Albert Einstein (1907) studied the effects within a uniformly accelerated frame, obtaining equations for coordinate dependent time dilation and speed of light equivalent to (2c), and in order to make the formulas independent of the observer's origin, he obtained time dilation (2i) in formal agreement with Radar coordinates. While introducing the concept of Born rigidity, Max Born (1909) noted that the formulas for hyperbolic motion can be used as transformations into a "hyperbolically accelerated reference system" (German : hyperbolisch beschleunigtes Bezugsystem) equivalent to (2d). Born's work was further elaborated by Arnold Sommerfeld (1910) and Max von Laue (1911) who both obtained (2d) using imaginary numbers, which was summarized by Wolfgang Pauli (1921) who besides coordinates (2d) also obtained metric (2e) using imaginary numbers. Einstein (1912) studied a static gravitational field and obtained the Kottler–Møller metric (2b) as well as approximations to formulas (2a) using a coordinate dependent speed of light. Hendrik Lorentz (1913) obtained coordinates similar to (2d, 2e, 2f) while studying Einstein's equivalence principle and the uniform gravitational field.