Obtaining the Metric in a Boosted Observer Frame?

[Mentz114]

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

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

Boosting the coordinate frame basis by $\beta$ in the $\phi$ direction gives the new frame basis

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

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

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

The metric that arises from this is sort of plausible. I'd appreciate any pointers, particularly to any errors.

 

[Mentz114]

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

Let (E1, . . . , En) be any local frame for TM, that is, n smooth vector fields defined on some open set U such that (E1|p, . . . , En|p) form a basis for TpM at each point p ∈ U. Associated with such a frame is the dual coframe, which we denote (ϕ1, . . . , ϕn); these are smooth 1-forms satisfying ϕi(Ej) = δij.

Couldn't be simpler really.