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
Velocity-addition formula appear there must be inverted here: This matrix rotates the a pure time-axis vector to , and all its columns are relativistically orthogonal to one another, so it defines a Lorentz transformation. If a fly is moving with four-velocity in the ship frame, and it is boosted by multiplying by the matrix above, the new four-velocity in the shore frame is , Dividing by the time component and substituting for the components of the four-vectors and in terms of the components of the three-vectors and gives the relativistic composition law as The form of the relativistic composition law can be