Abstract
General Relativity is a very complex theory whose quantization remains elusive. A reduced version of the theory exists: 3-dimensional Einstein’s gravity to which this lecture is dedicated. As a toy-model, it is a very useful framework thanks to which we can experiment some techniques and derive features, some of which extend to the physical 4d case.
In this lecture, we will review the typical properties of 3d gravity, which are mostly due to the vanishing of the Weyl curvature. Next we will turn to the AdS 3 phase space: we will describe global features of AdS 3 itself, give several elements on the Brown-Henneaux boundary conditions and the resulting asymptotic symmetry group, and finally discuss BTZ black holes. We will show that the asymptotically flat phase space can be obtained from the flat limit of the asymptotically AdS 3 phase space. Finally, we will shortly present the Chern-Simons formulation of 3d gravity, which reduces the theory to the one of two non-abelian gauge vector fields.
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Notes
- 1.
Note that the application defined by the system above, and which sends AdS 3 into \(\mathbb {R}^{(2,2)}\), is not injective, because it contains 2π-periodic functions of τ. So it wraps AdS 3 around \(\mathcal {H}\) an infinity of times. But locally the injectivity is however ensured, so we talk about immersion rather than embedding.
- 2.
We save more detailed explanations for the lecture on 3d asymptotically flat spacetimes where conical defects also appear. Doing so we do not develop the same explanations twice!
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Compère, G. (2019). Three Dimensional Einstein’s Gravity. In: Advanced Lectures on General Relativity. Lecture Notes in Physics, vol 952. Springer, Cham. https://doi.org/10.1007/978-3-030-04260-8_2
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