An introduction to BKL theory

These notes summarize six hours of lectures given at the 20th edition of the Modave Summer School in Mathematical Physics on the topic of the Belinskii, Khalatnikov and Lifshitz (BKL) theory. Their work gives a description of the generic oscillatory behavior of solutions to Einstein's equations in the vicinity of a spacelike singularity. In these notes, we introduce the generalized Kasner solution and subsequently study the mixmaster or diagonal Bianchi IX geometry as a toy model for the general near-singularity oscillatory behavior. We give an overview of the cosmological billiard approach and apply it to the mixmaster universe. We briefly discuss the cosmological billiard description of the near-singularity description of inhomogeneous vacuum gravity in four spacetime dimensions. Finally, we derive the Wheeler-DeWitt equation obtained by canonically quantizing the cosmological billiard and give an overview of the peculiar symmetries that appear in this problem. It is an interesting open problem whether these symmetries point to any fundamental feature of (quantum) gravity.