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 imaginary mass have come to play an important role in modern physics and are discussed in popular books on physics. Under no circumstances do any excitations ever propagate faster than light in such theories – the presence or absence of a tachyonic mass has no effect whatsoever on the maximum velocity of signals (there is no violation of causality). While the "field" may have imaginary mass, any physical particles do not; the "imaginary mass" shows that the system becomes unstable, and sheds the instability by undergoing a type of phase transition called tachyon condensation (closely related to second order phase