- #1

vinicius_linhares

- 5

- 0

- TL;DR Summary
- Im having problem with finding the spin coefficients in the NP formalism

I need to use the N-P formalism to apply in my work so I'm trying first to apply in a simple case to understand better. So in this article ( https://arxiv.org/abs/1809.02764 ) which I'm using, they present a null tetrad for the Schwarszchild metric in pg.14 (with accordance with the Chandrasekhar tetrad in "mathematical theory of black holes pg.134-135):

$${e_{(A)}}^\mu=\frac{1}{\sqrt2}\begin{bmatrix}

1/X & 1 & 0 & 0\\

1 & -X & 0 & 0 \\

0&0&1/r&\frac{i}{rsin\theta}\\

0&0&1/r&-\frac{i}{rsin\theta}

\end{bmatrix},\;\;A=1,2,3,4;\;\;\mu=t,r,\phi,\theta;\;\;X=1-\frac{2M}{r}$$

For calculating the spin coefficient (one of them for example): $$\mu=\gamma_{243}=\frac{1}{2}\big(C_{243}+C_{432}-C_{324}\big);\;\;\;{C^D}_{AB}=({e^D}_{\alpha,\beta}-{e^D}_{\beta,\alpha}){e_A}^\alpha {e_B}^\beta$$

$$C_{ABC}=\eta_{AD}{C^D}_{BA}.$$

Then I found the inverse needed above:

$${e^{(A)}}_\mu=\frac{\sqrt2}{2}\begin{bmatrix}

X & 1 & 0 & 0\\

1 & -1/X & 0 & 0 \\

0 & 0 & r & -irsin\theta\\

0 & 0 & r & irsin\theta

\end{bmatrix}.$$

Now when I go for the calculation, I don't get to the mentionated results: $$\mu=-\frac{X}{\sqrt2r}$$.

The books change index meaning throughout the texts and I am very confused. In Chandrasekhar pg.81 there is a tetrad and the inverse and there I can reproduce the results, I just don't know why in the other case I'm getting wrong.

$${e_{(A)}}^\mu=\frac{1}{\sqrt2}\begin{bmatrix}

1/X & 1 & 0 & 0\\

1 & -X & 0 & 0 \\

0&0&1/r&\frac{i}{rsin\theta}\\

0&0&1/r&-\frac{i}{rsin\theta}

\end{bmatrix},\;\;A=1,2,3,4;\;\;\mu=t,r,\phi,\theta;\;\;X=1-\frac{2M}{r}$$

For calculating the spin coefficient (one of them for example): $$\mu=\gamma_{243}=\frac{1}{2}\big(C_{243}+C_{432}-C_{324}\big);\;\;\;{C^D}_{AB}=({e^D}_{\alpha,\beta}-{e^D}_{\beta,\alpha}){e_A}^\alpha {e_B}^\beta$$

$$C_{ABC}=\eta_{AD}{C^D}_{BA}.$$

Then I found the inverse needed above:

$${e^{(A)}}_\mu=\frac{\sqrt2}{2}\begin{bmatrix}

X & 1 & 0 & 0\\

1 & -1/X & 0 & 0 \\

0 & 0 & r & -irsin\theta\\

0 & 0 & r & irsin\theta

\end{bmatrix}.$$

Now when I go for the calculation, I don't get to the mentionated results: $$\mu=-\frac{X}{\sqrt2r}$$.

The books change index meaning throughout the texts and I am very confused. In Chandrasekhar pg.81 there is a tetrad and the inverse and there I can reproduce the results, I just don't know why in the other case I'm getting wrong.

Last edited by a moderator: