# 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?

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+1, I also saw it when I was looking up the surface gravity of a black hole. Good question. – ja72 Oct 5 '12 at 12:50

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please could you summarise the initial research effort you made to find out what a killing vector is? – EnergyNumbers Oct 5 '12 at 12:55

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– benshepherd Oct 5 '12 at 13:00

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Note Killing is a name associated with the concept, as a quick look at Wikipedia will inform you. – Emilio Pisanty Oct 5 '12 at 13:17

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Wikipedia is generally a trustworthy source of information for scientific concepts. If you want to know about the definition of something, we do expect you to check there before posting a question here. If there is an article that directly answers your question, as in this case, it's not really a good question for this site. – David Zaslavsky♦ Oct 5 '12 at 17:22
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## 3 Answers

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

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It should also be noted that one can use that fact to construct conserved quantities and sometimes make a system integrable. In the case of the Schwarzschild Metric, those killing fields relate to conservation of energy and angular momentum (respectively). – Benjamin Horowitz Feb 12 at 18:36

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

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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.
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