Einstein's equations in electromagnetic media

Abstract

In this paper, we extend Plebanski's mapping to encode the Einstein equations in Arnowitt-Deser-Misner (ADM) form within a bianisotropic electromagnetic medium. We realise this by translating the ADM constraints and evolution equations into dynamical conditions on the medium's constitutive parameters. These transformed equations are then linearised in vacuum to derive gravitational-wave analogues as perturbations of the optical medium.

Introduction

In 1960, Jerzy Plebanski, building on earlier work and revisited in various forms in numerous follow-up studies, discovered the mapping given in equation 1: the relative permittivity and permeability tensors of an effective medium are both set equal to minus the square root of minus the metric determinant times the spatial metric components divided by the time-time metric component, while the magneto-electric vector is built from the time-space components divided by the same time-time component. This mapping expresses Maxwell's equations in a curved-spacetime metric as Maxwell's equations in a flat background but filled with a bianisotropic dielectric medium. This "material of space" obeys the constitutive relations of equation 2: in SI units the electric-displacement field equals the vacuum permittivity times the relative permittivity acting on the electric field, plus the magneto-electric vector divided by the speed of light crossed into the magnetic-field intensity; and the magnetic flux density equals the vacuum permeability times the relative permeability acting on the magnetic-field intensity, minus the same cross term taken with the electric field.

Several decades later, the field of transformation optics developed, in which coordinate transformations were shown to be a powerful design tool.

However, Plebanski's work, as well as the various analogue-gravity frameworks in optical media that subsequently emerged, did not assume that the metric solves Einstein's equations in vacuum; instead, the metric was taken as the background geometry in which Maxwell's equations are written.

In the present paper, we move beyond the purely kinematic aspects of this mapping and explore the conditions necessary to encode the dynamical structure of Einstein's equations within dielectric media. We thus require that the transformed optical metric also satisfy the gravitational constraints. This step is motivated by a conceptual similarity between transformation optics and the ADM formalism.

Both in the ADM formalism and transformation optics, the object described by the metric is a three-dimensional slice representing a spatial hypersurface. However, in the ADM formalism, one explicitly evolves these spatial slices — called foliations — through time, whereas in transformation optics the analysis is typically confined to a single effective optical slice. Hence the idea is to extend Plebanski's mapping to the whole ADM system.

It is important to note that this is an analogue encoding at the level of the three-plus-one field equations; we do not claim a canonical equivalence between the fundamental Hamiltonian phase spaces of general relativity and Maxwell theory. Therefore, the mapped ADM equations should be read as design constraints on constitutive parameters that vary in space and time, and may have to be implemented through active or externally controlled media, rather than as a microphysical derivation of a material whose internal dynamics enforces these conditions.

Electromagnetic ADM three-plus-one system

The Arnowitt-Deser-Misner formalism, also known as the Hamiltonian formalism, is an equivalent formulation of general relativity. Unlike the usual covariant picture, this formulation reduces the spacetime manifold to a trivial fibre bundle, equation 3, whose projection sends a spacetime point to its time coordinate. Here the base is the real line, the global Cauchy time, while the leaves of the foliation play the role of the fibre and are homeomorphic to a Cauchy surface. Figure 1 of the article visualises the ADM foliation as a stack of spatial slices.

In general relativity, the collection of fibres through the base space can be viewed as the collection of inertial frames, where a suitable choice of coordinates — an orientation of the foliation — can always flatten each fibre into an inertial one. The key difference between the trivial bundles of general relativity and those of Newton-Cartan theory is an extra degree of freedom that allows the spatial foliation to twist and curl. Hence, Newtonian foliations are absolute entities; no gravitational waves arise there, lacking the radiative transverse-traceless part of the curvature. Relatedly, both Newtonian and Einsteinian bundles differ from the Galilean bundle in that the extra freedom now resides in the fibres themselves.

