Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ matrices with positive-definite imaginary part. Royden in his article Intrinsic Metrics on Teichmüller Space gives the inifinitesimal form of the Kobayashi distance (metric) on $\mathbb{H}_{g}$ as the operator norm $\frac{1}{2}||Y^{-1/2} dZ Y^{-1/2}||$.
My question is what is the Kobayashi distance between two given points on $\mathbb{H}_{g}$?
For example, when $g=1$ (i.e., $\mathbb{H}_{g}=\mathbb{H}_{1}$ is the upper half plane), it is well-konwn that the Kobayashi distance between $z$ and $w$ in $\mathbb{H}$ is the usual hyperbolic distance $d_{\mathbb{H}}(z,w)$. What is the generalization of this to the upper half space?