In this article we shall discuss the theory of geodesics in information geometry, and an application in astrophysics. We will study how gradient flows in information geometry describe geodesics, explore the related mechanics by introducing a constraint, and apply our theory to Gaussian model and black hole thermodynamics. Thus, we demonstrate how deformation of gradient flows leads to more general Randers-Finsler metrics, describe Hamiltonian mechanics that derive from a constraint, and prove duality via canonical transformation. We also verified our theories for a deformation of the Gaussian model, and described dynamical evolution of flat metrics for Kerr and Reissner-Nordstr\"om black holes.

翻译：本文讨论信息几何中测地线理论及其在天体物理学中的应用。我们将研究信息几何中梯度流如何描述测地线，通过引入约束条件探索相关力学机制，并将该理论应用于高斯模型与黑洞热力学。由此，我们论证了梯度流形变如何导出更广义的Randers-Finsler度量，描述了源于约束条件的哈密顿力学，并通过正则变换证明了对偶性。我们还对高斯模型的形变进行了理论验证，并描述了Kerr和Reissner-Nordström黑洞平坦度量的动力学演化过程。