The line element in ADM coordinates adapted to the slicing defined by constant time reads equation 4: minus the lapse squared times the time interval squared, plus the spatial metric contracted with the coordinate displacements each shifted by the shift vector times the time interval. Equations 5 and 6 write the metric tensor and its inverse in matrix form. Here the spatial metric lives on the constant-time slices. The dynamism of this system is encoded in the lapse and the shift, both one-forms. While the net evolution vector, called the time vector, drags a point through the full three-plus-one-dimensional spacetime, it combines both effects, equation 7: the time vector equals the lapse times the future-pointing unit normal to each slice, plus the shift.

Shift as medium drag

Shift vectors gauge how points move within a spatial slice when we advance the coordinate time, equation 8: the spatial displacement equals the shift times the time interval, evaluated on the slice. Extending Plebanski's mapping and substituting the ADM decomposition of the time-space metric components, we obtain equation 9: the magneto-electric vector equals the spatial metric contracted with the shift, divided by the time-time metric component. Note that the magneto-electric vector, carrying one spatial index, is not a component of the metric. Raising the spatial index gives equation 10, and inverting gives equation 11: the shift equals the time-time metric component times the raised magneto-electric vector.

Hence a nonzero shift is realised optically as a 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. Equivalently, the shift can be regarded as an effective medium velocity proportional to that vector, which induces the very Lorentz-type mixing of the electric and magnetic fields that a real boost generates in free space.

Lapse as conformal factor

The lapse is a one-form that measures proper time between adjacent slices, equation 12: proper time equals the lapse times coordinate time. Since the source-free Maxwell equations in four dimensions are invariant under local conformal rescalings of the metric, we can factor out a positive scalar from the metric, equation 13. A convenient choice, equivalent to taking the constant-time slicing, is to set the time-time metric component to minus one in the ADM line element, which fixes the conformal factor squared to equal the lapse squared minus the shift contracted with itself, equation 14; taking the positive square root gives equation 15. When the shift vanishes, this reduces to the conformal factor being the lapse itself, matching the standard result in transformation optics: changing the lapse is merely a conformal rescaling of the metric.

The index-lowered vacuum expressions for the Plebanski constitutive tensors with this identification read equation 16: both the permittivity and the permeability equal the square root of the spatial metric determinant divided by the lapse, times the spatial metric. Under the conformal rescaling the ADM variables transform as equation 17, and substituting these into the definition of the permittivity gives equations 18 to 20: under the same rescaling the Plebanski constitutive tensors acquire four powers of the conformal factor. As expected, the source-free Maxwell equations are therefore insensitive to the overall scale.

It is worth noting that conformal transformations play a different role in lower-dimensional systems. In two spatial dimensions, every metric is locally conformally flat, so the Plebanski map can be satisfied with an isotropic medium, equation 21, making conformal methods a powerful design tool in transformation optics. By contrast, in three spatial dimensions local conformal flatness is dictated by the Cotton tensor, whereas in the full three-plus-one-dimensional spacetime conformal curvature is characterised by the Weyl tensor. In this setting, conformal rescalings do not generate an arbitrary spatial metric and are therefore not a viable general design tool. Hence, in the present work we analyse the conformal invariance of the source-free Maxwell equations rather than conformal flatness of the metric itself. This amounts to a gauge redundancy in the Plebanski map, which is then reduced once the ADM dynamical conditions are imposed.

The decomposition remains valid provided the time-time metric component is negative, that is, provided the lapse squared exceeds the shift contracted with itself — a condition automatically satisfied for any space-like foliation.

Electromagnetic ADM constraints and dynamics

The three-plus-one split of spacetime entails two constraint equations that determine the intrinsic structure of each hypersurface, namely the momentum and Hamiltonian constraints, equations 22a and 22b. The Hamiltonian constraint sets the intrinsic spatial curvature scalar plus the square of the trace of the extrinsic curvature minus the extrinsic curvature contracted with itself equal to sixteen pi times Newton's constant times the energy density. The momentum constraint sets the divergence of the trace-reversed extrinsic curvature equal to eight pi times Newton's constant times the momentum density. Figure 2 of the article visualises the ADM constraint equations.

