What is the significance of the weak field approximation in general relativity?

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SUMMARY

The weak field approximation in general relativity (GR) is crucial for demonstrating that GR reduces to Newtonian gravity in low velocity scenarios. This approximation allows for simplifications in the equations, particularly when the metric remains static over time, as discussed in sections 17.19 and 17.25 of the referenced material. The derivations serve as essential practice for understanding Einstein's equations, the connection, and the metric, reinforcing the consistency of GR as a theory. The discussion highlights the importance of these derivations despite their seemingly complex nature.

PREREQUISITES
  • Understanding of Einstein's equations in general relativity
  • Familiarity with the concept of metrics in spacetime
  • Knowledge of Newtonian mechanics and its principles
  • Basic mathematical skills for working with differential equations
NEXT STEPS
  • Study the derivation of the Newtonian limit from general relativity
  • Explore the role of metrics in general relativity
  • Learn about the significance of weak field approximations in gravitational theories
  • Investigate various textbooks on general relativity that cover these derivations
USEFUL FOR

Students of physics, particularly those studying general relativity, researchers in gravitational theories, and educators looking to deepen their understanding of the relationship between GR and Newtonian mechanics.

Thrice
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Looks like I really don't have a feel for it. So I was working on this the other day.

(arranged in order)
http://img218.imageshack.us/img218/9613/1gx9.jpg
http://img68.imageshack.us/img68/4677/2te4.jpg
http://img68.imageshack.us/img68/7853/3jg9.jpg
http://img68.imageshack.us/img68/6273/4qw8.jpg
http://img68.imageshack.us/img68/273/5hu6.jpg

It's fairly straightforward, but I think I'm just not used to the style. For example in 17.19 they took only the spatial equations because the metric doesn't change with time. Well just going by the math, I don't see any constraints on n. I see the constraints on k,j,p though. Do they translate to n as well? Same thing happens at 17.25. I figured you can choose to consider any parts of your system for whatever reason.

Then at 17.36 when they just dropped that entire term, but chose not to do the same with the 17.35 term. It works, though. The solution at the end is correct.

So I got to thinking .. What role do these derivations really play? Does it really matter how you show that GR reduces to Newtonian mechanics? GR is correct whether you do or not, right? I even saw a place where the author started with the metric for the Newtonian limit and 'derived' f=ma. It just seems like so much handwaving smoke and mirrors.

Of course I'm still new to all this so it's possible I didn't pay attention a few pages back. Thoughts?
 
Last edited by a moderator:
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GR has to reduce to Newtonian gravity in the low velocity, weak field limit to be a consistent theory. The derivations are also good practice in working with Einstein's equation, the connection, and the metric. What book is this from?
 
Daverz said:
GR has to reduce to Newtonian gravity in the low velocity, weak field limit to be a consistent theory. The derivations are also good practice in working with Einstein's equation, the connection, and the metric. What book is this from?
https://www.amazon.com/gp/product/012200681X/?tag=pfamazon01-20

If it's correct (and i figure there are other ways to verify that it's correct) then we already know that it reduces. But yeah it is good practice.
 
Last edited by a moderator:
Also, just about every GR book goes through this derivation. It's sort of traditional.
 

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