GR weak field approx. problem with Dunsby's notes

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Discussion Overview

The discussion revolves around a specific step in the derivation of the weak field approximation in General Relativity (GR), as presented in Dunsby's notes. Participants explore the assumptions and implications of approximating the metric and the treatment of derivatives within this context.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions the neglect of the time derivative in the weak field approximation, specifically in the context of the metric approximation g[a][b]=n[a][b]+epsilon h[a][b].
  • Another participant outlines the conditions for the Newtonian limit, emphasizing the need for the test particle's velocity to be much smaller than the speed of light, the gravitational field to be weak, and the field to be static.
  • A third participant suggests consulting additional resources, such as Sean Carroll's notes and Woodhouse's work, for further insights into the geodesic equation of motion and its relation to the weak field approximation.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions made in the weak field approximation, particularly regarding the treatment of time derivatives. There is no consensus on the necessity or implications of these assumptions.

Contextual Notes

Participants note that the assumptions of static fields and the conditions under which the Newtonian limit applies may not be explicitly stated in Dunsby's notes, leading to potential misunderstandings.

enomanus
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Hi again,
Thaks for previous help. I am self studying GR using Schutz and Dunsby's webpages & notes.
I am stuck on a step in the derivation of the weak field approximation.
They approximate the metric with g[a][/b]=n[a]+epsilon h[a]
In step 7.30 the partial derivative h[0],[0]is taken as zero to get (7.30), -1/2 epsilon h00,i
He has neglected the time derivative!
Can you explain Why??
Thanks for any help!
 
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Hi enomanus,

the Newtonian limit consists of the following:

-Take the velocity of the testparticle much smaller than c.
-The gravitational field is weak, so we can make an expansion around the vacuum (in your case minkowski space time).
-The field is static.

This last assumption should be mentioned somewhere in your notes, otherwise you could check the notes of Sean Carroll,

http://arxiv.org/PS_cache/gr-qc/pdf/9712/9712019v1.pdf

chapter 4 page 105.
 
You can also try p47 of Woodhouse's http://people.maths.ox.ac.uk/~nwoodh/gr/index.html . The geodesic equation of motion of the time coordinate ends up second order in the small parameter.
 
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Thanks! I've got the idea now!
 

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