Together with the evolution equations 23a and 23b — the time derivative of the spatial metric written in terms of the extrinsic curvature and the symmetrised covariant derivative of the shift, and the time derivative of the extrinsic curvature written in terms of the covariant Hessian of the lapse, the intrinsic Ricci tensor, quadratic curvature terms, the matter stress term and the Lie-drag terms along the shift — one has the full dynamical system. Here the spatial electromagnetic stress tensor is given by equation 24. Since we confine ourselves to electromagnetic fields and the Maxwell stress-energy tensor is traceless, one term of the evolution equation vanishes and the evolution equation retains only the electromagnetic back-reaction, which introduces nonlinearities into the sources discussed in the next section.

Solving the first evolution equation for the extrinsic curvature gives equation 25. Reorganised, the Plebanski map reads equation 26: the spatial metric equals minus the time-time metric component divided by the square root of minus the metric determinant, times the permittivity; and the shift equals the time-time component times the magneto-electric vector. Substitution yields equation 27, in which the first term captures explicit temporal modulation of the permittivity and the remaining covariant derivatives describe the stationary drag induced by the gauge fields. Both contributions appear quadratically in the Hamiltonian constraint.

Because the shift is the time-time metric component times the magneto-electric vector, the defining relation for that component becomes a quadratic equation whose negative-root solution is equation 28. Equations 27 and 28 express the extrinsic curvature entirely in terms of the permittivity, the magneto-electric vector, the lapse and the shift.

The intrinsic geometry of each optical slice is encoded in the spatial metric, from which one builds the intrinsic Ricci tensor and its scalar, equation 29. Under a conformal rescaling one obtains equation 30, and choosing the conformal factor squared to equal minus the time-time metric component gives equation 31, in which an effective lapse appears, defined as the square root of the lapse squared minus the shift contracted with itself. The numerator depends only on the spatial permittivity, whereas lapse and shift enter through the conformal prefactor and its gradients. Henceforth, every appearance of the spatial metric, the extrinsic curvature, the intrinsic Ricci tensor or the spatial covariant derivative refers to the optical metric built from the permittivity and permeability.

Electromagnetic ADM sources

Given mass sources and currents, the ADM constraints and evolution equations encode Einstein's equations. Using the identifications above, the entire system can be written in the language of the effective dielectric. To complete the analogue system, we take both the mass density and the momentum density to be provided solely by the electromagnetic field. This would describe the self-gravity of light. Note that this is not an entirely artificial situation. For example, the early Universe was radiation-dominated; black-hole magnetospheres can be modelled electromagnetically; and certain black-hole metrics can be generated from colliding electromagnetic waves through the Chandrasekhar-Xanthopoulos duality.

With this choice our source expressions coincide with those obtained by Thorne, Price and Macdonald in their three-plus-one formulation of electrodynamics. In the notation we already established, Maxwell's equations in the three-plus-one formalism read equations 32a to 32d: the divergence of the magnetic flux density vanishes; its evolution equation carries the shift-dragged flux and the curl of the electric field weighted by the lapse; the divergence of the electric displacement equals the charge density measured by an Eulerian observer; and its evolution equation carries the shift-dragged displacement, the curl of the magnetic-field intensity and the physical current.

The individual expressions can be understood within our framework when the identifications above are applied. Projecting the electromagnetic stress-energy tensor, one finds that, in the analogy, the mass density is identified with the energy density and the current with the Poynting vector, both divided by the speed of light squared, equations 33a and 33b.

In this way, we describe the nonlinear gravitational dynamics of light through the dielectric-medium equations; solving them yields the full dynamics of an optical medium that is the analogue of a self-gravitating spacetime.

