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Projection operator

  1. Oct 21, 2009 #1
    Another question I have from Schutz (CH3, 31 (c)), where he defines the Projection tensor as

    [tex] P_{\vec{q}}=g+\frac{\vec{q} \otimes \vec{q}}{\vec{q} \cdot \vec{q}} [/tex]

    This can be written in component form (or rather the associated (1 1) tensor can after operating a few times on it with the metric) as:


    [tex] P^{\alpha}{}_{\beta}=\eta^{\alpha}{}_{\beta}+\frac{q^{\alpha}q_{\beta}}{q^{\gamma}q_{\gamma}} [/tex]

    This obviously takes a vector V and produces another vector , i.e.


    [tex] V^{\alpha}{}_{\perp}=P^{\alpha}{}_{\beta} V^{\beta}=(\eta^{\alpha}{}_{\beta}+\frac{q^{\alpha}q_{\beta}}{q^{\gamma}q_{\gamma}})V^{\beta}=V^{\alpha}+\frac{q^{\alpha}q_{\beta}V^{\beta}}{q^{\gamma}q_{\gamma}}[/tex]

    The task is then to show that [tex] V^{\alpha}{}_{\perp} [/tex] is indeed orthoganal to [tex] \vec{q} [/tex], provided q is non-null.

    So I start of by taking the dot product of [tex] V^{\alpha}{}_{\perp} [/tex] and [tex] \vec{q} [/tex]:

    [tex] \vec{q} \cdot \vec{V}_{\perp}=\eta_{\alpha \beta} q^{\alpha}V^{\beta}{}_{\perp}= \eta_{\alpha \beta}q^{\alpha}(V^{\beta}+\frac{q^{\beta}q_{\sigma}V^{\sigma}}{q^{\gamma}q_{\gamma}})=q_{\beta}V^{\beta}+\frac{\eta_{\alpha \beta}q^{\alpha}q^{\beta}q_{\sigma}V^{\sigma}}{q^{\gamma}q_{\gamma}}=q_{\beta}V^{\beta}+\frac{\vec{q} \cdot \vec{q} (q_{\sigma}V^{\sigma})}{\vec{q} \cdot \vec{q} }=q_{\beta}V^{\beta}+q_{\sigma}V^{\sigma}=2q_{\sigma}V^{\sigma}[/tex]

    Which is not generally equal to zero. In Schutz's previous example he used the four velocity as [tex] \vec{q} [/tex] which obviously has magnitude of -1. Also he used the projection operator as

    [tex] P_{\vec{q}}=g+\vec{U} \otimes \vec{U} [/tex]

    Then everything works out fine, and you can easily show this does produce vectors orthoginal to [tex] \vec{U} [/tex] since following a similar derivation to above you end up with [tex]=U_{\beta}V^{\beta}+(\vec{U} \cdot \vec{U}) U_{\sigma}V^{\sigma}=U_{\beta}V^{\beta}- U_{\sigma}V^{\sigma}=0 [/tex]

    So I don't believe this thing above really is the projection operator for arbitrary [tex] \vec{q} [/tex], although if we instead defined
    [tex] P_{\vec{q}}=g-\frac{\vec{q} \otimes \vec{q}}{\vec{q} \cdot \vec{q}} [/tex]
    Then this would work I think?
     
    Last edited by a moderator: Oct 21, 2009
  2. jcsd
  3. Oct 21, 2009 #2

    cristo

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    I think you're right: the projection tensor should be [tex]P_{\alpha\beta}=\eta_{\alpha\beta}-\frac{q_\alpha q_\beta}{q^\gamma q_{\gamma}}[/tex]
     
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