Monday, January 28, 2019

differential geometry - What is a Killing vector field?


I recently read a post in physics.stackexchange that used the term "Killing vector". What is a Killing vector/Killing vector field?



Answer



I think https://en.wikipedia.org/wiki/Killing_vector_field answers your question pretty good:


"Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point on an object the same distance in the direction of the Killing vector field will not distort distances on the object."


A killing vectorfield X fulfills LXg=0 where L is the Lie derivative or more explicit μXν+νXμ=0.


So in a layback manner: When you move the metric g a little bit by X and g doesn't change, X is a killing vectorfield.


For example the Schwarzschildmetric https://en.wikipedia.org/wiki/Schwarzschild_metric has two obvious Killing vectorfields t and ϕ since g is independent of t and ϕ.


Edit: On recommndation I add a nice link to a discussion of how to use Killing vector fields: See the answer of Willie Wong at Killing vector fields



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