Relativistic Transformation of Spherical Co ordinates(t,r,{\theta},{\phi})

TL;DR

This paper develops a relativistic transformation for spherical coordinates to analyze rotational motion in special relativity, extending previous Cartesian and cylindrical transformations to better understand rotating frames.

Contribution

It introduces a novel relativistic transformation for spherical coordinates, enabling analysis of rotating frames along any axis within the framework of special relativity.

Findings

Relativistic effects in spherical coordinates are characterized.

Transformation applicable to rotational motion in relativistic regimes.

Enhanced understanding of rotating frames in special relativity.

Abstract

With the advent of relativistic mechanics, the Lorentz transformation replaced the Galilean transformation based on classical Newtonian mechanics among inertial frames at high uniform velocities, but both transformations are based on Cartesian coordinate system, hence position of particles obtaining linear velocities in space can be obtained. In case where frames are rotating with constant angular velocity, use of Galilean rotational transformation (GRT) is replaced by Franklin transformation, proposed by Philip Franklin in 1922. The modified transformation introduced the concept of rotational motion of points in a rigid body. Both the transformations are based on cylindrical coordinate system. Here we moved a step further for making a relativistic transformation using spherical coordinate system for understanding the behaviour of rotating frames along any axis in the space passing through the center of mass of a symmetrical object (Sphere). We finally came to an understanding about how Special Theory of relativity is found to be applicable in rotational motion using different co-ordinate system.