On Fri, 16 Jul 2004 03:29:33 -0700, paul wrote:

a little clarification - you don't take the covariant derivative w.r.t. a
vector, but of a vector w.r.t. a neighbourhood.

Ah -- I think my sloppiness in terminology has bitten me here.

When I talked about the covariant derivative of a vector, Y, with respect
to a vector, V, I meant

D_V Y^a = (@Y^a/@x^b)*V^b + C^a_bc * V^b * Y^c

which I would have described as the covariant derivative of Y with
respect to the vector V, taken at some point P (which doesn't appear in
the expression).

Of course, you need to know the metric on the neighborhood to evaluate
that, but 'V' can be defined at just one point -- there's no implication
that it's a member of a vector field.


you do take the Lie
derivative w.r.t. a one parameter group of diffeo's and this is related
to the vector field.

Right -- the vector field determines the "line" upon which you evaluate
it, as well as the linear backmap you use to pull the two tangent vectors
into the same space so you can take their difference.

all this can be found in wald's book, or nakahara as
mentioned before.

Duh -- I had totally forgotten that Wald covers this; his book was just
sitting on a shelf here! Thanks for the reminder.

As a rule, I've found it's easier to learn math from physics books, if any
happen to cover it. As Bilge has pointed out, physics books invariably
provide some sort of physical motivation for the math; OTOH math books
frequently don't.


a counter to the statemtent:

"Intuitively, it appears that the covariant derivative with respect to a
particular vector V is just the Lie derivative taken along a geodesic to
which V is tangent."

is simple one! consider the covariant derivative of the metric and the
Lie derivative of the metric and you get the point.

I dunno -- I'll have to think about that... Of course the covariant
derivative of the metric is zero.

But, consider the Lie derivative of the metric in a locally flat
coordinate system, taken along a path through the origin which is
parallel to a basis vector. Isn't that derivative necessarily zero, also,
as a result of the local flatness? The second derivatives shouldn't enter
into it, and the firsts are all zero.

That would seem to imply that whenever we take the Lie derivative of the
metric WRT a one-parameter family of diffeo's defined such that the path
we're evaluating it along is a geodesic, the result should be zero.

No...?


or, have a a look at equation 10.9.10 in weinberg's book and you'll see
that the Lie only reduces to the covariant in the special case where
certain terms vanish.

Well this is a first! I can't recall anyone ever posting a reference to
Weinberg in this newsgroup before.

His approach is a bit different -- I'll have to stare at this for a while
to make sense of it.


hope that helps.

Thanks. The picture is gradually getting clearer.


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