Because the Einstein Field Equations are computationally difficult and a tabletop experiment involves measurement of weak fields, Post-Newtonian approximations are helpful to understanding the scale of measurements of laboratory EM fields. Applicable approximations will also describe the nature of the EM field and give and indicate the geometry of gravitational warping. We found it necessary to focus on Post-Newtonian approximations that directly correlated an EM field to gravitational effects.
Mathematics describing the connection between neutron stars and gravitational waves is applicable to understanding similar relationships at a tabletop scale. Preston Jones has shown that a time varying spacetime metric gives rise to electromagnetic radiation (Jones, 2007) \[\nabla\cdot{S}+\frac{\partial U}{\partial t} = \frac{4}{c^4}U\frac{\partial \phi^2}{\partial t},\]
where use has been made of the Poynting vector, \(S = \frac{c}{4π}E × B\), and the factor of \(c\) has been reinstated to illustrate the order of magnitude of this effect. This demonstrates that in a region of nonzero internal energy density \(U\) and rapidly time varying gravitational potential, \(\phi\), electromagnetic radiation will be generated. Note, the approximation above only uses the covariant relationships.
We assume that the relationship can be reversed. By solving for the time varying gravitational potential, \(\phi\), we show that electromagnetic radiation gives rise to a time varying spacetime metric. \[\phi = \frac{c^2}{2}\sqrt{\int\frac{\nabla\cdot{S}}{U}\frac{\partial U}{\partial t}}\]
By having an equation that readily describes the relationship between spacetime warping and EM radiation, we can accurately predict results in experimental setups. This is also important because the calculations for White’s Alcubierre experiments were not readily available to assist estimations of laser warping and phase shift.