Flattening Electrodynamics in a curved space

Question

It is possible, apparently, to describe gravitational lensing as if gravitational potential induces an effective refractive index change in the vacuum, and spacetime is flat.

As pointed out by @AndrewSteane, this has nothing really to do with electromagnetism; it is merely a useful fiction that can be helpful in calculating null geodesics- which correspond, of course, to light rays in the vicinity of a gravitating mass.

I would like to know if something analogous can be done to describe the electric field in the vicinity of a charged mass, as if space were flat. I imagine this would have the form of an "effective permittivity field" around the mass, as a function of gravitational potential.

Edit 1/4/19: After a bit more thought -- there would likely be an "effective magnetic permeability field" as well; and both effective fields might well be anisotropic. This paper, "Effective refractive index tensor for weak-field gravity" appears to point in this direction.

Edit 1/16/19: After a lot more searching, this paper turned up. The authors, Isabel Fernandez-Nunez and Oleg Bulashenko, show that the trajectory of an electromagnetic wave is accurately described by specifying electric permittivity and magnetic permeability, dependent respectively on gravitational time dilation and spatial curvature. Moreover, the authors state:

Then, it can be shown that the covariant Maxwell’s equations written in curved coordinates can be transformed into their standard form for flat space but in the presence of an effective medium.

The "effective medium" is described by tensors dependent on the components of the space time metric.

The statement is not about electromagnetic waves; it is about Maxwell's equations. So-- if the authors are right -- my conjecture that something analogous can be done to describe the electric field in the vicinity of a charged mass, as if space were flat must be correct.

I'll welcome any informed comments. "Informed" means that the commenter has read and understood the two papers I've referenced.

Answer 1

Yes, this could be done for some cases. The approach known as the congruence approach is outlined in a book

• Landau L. D., Lifshitz E. M., The Classical Theory of Fields. (Pergamon Press, Oxford; New York, 1971), (Paragraph 90, and the problem immediately thereafter).

and is based on a $3+1$ splitting of spacetime. Within this approach the covariant equations of electrodynamics in a gravitational field can be reduced to a set of $3+1$ equations very similar to the usual Maxwell's equations in matter.

Another interesting paper containing applications specifically along the lines of flat space matter equivalent of a curved spacetime:

• Leonhardt, U., & Philbin, T. G. (2006). General relativity in electrical engineering. New Journal of Physics, 8(10), 247, doi:10.1088/1367-2630/8/10/247, arXiv:cond-mat/0607418

(and it has a LaTeX-ed equations):

… free-space Maxwell equations can be written in the macroscopic form \begin{equation} \quad \nabla\cdot\mathbf{D} = \rho \,,\quad \nabla\times\mathbf{H} = \frac{\partial \mathbf{D}}{\partial t} + \mathbf{j} \,,\quad \nabla\cdot\mathbf{B} = 0 \,,\quad \nabla\times\mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \,. \end{equation}

… In empty but possibly curved space, the electromagnetic fields are connected by the constitutive equations in SI units, \begin{equation} \mathbf{D} = \varepsilon_0\varepsilon\,\mathbf{E} + \frac{\mathbf{w}}{c}\times\mathbf{H} \,,\quad \mathbf{B} = \frac{\mu}{\varepsilon_0 c^2}\,\mathbf{H} - \frac{\mathbf{w}}{c}\times\mathbf{E} \,. \end{equation} with the symmetric matrices $ε$ and $μ$ and the vector $\mathbf{w}$ given as \begin{equation} \varepsilon = \mu = -\frac{\sqrt{-g}}{g_{00}}\,g^{ij} \,,\quad w_i = \frac{g_{0i}}{g_{00}} \,. \end{equation}

Note, that for a general spacetime purely spatial part of the metric would still be curved, (and the $\nabla$ operator would mean the covariant derivative with respect to this curved space). Also, constitutive relations generally contain electric/magnetic cross-terms. However, for static spacetimes ($\mathbf{w}=0$) with (conformally) flat spatial part it is potentially possible to construct dielectic medium in a flat spacetime with an effective “EM metric”, equivalent to such a curved space. Among the physically interesting spacetimes with this property are the Schwarzschild and Reissner–Nordström metrics. Note, that Kerr metric, however, does not admit conformally flat spatial slices.

---

Update: For Schwarzschild metric specifically, the task outlined above has been completed in the paper:

• Chen, H., Miao, R. X., & Li, M. (2010). Transformation optics that mimics the system outside a Schwarzschild black hole. Optics express, 18(14), 15183-15188, doi:10.1364/OE.18.015183, (open access).

In this paper the dielectric system obtained (as a simulation) is used not just for calculation of geodesics but for analyzing the interaction of EM beam with the “black hole”.

This work could be used as a basis for multiple black holes simulation, but only in cases where black holes are far away, so that their orbital motion is non-relativistic.