Killing's Equations in Flat 3 - space

[WannabeNewton]

Homework Statement

Could anyone show me how I would go about finding all the killing vector fields for:
ds^{2} = dx^{2} + dy^{2} + dz^{2}

Homework Equations

Requirement for a killing field:
\nabla_{b}X_{a} + \nabla_{a}X_{b} = 0
(Sorry the indexes should be lower indexes but latex is messing with me =p)

The Attempt at a Solution

I honestly have no idea how to solve this. The only thing I could figure out with my lacking brain is that since it is a flat space metric the Christoffel symbols vanish and the covariant derivatives simply reduce to ordinary ones. Any help would be appreciated =D.

 

[muppet]

My advice would be to first to consider the case a=b, and then think about the second derivatives

 

[dextercioby]

In my blog there's a link to a thread on PF which contains (in disguise, I would say) the solution to your problem. You should always use the search function on this forum, cause it's pretty likely that such a problem has been discussed over the past 8 years on PF.

 

[WannabeNewton]

Well I took a look at the thread and managed to reduce the situation down to:
X^{a} = \omega^{a}_{b}x^{b} + t^{a}

(again the 'b' on omega should be a lower index...latex hates me)

where \omega and t are just constants of integration

...but I still have no clue on how to find all the possible killing fields T_T

 

[dextercioby]

The 2 Killing vector fields are encoded there. One is t_a (generator of translations) and the other can be gotten from \omega_{ab} (click the image to see the code), as this is a 2-nd rank a-symm tensor.

 

[WannabeNewton]

The 2 Killing vector fields are encoded there. One is t_a (generator of translations) and the other can be gotten from \omega_{ab} (click the image to see the code), as this is a 2-nd rank a-symm tensor.

I understand the translation part of what you stated but the thing that is confusing me is that in my book there are six killing vectors for this case and they are represented in terms of the basis vectors (well in this book it represents them as \partial / \partialx^{a}) and I have no idea how to come to that. Sorry for all the questions =p.

 

[Dick]

I understand the translation part of what you stated but the thing that is confusing me is that in my book there are six killing vectors for this case and they are represented in terms of the basis vectors (well in this book it represents them as \partial / \partialx^{a}) and I have no idea how to come to that. Sorry for all the questions =p.

There aren't six Killing vectors. There are an infinite number of Killing vectors. But you can pick a basis of six of them that are linearly independent. From which you can generate all of them by linear combination. They are the three parameters in omega and t. And they are vector fields, functions of the coordinates x.

 

[WannabeNewton]

This is kinda off topic but could anyone explain why, for a 2 - sphere, when solving killing's equations for a = b = \theta which reduces to:
\partialX_{\theta} / \partial\theta = 0

why, once integrated, X_{\theta} is equal to some function of \phi?

(all the theta indexes should be lowered sorry I can never get this damn latex thing to work properly ><)

 

[WannabeNewton]

This is kinda off topic but could anyone explain why, for a 2 - sphere, when solving killing's equations for a = b = \theta which reduces to:
\partialX_{\theta} / \partial\theta = 0

why, once integrated, X_{\theta} is equal to some function of \phi?

(all the theta indexes should be lowered sorry I can never get this damn latex thing to work properly ><)

I don't know if I am right but is it simply because the right hand side was zero so if it is integrated with respect to theta it would have had to be a function of the other variable psi?

 

[muppet]

Basically, yes. It could still be a constant, of course, but you have to make allowances for the possibility that it depends on phi.