Decoupling of kinematical time dilation and gravitational time dilation in particular geometries

Two different forms of time dilation, namely, the kinematical time dilation of special relativity and gravitational red shift are coupled during observations of systems moving through a gravitational field. In the particular situation of free fall in a Schwarzschild geometry these two effects are decoupled and in consequence the time dilation, as observed by a distant observer, factorises. Such a factorization is not a universal feature. We define here a necessary and sufficient criterion for time dilation and gravitational red-shift decoupling. This property is manifested in a particular form of the Doppler shift in Schwarzschild geometry.