How to Find Killing Vectors for a Given Metric

[Airsteve0]

In my general relativity course we recently covered the definition of a killing vector and their importance. However, I am not completely comfortable calculating the killing vectors for a given metric (in a particular case, the 2-sphere), and would like to know if anyone knows of a good reference which may provide some examples of how they are calculated. Thanks in advance!

 

[WannabeNewton]

While I absolutely hate to reference this particular text, "General Relativity Demystified" - McMahon does a full calculation of the killing vectors for the 2 - sphere. In general you can take the condition for a vector field to be a killing field (the lie derivative of the metric tensor with respect to the vector field vanishing), express it in local coordinates, \triangledown _{\alpha }\xi _{\beta } + \triangledown _{\beta }\xi _{\alpha } = 0 and solve the pde's but as you can see from the calculation done in the aforementioned text, even for the metric tensor on the 2 - sphere this is quite a laborious task.