I'm reading Bridson & Haefliger's book on non-positively curved spaces. Specifically the parts in II.10 on $P(n,\mathbb{R})$. These are positive definite matrices, viewed as the image of symmetric matrices under matrix exponentiation, with a metric given by $d(e^A, e^B) = |\log(e^{-A/2} e^B e^{-A/2})|$, the usual trace norm after this isometry moving $e^A$ to the identity and going back to the symmetric matrices.
I'd like to understand this situation:
Let $e^D, e^A \in P(n,\mathbb R)$ with $D$ diagonal. Consider the pair of geodesics $e^{tD}$ and $e^{tA}$ going through the origin and these points. If $A$ commutes with $D$, then these geodesics diverge linearly, as in euclidean space.
However, when $A,D$ do not commute, the geodesics diverge faster than linearly. I would like to know how this divergence depends on the distance of $e^A$ to the set $C(e^D)$ of matrices commuting with $e^D$. My intuition is that the divergence should continuously increase with this distance. E.g., I imagine $d(e^{2D},e^{2A}) - 2\cdot d(e^{D}, e^{A})$ increases continuously as $d(e^A, C(e^D))$ increases.
Is this true? Better yet if it is, is it described somewhere, or clear from the definition of this metric and some relation between this distance and commutators? Bridson & Haefliger show that the differential of $e^{-X/2}e^{X+tY}e^{-X/2}$ is a matrix series $\sum_{k=0}^{\infty} \frac{ad^{2k}_{X/2}}{(2k+1)!}(Y)$ in order to prove $P(n,\mathbb R)$ is CAT(0). But I don't know enough to know if this can be pulled together to get what I'm asking.
Any pointers or pointing would be appreciated.