Find tetrad component from metric

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Discussion Overview

The discussion centers on how to derive tetrad components from a given metric, particularly focusing on the transition from diagonal metrics like Schwarzschild to non-diagonal metrics such as Kerr. Participants explore the mathematical relationships and transformations involved in this process.

Discussion Character

  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant states the relationship between tetrads and the metric, expressing familiarity with the diagonal case but questioning the approach for non-diagonal metrics.
  • Another participant suggests considering the metric as a 4x4 symmetric matrix and proposes diagonalizing it.
  • A different participant notes that diagonalizing the metric leads to the Lorentz metric.
  • One participant seeks clarification on whether finding eigenvectors is the intended approach.
  • Another participant reformulates the relationship between the metric and tetrads in matrix form, indicating that the tetrad components are necessary for transforming the metric into the Lorentz form.

Areas of Agreement / Disagreement

Participants present various methods and interpretations regarding the extraction of tetrad components, indicating that multiple approaches are being considered without a clear consensus on the best method.

Contextual Notes

The discussion does not resolve the specific steps required for non-diagonal metrics, and assumptions about the properties of the metrics and tetrads remain unaddressed.

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How can find components of tetrads from metric ?
i know the relation between tetrads and metric
g_{μ \nu}=η_{ab}e^{a}_{μ}e^{b}_{\nu}
where e^{b}_{\nu} are component of tetrads , in the case of Schwarzschild that metric is diagonal , it is a easy problem but what about non-diagonal metric like kerr ?
 
Last edited:
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Consider the metric as a 4x4 symmetric matrix. Diagonalize this matrix.
 
if we do that we get lorentz metric !
 
u mean find eigen vectors ?
 
gμν = ηabeaμebν

Write this as a matrix equation:

gμν = eaμ ηab ebν

g = eT η e

The tetrad components form the matrix necessary to transform g into η.
 
Thank u dude !
 

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