Gravitational Molecular Eigenstates

• Gravitational molecular eigenstates are quantized energy levels arising in systems dominated by long-range gravitational or effective confining potentials, with broad applications in quantum field theory, condensed matter physics, and astrophysics.
• They emerge through mechanisms such as non-relativistic reductions, harmonic oscillator potentials, and Schrödinger–Poisson dynamics, exemplified by the GMP spin-2 mode and multistate gravitational atoms.
• These eigenstates offer practical insights into novel gravitational wave polarizations, chaotic spectral splitting in three-body systems, and potential models for dark matter halo structures.

Gravitational molecular eigenstates are quantized energy levels that arise when systems of particles interact via gravitational or effective confining potentials in macroscopic or mesoscopic contexts. The term draws an analogy to traditional molecular eigenstates, with the key distinction being the dominance of gravitational or non-Coulombic long-range interactions. These eigenstates are relevant in quantum field theory, condensed matter physics, gravitational wave physics, and astrophysical models, and they reflect collective quantized modes—often involving higher spin or complex tensor degrees of freedom—localized or bounded by gravitational backgrounds or effective potentials.

1. Non-Relativistic Planar Spin-2 Eigenstates (GMP Mode)

The foundational microscopic description of gravitational molecular eigenstates stems from the non-relativistic limit of the massive spin-2 Fierz–Pauli equations in $(2+1)$ dimensions and from the null reduction of linearized Einstein equations in $(3+1)$ dimensions (Bergshoeff et al., 2017). The essential steps are:

• Fierz–Pauli Theory: The complexified, symmetric, traceless rank-2 tensor field $f_{(\mu\nu)}$ satisfies dynamical second-order equations under subsidiary constraints. Factoring out the rapidly oscillating rest energy,

$f_{(n)} = e^{-i(mc^2 - E_0)t} \Psi_{(n)},$

and sending $c \rightarrow \infty$ with fixed $\lambda = \frac{\hbar}{mc}$ yields:

$$i\hbar \partial_t \Psi[2] = H \Psi[2], \quad H = -\frac{\hbar^2}{2m}\nabla^2 + E_0,$$

reducing the problem to a Schrödinger equation for a single spin-2 degree of freedom, identified as the GMP gapped mode in fractional quantum Hall settings.

• Null Reduction of Einstein Equations: Using light-cone coordinates for the linearized 4D metric $h_{MN}$ and imposing $h_{M-} = 0$,

$$\partial_{-}\Psi_{(n)} = i\frac{m}{\hbar}\Psi_{(n)},$$

one finds that the surviving transverse-traceless component satisfies the same planar Schrödinger equation as above. The reduction breaks parity, paralleling the FP analysis.

These derivations establish that gravitational molecular eigenstates in a planar system follow the quantum mechanics of the GMP spin-2 mode, encapsulated by a non-relativistic Schrödinger equation.

2. Harmonic Oscillator Potential from Collective Gravitational Backgrounds

When considering a large, uniform distribution of spin-2 particles, the collective gravitational field background can be described by Brinkmann-wave solutions to Einstein’s equations (Bergshoeff et al., 2017):

$ds^2 = 2dx^+dx^- + 2v(x^i)(dx^-)^2 + dx^i dx^i,$

where $v(x^i)$ is a potential induced by the energy-momentum tensor source $T_{++} = \rho$. For a rotationally symmetric solution,

$v(x) \propto m\omega^2 |x - x_0|^2,$

the Hamiltonian for each excitation becomes

$$H = -\frac{\hbar^2}{2m}\nabla^2 + \frac{1}{2}m\omega^2 |x - x_0|^2,$$

with discrete energy levels—these quantized oscillator eigenstates constitute gravitational molecular eigenstates. The frequency $\omega$ relates to the density, e.g., $\omega = \sqrt{8\pi G\rho}$ (modulo factors).

This construction elucidates how macroscopic gravitational self-interaction can act as a confining mechanism, yielding quantized, localized eigenmodes within the many-body setting.

3. Gravitational Wave Propagation and Eigenstates in Structured Media

A distinct manifestation arises when gravitational waves propagate through a molecular medium—an aggregation of particles with inner structure—where averaging the microscopic stress-energy tensor induces an effective quadrupole polarization (Moretti et al., 2019):

$$\langle T_{\mu\nu} \rangle = T_{\mu\nu}^{(f)} + \frac{c^2}{2} \partial^\rho \partial^\sigma Q_{\mu\rho\nu\sigma}, \quad Q_{i0j0} = \varepsilon_g\left(\frac{2}{c^2}\phi_{,ij} + \frac{1}{2}h_{ij,00}\right),$$

with gravitational dielectric constant $\varepsilon_g = \frac{N L^5 c^2}{4G}$.

The wave equation governing metric perturbations becomes fourth order:

$$\partial^4_0 \overline{h}_{\mu\nu} + m^2\square \overline{h}_{\mu\nu} = 0, \quad m^2 = \frac{c^2}{4\pi G \varepsilon_g}.$$

This leads to dispersion and, crucially, the emergence of extra polarization modes beyond the standard two (plus and cross). Anomalous polarizations include cross modes in other planes and a breathing-longitudinal superposition, with amplitude ratio $\sim k^2/m^2$ with respect to standard modes.

