Historical research guide · 0539 · 2026
Einstein's equations in electromagnetic media
Eren Erberk Erkul · Ulf Leonhardt
In one minute
Eren Erberk Erkul and Ulf Leonhardt take a sixty-five-year-old idea and give it a moving part. In 1960 Jerzy Plebanski showed that light in curved spacetime behaves exactly like light in flat space filled with a particular material: bend the metric, or build the right glass, and you cannot tell the difference from the inside. That mapping was static. Erkul and Leonhardt extend it to the full time-evolving form of Einstein’s equations used in numerical relativity, where spacetime is sliced into three-dimensional sheets stacked through time. Every piece of the gravitational machinery gets a material counterpart: the lapse becomes a conformal factor, the shift becomes the magneto-electric coupling of a dragged medium, the spatial metric becomes the permittivity tensor, and the changing curvature becomes the changing permittivity. Linearise the result in vacuum and the equation for a gravitational wave and the equation for a ripple in the permittivity turn out to be the same equation. They also say how to build one, with an acousto-optic modulator on a pump beam.
Connection
Chapter 4 says the metric is something you engineer rather than something you are stuck with, and this paper writes down the dictionary: every term in Einstein’s evolution equations becomes a specification for a material’s permittivity, permeability and magneto-electric coupling. It is the clearest published statement of chapter 5’s picture too — the vacuum treated as a medium with optical properties — and it runs the correspondence in both directions, so a chosen spacetime becomes a design brief for a material and a material becomes a simulator for the spacetime.
From the source
Plebanski’s 1960 mapping expresses Maxwell’s equations in a curved spacetime metric as Maxwell’s equations in a flat background filled with a bianisotropic dielectric medium, whose relative permittivity and permeability tensors and magneto-electric coupling vector are read directly off the metric components — a material of space obeying constitutive relations in which the electric displacement and the magnetic flux density each pick up a cross term from the other field.Introduction, equations 1 and 2
The paper’s new step is to extend that mapping from the metric alone to the full Arnowitt-Deser-Misner three-plus-one system, so that the Hamiltonian and momentum constraints and the evolution equations of general relativity become dynamical conditions on the medium’s constitutive parameters: the lapse maps to a conformal factor, the shift to the magneto-electric coupling, the spatial metric to the permittivity tensor, the extrinsic curvature to the time derivative of the permittivity, the matter sources to the electromagnetic stress-energy, and gravitational waves to dielectric perturbations.Table 1, ADM-optical correspondence
A nonzero shift vector is realised optically as the cross-coupling term in the constitutive relations, which is exactly what one obtains for a dielectric moving with a physical three-velocity proportional to the magneto-electric vector; the shift can therefore be regarded as an effective medium velocity that induces the same Lorentz-type mixing of electric and magnetic fields that a real boost generates in free space.Shift as medium drag, equations 8 to 11
In the linearised vacuum limit with the transverse-traceless gauge, the wave equation for the transverse-traceless metric perturbation and the wave equation for the transverse-traceless part of the permittivity perturbation are the same partial differential equation, and the two independent components of the amplitude correspond to the plus and cross polarisation states of a gravitational wave — so the dielectric perturbation is a direct analogue of a gravitational wave propagating through the optical medium.Analogue of gravitational waves, equations 42, 48, 49 and 50
The authors name the hardware: a medium satisfying that wave equation can be implemented using the Kerr effect, by generating a sinusoidal modulation on a continuous-wave pump beam with an acousto-optic modulator — the same route used for the optical analogue of an event horizon — or by electrically tuning the permittivity profile of a spacetime metasurface.Analogue of gravitational waves, discussion following equation 49
The correspondence runs both ways and enables inverse design: starting from a chosen foliation or a target evolution of spacetime, one translates it into the required space- and time-dependent profile of permittivity and magneto-electric vector, so engineers can draw on the library of general-relativity solutions; conversely the same dictionary gives an electromagnetic reading of gravitational phenomena and offers an alternative simulation framework for numerical relativity.Conclusion
The way in
https://doi.org/10.1209/0295-5075/ae58cfHow to cite it
Eren Erberk Erkul, Ulf Leonhardt (2026) Einstein's equations in electromagnetic media doi:10.1209/0295-5075/ae58cf