Newman Black Hole
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Optical analogues to the equatorial Kerr–Newman black hole The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Tinguely, Roy Alexander et al. "Optical analogues to the equatorial Kerr–Newman black hole." Communications Physics 3, 1 (July 2020): 120 © 2020 The Author(s) As Published http://dx.doi.org/10.1038/s42005-020-0384-5 Publisher Springer Science and Business Media LLC Version Final published version Citable link https://hdl.handle.net/1721.1/126571 Terms of Use Creative Commons Attribution 4.0 International license Detailed Terms https://creativecommons.org/licenses/by/4.0/ ARTICLE 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 iso- tropic 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] COMMUNICATIONS PHYSICS | (2020) 3:120 | https://doi.org/10.1038/s42005-020-0384-5 | www.nature.com/commsphys 1 ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0384-5 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 2 COMMUNICATIONS PHYSICS | (2020) 3:120 | https://doi.org/10.1038/s42005-020-0384-5 | www.nature.com/commsphys COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-020-0384-5 ARTICLE 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 .