Tensor algebra and the Projection Tensor

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Homework Statement


(firstly, Apologies for having to use a picture..)

If [tex]u^{i}[/tex] is the 4-velocity of a point on a manifold, then we use affine parameterisation [tex]g_{ij}u^{i}u^{j}=1[/tex].
The attached picture shows our rest frame, ie [tex]x^{0}=const[/tex] and a point ("us") on this surface. If our velocity is [tex]u^{i}[/tex] we want to describe the projection of another vector [tex]v^{i}[/tex] on to our velocity, [tex]u^{i}[/tex].

We define [tex]v^{i}=v^{i}_{para}+v^{i}_{perp}[/tex] the parallel and perpendicular components.


Homework Equations


(Mainly just algebra of tensors which I don't understand)

The Attempt at a Solution


This is a situation from my notes, so I have the answer but i don't understand the working...

[tex]v^{i}=u^{i}u_{j}v^{j}[/tex] (I already don't understand this line - why is this so? Isn't the contraction [tex]u_{j}v^{j}[/tex] on the RHS already a scalar product and hence a projection?)
gives [tex]v^{j}u_{j}=g_{jk}u^{k}v^{j}[/tex] (i think using [tex]g_{ij}u^{i}u^{j}=1[/tex]?)
Now we write [tex]v^{i}_{para}=h^{i}_{j}v^{j}[/tex] where we define the projection [tex]h^{i}_{j}=\delta^{i}_{j}-u^{i}u_{j}[/tex].

Could someone run me through this working please? There is scarecly a line where I can see where it came from... And perhaps then, if i understand it, I could find [tex]v^{i}_{perp}[/tex].

Thank you very very much to any helpers.
 

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