What is a Killing vector field?

Question

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

Answer 1

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 $L_X g=0$ where $L$ is the Lie derivative or more explicit $ \nabla_\mu X_\nu + \nabla_\nu X_\mu =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 $\partial_t$ and $\partial_\phi$ since $g$ is independent of $t$ and $\phi$.

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

Answer 2

Another definition is;

If $V$ is a vector field whose flow $\phi$ is a one parameter group of isometries, then $V$ is called a Killing vector field (or just a Killing vector).

$V$ is a killing vector if and only if $L_vg=0$ ; where Lie derivative.

Here I am giving you a good paper for reference: http://www.physics.ohio-state.edu/~mathur/grnotes2.pdf

Answer 3

If any set of points is displaced by $x^i dx_i$ where all distance relationships are unchanged (i.e., there is an isometry), then the vector field is called a Killing vector.

For more,click the link below

http://mathworld.wolfram.com/KillingVectors.html