# Gravitational Stark Effect

Could gravity induce line splittings in the optical spectrum of a molecule similar to the Stark or Zeeman Effects?

Naively, a gravitational potential would be a simple addition to the Hamiltonian that should behave similarly to a constant electric field (i.e. the simplest condition for the Stark Effect). Is there a reason that things should not be so simple?

If such splittings do occur, what would the typical energy difference be between the states? Would they be observable at tenable energies?

Unlike electrical or magnetic field, which acts differently on particles of different charge, for the gravitational field there is equivalence principle, which means that electrons and nuclei would experience the same acceleration due to gravity.

There is, of course, the overall shift of energy level if the atom is observed from the place with the different gravitational field. This shifts would be the same for all spectral lines and can be measured in laboratory (see a page on Pound–Rebka experiment).

The splitting of levels does occur if we consider the tidal forces inside the atom. Because the gravitational field is non-uniform at different points, the gravitational acceleration would be slightly different for electron and nucleus. The general relativistic treatment of the effect is given in geodesic deviation equation. The effect is proportional to the spatial derivatives of the gravitational acceleration. So for the gravitational field of a single mass $M$ we can obtain an estimate for the shift for $1s$ level: $$E^{(1)} \sim \frac{G M}{r^3} a_0^2 m_e,$$ where $a_0$ is Bohr radius. Such shift would be different for various atomic levels. The absolute value of such a shift is extremely small: for an atom near the horizon of a solar mass black hole this produces the $E^{(1)} \sim 10^{-22} \,\text{eV}$, well beyond detectable limits. Using Rydberg atoms with large $n$ could greatly increase the effect: since the size of the atom increases as $n^2$, the energy shift would increase as $n^4$. Also, for microscopic size black holes, the effect could potentially be detectable.

Reference:

1. Gill, E., Wunner, G., Soffel, M., & Ruder, H. (1987). On hydrogen-like atoms in strong gravitational fields. Classical and Quantum Gravity, 4(4), 1031. doi:10.1088/0264-9381/4/4/033.