How to calculate the outer product in GR?

[Haorong Wu]

Homework Statement

Show that the scalar product of two vectors is not altered as they are both Fermi-Walker transported along a curve $C$

Relevant Equations

If a curve is timelike, $u$ is its tangent vector, and $a=\nabla_u u=Du/d \tau$, then a vector $V$ is said to be Fermi-Walker transported along $u$ if $$ \nabla_u V=(u \otimes a- a \otimes u)\cdot V .$$

I will post the answer here, part of which I do not follow.

Let the two vectors be $x$ and $y$. Then the Fermi-walker transport law reads$$\nabla_u x=(u \otimes a -a \otimes u) \cdot x$$ $$\nabla_u y=(u \otimes a- a \otimes u)\cdot y ,$$ where $u$ is the tangent vector to the curve $c$ and $a=\nabla_u u$. Using the product rule we evaluate the change in the dot product along the curve $$\begin{align} \nabla_u ( x \cdot y)&= (\nabla_u x)\cdot y+ x \cdot(\nabla_u y) \nonumber \\& =(a \cdot x) (u \cdot y) -(a \cdot y) (u \cdot x) +(u \cdot x) (a \cdot y) -(a \cdot x) (u \cdot y) =0 . \nonumber \end{align}$$ The scalar product is unaltered.

I do not follow the outer-product part. I know that I should multiply two terms together if they are in the same space. However, in this problem, I do not know how to determin which term belongs to which space. It seems, sometimes $x$ and $u$ are in the same space, and sometimes $x$ and $a$ are in the same space. I am stuck.

 

[etotheipi]

By definition, let $S:= u \otimes a - a \otimes u$, then $S(\alpha, \beta) = (u \cdot \alpha)(a \cdot \beta) - (a \cdot \alpha)(u \cdot \beta)$