Analogue of gravitational waves

Now that we have the machinery at hand, let us consider its implications for the dielectric by examining the linearised regime of general relativity, which is as simple as it gets, but not simpler. Assume the time-time metric component is minus one, so the lapse is one and the shift vanishes throughout this section. For the dielectric tensors we obtain, from the Plebanski map, equation 34: the spatial metric equals the determinant of the permittivity raised to the power minus one fifth, times the permittivity — a form that also arises in many applications of transformation optics.

In the regime of weak gravity, the spatial metric is the identity plus a small perturbation, equation 35. At linear order, the matter stress term in the second evolution equation is second order in the perturbation and is therefore neglected here. The evolution equations reduce to equations 36a and 36b: the extrinsic curvature is minus one half the time derivative of the perturbation, and the time derivative of the extrinsic curvature equals the linearised Ricci tensor, which is given by equation 37 in terms of the Laplacian of the perturbation, the double gradient of its trace and the symmetrised divergence terms.

Assuming vacuum and the weak-gravity regime, the Hamiltonian constraint yields equation 38 and the momentum constraint gives equation 39. Under the transverse-traceless conditions, equation 40 — the perturbation is divergence free and traceless — both are trivially satisfied. Continuing in this gauge, the linearised Ricci tensor reduces to minus one half the Laplacian of the perturbation, equation 41. Finally, substituting the two evolution results, we isolate the two radiative degrees of freedom and obtain equation 42: the ordinary wave equation for the transverse-traceless metric perturbation, with the second time derivative divided by the speed of light squared minus the Laplacian equal to zero.

We write the permittivity as a small perturbation about the identity, equation 43. Expanding the determinant to first order gives equation 44, and inserting these into the constitutive relation gives equations 45 and 46: the metric perturbation equals the permittivity perturbation minus one fifth of the identity times the trace of that perturbation. Any symmetric tensor can be decomposed into trace, longitudinal, and transverse-traceless parts, equation 47. Applying this projector and using the gauge conditions, we identify the dielectric perturbation with the metric wave, equation 48: the transverse-traceless metric perturbation equals the transverse-traceless permittivity perturbation. Substituting into the wave equation gives equation 49, the same wave equation now written for the permittivity perturbation.

The vacuum speed of light appears in equation 49 because the linearisation is done with respect to flat space, which corresponds to a background vacuum dielectric with permittivity equal to the identity. Physically, a medium satisfying equation 49 can be implemented using the Kerr effect in an optical medium, similar to the implementation of the optical analogue of the event horizon. All one has to do is to generate a sinusoidal modulation on a continuous-wave pump beam using, for example, an acousto-optic modulator. For a planar pump wave, the modulation would naturally propagate according to equation 49 except that the background medium has an effective refractive index other than one, so the vacuum speed is replaced by the speed in the medium, but the mathematical structure of the wave equation is unchanged.

Equation 49 admits the usual plane-wave solution, equations 50a and 50b, in which the permittivity perturbation, or equivalently the permeability perturbation, is the real part of a constant symmetric traceless amplitude tensor times a plane-wave phase, with the amplitude tensor transverse to the propagation direction. For a given propagation direction, the transverse and traceless conditions reduce the amplitude to two linearly independent components, corresponding to the plus and cross polarisation states of the gravitational wave. Since equations 42 and 49 are the same partial differential equation, the optical correspondence inherits the solutions of the gravitational Cauchy problem with the transverse-traceless conditions holding for both polarisations. Thus, equation 50 is a direct analogue of a gravitational wave propagating through the optical medium and can be used as a theoretical model for spatio-temporal perturbations in engineered media. In particular, the permittivity profile of a spacetime metasurface can be electrically tuned to exhibit the wave-like behaviour described by equation 49. In this sense, this model serves as a design tool for dynamic media, in the same spirit as Plebanski's mapping, which provides a design specification for the static case. The established framework can be extended beyond the linear regime using analytic and numerical methods, and may serve as a modelling tool to elaborate time-varying media such as photonic time crystals.

