2020 was a bit of a disaster. There were two things that motivated me to get out of bed every morning though: my hungry cat, and this research task. Supervised again by Dr. Shinji Hirano, this project threw me head first into the wild world of general relativity and horizon radiation — and I loved it all.
Much like my writings about my other bachelor’s project, this page will first go over a technical summary of the report. After this is a section with pedagogical explanations of the ideas involved, which will take absolute ages to write — so check back later if they are not yet complete.
High-level summary
After introducing the laws of black hole thermodynamics and discussing their history, the introductory section then also introduces the notion of cosmological horizons in de Sitter space and shows how Wick rotating the metric implies that both of these have characteristic temperatures. Also briefly discussed are the issues pertaining to a particularly confusing minus sign in the calculation of the 1st law of thermodynamics for de Sitter universes.
To perform calculations involving cross-horizon dynamics we introduce Painleve coordinates for Schwarzschild and de Sitter spacetimes, which are regular at the respective horizons. Then, using the WKB approximation, a cross-horizon tunnelling rate is derived and is found to be expressible as $\Gamma \sim e^{\Delta S_\text{BH}}$. This is argued to align well with ideas of quantum unitarity, and shown to agree with Hawking’s predictions of temperature to leading order in the emitted particle’s frequency at the horizon for both black holes and de Sitter universes.
A very small note at the end of the paper talks about perhaps being able to extend this using a method known as the “exact WKB procedure”, which is a lovely little piece of foreshadowing.