MSc defense by Jørgen Sandøe Musaeus

2+1 Dimensional Non-Relativistic Gravity - In Search of a Theory with BTZ Solutions

The simplifying case of 2+1 dimensional general relativity has proven to be a useful tool for gaining insights into certain aspects of gravity. These include holography and black hole physics, for which the BTZ geometry has been instrumental. In this thesis we take the first step in an attempt to replicate this success for the field of non-relativistic gravity.

The method for attaining type 2 Newton-Cartan theory from a $1/c^2$-expansion of general relativity in $D\geq 3$ spacetime dimensions is reviewed and some of the important features of the theory are showcased. Similar methods are used to perform a $1/c^2$-expansion of the Chern-Simons action for 2+1D general relativity with a negative cosmological constant. It is shown that the expansion depends on certain scaling parameters from the relativistic theory which we argue are not uniquely determined. A specific choice of scaling parameters is found and the corresponding non-relativistic action is constructed to arbitrary order. We show that the expanded action to any finite order cannot be expressed as a Chern-Simons action. The equations of motion are computed and shown to contain the twistless torsional Newton-Cartan condition at any order. Furthermore, a non-relativistic expansion of the BTZ geometry that allows for strong gravitational effects is found and shown to be a solution of the proposed non-relativistic 2+1D theory.

Finally, a $1/c^2$-expansion of the 2+1 dimensional Einstein-Hilbert action with zero cosmological constant is constructed and expressed as a Chern-Simons action to any finite order with the corresponding gauge group being attained through an expansion of the Poincaré algebra. The non-relativistic action is shown to have angular deficit geometries as solutions with boundary charges corresponding to rotations at different orders in the geometry.