Newman Black Hole

Optical analogues to the equatorial Kerr–Newman black hole

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https://doi.org/10.1038/s42005-020-0384-5 OPEN Optical analogues to the equatorial Kerr–Newman black hole ✉ ✉ R. A. Tinguely 1 & Andrew P. Turner 2 1234567890():,; Optical analogues to black holes allow the investigation of general relativity in a laboratory setting. Previous works have considered analogues to Schwarzschild black holes in an isotropic coordinate system; the major drawback is that required material properties diverge at the horizon. We present the dielectric permittivity and permeability tensors that exactly reproduce the equatorial Kerr–Newman metric, as well as the gradient-index material that reproduces equatorial Kerr–Newman null geodesics. Importantly, the radial profile of the scalar refractive index is finite along all trajectories except at the point of rotation reversal for counter-rotating geodesics. Construction of these analogues is feasible with available ordinary materials. A finite-difference frequency-domain solver of Maxwell’s equations is used to simulate light trajectories around a variety of Kerr–Newman black holes. For rea- sonably sized experimental systems, ray tracing confirms that null geodesics can be well- approximated in the lab, even when allowing for imperfect construction and experimental error.

1 Plasma Science and Fusion Center, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. 2 Center for Theoretical ✉ Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. email: [email protected]; [email protected]

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n recent years, there has been a great amount of interest in allows one to more easily keep track of the relationship between Iprecisely controlling the electromagnetic response of artificial real space coordinates and the spacetime coordinates they materials. By introducing subwavelength structural features, represent. the permittivity and permeability tensors of the medium can be tuned to exhibit a wide range of interesting and useful phe- – The Schwarzschild black hole. We will begin by studying the nomena, such as cloaking1 7, negative refraction1,8,9, and sub- Schwarzschild black hole and various optical analogues thereof. wavelength microscopy with superlenses10–13. – The Schwarzschild metric describes the spacetime geometry of a Analogue spacetimes1,2,14 19 use optical materials to imple- static, uncharged black hole of mass M, and is given in dimen- ment coordinate transformations between a physical space and a sionless Schwarzschild coordinates ^s;^t; ρ; θ; ϕ (related to the usual “ ” virtual electromagnetic space , via the formal equivalence dimensionful quantities via s ¼ M^s, t ¼ M^t, r = Mρ) by ref. 52 between Maxwell’s equations in curved spacetime and those in flat spacetime within a corresponding bianisotropic medium19–23. À1 ÀÁ ^2 2 ^2 2 ρ2 ρ2 θ2 2 θ ϕ2 : This allows one to build optical analogues to gravitational sys- ds ¼À 1 À ρ dt þ 1 À ρ d þ d þ sin d tems24–44. In particular, there has been a fair amount of interest in reproducing the metrics of black holes45–50. The null geodesics ð1Þ and polarizations of light moving in the spacetime metric can be ρ ~ρ 1 2 Making the coordinate transformation ¼ ð1 þ 2~ρÞ , the reproduced exactly within a fully bianisotropic material; if one Schwarzschild metric (1) can be written in the form52 simply wishes to reproduce the null geodesics of the metric, 2 however, it is much simpler to use an appropriately designed 1 1 À ~ρ 4ÀÁ gradient-index material that is easier to construct experimentally. 2 2 2 1 2 2 2 d^s ¼Àd^t þ 1 þ d^x þ d^y þ d^z ; ð2Þ In this paper, we discuss the bianisotropic and gradient-index 2 2~ρ 1 þ 1 materials that imitate the exterior equatorial Kerr–Newman black 2~ρ hole solution. We first carry out the analysis for optical systems where the spacetime isotropic coordinates ðÞ^x; ^y; ^z are related to reproducing the null geodesics of the Schwarzschild black hole. ~ρ; θ; ϕ We recover the familiar results for the permittivity and perme- the transformed Schwarzschild coordinates ðÞvia the transformation from Cartesian to spherical coordinates. ability tensors and scalar refractive index reproducing the metric fi in isotropic coordinates, as well as the permittivity and perme- We rst replicate the metric in isotropic coordinates, given in Eq. (2), in order to make contact with existing literature. As ability tensors reproducing the metric in the Schwarzschild 15,19 coordinates20,46,47,51. We then present the scalar index that discussed in refs. , there is a formal equivalence between the equations of electrodynamics in a curved spacetime and those in reproduces the null geodesics for Schwarzschild coordinates, fl fi which, by comparison with the isotropic result, has the significant at space in a macroscopic medium. Speci cally, the behavior of fi fi light in a curved spacetime background described by metric gμν is experimental bene t of remaining nite all the way to the hor- fl izon. We then carry out these same analyses for the equatorial reproduced in at space within an impedance-matched bianiso- – – tropic medium with permittivity ϵij, permeability μij, and Kerr Newman metric in Boyer Lindquist coordinates, reprodu- α cing the metric within a fully bianisotropic material23, and magnetoelectric coupling i given by fi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi nding the scalar index required to reproduce the null geodesics. À det g g We use finite-difference frequency-domain simulations of sys- ϵij ¼ μij ¼À pffiffiffiffiffiffiffiffiffiffi gij ; α ¼ p0ffiffiffiffiffiffiffiffiffiffii ; ð3Þ γ i γ tems that approximate the gradient-index solutions of the g00 det g00 det Schwarzschild and Kerr–Newman black holes with concentric where γ is the three-dimensional metric tensor of the real space circular shells of constant index, and use ray tracing to perform ij coordinate system in which we construct the medium, onto which an analysis of the error sensitivity of such systems. These analyses we map the spatial components g . Here, g and γ denote the demonstrate that these approximate gradient-index systems, ij determinants of gμν and γ , respectively. The macroscopic fields which are far simpler to construct than true gradient-index sys- ij D, H are related to the microscopic fields E, B via tems or full bianisotropic media, can adequately reproduce null geodesics and are forgiving to fabrication and experimental error D ¼ ϵE þ α ´ H ; B ¼ μH À α ´ E : ð4Þ for reasonable geodesics. As such, they are practical tabletop analogues for charged and/or rotating black holes. As discussed in ref. 53, this choice of identification between the spacetime geometry and the electromagnetic analogue, elaborated first in ref. 19, is not unique, and cannot reproduce all measurable Results properties of light moving in the spacetime metric. However, it is Throughout this paper we use Gaussian Planck units, with c = ℏ sufficient to reproduce both the null geodesic trajectory and the = G = 4πϵ = 1. Greek indices range over temporal and spatial 0 polarizations of light moving along these geodesics, which makes coordinates, e.g., μ = 0, …, 3, while Roman indices range over analogues produced with this identification worthy subjects only spatial coordinates, e.g., i = 1, …, 3. We use uppercase Greek of study. and Roman letters to indicate variables related to the optical Using Eq. (3) to map the dimensionless spacetime isotropic system, while we use lowercase letters to indicate variables related coordinates ðÞ^x; ^y; ^z onto theÀÁ corresponding dimensionless real to the spacetime metric that it is replicating. We refer to these ^; ^; ^ respectively as “real space” and “spacetime” variables. We typi- space Cartesian coordinates X Y Z (and thus mapping the ~ρ cally use hats to indicate the dimensionless versions of variables. dimensionless spacetime isotropic radial coordinate onto the fi When we map spacetime coordinates onto real space coordinates, dimensionless real space radial coordinate P), we nd that the we always do so by equating the dimensionless coordinates. behavior of light in the Schwarzschild metric (2) is reproduced in fl Spacetime variables are dedimensionalized via multiplication by at space within a medium described by the appropriate power of the black hole mass M. Real space 3 ð2P þ 1Þ ÈÉ dimensionless variables are then dimensionalized by a convenient ϵij μij 1ij ; ; ^; ^; ^ : ¼ ¼ 2 i j 2 X Y Z ð5Þ length scale for construction. Using this matching of coordinates 4P ð2P À 1Þ

