Special relativity effects in general relativity

[WannabeNewton]

Ok, I guess I'm wrong, but kinematical time dilation and gravitational time dilation do add up?

Sort of, at least in simple circumstances. For example if we have an observer in circular free fall orbit in Schwarzschild space-time then the observer's "gamma factor" in Schwarzschild coordinates will be $\gamma = (1 - 2M/R - R^2 \omega^2)^{-1/2}$ where the $2M/R$ accounts for the gravitational part and the $R^2 \omega^2$ accounts for the kinematical part. Then $\gamma^{-1}$ represents the rate at which the circularly orbiting observer's clock ticks relative to a clock at spatial infinity.