There's a question in Schnutz - A first course in special relativity(adsbygoogle = window.adsbygoogle || []).push({});

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

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# I Schnutz Special Relativity Tensors Question

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