Wave propagation in metamaterials mimicking spacetime geometry: black holes and cosmic strings

Abstract

In physics, it is common to find different phenomena being described by similar equations. A good analogy can make us look at a problem from a different point of view. In that way, ideas may be transferred from one field of science to another, allowing to model new phenomena after previous, well-studied ones. In the case of the field of analogue gravity, systems that mimic certain aspects of the physics of curved spacetimes are studied. In this thesis, we are interested in the analogy between geometry and media. It has been known for several decades that light propagation in a gravitational field is formally equivalent to that in a bianisotropic medium. On the one hand, ray paths are bent due to spacetime curvature. On the other hand, spatial variations of the permittivity and permeability of a material can make light follow curved trajectories. These two phenomena can be related mathematically in the context of transformation optics, which provides the tools to determine the medium parameters necessary to mimic a certain coordinate transformation.

Materials with these specific properties are not naturally occurring, therefore, the emergence of metamaterial science at the beginning of the century was needed to realize them. Metamaterials are artificial composite materials with sub-wavelength constitutive elements that exhibit exotic properties. They have been one of the hot topics of the past years given the variety of opportunities they offer: negative refraction, superlenses, indefinite dispersion, invisibility, among many others.

In this thesis we study the analogues of two static spacetimes from the point of view of transformation optics: one with spherical symmetry and one with conical geometry. Both cases are inspired by solutions to Einstein’s equations: the Schwarzschild black hole and the cosmic string, respectively. For each case, we derive the permittivity and permeability of the analogous material using Plebanski’s formulation of the electromagnetic constitutive equations. We solve numerically the wave equation in the metamaterial and compare the results with analytical theories.

We find that the spherically symmetric spacetime can be mimicked by either an anisotropic or isotropic medium due to its rotational symmetries. This is achieved by performing a coordinate transformation of the general metric to a conformally flat form. We obtain the medium parameters for both cases and apply the results to the case of the Schwarzschild black hole. We simulate the propagation of a Gaussian beam in the two materials and compare the numerical results with the null-geodesics in the Schwarzschild spacetime, finding a good agreement.

The cosmic string is an example of a topological defect with conical geometry. A conical space can be interpreted as flat space with a wedge removed. We make use of this transformation to study the wave equation in the cosmic string background. We apply asymptotic diffraction theories to obtain analytical models that describe wave propagation of electromagnetic or gravitational waves (in a certain gauge). We find that our expressions reproduce accurately the results of the numerical simulations in the analogous metamaterial. Moreover, with our models, we can understand the observed diffraction pattern as the interference of four characteristic waves. With this interpretation we can introduce the Fresnel observation zones, which are related to the diffraction maxima. They help localize the regions – in either space or frequency – where the wave effects are more significant. In fact, in the diffraction by a non-compact object such as the cosmic string, we find that the contribution to the field of wave effects such as interference or diffraction can be of the same order as the geometrical optics terms. Furthermore, the conical topology also appears in condensed matter systems as disclinations or wedge dislocations, therefore we expect our results to be applicable in those systems as well.

La investigación en gravitación análoga consiste en el estudio de sistemas físicos donde se pueden reproducir algunos de los fenómenos propios de relatividad general. Esta tesis se centra en la conocida propagación análoga de las ondas electromagnéticas en un espacio-tiempo curvado y en un medio con una permitividad y permeabilidad generalmente anisótropas. Para la realización de este tipo de medios, fue necesaria la aparición de los metamateriales, materiales artificiales diseñados para tener propiedades electromagnéticas fuera de lo común. En este contexto, se estudian modelos análogos a dos objetos estáticos con distinta simetría: uno con simetría esférica y otro con topología cónica. Ambos casos están motivados por soluciones a las ecuaciones de Einstein: el agujero negro de Schwarzschild y la cuerda cósmica, respectivamente. A través de las ecuaciones de óptica de transformación, determinamos los parámetros de los medios análogos a estos espacio-tiempos. Estudiamos la propagación de ondas electromagnéticas en los materiales obtenidos mediante simulaciones numéricas y comparamos los resultados con teorías analíticas, encontrando muy buen acuerdo. Por un lado, los contrastamos con las geodésicas en los espacio-tiempos considerados a través del formalismo Hamiltoniano. Por otro lado, desarrollamos modelos analíticos para describir la difracción de una onda (en principio tanto electromagnética como gravitatoria) debido a la cuerda cósmica. Para ello, usamos teorías asintóticas de difracción en un espacio virtual plano con un déficit de ángulo, ya que es una representación equivalente a la geometría cónica de la cuerda. De este modo, se obtienen expresiones que nos permiten explicar detalladamente los fenómenos ondulatorios de interferencia y difracción que se producen en este espacio-tiempo. Observamos que estos efectos pueden ser comparables a los términos de óptica geométrica: añaden una modulación en la amplificación del campo relacionada con la formación de imágenes dobles propia de la topología de la cuerda. Cabe destacar que estos fenómenos son conceptualmente distintos a los que se podrían esperar en la difracción sobre un objeto sólido como una barra. Utilizando nuestros modelos analíticos, obtenemos el patrón de difracción característico de la cuerda cósmica, que podría ser de interés para su detección.