where x = 1 / α α is constant and α α τ τ is variable, with the worldline resembling the hyperbola X 2 − − T 2 = x 2 . Sommerfeld showed that the equations can be reinterpreted by defining x as variable and α α τ τ as constant, so that it represents the simultaneous "rest shape" of a body in hyperbolic motion measured by a comoving observer. By using the proper time of the observer as the time of the entire hyperbolically accelerated frame by setting τ τ = t , the transformation formulas between the inertial coordinates and the hyperbolic coordinates are consequently: