Question

This problem is " Spacetime and Geometry, An Introduction to General Relativity by Sean Carroll"
Chapter 3 - Problem 14

14. Consider the three Killing vectors of the the two-sphere. (3.188). Show that their com mutators satisfy the following algebra: IR, S T IS, TI R IT, RI S. (3.225)

Please explain every thing.
Please write in the paper and then take a photo.

Answer 1

[R, S] = T [S, T] = R [T, R] = S Assume R = k_1: S = k_2 and T = k_3 So, [k_1, k_2] = k_3 [k_2, k_3] = k_1 [k_3, k_1] = k_2 For the first commutor we have, [k_1, k_2] = [cos phi partial differential_v - cot phi partial differential_y, - sin phi partial differential_v - cot v cos phi partial differential_y] = cos phi partial differential_v [-sin phi partial differential_v - cot v cos phi v] = cot v sin phi partial differential_phi [-sin phi partial differential_v - cot v cos phi partial differential_phi] - cot v sin phi partial differential_phi [-sin phi partial differential_v - cot v cos phi partial differential_phi] + sin phi partial differential_v [cos phi partial differential_v - cot v sin phi partial differential_phi] + cot v cos phi partial differential_phi [cos phi partial differential_v - cot v sin phi partial differential_phi] = -cos phi sin phi partial differential_phi^2 - cos^2 phi (partial differential_v cot v] partial differential_phi - cot v cos^2 phi partial differential_v partial differential_phi