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In this case, the medium is isotropic, and the scalar index can We then make use of the spacetime impact parameter ^ be read off immediately from Eq. (5)as bðρÞ¼ρ sin β, where β is defined by the relation ð2P þ 1Þ3 dϕ nðPÞ¼ : ð6Þ ρ ¼Àtan β : ð11Þ 4P2ð2P À 1Þ dρ Note that the results of Eqs. (5) and (6) are well-established in Plugging this relation into Eq. (10) and making the sign choice 20,46,47,51 fi consistent with our definition of β,wefind that the literature . Equation (6) has the bene t that there is a single scalar index that reproduces all null geodesics and the À1=2 ^ ρ ^À2 ρÀ3 ; ð12Þ polarization of light moving along these geodesics, and as the bð Þ¼ b1 þ 2 material is isotropic (though still inhomogeneous) it is thus easier fi ^ ‘=^ ^ε ’ to construct experimentally. However, this refractive index where we have de ned b1 ¼ . Fermat s principle relates the real space impact parameter and index of refraction by diverges approaching the horizon, i.e., as P ! 1, so it is not 2 ^ À1 useful for investigating geodesics in the vicinity of the horizon. nðPÞ/BðPÞ : ð13Þ Another approach is to instead use the Schwarzschild Equation (12) is then taken as input to Eq. (13) by equating the coordinates (1), which produces an anisotropic medium distinct spacetime coordinates ðÞρ; ϕ with the real space coordinates from Eq. (5). As before, we use Eq. (3) to map the dimensionless ðÞP; Φ , which also equates the dimensionless spacetime impact spacetime Schwarzschild coordinates ðÞρ; θ; ϕ (now with the ^ Schwarzschild radial coordinate ρ rather than the isotropic radial parameter b with the dimensionless real space impact parameter ~ρ B^. This yields coordinate ) onto the corresponding dimensionless real space qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi spherical coordinates ðÞP; Θ; Φ , yielding ^À2 À3 : ð14Þ 0 1 nðPÞ/ b1 þ 2P 10 0 B C This solution has a number of noteworthy features. First, we ij ij B 0 1 0 C ϵ ¼ μ ¼ @ PðPÀ2Þ A ; i; j 2 fgP; Θ; Φ : ð7Þ reiterate that Eq. (14) only reproduces the geodesic trajectories of 00 csc2Θ light moving in the Schwarzschild metric (1), but does not PðPÀ2Þ faithfully reproduce its polarizations. The radial profile depends ^ This same system isÀÁ described in dimensionless real space on the initial condition b , which is related to initial angle β and ^ ^ ^ 1 Cartesian coordinates X; Y; Z by initial radius P0 by 0 1 2 ^ 2 3 ^ ^ ^ ^ ^ P 2X À P 2XY 2XZ b2 ¼ 0 : ð15Þ 1 B C 1 2β À1 ϵij μij @ ^ ^ ^ 2 3 ^ ^ A csc 0 À 2P0 ¼ ¼ 2 2XY 2Y À P 2YZ P ð2 À PÞ 2 This is somewhat inconvenient for experimental application, 2X^Z^ 2Y^Z^ 2Z^ À P3 as it means that a different apparatus must be constructed for i j 31ij ÈÉ 2P P À P ; ; ^; ^; ^ ; each family of geodesics; to address this, one could in principle ¼ 2 i j 2 X Y Z ^ P ð2 À PÞ construct a cylinder, where b1 varies along the cylinder axis ð8Þ and each 2D slice recreates the corresponding family of null ÀÁ geodesics. A significant benefitofthiscoordinatesystem, i ^; ^; ^ fi → where P ¼ ÀÁX Y Z and the dimensionless real space Cartesian however, is that n(P)approachesa nite value as P 2solong ^ ^ ^ ^ ≠ coordinates X; Y; Z are related to the real space spherical as b1 0, which means that geodesics can be studied in the coordinates ðÞP; Θ; Φ in the usual way. This result matches those vicinity of the event horizon, in contrast to the solution (6). The presented in refs. 46,47. constant of proportionality in Eq. (14)allowsustotunethe If we only wish to reproduce the trajectories of light in the scalar index at the initial P0 to the most feasible value for Schwarzschild metric (and not the proper polarizations), then a construction. radially varying scalar index n(P) is sufficient. In Schwarzschild Finally, note that we can relate conserved quantities ε, ‘ in coordinates, we will find that the radial profile depends on the spacetime to E; L in real space in the following way: in flat space, a initial conditions defining the geodesic. We consider null photon with frequency f and wavelength λ has energy E¼2πf geodesics of the metric (1); all such geodesics are planar, and and angular momentum L = 2πB/λ, with B the dimensionful real θ = π = ^= so the spherical symmetry allows us to take /2 without loss space impact parameter. Thus, L ERS ¼ nB 2 ¼ constant. Set- of generality. Such null geodesics of the SchwarzschildÀÁ metric are ting n = 1atP → ∞ in Eq. (14) and equating the real space and fi ε 2M dt ^ ^ parametrized by a conserved energy at in nity, ¼ 1 À r dσ, spacetime dimensionless impact parameters, B ¼ b, yields dϕ = ‘=ε = ‘ 2 σ fi L ERS ¼ rS. Here, rS 2M, and RS is the real space radius and the conserved angular momentum, ¼ r dσ, with the af ne parameter of the geodesic. Dedimensionalizing these parameters onto which rS is mapped. ‘ ‘^ σ σ^ via ¼ M and ¼ M (note that the energy is already – dimensionless, ^ε ¼ ε), null geodesics satisfy the geodesic equa- The Kerr Newman black hole. We now apply the same 52 approaches to investigate optical analogues of the Kerr–Newman tion – black hole, of which the Kerr, Reissner Nordström, and 2 ^2 dρ 2 ‘ Schwarzschild results are special cases. We will restrict our À^ε2 þ þ 1 À ¼ 0 : ð9Þ σ^ ρ ρ2 attention to equatorial null geodesics. d – 2 The Kerr Newman metric describes the spacetime geometry dϕ ^2 4 Combining this equation with σ^ ¼ ‘ =ρ yields surrounding a black hole of mass M, angular momentum per unit d = À1=2 mass a J/M, electric charge Q, and magnetic charge Qm. dϕ ^ε2 ^ ^ ¼ ± ρ4 À ρ2 þ 2ρ : ð10Þ Dedimensionalizing the quantities via a ¼ Ma, Q ¼ MQ, ρ ^2 ^ d ‘ Qm ¼ MQm, the metric is given in dimensionless

