There exist unresolved problems such as Abraham-Minkowski controversy (A-MC) and possibility of superluminal phenomena, which relate to applicability of Lorentz transformations (LTs) in materials. In this work, we shall show that, usually, expression(s) of optical Doppler shift (ODFS) may be obtained on the basis of LTs, which predicts that, in a zero-index metamaterial (ZIM), ODFS holds a nonzero second-order small quantity. Noting the report that, in a ZIM, there is no ODFS observed for a beam of microwave reflected by a moving surface, which indicates preliminarily that applicability of LTs in materials need to be further investigated. On the other hand, drawing lessons from general relativity and combining Fermat’s principle, it is proposed that LTs may be modified as that speed of light in free space should be replaced by phase velocity of light in a medium. Thus another expression of ODFS is obtained, which predicts that, in a ZIM, ODFS equals to zero strictly. This work may provide a new possible way to examine applicability of LTs in materials, and deepen the understanding of foundations of relativity.
PACS: 03.30. +p, 03.50.De