I'm trying to get a metric in the frame of a boosted observer. The spacetime in question has coframe and frame basis vectors(adsbygoogle = window.adsbygoogle || []).push({});

[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.

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# Boosting the frame basis

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