# Tensor algebra and the Projection Tensor

## Homework Statement

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

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

We define $$v^{i}=v^{i}_{para}+v^{i}_{perp}$$ 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...

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

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 $$v^{i}_{perp}$$.

Thank you very very much to any helpers.

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