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Recently in a paper $[1]$ the researchers presented a rotating traversable wormhole solution using the famous Newman-Janis Algorithm $[2]$. But something is anoying me. In $[1]$ they presented the Morris-Throne metric in the following way (section 4):

(...) we rewrite the spherical symmetry wormhole spacetime metric as follows: $$ ds^{2} = -f(r)dt^{2} + g(r)^{-1}dr^{2} + r^{2}d\theta^{2}+r^{2}sin^{2}(\theta)d\phi^{2} \tag{1}$$

where the metric coefficients are $f(r)= e^{2\Phi(r)}$ and $g(r) = 1-\frac{b(r)}{r}$

And later in the paper they derived the Newman-Janis Algorithm using the metric $(1)$, of course. But, metric $(1)$ is in some way more general than:

$$ ds^{2} = -e^{2\Phi(r)}dt^{2} + \Bigg[1-\frac{b(r)}{r}\Bigg]^{-1}dr^{2} + r^{2}d\theta^{2}+r^{2}sin^{2}(\theta)d\phi^{2} \tag{2}$$

Of course that in the paper the authors used the notation given in $(1)$ just to avoid messy expressions but I think that if you start just with metric $(1)$ with $f(r)$ and $g(r)$ arbitrary, the Newman-Penrose Algorithm will give the same results.

My question is:

If I leave the coeficients $f(r)$ and $g(r)$ arbitrary, the Null Tetrad will still be the same as calculated by $[1]$?

The null tetrad are:

$\ell ^{\mu} = \delta _{r}^{\mu} $

$n^{\mu} = \sqrt{\frac{f(r)}{g(r)}}\delta^{\mu}_{\mu} - \frac{f(r)}{2}\delta^{\mu}_{r}$

$m^{\mu} = \frac{1}{\sqrt{2}r}\delta^{\mu}_{\theta} + \frac{i}{\sqrt{2}r sin(\theta)}\delta^{\mu}_{\phi}$

$\bar{m}^{\mu} = \frac{1}{\sqrt{2}r}\delta^{\mu}_{\theta} - \frac{i}{\sqrt{2}r sin(\theta)}\delta^{\mu}_{\phi}$

$$ --\circ-- $$

$[1]$ XU.Z; TANG.M; CAO.G; ZHANG.S; Possibility of traversable wormhole formation in the dark matter halo with istropic pressure. The European Pheysical Journal C. 2020.

$[2]$DELOSHAN. N. Complex Spacetimes and the Newman-Janis Trick. Chapter 5. http://researcharchive.vuw.ac.nz/handle/10063/4938

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In fact an Professor gave me a excellent paper which answers this question of mine.

The paper is: Generating rotating regular black hole solutions without complexification by Mustapha Azreg-Ainou https://arxiv.org/abs/1405.2569

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