Question

In the light clock used to derive the Minkowski line element, events A and C are separated by the displacement vector (cAt, Ax)(2L, 0) in the rest frame of the mirrors. In the moving frame, the displacement vector is (cAt, Ar) with Solve the two equations in 1) for cAt and Ar in terms of L, v and c. Confirm that (СДt,. ДХ') is obtained from (câ.Ar) by a Lorentz transformation with velocity v. ct

Answer 1

"Given that: Delta X prime = -V Delta - t prime C Delta t prime = 2 squareroot t^2 + (Delta X prime/2)^2 = 2 squareroot L^2 + (-v Delta t prime/2)^2 (C Delta t prime/2)^2 = L^2 + (v Delta t prime/2)^2 Delta t prime^2 (c^2 - v^2) = 4L^2 C Delta t prime = 2 L/ squareroot 1 - (v/c)^2 Delta x prime = -v Delta t prime = 2 L (v/c)/ squareroot t (v/c)^2 C Delta t prime = 2 L/ squareroot 1 - (v/c)^2 = C Delta t - V x