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Timelike and spacelike vectors

  1. Sep 16, 2015 #1
    1. The problem statement, all variables and given/known data
    Show that if [itex]x^{\mu}[/itex] is timelike and [itex]x^{\mu}y_{\mu}=0[/itex], [itex]y^{\mu}\neq 0[/itex], then [itex]y^\mu[/itex] is spacelike.

    2. Relevant equations
    [itex]ds^2=\\>0\hspace{0.5cm}\text{timelike}\\<0\hspace{0.5cm}\text{spacelike}\\0\hspace{0.5cm}\text{lightlike}[/itex]
    metric is [itex]diag (+---)[/itex]
    3. The attempt at a solution

    Don't know if this is the correct way, but here goes: assuming that [itex]x^\mu[/itex] is timelike, we can pick a reference frame in which [itex]x^\mu=(x^0,\vec{0})[/itex], so due to invariance [itex]x^{\mu}y_{\mu}=x^0 y_0=0\rightarrow y^0=0[/itex], but, since it is stated that [itex]y^{\mu}\neq 0[/itex], then [itex]\vec{y}\neq 0[/itex], and hence we have [itex]y^\mu y_\mu=y^i y_i=-(\vec y)^2<0[/itex], making [itex]y^\mu[/itex] indeed spacelike. Am I missing something here or is this the right procedure?
     
  2. jcsd
  3. Sep 16, 2015 #2

    Orodruin

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    Your procedure is fine.
     
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