Another gedanken on relativity

DOPPLER SHIFT AND HAWKING RADIATION

Consider an event of pair production by a photon with energy > 1.02 Mev.
(NOTE-Pair production can take place by a single photon when it interacts with a massive( as compared to itself)body.)
Now consider the same event to be viewed from a frame in which the original photon was red shifted so that its energy was < 1.02 MeV. In this frame the pair production could have taken place only if the excess energy of the electron and the positron would have been negative.This leads to the conclusion that either there exists a potential energy function which has different values in different inertial frames or the event would not have taken place i.e. the electron and the positron will still be seen as a photon from the other frame(If the second assumption is assumed to be correct also consider the case when the pair production results in Hawking radiation.)Which of the above conclusions is correct?

Priyank