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

In summary, the conversation discusses the process of deriving Newtonian mechanics from general relativity, with the focus on the role and importance of these derivations. The speaker is unsure about the relevance of these derivations and questions their necessity, but acknowledges that they serve as good practice in working with Einstein's equation, the connection, and the metric. The conversation also mentions the requirement for general relativity to reduce to Newtonian gravity in the low velocity, weak field limit to be a consistent theory.
  • #1
Thrice
258
0
Looks like I really don't have a feel for it. So I was working on this the other day.

(arranged in order)
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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?
 
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  • #2
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?
 
  • #3
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.
 
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  • #4
Also, just about every GR book goes through this derivation. It's sort of traditional.
 

What is the weak field approximation?

The weak field approximation is a mathematical technique used to simplify the analysis of a physical system that is affected by a weak gravitational or electromagnetic field. It assumes that the field is small enough that the effects can be approximated using linear equations.

When is the weak field approximation used?

The weak field approximation is commonly used in astrophysics and cosmology to study the behavior of objects in the presence of gravitational fields. It is also used in electromagnetism to analyze the behavior of particles in weak electromagnetic fields.

How accurate is the weak field approximation?

The accuracy of the weak field approximation depends on the strength of the field and the precision needed for the analysis. In general, it is more accurate for weaker fields and less accurate for stronger fields.

What are the limitations of the weak field approximation?

The weak field approximation is only valid for small fields and cannot be used to analyze systems with strong gravitational or electromagnetic fields. It also does not take into account non-linear effects, so it may not be accurate for highly complex systems.

How is the weak field approximation calculated?

The weak field approximation is calculated by using the linearized version of the field equation, which simplifies the equations and makes them easier to solve. This linearized equation is then used to approximate the behavior of the system in the presence of a weak field.

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