# Newmann-Penrose Spin coefficients for Schwarschild metric

• A
• vinicius_linhares
In summary, the Newmann-Penrose spin coefficients are used to describe the spin and rotation of particles in the Schwarschild metric, which is a mathematical model used to describe the curvature of space-time around a non-rotating spherical mass. These coefficients provide a way to calculate the angular momentum and energy of particles in this metric, allowing for a better understanding of the behavior of matter in the presence of a massive object such as a black hole. They play a crucial role in the study of general relativity and have been used in various astrophysical and cosmological applications.
vinicius_linhares
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.

Last edited by a moderator:

Firstly, it is important to note that the Newmann-Penrose (N-P) formalism is a powerful mathematical tool for studying the properties of spacetime, particularly in relation to black holes. It was developed by Roger Penrose and Ezra T. Newmann in the 1960s and has been widely used in the field of general relativity.

In the context of the Schwarschild metric, the N-P formalism involves the use of a null tetrad, which is a set of four linearly independent vectors that are all null (meaning they have zero length) and orthogonal to each other. This tetrad is used to decompose the spacetime into two null directions and two spacelike directions, allowing for a more manageable analysis of the metric.

In the article you have referenced, the authors have presented a null tetrad for the Schwarschild metric, which is in accordance with the Chandrasekhar tetrad. This tetrad is given by the matrix ${e_{(A)}}^\mu$, where A=1,2,3,4 and $\mu=t,r,\phi,\theta$. It is important to note that the indices A and $\mu$ have different meanings - A corresponds to the tetrad basis vectors, while $\mu$ corresponds to the spacetime coordinates.

To calculate the spin coefficient $\gamma_{243}$, the authors have used the formula: $$\gamma_{243}=\frac{1}{2}\big(C_{243}+C_{432}-C_{324}\big)$$ where $C_{ABC}$ is the N-P curvature coefficient, given by the formula ${C^D}_{AB}=({e^D}_{\alpha,\beta}-{e^D}_{\beta,\alpha}){e_A}^\alpha {e_B}^\beta$. Here, ${e^{(A)}}_\mu$ is the inverse of the tetrad matrix ${e_{(A)}}^\mu$, and is given by the matrix in your question.

It is important to note that the authors have used the Minkowski metric $\eta_{AB}$ (with signature +2) to calculate the N-P curvature coefficient, as shown in the formula $C_{ABC}=\eta_{AD}{C^D}_{BA}$. This is different from the metric used in the Chandrasekhar tetrad, which is the Schwarschild metric (with signature -2). Hence, the results obtained using the

• Special and General Relativity
Replies
5
Views
461
• Calculus and Beyond Homework Help
Replies
18
Views
2K
• Special and General Relativity
Replies
11
Views
569
• Quantum Physics
Replies
3
Views
953
• Special and General Relativity
Replies
10
Views
1K
• Special and General Relativity
Replies
2
Views
1K
• Classical Physics
Replies
3
Views
2K
• Set Theory, Logic, Probability, Statistics
Replies
1
Views
2K
• MATLAB, Maple, Mathematica, LaTeX
Replies
6
Views
6K
• Special and General Relativity
Replies
7
Views
2K