# Doubt on Newman-Janis algorithm for a traversable Wormhole

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