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Boyer–Lindquist coordinates by52 We proceed by equating spacetime coordinates (ρ, θ, ϕ) and ^ ^ 2 ^ ÀÁreal space coordinates (P, Θ, Φ), which sets BðPÞ¼bðρ ¼ PÞ. dρ Δ 2 d^s2 ¼ Σ^ þ dθ2 À d^t À â sin2 θ dϕ The scalar index for an optical Kerr–Newman black hole is again Δ^ Σ^ ð16Þ given by 2 ÂÃÀÁ sin θ 2 þ ρ2 þ a^2 dϕ À a^ d^t ; ^ À1 Σ^ nðPÞ/BðPÞ : ð23Þ where An optical system with this scalar index reproduces the equatorial – Σ^ ¼ ρ2 þ â2cos2θ ; null geodesic trajectories of the Kerr Newman metric. Unlike the Schwarzschild case, this scalar index is not always Δ^ ρ2 ρ ^2 ρ2 ; ¼ À 2 þ a þ Q ð17Þ sufficient to fully reproduce the given family of Kerr–Newman ρ2 ^ 2 ^ 2 : geodesics. This can be seen immediately by noting that initially Q ¼ Q þ Qm ^ counter-rotating geodesics (those with ‘ of opposite sign to â) Here, M is the total mass-equivalent, which contains contribu- must turn around and become co-rotating before crossing into tions from the irreducible mass, the rotational energy, and the dϕ 54 the ergosphere; such a reversal of the sign of dρ is not possible Coulomb energy of the black hole . fi After setting θ = π/2 and dθ = 0 to restrict to the equatorial with a nite (and positive) scalar index. This shortcoming case, we use Eq. (3) to map the dimensionless spacetime manifests itself as a divergence of the scalar index; the outermost coordinates (ρ, θ, ϕ) onto the dimensionless real space divergence occurs at radius coordinates (P, Θ, Φ), as before, yielding rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 1 ^À1 ^ ^ Δ^ 0 1 P ¼ 1 À âb þ 1 À âbÀ1 1 À a^bÀ1 À ρ2 : ð24Þ 00 0 Ã 1 1 1 Q B Δ^Àâ2 C B C B C ϵij ¼ μij ¼ 0 1 0 ; α ¼ @ 0 A ; @ Δ^Àâ2 A i This is a removable pole in the Schwarzschild and ^ 1 1 1 a Δ^ ^2 À 2 – 00Δ^ Àa P Reissner Nordström cases. For rotating black holes, the divergence occurs at the point in the trajectory where the direction of ð18Þ rotation reverses, consistent with the above observation that a fi fi where Δ^ should now be interpreted as a function of P. The nite radially varying scalar index is insuf cient to implement derivation is given in full in the “Methods” section. The equatorial such a reversal. Thus, the pole only affects initially counter- geodesics and polarizations of the Kerr–Newman metric are rotating geodesics that enter the ergosphere. exactly reproduced in flat space within a medium with these properties23. There is a subtlety here—although the radial and Simulations of constructible optical black holes. Optical ana- azimuthal components P, Θ appear to diverge at the ergosphere logues to black holes are particularly useful if their constructions Δ^ ¼ â2, this is a spurious divergence. As discussed in refs. 23,55,56, are realizable. In the following sections, we model optical black the physically relevant covariant quantity is the tensor χ defined holes with radially varying scalar refractive indices n(P), as given therein, which relates the macroscopic and microscopic fields. by Eqs. (14) and (23). For Schwarzschild (and many – This quantity diverges only at the horizon Δ^ ¼ 0. Kerr Newman) black holes, n(P) is maximal at the horizon. Because the impact parameter of light on the optical black hole As before, we can also replicate equatorial null geodesics of the “ ” Kerr–Newman metric using only a scalar index. As in the must be less than or equal to the radius of the edge of the ^ ≤ ≤ Schwarzschild case, these geodesics are parametrized by the system,pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i.e., B P0, it is found that n(P) c0n0P0, where c0 ¼ = : = dimensionless conserved energy at infinity and conserved angular 31 108 0 54 and n0 n(P0). Thus, the construction of an momentum, given in this case by optical Schwarzschild black hole with n = 1 and moderate P ≤ 6 0 0 ρ2 ^ ρ2 â ϕ is plausible and achievable with indices of refraction in the range ^ε 2 Q dt 2â Q d ; ¼ 1 À ρ þ ρ2 σ^ þ ρ À ρ2 σ^ of ordinary materials, such as water, glass, and plastic. (As will be d d 2 ð19Þ – ρ2 â ^ 2 ρ2 â ϕ seen, many optical Kerr Newman black holes are also con- ‘^ 2a^ Q dt ρ2 ^2 2â Q d : fi ¼À ρ À ρ2 dσ^ þ þ a þ ρ À ρ2 dσ^ structible.) True gradient-index pro les of the form (14) could perhaps be achieved with metamaterials; however, it is not clear The geodesic equations describing the equatorial motion are hi how easily realizable such systems are, so in this work we dϕ ρ2 ^ ^ ρ2 a^ approximate the profiles with concentric annuli of constant ¼ 1 1 À 2 þ Q ‘ þ 2a À Q ε ; dσ^ Δ^ ρ ρ2 ρ ρ2 scalar index. ÂÃð20Þ 2 ρ2 ^2 2 ^2Δ^ ‘^2 For a system size in which the wavelength of the source light is dρ ðÞþa Àa ^À1 ^ ^À1 ^ ; dσ^ ¼ ρ4 b1 À Vþ b1 À VÀ much smaller than the gradient length scale of the scalar-index profile, i.e., λ n=kk∇n , a highly localized and highly directional where pffiffiffiffi light source, like a laser, would nearly approximate the geodesics ^ ρ ρ2 ‘^ ρ2 Δ^ of Eqs. (10) and (20). Simulating these trajectories amounts to ray a 2 À Q ± sgnð Þ fi ^ ÀÁ : ð21Þ tracing, which we pursue in the following section. Speci cally, we V ± ¼ 2 ρ2 þ a^2 À â2Δ^ investigate the number of annuli needed to sufficiently mimic the true scalar-index profile and explore the impact of imperfect fi ^ ρ ρ β As before, we nd the impact parameter bð Þ¼ sin by construction and experimental error on the deviation of the ray dϕ dϕ=dσ^ plugging Eq. (11) into dρ ¼ dρ=dσ^, which yields trajectory from the geodesic. However, in the following section, hiÀÁ ^ we will first consider the case in which the source wavelength is ρ2 Δ^ À ^2 þ ρ À ρ2 ^ À1 ^ a 2 Q ab1 bðρÞ¼rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihi hiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi: similar to the size of the optical black hole, i.e., M/λ ~ O(10). This ÀÁ 2 2 ÀÁ ρ2 Δ^ ^2 ρ ρ2 ^^À1 Δ^ ρ2 ^2 2 ^2Δ^ ^À1 ^ ^À1 ^ À a þ 2 À Q ab1 þ þ a À a b1 À Vþ b1 À VÀ is done to demonstrate the strengths and limitations of this study’s approach, as well as to be consistent with previous studies ð22Þ such as refs. 28,29,31,32,38,41,46,47,57.

