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Pulsar timing arrays (PTA) have the promise to detect gravitational waves(GWs) from sources which are in a unique frequency range of 10^-9 - 10^-6 Hz.This in turn also provides an opportunity to test the theory of generalrelativity in the low frequency regime. The central concept of the detection ofGWs with PTA lies in measuring the time of arrival difference of the pulsarsignal due to the passing of GWs i.e. the pulses get red-shifted. In this paperwe provide a complete derivation of the redshift computation for all sixpossible polarizations of GW which arise due to the modifcations to generalrelativity. We discuss the smoothness of the redshift and related properties atthe critical point, where the GW source lies directly behind the pulsar. Fromour mathematical discussion we conclude that the redshift has to be splitdifferently into polarization part (pattern functions) and interference part,to avoid discontinuities and singularities in the pattern functions. Thischoice of pattern functions agrees with the formula one uses forinterferometers with a single detector arm. Finally, we provide a generalexpression which can in principle be used for pulsars and GW of any frequencywithout invoking the low frequency assumption and using said assumption wedevelop the expression up to first order in the strain and find correctionterms to the canonical redshift formula.

Pulse redshift of pulsar timing array signals for all possible gravitational wave polarizations in modified general relativity