We study the geodesic equation for the Dirichlet (gradient) metric in the space of Kähler potentials. We first solve the initial value problem for the geodesic equation of the combination metric, including the gradient metric. We then discuss a comparison theorem between it and the Calabi metric. As geometric motivation of the combination metric, we find that the Ebin metric restricted to the space of type II deformations of a Sasakian structure is the sum of the Calabi metric and the gradient metric.

S. C. is supported by the PRIN Project "Varietà reali e complesse: geometria, topologia e analisi armonica'', by SIR 2014 AnHyC "Analytic aspects in complex and hypercomplex geometry" (code RBSI14DYEB), and by GNSAGA of INdAM.
D. P. is supported by the Research Training Group 1463 "Analysis, Geometry and String Theory'' of the DFG, as well as the GNSAGA of INdAM.
The work of K. Z. has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 703949, and was also partially supported by the Engineering and Physical Sciences Research Council (EPSRC) on a Programme Grant entitled "Singularities of Geometric Partial Differential Equations" reference number EP/K00865X/1.