observables are expressed in the local Lorentz frame, restoring equivalence-principle compatibility at leading order (Jentschura, 2018). In perturbative gravity, localization itself must be reformulated because gauge-invariant dressings extend to infinity (Donnelly et al., 2018).

Third, the observable can be geometric, spectroscopic, dynamical, or algebraic. Epicyclic splitting is read off from orbital frequencies and secular drift (Idrissov et al., 22 Jun 2026). Ultracold-neutron splittings appear as resonant frequencies (Landry et al., 2016). Schrödinger–Newton splittings appear as relative phases between interferometric branches (Großardt, 2023). Gravitational Zeeman and curved-space Dirac splittings are energy-level separations (Mukhopadhyay et al., 2018, Jentschura et al., 2015). Fractional quantum Hall graviton splitting is read from the dynamical structure factor (Balram et al., 2024). Perturbative gravitational splitting is a property of Hilbert subspaces (Donnelly et al., 2018).

Finally, observability depends strongly on regime. Newtonian quadrupole splittings can dominate relativistic Shirokov splitting for Earth satellites (Idrissov et al., 22 Jun 2026). Ultracold-neutron gravitational transitions are small but not infinitesimal and can be driven resonantly (Landry et al., 2016). Molecular gradient-induced gravitational shifts reach the millihertz scale on Earth (Jentschura, 2018). By contrast, gravitational fine structure for microscopic gravitationally bound systems is suppressed by $$V_{\rm eff}(r,\theta)=\Phi(r,\theta)+\frac{\ell^2}{2r^2\sin^2\theta},$$53 and is effectively negligible (Jentschura et al., 2015). This suggests that the most practically relevant gravitational splittings are those tied to collective motion, external gradients, or engineered coherent accumulation, rather than those controlled solely by microscopic gravitational binding.