- #1
FreeThinking
- 32
- 1
Homework Statement
(Self study.)
Several sources give the following for the Riemann Curvature Tensor:
The above is from Wikipedia.
My question is what is [itex] \nabla_{[u,v]} [/itex] ?
Homework Equations
[A,B] as general purpose commutator: AB-BA (where A & B are, possibly, non-commutative operators),
[A,B] as Lie bracket, which is a directional derivative: [itex] \nabla_A B - \nabla_B A [/itex],
The directional derivative: [itex] \nabla_A B = \frac {\partial B} {\partial a} [/itex] where a is the parameter for the curve A, and
Additivity: [itex] \nabla_{gA + hB} C = g \nabla_A C + h \nabla_B C[/itex] (MTW p 252, equation (10.2d)). This shows how MTW "moves" the coefficients of the subscripts "up" to coefficients of the differentials as well as the addition operation. This is just a notational issue.
3. The Attempt at a Solution
According to the Wikipedia page, the [u,v] as subscripts on the last term of Riemann is a Lie derivative. That would make it: [itex] \nabla_{[u,v]} w = \nabla_{(\nabla_u v - \nabla_v u)} w = \nabla_{\nabla_u v} w - \nabla_{\nabla_v u} w [/itex].
But, what does it mean to take a directional derivative with respect to a directional derivative? I get what it means to take a directional derivative, along one direction, of a directional derivative along a, possibly different, direction, but is a directional derivative always a direction? I would think not. In general, a directional derivative may not, itself, be a direction, so what does the [itex] {\nabla_u v} [/itex] in the bottom mean?
In an attempt to make sense of this, let's expand the nablas. We get [itex] \nabla_{(\frac {\partial v} {\partial u})} w - \nabla_{(\frac {\partial u} {\partial v})} w [/itex] = [itex] \frac {\partial w} {\partial (\frac {\partial v} {\partial u})} - \frac {\partial w} {\partial (\frac {\partial u} {\partial v})} [/itex].
I realize we're headed for a second derivative, and I would think it would be [itex] \frac {\partial^2 w} {{\partial v} {\partial u}} - \frac {\partial^2 w} {{\partial u} {\partial v}} [/itex] (noting the order of the bottom factors since the "u" is the "lowest" on the first nabla & the right-most partial on the bottom is the first applied), but I've been unable to find anything in my books or on the inet that would justify such a step.
So am I completely misunderstanding this? Is this really the correct form of the equation? Does anyone have any references that explain this notation to really, really stupid people?
Thanks.