Effective mass (solid-state physics) from both sides gives where is the th element of , is the th element of , and are the th and th elements of , respectively, and is the total energy of the particle according to the Planck–Einstein relation. The index is contracted by the use of Einstein notation (there is an implicit summation over ). Since Newton's second law uses the inertial mass (not the gravitational mass), we can identify the inverse of this mass in the equation above as the tensor This tensor expresses the change in group velocity due to a change in crystal momentum. Its
Mass in special relativity derived form formula_44 is not valid because formula_45 in formula_46 is generally not a constant (see the section above on transverse and longitudinal mass). Even though Einstein initially used the expressions "longitudinal" and "transverse" mass in two papers (see previous section), in his first paper on formula_47 (1905) he treated "m" as what would now be called the "rest mass". Einstein never derived an equation for "relativistic mass", and in later years he expressed his dislike of the idea: Okun and followers reject the concept of relativistic mass. Also Arnold B. Arons has argued against teaching the concept of relativistic