[Seminar] "Theory of Thermal Hall Effect" by Dr. Atsuo Shitade from RIKEN
Thermal Hall effect is a powerful experimental tool to study effects of inelastic scattering on the anomalous Hall effect in electron systems and nontrivial properties of neutral excitations in spin systems. Theoretically, we have to introduce Luttinger’s gravitational potential and calculate a magnetization correction called heat magnetization. This is a heat analog of the orbital magnetization and characterizes the circulating heat current.
In this seminar, we discuss why Luttinger’s gravitational potential is gravitational [1,2]. By gauging the time translation symmetry, we introduce a gravitational gauge field that is coupled to the heat current. This induces a field strength; a gravitational electric field corresponds to a temperature gradient, while a magnetic one defines the heat magnetization.
 A. Shitade, Prog. Theor. Exp. Phys. 2014, 123I01 (2014)
 A. Shitade, J. Phys. Soc. Jpn. 86, 054601 (2017).
P.S. In terms of a theory of gravity, the time translation symmetry is gauged by the general covariance principle. Gravitational gauge field and field strength are identified as vielbein and torsion, respectively. However, you do not have to worry about a bunch of indexes.