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It is important to note here that this geometry was chosen for simplicity in the ﬁnite-difference frequency-domain simulations of the next section, in which dimensions are constrained by the wavelength. For consistency, the same geometry is scaled linearly for the ray tracing analyses that follow. In practice, non-uniform annulus thicknesses could be used to minimize steps Δn in dn regions of high dR and to reduce light scattering at each boundary, but this is left for future work.

Finite-difference frequency-domain simulations. In this section, the trajectory of light around an optical black hole is modeled using a finite-difference frequency-domain (FDFD) solver58,59 of Maxwell’s equations. Simulation details are provided in the “Methods” section. Figure 1 shows the profiles n(P) used when modeling light incident on an optical Schwarzschild black hole ^ with four different impact parameters, b1 ¼ 2, 3, 4, and 5; the resulting FDFD simulations are shown in Fig. 3, with the wave- λ = μ = length of light 0.5 m and optical Schwarzschild radius RS 2M = 5 μm. Fig. 1 Scalar refractive index of simulated optical Schwarzschild black Consider the simulation shown in Fig. 3a, for which the holes. Radial profiles of the scalar refractive index used for simulations of ^ dedimensionalized impact parameter at infinity is b ¼ 2. The optical Schwarzschild black holes with impact parameters ^b a 2, b 3, 1 1 ¼ peak of the electric field normalized to its maximum, jjE = maxjjE , c 4, and d 5. The outer radius is R /M = 6 with M the black hole mass. Note pffiffiffiffiffiffiffiffi 0 follows the path of the geodesic quite closely. Here, jjE ¼ EEÃ, the logarithmic scale of the vertical axis. * with E the complex conjugateÀÁ of E. Time-averaged Poynting 1 ´ Ã ∝ vectors are calculated as Re 2 E H and scaled by 1/R in the figures. Those with largest magnitude point mostly along the geodesic, and much of the energy flux is directed into the optical black hole. The same spatial trend is seen in Fig. 3b, for which ^ fi b1 ¼ 3. Note in Fig. 1 how the pro le of scalar index n(P) increases in amplitude as the initial impact parameter increases, in order to further bend light toward the horizon. ^ For b1 ¼ 4 and 5, seen in Fig. 3c, d, respectively, the brightest regions of jjE =maxjjE (and longest Poynting vectors) predomi- nantly follow the geodesics. This is actually seen more clearly in the energy contained in the electric field (/jjE 2); however, only the electric field amplitude is shown here for better visualization of both small and large amplitude features. Agreement between the simulated light path and actual geodesic is expected to improve as the wavelength and beam width decrease relative to the size of the optical black hole, as described in the following section. Another interesting effect is observed in Fig. 3d: the FDFD simulation does not show light following the geodesic all the way to the horizon. Instead, light begins to orbit at the photon sphere, = = μ Fig. 2 Scalar refractive index of simulated optical Kerr–Newman black R 3M 7.5 m. This results because the impact parameter is nearly equal to that at which light becomes trapped, holes. Radial profiles of the scalar refractive index n used for simulations of pffiffiffi ^ ^ : “ ” optical Kerr–Newman black holes: a maximally co-rotating (a ¼ 1; ρQ ¼ 0); b1 ¼ 3 3 5 2. Only traces of the photon ring are resolved fi b maximally charged (â ¼ 0; ρQ ¼ 1); c charged and co-rotating ðâ ¼ 2=5; in Fig. 3d. Higher delity simulations, with the optical black hole ρQ ¼ 4=5Þ;andd charged and counter-rotating ðâ ¼À2=5; ρQ ¼ 4=5Þ.The comprising many more annuli, would likely be required to ^b R M = M impact parameter is 1 ¼ 3, and outer radius is 0/ 6, with the black properly simulate and study this phenomenon. This is left to hole mass. Note the logarithmic scales and limits of the vertical axes. future work. Several optical Kerr–Newman black holes are also simulated, fi In this study, all optical black holes are modeled with with pro les n(P) in Fig. 2 corresponding to the FDFD solutions = ^ dimensionless outer radius P0 6. The system comprises either in Fig. 4. For