- #1

- 19

- 1

Consider a Velocity Four Vector U , and the tensor P whose components are given by

Pμν = ημν + UμUν .

(a) Show that P is a projection operator that projects an arbitrary vector V into one orthogonal to U . That is, show that the vector V⊥ whose components are

Vα ⊥ = P

^{α}

_{β}V

^{β}= (η

^{α}

_{β}+ U

^{α}U

_{β})V

^{β}is

(i) orthogonal to U

Now I've attempted the solution and it is the following

PβαVα = V

^{β}+U

^{β}U

_{α}V

^{α}

So now if I calculate

Vα ⊥ ⋅ U = V

_{α}U

^{α}+U

_{α}U

^{α}U

_{α}V

^{α}

which is orthogonal if c=1 ... as |U|^2= -c^2

but.. this is just in the metric -+++ , if I change metrics to +--- then it won't be orthogonal? Also it's not orthogonal if c=/=1 .. which doesn't seem right to me either

how can that be?

Thank for you help!

Adam