# 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?

• +1, I also saw it when I was looking up the surface gravity of a black hole. Good question. Commented Oct 5, 2012 at 12:50
• please could you summarise the initial research effort you made to find out what a killing vector is? Commented Oct 5, 2012 at 12:55
• en.wikipedia.org/wiki/Killing_vector_field Commented Oct 5, 2012 at 13:00
• Note Killing is a name associated with the concept, as a quick look at Wikipedia will inform you. Commented Oct 5, 2012 at 13:17
• 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. Commented Oct 5, 2012 at 17:22

I think Wikipedia 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 vector field $$X$$ fulfills $$L_X g=0$$ where $$L$$ is the Lie derivative or more explicitly $$\nabla_\mu X_\nu + \nabla_\nu X_\mu =0$$

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 vector field.

For example the Schwarzschild metric has two obvious Killing vector fields $$\partial_t$$ and $$\partial_\phi$$ since $$g$$ is independent of $$t$$ and $$\phi$$.

Edit: On recommendation 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.

• 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). Commented Feb 12, 2013 at 18:36
• It feels incorrect to say 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." Consider moving an object along $\partial_\phi$ in the Schwarzschild metric. Parts of the object that are close to the axis of rotation will be rotated a greater angle around the axis than parts of the object farther away; hence the object is going to be sheared. In the extreme case, the parts closest to the axis of rotation will rotate many times around it while parts farther away won't rotate even once around it. Commented May 10, 2021 at 20:27

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

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