each case, the impact parameter is b1 ¼ 3, and the = 16 or 21 concentric annuli, with the number depending on outer edge of the optical black hole is again at radius P0 6. acceptable annulus thickness (i.e., greater than the wavelength) These can be compared to the optical Schwarzschild black hole of and the minimum modeled radius Pmin. The innermost and Fig. 3b. The innermost modeled radius varies for each simulation, outermost annuli each have half the width of each interior depending on whether n(P) diverges outside of the horizon. Each annulus. The scalar index of each annulus is uniform, so that the simulation is described in detail below. fi ^ ; ρ simulated n(P) pro les are piecewise functions, as shown in An extremal Kerr black hole (a ¼ 1 Q ¼ 0), with beam Figs. 1 and 2. The values of n for the innermost and outermost trajectory co-rotating with the black hole spin, is shown in Fig. 4a. annuli are taken as n(P) at the minimum and maximum radii, Here, n(P) diverges at P* = 4/3; however, the true geodesic dr = respectively; the refractive index of an inner annulus is taken as escapes the black hole with dσ ¼ 0atP 2. Therefore, though the = the value of n(P) at its center. horizon is at Ph 1, the system is modeled with innermost radius

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Fig. 3 Numerical simulations of optical Schwarzschild black holes. Finite-difference frequency-domain simulations of light incident on an optical ^b Schwarzschild black hole with impact parameters 1 ¼ a 2, b 3, c 4, and d 5. True geodesics are plotted as thick lines. Poynting vectors (white arrows) are scaled by ∝1/R. Edge radii and Schwarzschild radii (RS) are solid circles. Each interior annulus's edge is marked. Color scales for the normalized electric ﬁeld amplitude jjE =maxjjE are the same for each subplot.

P = 1.4. Comparing to the Schwarzschild case in Fig. 3b, we see 1.96) ≈ 6. Comparing the co- and counter-rotating black holes, we that light is “dragged” further around the co-rotating black hole, see that light travels further in the Φ-direction for the former as expected. In addition, more light “escapes”, although not all is system, as expected. directed along the geodesic. Inevitably, some energy flux is As described in this section, a variety of optical Schwarzschild directed into the optical black hole, as indicated by the Poynting and Kerr–Newman black holes can be constructed feasibly with vectors; this is partially due to the finite width of the beam, and low indices of refraction. If such systems are built at a small scale, partially to the discrete annular approximation of the true FDFD simulations show that the trajectories of light behave as gradient-index profile. expected, mostly following the true geodesics despite the discrete In contrast to Fig. 4a, an extremal Reissner–Nordström black approximation to the proper gradient-index profile. The benefits ^ ; ρ hole (a ¼ 0 Q ¼ 1) is simulated, with FDFD results depicted in of building larger systems are discussed in the next section. Fig. 4b. Here, the optical black hole could be modeled completely = μ = to the horizon at Rh 2.5 m. In general, the peak of jjE maxjjE follows the geodesic to the horizon. Little difference is seen when Ray tracing calculations. In principle, the optical black holes of comparing to the Schwarzschild case of Fig. 3b, except that light the previous section could be scaled in size from micrometer to = μ now propagates within RS 5 m. centimeter or larger. This would simplify not only the construc- Two non-extremal Kerr–Newman black holes, with the same tion of the optical black hole but also the calculation of light ρ = ^ = charge ( Q 4/5) but opposite spins (a ¼ ±2 5), are also propagation, since the wavelength and width of the light source simulated and shown in Fig. 4c, d. Thepffiffiffiffiffiffiffiffi co-rotating black hole is would be much smaller than the system size and related gradient = : modeled to the horizon at Ph ¼ 1 þ 1 5 1 45. Compared to length scales. The minimum gradient scale length of the scalar- the extremal Kerr black hole in Fig. 4a, light is not dragged as far index profiles in Figs. 1 and 2 is n=kk∇n 0:6 μm, so around the black hole. λ

