# Gauge freedom in the tetrad

+ 3 like - 0 dislike
144 views

I'm reading the following paper about Petrov type D space times called "Type D vacuum metrics":

http://scitation.aip.org/content/aip/journal/jmp/10/7/10.1063/1.1664958

by Kinnersley. I have a question about his choice of gauge.

In particular, starting with the tetrad $\lbrace l^{\mu},n^{\nu},m^{\mu},\overline{m}^{\mu}\rbrace$ where $l^{\mu},n^{\mu}$ are null, he then makes an argument that we can pick a scaling $A$ such that $l^{\mu}\rightarrow Al^{\mu}$ and $n^{\mu}\rightarrow (1/A)n^{\mu}$, and make $\nabla_{l}l=0$ (i.e. $l^{\mu}$ a geodesic vector field). He then picks a coordinate system $(x^{1},r,x^{2},x^{3})$ such that $l^{\mu}=(0,1,0,0)$. That's what he explains at the beginning of his section 2, and it's clear to me.

What confuses me is what he writes at the beginning of section 3C: he says that there is still freedom left if we choose the scaling, call it $A^{0}$, to be independent of $r$. Now, I understand that such choice will not violate the condition $\nabla_{l}l=0$. However, my question is: why doesn't it violate the already chosen $l^{\mu}=(0,1,0,0)$?

What am I missing? Why is one allowed to scale once more?

Thanks for any help.

This post imported from StackExchange MathOverflow at 2015-07-05 20:45 (UTC), posted by SE-user GregVoit
 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.