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：本文探讨信息几何中的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的梯度流如何描述测地线，通过引入约束条件探究相关力学机制，并将理论应用于高斯模型与黑洞热力学中。由此，我们展示了梯度流形变如何导出更一般的Randers-Finsler度量，描述了源自约束的哈密顿力学，并通过正则变换证明了对偶性。我们还针对高斯模型的形变验证了理论，并描述了Kerr黑洞与Reissner-Nordström黑洞平直度量的动力学演化过程。