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– ^b Fig. 4 Numerical simulations of optical Kerr Newman black holes. Finite-difference frequency-domain simulations of light incident ð 1 ¼ 3Þ on four optical Kerr–Newman black holes, which are a maximally co-rotating (â ¼ 1; ρQ ¼ 0); b maximally charged (â ¼ 0; ρQ ¼ 1); c charged and co-rotating (â ¼ 2=5; ρQ ¼ 4=5); and d charged and counter-rotating (â ¼À2=5; ρQ ¼ 4=5). True geodesics are plotted as thick lines. Poynting vectors (white arrows) are scaled by ∝1/R. Maximum and minimum radii are solid circles; radii of interest, such as the horizon radius (Rh) or Schwarzschild radius (RS), are also plotted and labeled. Each annulus's edge is marked. Color scales for the normalized electric field amplitude jjE = maxjjE are the same for each subplot.

It is of interest to calculate the deviation of a ray trajectory 50 30 around the optical black hole from the true geodesic. These deviations could occur for a number of reasons: for instance, the 40 24 discretization of n(R) due to the finite number of annuli; manufacturing error, leading to an offset Δn of the desired scalar Δ 30 18 index; or experimental error, resulting in a deviation B0 from the desired initial impact parameter B0. We explore the impacts of these below for light incident on an optical Schwarzschild black 20 12 = hole with outer radius P0 6. First, we investigate the number of annuli (with uniform 10 6 thicknesses) needed to sufficiently approximate the scalar-index profile for a range of initial impact parameters. We define our 0 performance metric as the deviation of the ray trajectory from the 012345 geodesic, quantified by the difference in azimuthal angle ΔΦ = Φ − Φ fi ray geo. Note that this performance metric is design-speci c Fig. 5 Impact of annulus number on ray trajectories. The angular deviation and does not account for scattering, whereas the semi-classical of the ray trajectory (Φ ) from the geodesic (Φ ) at the horizon for an calculations of refs. 29,32 do. However, as there is no analytic ray geo optical Schwarzschild black hole with outer radius P0 = 6, as a function of solution to the wave equation for the system under consideration, ^b the initial impact parameter 1 and number of annuli used in the the pursuit of a more appropriate metric is left to future work. construction. Here, we are concerned with the deviation at the horizon. This ^ ; value is shown in Fig. 5 for b1 2½0 5 and the number of annuli ^ ΔΦ ≤ ∘ ranging 1–50. We see that only 25 annuli are needed to reproduce However, even the trajectory with b1 ¼ 5 can achieve 3 ^ ≤ ΔΦ = ∘ ΔΦ with 1000 annuli. trajectories with b1 3 to within 3 . As expected, Next, we consider the scenario in which the scalar-index profile ^ fi increases rapidly for large b1 and xed annulus number. is imperfect, offset by a constant Δn due to some manufacturing

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Fig. 6 Effects of construction and experimenter errors on ray trajectories. a, b A uniform offset Δn from the true scalar refractive index profile n(R). c, d A deviation ΔB0 from the desired impact parameter B0. a, c Ray trajectories in real space, computed from Eq. (25), compared to the true geodesic (gray dashed). b, d Angular deviation of the ray trajectory (Φray) from the geodesic (Φgeo) versus radius R normalized to the black hole mass M. Both scans use ^b the optical Schwarzschild black hole with 1 ¼ 3. The scale of the color bar of subplot a is the same as the scale of the vertical axis of subplot b; the same is true for subplots c and d. error. We choose a specific trajectory with impact parameter black holes in an isotropic coordinate system. In this paper, we ^ have calculated the dielectric permittivity and permeability ten- b1 ¼ 3 to connect with the FDFD simulations of the previous section. The range spans Δn ∈ [0, 0.5] in Fig. 6; this is a sors ϵ, μ that reproduce the equatorial null geodesics and polar- significant percent change compared to profile b in Fig. 1.In izations of light moving in the metric of spinning, charged Fig. 6a, we see that the ray trajectory skews radially outward as Δn (Kerr–Newman) black holes. Furthermore, we have conceived, increases. We are again interested in the deviation of the ray for the first time, a gradient-index material that exactly repro- – trajectory from the geodesic, ΔΦ = Φray − Φgeo, shown in Fig. 6b duces families of equatorial Kerr Newman null geodesics in as a function of radius P = R/M. Most trajectories follow the almost all cases. Importantly, the radial profile of the scalar geodesic closely, within ΔΦ ≤ 2∘, for P > 3; however, within P <3, refractive index n(R)isfinite along the entire trajectory (even to ΔΦ grows rapidly. The small gray region, near P ≈ 2 and Δn ≈ 0.5, the horizon, if applicable), except at the point of rotation reversal indicates that the ray trajectory escapes the black hole, so that ΔΦ for initially counter-rotating null geodesics. Values of n ≲ 6 can diverges. In this case, if errors of ΔΦ ≤ 5∘ were allowable, then be achieved for many trajectories of interest, meaning that such n(R) must be constrained with Δn ≤ 0.1. gradient-index optical analogues could be constructed with con- In addition, a scan in initial impact parameter is performed to ventional materials and metamaterials. assess how experimental error would affect the ray trajectory. The Simulations of a variety of optical black holes were performed, Δ ratio B0/B0 is varied within ±10%, with results shown in Fig. 6. each with n(R) approximated by concentric circular annuli of Δ constant scalar index. First, a FDFD solver of Maxwell’sequations The ray trajectories (Fig. 6c) vary as expected: as jjB0 increases, the ray path moves farther from the true geodesic, but keeps the same was used to simulate the path of light incident on a Schwarzschild ^ = general shape. Again, the deviation in azimuthal angle is shown in black hole with varying impact parameter b1 ¼ b1 M. Good Δ = ΔΦ agreement was observed between the light trajectory (indicated by Fig. 6d. For large jjB0 B0, increases rapidly as the trajectory approaches the horizon. The deviation can be as large as ΔΦ = 30∘ maximum values of the electric field and Poynting vectors) and = Δ ≈ ΔΦ ^ = – at P 2 when B0/B0 10%. Interestingly, the contours of geodesic for low impact parameters b1 2 3, but the discrepancy Δ Δ = ^ = – ^ versus P and B0/B0 are not symmetric about B0/B0 0inFig.6d. grew for b1 4 5. Interestingly, for b1 ¼ 5, some features of light This results from the discretization of n(R). Therefore, if the orbiting at the photon sphere were observed. Utilizing the same number of annuli cannot be increased, it could actually be beneficial FDFD framework, several optical Kerr–Newman black holes were Δ – to purposefully shift the impact parameter ( B0/B0 <0,inthiscase) simulated: extremal Kerr, extremal Reissner Nordström, and non- to better match the light trajectory with the true geodesic. extremal Kerr–Newman with initially co- and counter-rotating trajectories. Each of these optical systems was simulated within the Discussion Schwarzschild radius, some even to the horizon. While there exist The application of analogue spacetimes to the study of general some discrepancies between the simulated light trajectories and true relativity has seen a resurgence in theory, simulation, and geodesics, the qualitative feature of light “dragged” in the direction experiment in the past two decades. Many recent works have of the black hole’s spin was observed. The three co-rotating cases focused on optical analogues to static, uncharged (Schwarzschild) require n ≲ 3, meaning that constructions of these optical

