Transformation equation for frames undergoing non-uniform acceleration such as SHM and rotational motion

Lorentz transformation equations provide us a set of relations between the spacetime coordinates as observed from two different inertial frames. In case, one of the frames is moving with a uniform rectilinear acceleration we have Rindler’s transformation equations under such a situation. In the present work, we extend the Rindler’s equations to a situation where we have in general non-uniform acceleration. After that we consider the non-inertial frame to undergo simple harmonic motion (SHM) and as a second case we consider the non-inertial frame to move uniformly along a circle. This set of transformation equations will have applications in various branches of Physics and in general in Astrophysics.