Formulations of General Relativity and their Applications to Quantum Mechanical Systems (with an emphasis on gravitational waves interacting with superconductors)
The linearization of General Relativity leads to various formulations of gravity often referred to as gravito-electromagnetism due to its resemblance to electromagnetism. Three methods are compared: (i) the harmonic gauge approach; (ii) the Parameterized Post-Newtonian (PPN) approach; and (iii) the Helmholtz Decomposition (HD) approach. New relationships are developed that are not generally found in the literature. These include the use of the linearized Bianchi identity, the Landau-Lifshitz pseudotensor, the Isaacson power formula, the geodesic equation of motion, and the geodesic deviation equation. The formalism is applied to examples such as a mass-solenoid and a gravitational mutual inductance system.
The HD approach is shown to be the most favorable of the three methods due to being gauge-invariant (to linear order in the metric), and because it shows explicitly that the transverse-traceless part of the metric contains the only radiative degrees of freedom. This is similar to the transverse-traceless (TT) gauge except that the HD formulation is fully valid in matter. Therefore, unlike the TT gauge, the HD formulation can be used to describe how gravitational waves interact with various types of material. Traditionally, it is believed that all known materials are essentially transparent to gravitational waves. However, this conclusion relies on a classical treatment which describes how gravitational waves (originating from astrophysical sources) are passively detected with no affect on the wave itself. As an alternative, we consider how gravitational waves could be coupled to quantum systems which may be used for detection as well as reflection and even generation of gravitational waves.
To investigate this possibility, a classical Hamiltonians is developed which describes the kinematics of charged, relativistic, massive particles in curved space-time. The coupling of quantum matter to gravitational fields is then described by quantizing the Hamiltonian. This leads to various gravitational quantum effects such gravitational Aharonov-Bohm effects, gravitational Casimir effects, and various time-holonomies. Furthermore, developing a quantized stress tensor and taking the expectation value allows the Einstein field equation to predict how quantum matter can produce classical gravitational fields. This semi-classical approach is used to describe how superconductors interact with gravitational waves. A London-like constitutive equation describes the response of the superconductor in terms of a "gravitational shear modulus" analogous to the standard shear modulus of elastic mechanics. Also using a "gravitational permeativity" (analogous to the magnetic permeability) leads to a gravitational plasma frequency, index of refraction, penetration depth, and impedance. The same analysis also done for a normal conductor using a gravitational Ohm-like constitutive equation, however, it is shown that a superconductor exhibits a gravitational Meissner-like effect, while a normal conductor does not.
For the case of a superconductor, the Cooper pairs are described by the Ginzburg-Landau free energy density embedded in curved spacetime. This leads to a new gravito-London gauge condition and a predicted graviton mass within the superconductor. Next, the ionic lattice is modeled by an ensemble of quantum harmonic oscillators coupled to gravitational waves and characterized by quasi-energy eigenvalues for the phonon modes. This formulation predicts a gravitationally-induced dynamical Casimir effect within the ionic lattice since the zero-point energy of the phonon modes is modulated by the gravitational wave. Applying periodic thermodynamics and the Debye model in the low-temperature limit leads to a free energy density for the ionic lattice. From these results it is shown that the response to a gravitational wave is far less for the Cooper pair density than for the ionic lattice. This predicts a charge separation effect which can be used to detect the passage of a gravitational wave, and the possibility of reflection of gravitational waves by a superconductor. Lastly, a long-range communication system is proposed based on the coupling of gravitational and electromagnetic waves via ellipsoidal superconducting cavities.