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Kerr–Newman black holes are feasible; the counter-rotating case where μ, ν run over ^t; ρ; θ; ϕ. This metric has inverse

≲ 0 2 1 requires n 6, which might be realized with more exotic materials â2Δ^ÀðÞρ2 þâ2 a^ðÞΔ^Àρ2Àa^2 B ρ2Δ^ 00 ρ2Δ^ C like metamaterials. B C Finally, we have investigated the number of annuli used in B Δ^ C μν B 0 ρ2 00C g ¼ B C ð28Þ construction as well as the effects of fabrication and experimental B 1 C B 002 0 C errors on these optical black holes. The results demonstrate that @ ρ A ^ Δ^ ρ2 ^2 aðÞÀ Àa Δ^Àa^2 with a modest number of annuli, the approximate gradient-index ρ2Δ^ 00 ρ2Δ^ systems adequately reproduce null geodesics and are robust to and determinant det g ¼Àρ4. We map the curved spacetime coordinates small variations in refractive index and impact parameter. As ð^t; ρ; θ; ϕÞ onto the flat spacetime spherical coordinates ðT^; P; Θ; ΦÞ, so the flat these systems are far easier to manufacture than true gradient- space coordinate metric is in this case index or bianisotropic media, they are thus practical tabletop 0 1 10 0 analogues for equatorial Kerr–Newman black holes. B C γ @ 2 A ij ¼ 0 P 0 ð29Þ 2 2Θ Methods 00P sin Numerical simulations. The trajectories of light around optical black holes are with determinant det γ ¼ P4sin2Θ. Because we have restricted our attention to modeled using a FDFD solver of Maxwell’s equations58,59. The wavelength of light θ = π/2, we similarly have Θ = π/2, and so this simply becomes det γ ¼ P4. After is chosen to be λ = 0.5 μm. The 2D simulation domain is modeled as a vacuum, this coordinate matching, we have ϵ = μ = = λ λ 0 1 with scalar properties n 1 and size 60 ×60 ; a perfectly matching layer Δ^ 2 00 of width λ/5 is applied at its boundary. A Gaussian beam of light is approximated B P C λ = fi ij B 1 C ; as an array of line sources, each of width /25 20 nm and electric eld amplitude g ¼ @ 0 P2 0 A 2 = δ2 Δ^ ^2 calculated from a Gaussian envelope of the form expðÀðX À B0Þ 2 Þ. Here, B0 is 00 Àa P2Δ^ the dimensionful real space impact parameter at P0, and δ = λ/2 so that the beam fi 60 Δ^ ^2 ð30Þ satis es the paraxial approximation . The total width of the beam is truncated at À a ; λ ϵ = − π g00 ¼À 2 2 by imposing two absorbing ( 1 i ) boundaries as vertically aligned P “waveguides” of the light from the edge of the domain to the edge of the optical ^ Δ^ 2 ^2 g ¼ aðÞÀP Àa ; black hole. These restrict the beam to travel along a straight path in free space, as a 0i 00 P2 directional light source would in the laboratory. Note that the factor of −π is det g ¼ÀP4; arbitrarily chosen for the imaginary (damping) component. ^ Each simulated optical black hole is centered in the domain, with the where Δ is now interpreted as a function of P, as opposed to ρ. Plugging these Schwarzschild radius always R = 10λ = 5 μm(M = 2.5 μm) and edge at R = 30λ values into Eq. (3), we arrive at S 0 0 1 = 15 μm. The Gaussian light source propagates in the vertical direction toward the ^ 0 1 Δ 00 0 black hole. For all simulations, the region within the minimum radius (oftentimes B Δ^Àa^2 C ij ij B C B C the horizon radius R ) is modeled as a disc with dielectric permittivity ϵ = ϵ − iπ. ϵ ¼ μ ¼ 0 1 0 ; α ¼ @ 0 A : ð31Þ h in @ Δ^Àa^2 A i ϵ ϵ = 2 ^ 1 1 Here, in is the scalar permittivity ( n ) of the innermost annulus, and a factor 1 a À 2 00 Δ^Àâ2 P of −π is used for the imaginary (damping) component, as with the aforementioned Δ^ “waveguides.” Data availability Ray tracing algorithm. Consider an optical system consisting of N concentric Data is available from the corresponding author upon request. annuli. Let the radii bounding each annulus i be Ri < Ri−1, so that the annuli are … numbered 1, 2, , N from the outside in, and the outer edge of the system is at R0. ≤ = Code availability The scalar index of each annulus is n(Ri < R Ri−1) ni, which monotonically → Code for the simulations shown here is available from the corresponding author upon increases from annulus 1 N,soni < ni+1. Let the scalar index for R > R0 be n0. + ≤ ≤ fi For a light ray incident on annulus (i 1) (propagating in the region Ri R request. The nite-difference frequency-domain solver used in this work is available at Φ Φ https://github.com/wsshin/maxwellfdfd. Ri−1), let the impact parameter be Bi ¼ Ri sin i, where i is the azimuthal angle at which the ray intersects the annulus at Ri. Then, the azimuthal angle at which the + light ray intersects the next annulus (i 2) at Ri+1 is given by Received: 14 December 2019; Accepted: 