Let $SH$ be the unit tangent bundle of upper half plane defined by the pair $(z,\eta),z\in H,\eta\in C,|\eta|=Im(z)$. Define group action $SL_2(R)$ on $SH$ by $g=\frac{az+b}{cz+d}\in SL_2(R), g(z,\eta)=(g(z),\frac{\eta}{(cz+d)^2})$. On $H$ there is a canonical hyperbolic metric $ds$ and volume $dv$ invariant under $SL_2(R)$ action.
Consider $\phi=\frac{1}{2\pi}Arg(\eta)$ where $\eta$ is defined as before as an element of $S^1$.
Define $dv'=dvd\phi$ and $ds'=\sqrt{ds^2+d\phi^2}$. I knew $ds^2$ and $dv$ part invariant under $SL_2(R)$ action for sure. Then the author claims $dv'$ and $ds'$ are invariant under $SL_2(R)$ action.
If I take a path on $SH$(i.e. $(z(t),\eta(t))$ parametrized by $t$ on $SH$), then $T\eta(t)=\frac{\eta(t)}{(cz(t)+d)^2}$ for $T=\frac{az+b}{cz+d}$. Now there is possible change of argument of $cz(t)+d$ part. If I take argument it should become $Arg(\eta(t))-2Arg(cz(t)+d)$. There is no reason that second term should not contribute if I take $\frac{d}{dt}$ acting upon.
Q: What have I done wrong here? I feel something wrong with my understanding here. I should have $\frac{d\phi}{dt}$ invariant.