# Black Hole Evaporation: Sparsity in Analogue and General Relativistic Space-Times

Our understanding of black holes changed drastically, when Stephen Hawking discovered their evaporation due to quantum mechanical processes. One core feature of this effect, later named after him, is both its similarity and simultaneous dissimilarity to classical black body radiation as known from thermodynamics: A black hole’s spectrum certainly looks like that of a black (or at least grey) body, yet the number of emitted particles per unit time differs greatly. However it is precisely this emission rate that determines — together with the frequency of the emitted radiation — whether the resulting radiation field behaves classical or non-classical. It has been known nearly since the Hawking effect’s discovery that the radiation of a black hole is in this sense non-classical (unlike the radiation of a classical black or grey body). However, this has been an utterly underappreciated property. In order to give a more readily quantifiable picture of this, we introduced the notion of ‘sparsity’, which is easily evaluated, and interpreted, and agrees with more rigorous results despite a semi-classical, semi-analytical origin. Sadly, and much to relativists’ chagrin, astrophysical black holes (and their Hawking evaporation) have a tendency to be observationally elusive entities. Luckily, Hawking’s derivation lends itself to reformulations that survive outside its astrophysical origin — all one needs, are three things: a universal speed limit (like the speed of sound, the speed of light, the speed of surface waves, . . . ), a notion of a horizon (the ‘black hole’), and lastly a sprinkle of quantum dynamics on top. With these ingredients at hand, the last thirty-odd years have seen a lot of work to transfer Hawking radiation into the laboratory, using a range of physical models. These range from fluid mechanics, over electromagnetism, to Bose–Einstein condensates, and beyond. A large part of this thesis was then aimed at providing electromagnetic analogues to prepare an analysis of our notion of sparsity in this new paradigm. For this, we developed extensively a purely algebraic (kinematical) analogy based on covariant meta-material electrodynamics, but also an analytic (dynamical) analogy based on stratified refractive indices. After introducing these analogue space-time models, we explain why the notion of sparsity (among other things) is much