5 June 2020; Φ Φ Biþ1 Biþ1 ; iþ1 À i ¼ arcsin À arcsin ð25Þ Riþ1 Ri ≤ fi provided that Bi+1 Ri+1. Note that the impact parameter always satis es fi fi niBi ¼ constant. Thus, given an optical system with a well-de ned pro le n(R) and an initial impact parameter B , the trajectory of a light ray can be iteratively References 0 fi computed via Eq. (25) until the ray reaches its minimum radius. Note that only in- 1. Pendry, J. B., Schurig, D. & Smith, D. R. Controlling electromagnetic elds. – going trajectories are considered here, so light escaping the optical black hole is not Science 312, 1780 1782 (2006). – modeled. Furthermore, it is assumed that all light is transmitted at each boundary; 2. Leonhardt, U. Optical conformal mapping. Science 312, 1777 1780 (2006). absorption and reflection are left for future work. 3. Leonhardt, U. 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Mimicking celestial mechanics in tition. The authors thank K. R. Moore for inspiration; R. Bekenstein, S. G. Johnson, N. metamaterials. Nat. Phys. 5, 687 (2009). Rivera, and S. P. Robinson for fruitful discussions; A. Patterson and the MIT Department 29. Narimanov, E. E. & Kildishev, A. V. Optical black hole: broadband of Physics. A.P.T. gratefully acknowledges the Tushar Shah and Sara Zion Fellowship for omnidirectional light absorber. Appl. Phys. Lett. 95, 041106 (2009). funding and was partially supported by DOE grant DE-SC00012567. The authors are also 30. Anderson, T. H., Mackay, T. G. & Lakhtakia, A. Ray trajectories for a spinning grateful to the MIT Open Access Article Publication Subvention Fund, the MIT Plasma cosmic string and a manifestation of self-cloaking. Phys. Lett. A 374, Science and Fusion Center, and the MIT Center for Theoretical Physics for their support. 4637–4641 (2010). 31. Cheng, Q., Cui, T. J., Jiang, W. X. & Cai, B. G. 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Mackay, T. G. & Lakhtakia, A. Towards a realization of Schwarzschild-(anti-) Correspondence de Sitter spacetime as a particulate metamaterial. Phys. Rev. B 83, 195424 and requests for materials should be addressed to R.A.T. or A.P.T. (2011). Reprints and permission information 37. Smolyaninov, I. I. & Hung, Y.-J. Modeling of time with metamaterials. J. is available at http://www.nature.com/reprints Optical Soc. Am. B 28, 1591–1595 (2011). Publisher’s note 38. Wang, H.-W. & Chen, L.-W. A cylindrical optical black hole using graded Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. index photonic crystals. J. Appl. Phys. 109, 103104 (2011). 39. Smolyaninov, I. I., Hung, Y.-J. & Hwang, E. Experimental modeling of cosmological inflation with metamaterials. Phys. Lett. A 376, 2575–2579 (2012). Open Access This article is licensed under a Creative Commons 40. Yang, Y., Leng, L. Y., Wang, N., Ma, Y. & Ong, C. K. Electromagnetic field Attribution 4.0 International License, which permits use, sharing, attractor made of gradient index metamaterials. J. Optical Soc. Am. A 29, adaptation, distribution and reproduction in any medium or format, as long as you give 473–475 (2012). appropriate credit to the original author(s) and the source, provide a link to the Creative 41. Sheng, C., Liu, H., Wang, Y., Zhu, S. N. & Genov, D. A. Trapping light by Commons license, and indicate if changes were made. The images or other third party mimicking gravitational lensing. Nat. Photonics 7, 902–906 (2013). material in this article are included in the article’s Creative Commons license, unless 42. Yin, M., Tian, X. Y., Wu, L. L. & Li, D. C. A broadband and omnidirectional indicated otherwise in a credit line to the material. If material is not included in the electromagnetic wave concentrator with gradient woodpile structure. Opt. article’s Creative Commons license and your intended use is not permitted by statutory Express 21, 19082–19090 (2013). regulation or exceeds the permitted use, you will need to obtain permission directly from 43. Bekenstein, R., Schley, R., Mutzafi, M., Rotschild, C. & Segev, M. Optical the copyright holder. To view a copy of this license, visit http://creativecommons.org/ simulations of gravitational effects in the Newton-Schrödinger system. Nat. licenses/by/4.0/. Phys. 11, 872–878 (2015). 44. Patsyk, A., Bandres, M. A., Bekenstein, R. & Segev, M. Observation of accelerating wave packets in curved space. Phys. Rev. X 8, 011001 (2018). © The Author(s) 2020 45. Leonhardt, U. Quantum physics of simple optical instruments. Rep. Prog. Phys. 66, 1207–1249 (2003).

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