Optical second harmonic generation near a black hole horizon as possible source of experimental information on quantum gravitational effects

Abstract

Optical second harmonic generation near a black hole horizon is suggested as a source of experimental information on quantum gravitational effects. While absent in the framework of general relativity, second harmonic generation appears in the toy models of sonic and electromagnetic black holes, where spatial dispersion at high frequencies for waves boosted towards the horizon is introduced. Localization effects in the light scattering from random fluctuations of matter fields and space-time metric near the black hole horizon produce a pronounced peak in the angular distribution of second harmonics of light in the direction normal to the horizon. Such second harmonic light has the best chances to escape the vicinity of the black hole. This phenomenon is similar to the well-known strong enhancement of diffuse second harmonic emission from a randomly rough metal surface in the direction normal to the surface. PACS no.: 78.68.+m; 42.65.Ky Wave propagation and localization phenomena in random media have been the topic of extensive studies during the last years [1]. One of the most striking examples of such phenomena is the strong and narrow peak of diffuse second harmonic light emission observed in the direction normal to a randomly rough metal surface (see Fig.1(a)). This peak is observed under the coherent illumination at any angle. This effect was initially predicted theoretically [2] and later observed in the experiment [3]. The enhanced second harmonic peak normal to the mean surface arises from the fact that a state of momentum k introduced into a weakly localized system will encounter a significant amount of backscattering into states of momentum centered about -k. When these surface k and -k modes of frequency ω interact through an optical nonlinearity to generate 2ω radiative modes, the 2ω light has nonzero wave vector components only perpendicular to the mean surface. The angular width of the normal peak can be as small as a few degrees, and its amplitude far exceeds the diffuse omnidirectional second harmonic background.Very recently it was realized [4] that many results obtained in the optics of random media may be applicable to the case of light propagation in stochastic space-time metrics. This is possible because of an analogy between the propagation of light in matter and in curved spacetime. It is well known that Maxwell equations in a general curved space-time background g ik (x, t) are equivalent to the phenomenological Maxwell equations in the presence of matter background with nontrivial electric and magnetic permeability tensors ǫ ij (x, t) and µ ij (x, t) [5]. In this analogy, the event horizon corresponds to a surface of singular electric and magnetic permeabilities. There are quite a few papers which consider solidstate toy models of electromagnetic [6] and sonic [7] black holes. In the absence of established quantum gravitation theory such toy models are helpful in understanding electromagnetic phenomena in curved space-time, such as Hawking radiation [8] and Unruh effect [9]. Introducin...

Surface Electromagnetic Waves near a Black Hole Event Horizon and Their Observational Consequences

“…This potential (plotted schematically in Figure 2) appears to be real and well-behaved even below the horizon, even though the effective ε and µ parameters themselves are imaginary (as defined by Equation ( 5)). Moreover, as noted in [4], the divergence of this potential at ρ = 0 is supposed to be tamed by the quantum gravity effects. Note that such a situation is not unusual in macroscopic electrodynamics.…”

“…where F ik is the electromagnetic field tensor. Consideration of electromagnetic wave propagation near a black hole event horizon in [4] was based on the well-known analogy between these Maxwell equations in a curvilinear spacetime metric g ik (x, t) and the macroscopic Maxwell equations describing electromagnetic fields in the presence of matter background with some non-trivial electric permittivity tensor ε ij (x, t) and magnetic permeability tensors µ ij (x, t) [7]. For example, the equations of electrodynamics in the presence of a static gravitational field look exactly like Maxwell's equations in some macroscopic electrodynamic medium, in which…”

“…However, many more experimental tools will be required in the future to observe and study the complicated physics of these highly non-trivial spacetime regions in more detail. In this communication, we will concentrate on the further development of a recent proposal [4] to use optical second-harmonic (SH) generation as an alternative observational tool to explore gravitational physics in the immediate vicinity of an event horizon of a black hole. In ref.…”

“…In ref. [4], it was demonstrated that SH generation must be strongly enhanced when light from a distant star interacts with random fluctuations of various fields (such as matter fields and fluctuating spacetime metrics) near an event horizon. This strongly directional SH emission occurs due to localization phenomena in light scattering from these fluctuations.…”

“…According to experimental observations [6], the angular width of this SH peak can be as small as a few degrees with respect to the mean normal, and its intensity appears to be much larger compared to the diffuse omnidirectional SH background. While potential observability of this very interesting effect has been established in [4], the theoretical consideration in that work was limited only to the case of the Rindler metric, which describes the immediate vicinity of the black hole event horizon. In this communication, we perform a more realistic study of surface electromagnetic wave solutions near the horizon of a Schwarzschild metric, describing a "real" astronomical black hole in vacuum.…”