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Boosting the frame basis

  1. May 27, 2010 #1

    Mentz114

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    Gold Member

    I'm trying to get a metric in the frame of a boosted observer. The spacetime in question has coframe and frame basis vectors

    [tex]
    \begin{align*}
    \vec{\sigma}^0 = \frac{-1}{\sqrt{F}}dt\ \ \ \ & \vec{e}_0 = -\sqrt{F}\partial_t \\
    \vec{\sigma}^1 = \sqrt{F}dz\ \ \ \ & \vec{e}_1 = \frac{1}{\sqrt{F}}\partial_z \\
    \vec{\sigma}^2 = \sqrt{F}dr\ \ \ \ & \vec{e}_2 = \frac{1}{\sqrt{F}}\partial_r \\
    \vec{\sigma}^3 = r\sqrt{F}d\phi\ \ \ \ & \vec{e}_3 = \frac{1}{r\sqrt{F}}\partial_\phi
    \end{align*}
    [/tex]

    Boosting the coordinate frame basis by [itex]\beta[/itex] in the [itex]\phi[/itex] direction gives the new frame basis

    [tex]
    \begin{align*}
    \vec{f}_0 &= -\gamma\sqrt{F}\partial_t + \gamma\beta \frac{1}{r\sqrt{F}}\partial_\phi \\
    \vec{f}_1 &= \frac{1}{\sqrt{F}}\partial_z \\
    \vec{f}_2 &= \frac{1}{\sqrt{F}}\partial_r \\
    \vec{f}_3 &= \gamma\frac{1}{r\sqrt{F}}\partial_\phi + \gamma\beta \sqrt{F}\partial_t
    \end{align*}
    [/tex]

    Now, my problem is reading off the new coframe basis [itex]s[/itex]. My attempt is below, but I'm only 50% confident it's right.

    [tex]
    \begin{align*}
    {\vec{s}}^0 &= (\gamma\sqrt{F})^{-1}dt+(\gamma\beta)^{-1}r\sqrt{F}d\phi \\
    {\vec{s}}^1 &= \sqrt{F}dz \\
    {\vec{s}}^2 &= \sqrt{F}dr \\
    {\vec{s}}^3 &= \gamma^{-1}r\sqrt{F}d\phi + (\gamma\beta)^{-1}\sqrt{F}dt
    \end{align*}
    [/tex]

    The metric that arises from this is sort of plausible. I'd appreciate any pointers, particularly to any errors.
     
  2. jcsd
  3. Jun 1, 2010 #2

    Mentz114

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    Gold Member

    From Lee's book "Riemanian Manifolds : An Introduction to Curvature" ( page 30)

    Couldn't be simpler really.
     
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