Gravitational metamaterials from optical properties of spacetime media

TL;DR

The paper develops a comprehensive framework to treat spherically symmetric spacetimes as optical media, introducing two compatible definitions of the gravitational refractive index $n_O$ and exploring their equivalence via coordinate transformations. It analyzes how $n_O$ behaves across Schwarzschild, charged, and de Sitter extensions, as well as regular black holes and horizonless spacetimes, highlighting the possibility of negative refraction and its connection to Misner–Sharp mass and redshift. By incorporating electromagnetic fields, the work shows an emergent anisotropic $n_O$ that aligns with the purely geometric results in appropriate limits, and uses Snell’s law in the radial gravity approximation to connect optical indices with deflection angles. The authors then propose gravitational metamaterials—spacetimes with $n_O<0$—as potential particle-like or dark-matter candidates, presenting three toy models (conformal, Newtonian, and Simpson–Visser regularizations) and outlining a quantum extension via geometric quasiparticles. Overall, the study offers a novel, gravity-centered metamaterial perspective that links optical propagation in curved spacetime to dark matter phenomenology and lays groundwork for future observational and theoretical explorations of geometric quasiparticles and horizonless compact objects.

Abstract

Gravitational optical properties are here investigated under the hypothesis of spherically-symmetric spacetimes behaving as media. To do so, we first consider two different definitions of the refractive index, $n_O$, of a spacetime medium and show how to pass from one definition to another by means of a coordinate transformation. Accordingly, the corresponding physical role of $n_O$ is discussed by virtue of the Misner-Sharp mass and the redshift definition. Afterwards, we discuss the inclusion of the electromagnetic fields and the equivalence with nonlinear effects induced by geometry. Accordingly, the infrared and ultraviolet gravity regimes are thus discussed, obtaining bounds from the Solar System, neutron stars and white dwarfs, respectively. To do so, we also investigate the Snell's law and propose how to possibly distinguish regular solutions from black holes. As a consequence of our recipe, we speculate on the existence of \emph{gravitational metamaterials}, whose refractive index may be negative and explore the corresponding physical implications, remarking that $n_O<0$ may lead to invisible optical properties, as light is bent in the opposite direction compared to what occurs in ordinary cases. Further, we conjecture that gravitational metamaterials exhibit a particle-like behavior, contributing to dark matter and propose three toy models, highlighting possible advantages and limitations of their use. Finally, we suggest that such particle-like configurations can be ``dressed" by interaction, giving rise to \emph{geometric quasiparticles}. We thus construct modifications of the quantum propagator as due to nonminimal couplings between curvature and external matter-like fields, finding the corresponding effective mass through a boson mixing mechanism.

Figures (4)

• Figure 1: (Top.) Plots of Eqs. \ref{['nS']}, varying the angle $\sigma$. The thick curve corresponds to the case $\sigma=0$, namely a radial approximation. The dashed, dotdashed and dotted curves are respectively displayed with $\sigma=\frac{\pi}{3}$, $\sigma=\frac{\pi}{4}$ and $\sigma=\frac{\pi}{6}$, all with unitary mass. The output shows that only the amplitude of $n_O$ is influenced by varying $\sigma$, while the shape remains unaltered. (Bottom.) Plots of the two definitions of refractive indexes, Eqs. \ref{['indicedirifrazione1']} and \ref{['opticalbis']}, corresponding to the thick and dashed lines, respectively. The two shapes are essentially indistinguishable, however with different magnitudes.
• Figure 2: Plots of $n_O$ for the black hole solutions. Precisely, the thick black line corresponds to the Schwarzschild black hole, while the dashed and dotted lines are respectively the Schwarzschild-de Sitter and Reisser-Nordstr$\ddot {\rm o}$m spacetimes, with indicative unitary mass and $Q=0.5$, $\Lambda=0.01$ and $\sigma=0$. Remarkably, close to the horizon, $r_H\sim 2M$, the refractive index tends to infinity as expected, while at larger distances it decreases to zero. Quite interestingly, the extreme case $M=Q$ is displayed in the dot-dashed curve, corresponding to the maximal departure of the Reissner-Nordstr$\ddot {\rm o}$m metric from the Schwazschild case.
• Figure 3: Plots of the refractive indexes for the regular black hole solutions, as reported in Eqs. \ref{['nORBH']}; (top figure). Plots of $n_O$, compared with the Schwarzschild black hole (thick gray line), beyond the horizon. The black thick, dotted, dashed and dot-dashed lines are respectively the Hayward, Bardeen, Dymnikova and Fang-Wang regular black holes; (bottom figure). Plot of $n_O$ within the horizon for all the solutions, excluding Schwarzschild that would tend to negative values. In all the plots, we adopted an indicative unitary mass and arbitrarily $\Lambda=\frac{1}{2}$ and $\sigma=0$. At $\sigma=0$ and $a^2\equiv M\Lambda^{-1}$, comparing the Hayward de Sitter core with the other three behaviors in Eqs. \ref{['RBHcores']}, the values of $q,l_D$ and $l_{FW}$ can be expressed in terms of the Hayward de Sitter phase, $\Lambda$, namely $q=\left(\frac{2}{\Lambda}\right)^{\frac{1}{3}}$, $l_{FW}=\left(\frac{2}{\Lambda}\right)^{{1\over3}}$ and $l_D=\left(\frac{8}{3\pi\Lambda}\right)^{1/3}$. The main departures from the Schwarzschild solution occur close to the horizon, as expected, while asymptotically the solutions tend to match.
• Figure 4: Plots of the refractive indexes for solutions with no horizon. The AdS and quasi AdS solutions are displayed in dashed and dotted black lines, respectively. Indicatively, we also reported the Schwarzschild black hole with $M\rightarrow-M$, in gray as dot-dashed curve, with $M=-0.5$. The action of $Q$ is portrayed in the light gray curve, where we plot the Reisser-Nordstr $\ddot{o}$m solution with $Q=3M$ and $M=0.5$. The behaviors of each solution is particularly different than singular and regular solutions, indicating superluminal regions. The indicative values, here used for the plots, are $\Lambda=0.5$ and $k_0=4\Lambda$. The AdS and quasi AdS solutions appear quite similar, as expected, conversely to the Schwarzschild solution with negative mass whose concavity appears quite different, as a consequence of $M$.