If, instead of fixing the shift to zero, we keep a uniform, time-independent shift, the ordinary time derivative in the wave equation is replaced by the convective derivative, equation 51, so the flat d'Alembertian becomes the boosted operator of equation 52 — the d'Alembertian expressed in Lorentz-boosted coordinates with velocity equal to the shift. Hence the plane-wave solution acquires an extra phase, equation 53, corresponding to the constant Doppler shift introduced by the boost.

When the shift vector stops being uniform in spacetime, the simple substitution no longer leaves the d'Alembertian unchanged: extra geometric terms enter the operator and couple different Fourier modes. As the Einsteinian system is nonlinear, these additional couplings do not obey superposition: every harmonic becomes entangled with the background curvature along with the other modes. Therefore, in strong regimes, although the equations can still be expressed analytically, the gravitational waves cannot be built from boosted plane waves. One must evolve the analogous three-plus-one system numerically, which is highly nontrivial.

Conclusion

Building on Plebanski's work, we have shown that electrodynamics in a bianisotropic medium not only has sufficient independent degrees of freedom to map the spacetime metric but also the dynamical structure to encode the ADM field equations of general relativity as conditions on the constitutive parameters. Table 1 of the article sets out the resulting ADM-optical correspondence: the lapse function corresponds to the conformal factor; the shift vector to the magneto-electric coupling; the spatial metric to the permittivity tensor, which equals the permeability tensor; the extrinsic curvature to the material evolution, that is the time derivative of the permittivity; the matter sources to the electromagnetic stress-energy; and the transverse-traceless gravitational waves to the transverse-traceless dielectric perturbations of permittivity and permeability.

Hence, this correspondence enables inverse design: starting from a chosen foliation or a target ADM evolution, one can translate it into conditions on a required space- and time-dependent profile of the permittivity and the magneto-electric vector. This would enable engineers to use the rich library of general relativity solutions in their designs, which might be a powerful tool for transformation optics. Conversely, the same correspondence provides an electromagnetic interpretation of gravitational phenomena. In particular, we demonstrate the simplest case in general relativity, namely that linear perturbations of the mapped constraints in the transverse-traceless gauge provide a toy model for cosmological gravitational waves in engineered media. This can potentially be extended to a wider class of cosmological and dynamical phenomena, and may serve as a useful benchmark for our theoretical understanding.

The dGREM formulation is a flux-conservative, first-order reformulation of the three-plus-one Einstein system, and it offers a natural companion framework for the field-equation-level mapping presented here. Thus, our theory can enrich numerical relativity and provide an alternative simulation framework, in which selected gravitational dynamics are realised through space- and time-dependent electromagnetic constitutive parameters.

Ever since Einstein's breakthrough, mappings between the classical field theories of general relativity and Maxwell's electrodynamics have repeatedly resurfaced, of which the independent foundational contributions of Plebanski and Tamm are prime examples. This persistent analogy continues to captivate the imaginations of successive generations of physicists. Ultimately, who can definitively say when an analogue system ceases to be a mere analogy and becomes an essential guide toward deeper layers of gravitation?

Acknowledgements

EEE thanks Adem Deniz Pişkin, Prof. Saul Teukolsky, and Prof. Kip Thorne for fruitful discussions on the ADM formalism, and Prof. Elias Most for introducing dGREM and whose thoughtful comments enriched the development of the ideas presented. Also, he is grateful to Prof. Mustafa Halilsoy and Prof. Mustafa Kuzuoğlu for their guidance. He also acknowledges the support of the Weizmann Institute of Science. UL is supported by the Murray B. Koffler Professorial Chair.

Data availability statement: All data that support the findings of this study are included within the article and any supplementary files.

(The reference list is omitted for length; the complete text, with its full bibliography, is at the source. Its anchors are Plebanski's 1960 Physical Review paper, the earlier work of Gordon, Tamm and Skrotskii, the Arnowitt-Deser-Misner original, the transformation-optics papers of Leonhardt and of Pendry, Schurig and Smith, the Thorne-MacDonald three-plus-one electrodynamics, and the numerical-relativity texts of Gourgoulhon and of Baumgarte and Shapiro.)