# Evaluating Tensor in Special Relativity

## Homework Statement

Using the dyadic form of the velocity dependent mass, find the acceleration vector a of an electron whose initial velocity is v =0.9c i at the instant a force is applied given by:

F = F0 (i + j) / $$\sqrt{}\pi$$

F0 = 106 N

2. The attempt at a solution

I feel like I've got this to a point, and then I'm not sure how to evaluate the dyadic. So here it is:

After a good bit of math I arrive at:

dv/dt = F c2 /E - v / E (F * v)

which in dyadic form is dv/dt = (c2 / E) F (I - vv/c2 )

That form is the acceleration and is where I draw a blank. The first step would seem to be to plug in the given values for velocity and F , but it still leaves the identity dyad which is what I'm not totally sure how to handle. So; am I going at this correctly, and how do I evaluate the dyad?

Related Advanced Physics Homework Help News on Phys.org
Dyads are ugly old names for tensors of second rank, using index calculus makes everything much simpler. But well. This "identity dyad" is just a unit matrix, and the dyad "$$\mathbf{v} \mathbf{v}$$" is better written $$\vec{v} \otimes \vec{v}$$ -- it is the tensorial or outer product, which in 3D reads
$$(v1, v2, v3) \otimes (v1,v2,v3) = \begin{pmatrix} v1 v1 & v1 v2 & v1 v3 \\ v2 v1 & v2 v2 & v2 v3 \\ v3 v1 & v3 v2 & v3 v3 \end{pmatrix}$$. The one important rule for that product is $$\vec{F} ( \vec{v} \otimes \vec{v} ) = \vec{v} ( \vec{F} \cdot \vec{v} )$$, where the central dot is the usual scalar product of two vectors. I think you can go on now yourself :)

I hope you can excuse my ignorance on tensors... one of these days it will click, I hope.

Do I need to do anything with the subtraction of the identity dyad and the velocity dyad, or can I just jump right to the dot product as usual and get the very straightforward result?

Of course you have the linearity rule, so F(1-vv) = F1 - Fvv. Straightforward if that's what you meant.

As straightforward as tensors get I suppose :)