(Solved): The line element in Minkowski space-time in polar coordinates is given by ds = dt + dr + ...

The line element in Minkowski space-time in polar coordinates is given by ds² = ?dt² + dr² + r²d², where d?² = d0² + sin²0 dp² denotes the line element in the unit 2-sphere. Use the null coordinates defined by u = t - r and v = t + r, then the compactified coordinates p = tan-¹u and q tan¯¹v, and finally, T = q + p and R = q - p. Show that the line element can be written in the form = ds² ?-2ds2, = ds² " where = 2 cos p cos q is known as a conforming factor and = ?dT² +dR² + sin² Rd².