Finding the mass-energy counterpart to the transverse Doppler shift:   Avoiding de Broglie's paradox
  This paper identifies specific angles of emission and reception of light for which there exists a mass-energy counterpart to the well known transverse Doppler shift. At these angles, the relationship of proper and relative frequency is the same as that for proper and relative mass-energy of a source. Paradoxically, the transverse Doppler shift is often used to demonstrate that for specific angles of emission and reception the relationship of proper and relative frequency of light is the same as that for proper and relative time. But by carefully defining angles, this apparent paradox is resolved. 
Steven M Taylor