The first three terms, excepting the factor of γ γ ( v ) , is the velocity as seen by the observer in their own reference frame. The γ γ ( v ) is determined by the velocity v between the observer's reference frame and the object's frame, which is the frame in which its proper time is measured. This quantity is invariant under Lorentz transformation, so to check to see what an observer in a different reference frame sees, one simply multiplies the velocity four-vector by the Lorentz transformation matrix between the two reference frames.