In Schwarzschild coordinates the photon sphere is located at an "r-parameter" of $r=3GM/c^2$. If we are watching the photon sphere from infinity using a telescope such as the Event Horizon Telescope there will be a gravitational lensing effect making the photon sphere look larger. The expression for this is $r_{obs}=r(1-\frac{2GM}{rc^2})^{-0.5}$. I do not know if this formula for gravitational lensing enlargement is specific to Schwarzschild coordinates or if you get the very same expression using isotropic coordinates. Anyhow, using Schwarzschild coordinates the expression for the apparent radius of the photon sphere for an observer watching from infinity as a function of the central mass and the "r-parameter" is $r_{obs= \sqrt{27}GM/c^2}$.
Question: What is the expression for the apparent radius of the photon sphere for an observer watching from infinity in isotropic coordinates as a function of the central mass and the "isotropic r-parameter"?
The Schwarzschild isotropic metric looks like: $$ -c^2d\tau^2 = -\left(\frac{1-GM/2rc^2}{1+GM/2rc^2}\right)^2c^2dt^2 +(1+GM/(2rc^2))^4[dr^2 +r^2( d\theta^2 +\sin^2\theta d\phi^2)]$$
From this I am able to get that the time dilation formula must be:
$$d\tau=\sqrt{\frac{(1-\frac{GM}{2rc^2})^2}{(1+\frac{GM}{2rc^2})^2}-(1+\frac{GM}{2rc^2})^4\frac{v^2}{c^2}}dt$$
and the velocity of light:
$$c_{coordinate}=c\frac{(1-\frac{GM}{2rc^2})}{(1+\frac{GM}{2rc^2})^3}$$
I guess that you can somehow decide the orbital velocity of a circular orbit in isotropic coordinate and when this velocity becomes equal to the speed of light, that is where the radius of the photon sphere is. I really have no clue to what the expression for gravitational lensing enlargment will be using isotropic coordinates.
In solar system dynamics relativistic effects are accounted for by using the post-newtonian expansion which is a low order expansion of the isotropic metric and, as I understand it, treat the "r-parameter" as a real radial distance.
