Decomposition of metric tensor in thermodynamic geometry in terms of relaxation timescales

Geometrical methods are extensively applied to thermodynamics, including stochastic thermodynamics. In the case of a slow-driving linear response regime, a geometrical framework, known as thermodynamic geometry, is established. The key to this framework is the thermodynamic length characterized by a metric tensor defined in the space of controlling variables. As the metric tensor is given in terms of the equilibrium time-correlation functions of the thermodynamic forces, it contains the information on timescales, which may be useful for analyzing the performance of heat engines. In this paper, we show that the metric tensor for underdamped Langevin dynamics can be decomposed in terms of the relaxation times of a system itself, which govern the timescales of the equilibrium time-correlation functions of the thermodynamic forces. As an application of the decomposition of the metric tensor, we demonstrate that it is possible to achieve Carnot efficiency at finite power by taking the vanishing limit of relaxation times without breaking trade-off relations between efficiency and power of heat engines in terms of thermodynamic geometry.