# Flattening Electrodynamics in a curved space

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.

• I'd also argue that the "effective refractive index" thing only applies in a certain limit, becuase if the light ray gets close enough to the unstable $r=3M$ orbit, it can do an arbitrary number of loops around the horizon before escaping to infinity. – Jerry Schirmer Jan 4 at 23:33
• @JerrySchirmer, please elaborate. Is it impossible for that to happen, regardless of the form of the radial dependence of an "effective refractive index" field? – S. McGrew Jan 5 at 0:41
• Just to say that I think effective index method is exact throughout exterior Schwartzschild region, and can cope with the light sphere at $r=3M$ and things like that. However I couldn't find this elsewhere so developed my own proof (for my book) so might have got this wrong I suppose ... – Andrew Steane Jan 5 at 2:16
• Check out the comments by AFT and the links under this answer: physics.stackexchange.com/questions/429311/… – safesphere Jan 5 at 9:57
• @JerrySchirmer: light ray gets close enough … it can do an arbitrary number of loops around the horizon before escaping to infinity. It is certainly possible to wind a number of fiber optic loops around a point. So your counterargument to simulating metric with varying refractive index does not work. – A.V.S. Jan 6 at 21:16

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:

(and it has a LaTeX-ed equations):

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

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

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.

• Would it be possible to construct an effective "EM metric" in the space surrounding two gravitationally bound, mutually orbiting charged black holes? – S. McGrew Jan 7 at 4:29
• With a physical flat space? No, at least for the really interesting regimes, such as light ray orbit encircling both BH's. Note, that even Schwarzschild black holes when moving have ergoregions, so there would be large E-M mixing there. However, maybe it would be possible for the Majumdar-Papapetrou solution (when multiple charged black holes are in a static equilibrium when gravitational attraction balances electrostatic repulsion). – A.V.S. Jan 7 at 7:36
• my specific interest at the moment is actually the static interaction between charged massive particles, and whether or not it makes sense to describe the interaction in terms of "effective electric permittivity" and "effective magnetic permeability" induced by interaction between the gravitational field and the electromagnetic field. – S. McGrew Jan 8 at 21:48
• No it does not make sense, since there is no description of gravitational interaction in terms of effective permeability/permittivity alone, you still need to solve Einstein-Maxwell field equations to get the metric & EM fields. Only then it is possible to calculate effective embedding in flat space. – A.V.S. Jan 9 at 3:21