Say you have a scalar field defined on a manifold M. Then imagine evaluating the field at a point an infitesimal distance away from x at where X is a illing vector of the manifold. Then
to first order. Now the second term on the rhs is the Lie derivative of a scalar, so
Say now that is not a scalar, say maybe a vector of spinor or tensor. Does the same thing follow, that
Where now is not the appropriate Lie derivative of whatever object is.
Thanks very much for your quick reply.
The manifold is a spacetime, I forgot to mention.
Maybe I didn't explain very well. That expression is exactly the same as the expression I wrote, because that derivative should be a covariant derivative when operating on a vector and so the metric commutes through it and lowers the index on Is that right? That expression won't work for say a vector because we are comparing vectors in two different tangent spaces.
I meant would it still be the same expression, for say a vector
where now the expression for the Lie derivatie on a covariant vector is
if I got the indices right.
I mean roughly can the Lie derivative be thought of as a generalisation of the directional derivative in vector calculus to more general settings.
lol, I know. In the office last week we were discussing the CMB background and whether it violates relativity by defining a prefered reference frame. We decided it didn't, but later we were discussing the area of a paralelogram and the determinant of a two by two matrix, and also how to factorise cubics. We were totally lost.Well, it's been (regrettably) about 10 months since I even looked at my QFT books, and I came up with that expression on the fly, so I'm not particularly surprised that it wasn't quite right for your needs.
So on the same day we saved general relativity, but failed to remember some A-level algebra lol.
Thanks for your help. I'll just go with it.