Re: Connection not a tensor..
- From: "Igor Khavkine" <igor.kh@xxxxxxxxx>
- Date: Fri, 16 Dec 2005 20:56:19 +0000 (UTC)
mike.james wrote:
> I've been thinking about what it means that the connection coefficients
> don't transform like a tensor. Given that we start off with the idea
> that thing that have a co-ordinate independent reality are tensors i.e.
> transform like tensors does the fact that the connection isn't a tensor
> mean that it isn't a "physical thing" or does it mean that other
> transformation laws define physical things.
>
> Given that a connection does define things that are independent of
> co-ordinates such as curvature, parallel transport etc. presumably the
> connection is as real as a tensor and hence the transformation law that
> it obeys is "as good as" the tensor transformation law.
>
> Or is there a bigger picture in which the connection is part of a tensor?
Actually, contrary to popular belief, the connection coefficients (aka
Christoffel symbols) can be seen as tensors. This is explained in
Chapter 3 or Wald's book on General Relavitity. Unfortunately, you have
to read between the lines to get the full picture.
Here's the full story. There are many possible covariant derivaties
that can be defined on a given manifold. Let @ and & two of them. Wald
proves that the difference between can be expressed in terms of a
tensor:
@_a w_b - &_a w_b = -C^c_{ab} w_c.
That is, the difference between the two covariant derivatives of a
1-form w_b is proportional to w_c, with the tensor -C^c_{ab} serving as
the proportionality factor. This tensor can also be computed by looking
at the difference of the second covariant derivatives of a scalar
function:
C^c_{ab} f = &_a &_b f - @_a @_b f,
which implies that it is symmetric in the lower indices. This symmetry,
BTW, implies that the covariant derivatives in question are torsion
free. Once we know this C tensor, we can compute the action of @ on any
tensor if we know the action of & on them:
@ T = & T + C[contracted with each upper index]T - C[contracted with
each lower index]T
It is well known that given a (pseudo-)riemanninan metric g_{ab}, there
is a unique torsion free covariant derivative compatible with it. Let
it be denoted by @ and let & be *any other* covariant derivative. In
this case, what is this tensor C that relates @ to &? Wald proves that
it can be computed using the following formula:
C^c_{ab} = (1/2) g^{cd}[&_a g_{bd} + &_b g_{ad} - &_d g_{ab}].
If you're sharp, you've noticed by now that this C tensor looks an
awful lot like a Christoffel symbol. This confirmed by the above
formula, which is very reminiscent of the formula for a Christoffel
symbol Gamma^c_{ab}. And that is pretty much what it is. However, it
remains to make the connection precise.
The crucial thing you have to note is that a given coordinate system
uniquely defines a covariant derivative. It's not a very exciting one,
it just takes the derivative of each component of a tensor with respect
to this coordinate system. If you wish, you can think of it as the
unique covariant derivative associated with the flat metric that is
diagonal (with appropriate +-1's on the diagonal) in the given
coordinate system. If you let & denote the covariant derivative
associated to *this particular* coordinate system, the the tensor
C^c_{ab} is none other than the Christoffel symbol, and allows you to
express the action of the metric compatible covariant derivative @ in
*this particular* coordinate system. This point is a little subtle, and
it is indicated in Wald simply by a change from his "abstract index
notation" to the "coordinate index notation" in between equations
(3.1.29) and (3.1.30).
In other words, given a metric, for each coordinate system, there is
covariant derivative associated to it and a Christoffel symbol, which
is a tensor, relating it to the metric compatible covariant derivative.
Thus, for each different coordinate system, there is a different
Christoffel symbol tensor. When the transformation properties of the
components of a Christoffel symbol are given in text books, they
include second derivatives of the coordinates. This feature is cited as
evidence of their non-tensorial characteristics. However, what we
actually see are components of two different Christoffel tensors in two
different coordinate systems, hence the confusion.
Hope this helps.
Igor
.
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