The structured medium "quantizes" the gravitational perturbation, producing molecular eigenmodes whose character (e.g., longitudinal/breathing) is set by the medium’s constitutive relations. Even if quantitative deviations are minor ($\sim 10^{-14}$ to $10^{-8}$ depending on medium density and scale), they accumulate over astrophysical distances and are potentially observable in multi-messenger scenarios. This extends the concept of gravitational molecular eigenstates to the spectral modes arising from collective internal structure.

4. Formation and Structure of Multistate Gravitational Atoms via Schrödinger–Poisson Dynamics

In the context of self-gravitating Bose condensates, colliding ground-state solutions of the Schrödinger–Poisson (SP) system in orthogonal states dynamically form mixed-state equilibrium configurations with both spherical (monopolar) and dipolar components, interpreted as multistate gravitational atoms (Guzman, 2022). The governing equations are:

$$i\partial_t \Psi_{nlm} = -\frac{1}{2}\nabla^2 \Psi_{nlm} + V \Psi_{nlm}, \quad \nabla^2 V = \sum_{nlm} |\Psi_{nlm}|^2,$$

where $\Psi_{nlm}$ represents wavefunctions for quantum numbers $n,l,m$.

Simulated collisions, parameterized by mass ratios and head-on momenta, show that the less massive configuration pinches off from the denser core and redistributes along the collision axis, forming a stable mixed state upon time-averaging. The final density profiles exhibit characteristic monopolar and dipolar features matching stationary eigenstate expectations.

This process yields gravitational molecular eigenstates as attractors in the phase space of the SP system, suggesting a means of astrophysical structure formation that naturally generates multi-component eigenmodes. Such states underpin anisotropic gravitational potentials relevant for dark matter halo structure and the planar arrangements of satellite galaxies.

5. Quantum Three-Body Harmonic Molecule: Degeneracy, Level Splitting, and Chaos

The quantum three-body harmonic system, modeled by pairwise potentials with finite rest length $R$,

$$V_R = \frac{3}{2} m \omega^2 \left[(r_{12} - R)^2 + (r_{13} - R)^2 + (r_{23} - R)^2 \right],$$

analyzes the eigenstates of three identical masses (Olivares-Pilón et al., 2022). At $R=0$, the system is fully separable in Jacobi coordinates, exactly solvable, and exhibits maximal degeneracy,

$$\psi^G_N(r_{12}, r_{13}, r_{23}) = P_{N,g}(r_{12}^2, r_{13}^2, r_{23}^2)\exp\left[-\frac{\omega}{2}(r_{12}^2 + r_{13}^2 + r_{23}^2)\right],$$

with energy eigenvalues $E^N = 3\omega(2N+3)$, degeneracy $g=(N+1)(N+2)/2$. At $R>0$, separation is lost, the spectrum splits into sublevels, and the eigenvalue problem requires numerical solution (Lagrange-mesh method), with the system's classical analogue exhibiting chaotic dynamics.

These eigenstates represent vibrational modes of gravitational or effective molecules, realized in macroscopic clusters where binding is not Coulombic but modeled by confining potentials. The idealized $R=0$ case provides mathematical tractability; the physically richer $R>0$ regime introduces substantial complexity.

6. Physical Significance and Applications

Gravitational molecular eigenstates encapsulate discrete, quantized modes emerging from collective gravitational effects in systems with internal structure, higher spin (e.g., spin-2) dynamics, or effective confining interactions. Their theoretical construction spans quantum field limits, non-relativistic reductions, condensed matter analogs, gravitational wave propagation in structured media, and multi-component Schrödinger–Poisson dynamics.

In condensed matter, the GMP mode’s eigenstates govern collective excitations in fractional quantum Hall systems. In astrophysics, multistate gravitational atoms provide candidate structures for dark matter halos and satellite galaxy planar distributions. In gravitational wave astronomy, the presence of structured matter yields extra polarization channels—a consequence of molecular eigenmode coupling.

Methodologically, these systems blend field-theoretic reduction (Fierz–Pauli, Einstein equations), molecular moment averaging, numerical evolution (Runge–Kutta, Lagrange-mesh), and algebraic symmetry analysis (degeneracy and splitting). They thus supply an intensive laboratory for exploring quantized dynamics, symmetry breaking, collective localization, and chaos in gravitationally dominated or mimetic environments.

7. Summary Table: Main Systems and Mechanisms

Framework	Key Mechanism	Physical Manifestation
FP/Einstein Reduction	Non-relativistic Schrödinger equation	GMP spin-2 eigenstates, confining oscillator
Gravitational Waves in Medium	Quadrupole polarization, dispersion	Extra polarizations, molecular eigenmodes
Schrödinger–Poisson Collisions	Multistate formation through dynamics	Mixed monopolar/dipolar gravitational atoms
3-body Harmonic Molecule	Degeneracy, level splitting, chaos	Vibrational modes of effective molecules

Each entry reflects how quantized gravitationally-bound eigenmodes emerge due to collective dynamics, internal structure, or confining potentials. A plausible implication is that the rich spectrum of molecular eigenstates—linked to underlying symmetries, symmetry breaking, and collective excitation—extends traditional quantum mechanical intuition into gravitational and astrophysical domains.