How to obtain Kerr Metric via Spinors (N-P Formalism)

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Discussion Overview

The discussion revolves around obtaining the Kerr Metric using Spinors within the Newman-Penrose formalism. Participants express confusion regarding coordinate transformations and the formulation of null tetrads, as well as the implications of using complex coordinates in this context.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant questions the necessity of the coordinate transformation from 2r-1 to r-1 + r*-1, expressing confusion about its purpose.
  • Another participant seeks clarification on whether Ray d'Inverno discusses spinors in his book, specifically asking for page references.
  • A later reply provides a page number (250) where spinors are mentioned in d'Inverno's text.
  • One participant describes the nature of a null tetrad, suggesting that it consists of real null vectors and spurious components involving "i," and warns against the confusion this can cause in spacetime covariants.
  • Another participant proposes that using complex coordinates for the Kerr metric allows for a natural interpretation of the mass being located at a distance 'a' along the imaginary axis, where 'a' is the Kerr parameter, but questions the implications of this interpretation.
  • A follow-up question asks for further explanation on how the mass's position affects the extension of r, indicating a desire for deeper understanding.

Areas of Agreement / Disagreement

Participants express varying levels of confusion and curiosity about the concepts discussed, with no clear consensus on the necessity or implications of the coordinate transformation or the formulation of the null tetrad.

Contextual Notes

Participants exhibit uncertainty regarding the definitions and implications of complex coordinates and null tetrads, as well as the mathematical steps involved in the transformation process.

Who May Find This Useful

Readers interested in advanced topics in general relativity, particularly those exploring the Kerr metric, spinors, and the Newman-Penrose formalism may find this discussion relevant.

yicong2011
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How to obtain Kerr Metric via Spinors (Newman-Penrose Formalism)?

I am a bit confused with Ray d'Inverno's Book.

Why perform the coordinates transformation:

2r-1 -> r-1 + r*-1


I am bit confused of it.

And I am a bit confused, too, of how to write out null tetrad...


Is there any resource rendering in-detail discussion of it?
 
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Does D'Inverno talk about spinors? I would be thankful if you could point me to where in the text itself if he does.
 
WannabeNewton said:
Does D'Inverno talk about spinors? I would be thankful if you could point me to where in the text itself if he does.

Page 250...
 
A null tetrad is something of a ruse. Two of its legs are real null vectors whose inner product is 1, while the other two are formed by the introduction of a spurious "i" and have components that are real, orthogonal spacelike vectors:

(s1 + i s2)^2 = (s1^2 - s2^2) + 2i (s1.s2) = 0

..so you could take s1 = (1,0,0,0) and s2=(0,1,0,0) in the rest frame and boost them up to non-zero momentum. So a typical null tetrad is up to normalization,

[1,i,0,0], [1,-i,0,0], [0,0,1,1], [0,0,-1,1]

If one now wants the Pauli algebra (2nd-rank mixed spinorial) form of these basis vectors, you get up to normalization,

[0,1;0,0] [0,0;1,0] [1,0;0,0] [0,0;0,1]

This is what one is really after, and the tetrad itself is a side show.

Ad-hoc introduction of "i" in this way always leads to confusion of spacetime covariants. It is to be avoided.

-drl
 
Last edited:
yicong2011 said:
How to obtain Kerr Metric via Spinors
Why perform the coordinates transformation:
2r-1 -> r-1 + r*-1

Going to complex coordinates seems like a strange idea at first, but for the Kerr metric it turns out to be useful, because there's a natural interpretation in which the mass m is located not at the origin but at a distance 'a' along the imaginary axis, where 'a' is the Kerr parameter.
 
the mass m is located not at the origin but at a distance 'a' along the imaginary axis, where 'a' is the Kerr parameter.


Why is that?

How can it affect the extension of r

Can someone explain it?

Thanks.
 

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