Let $\text{SL}(m+n,\mathbb R)$ be equipped with a left or right invariant Riemannian distance $d$ (this is part of the question). Let $g_t=\text{diag}(e^{t/m}I_m,e^{-t/n}I_n) $ where $I_m, I_n$ denote the identity matrices. I have seen people using the following equivalence without proof quite often:
$$d(g_t, I_{m+n}) \asymp t$$
but I don't know why (the notation $\asymp$ means asymptotically up to constant multiples).
My question has two parts:
(1) What canonical distance $d$ are we using for $\text{SL}(m+n,\mathbb R)$? Please I know this is somehow related to the hyperbolic space $\mathbb H^{m+n-1}$ (the dimension might not be m+n-1 may not be correct and we may need to look into the unit tangent bundle) but I want a very explicit description of the distance $d$ and please make sure it is indeed left or right invariant
(2) How do we see $d(g_t, I_{m+n}) \asymp t$ under this metric?
For the case when $m=n=1$, I learnt from a friend that we can identify $\text{SL}(m+n,\mathbb R)$ with left-invariant Riemannian metric with $T^1(\mathbb H^1)$ (the metric on the latter should be additively equivalent geodesic distance).The higher dimensional case seems to be significantly harder.