I Geodesic in Weak Field Limit: Introducing Einstein's Relativity

GR191511
Messages
76
Reaction score
6
I'm reading《Introducing Einstein's Relativity_ A Deeper Understanding Ed 2》on page 180,it says:

since we are interested in the Newtonian limit,we restrict our attention to the spatial part of the geodesic equation,i.e.when a=##\alpha####\quad ##,and we obtain,by using ##\frac{d\tau}{dt}=1+O(\varepsilon^2)##:
##\frac {\partial^2 x^\alpha} {\partial t^2}+\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}(1+O(\varepsilon))=0##.##\quad ##I wonder how the auther gets "(1+O(##\varepsilon##))"?
 
Physics news on Phys.org
The authors (D'Inverno & Vickers) explained that on p179, just before eq(10.43). Do you not understand how eq(10.43) was obtained? Their preceding (unnumbered) equation is $$c^2 d\tau^2 ~=~ c^2 dt^2 (1 - \varepsilon^2) ~.$$ Take the square root of both sides. You might have to expand ##\sqrt{1 - \varepsilon^2}## as a Taylor series in ##\varepsilon## to see why it ends up as ##\,1 - O(\varepsilon^2)##.
 
  • Like
Likes PeroK, topsquark and vanhees71
strangerep said:
You might have to expand ##\sqrt{1 - \varepsilon^2}## as a Taylor series in ##\varepsilon## to see why it ends up as ##\,1 - O(\varepsilon^2)##.
Thanks!But it doesn't end up as"##1+O(\varepsilon^2)##",it is"##1+O(\varepsilon)##"
 
GR191511 said:
it doesn't end up as"##1+O(\varepsilon^2)##",it is"##1+O(\varepsilon)##"
Check your work. Doing a Taylor series expansion of ##\sqrt{1 + \varepsilon^2}## does not mean you take the square root of ##\varepsilon^2##.
 
  • Like
Likes topsquark and Vanadium 50
PeterDonis said:
Check your work. Doing a Taylor series expansion of ##\sqrt{1 + \varepsilon^2}## does not mean you take the square root of ##\varepsilon^2##.
I mean the geodesic "##\frac {\partial^2 x^\alpha} {\partial t^2}+\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}(1+O(\varepsilon))=0## "it ends up as ##(1+O(\varepsilon))##
 
GR191511 said:
I mean the geodesic "##\frac {\partial^2 x^\alpha} {\partial t^2}+\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}(1+O(\varepsilon))=0## "it ends up as ##(1+O(\varepsilon))##
You make it much harder to help you if you don't show your work. I'm guessing you haven't correctly followed the author's instructions after eq(10.46).
 
  • Like
Likes topsquark and vanhees71
strangerep said:
You make it much harder to help you if you don't show your work. I'm guessing you haven't correctly followed the author's instructions after eq(10.46).
##\frac {d^2 x^\alpha} {d \tau^2}+\Gamma^\alpha{}_{bc}\frac{dx^b}{d\tau}\frac{dx^c}{d\tau}=0\Rightarrow\frac{d}{d\tau}(\frac{dx^\alpha}{dt}\frac{dt}{d\tau})+\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}\frac{dt}{d\tau}\frac{dt}{d\tau}=0\Rightarrow##
##\left[\frac{d}{dt}(\frac{dx^\alpha}{dt}\frac{dt}{d\tau})\right]\frac{dt}{d\tau}+\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}\frac{dt}{d\tau}\frac{dt}{d\tau}=0\Rightarrow\frac{d}{dt}(\frac{dx^\alpha}{dt}\frac{dt}{d\tau})+####\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}\frac{dt}{d\tau}=0\Rightarrow\frac{d^2x^\alpha}{dt^2}\frac{dt}{d\tau}+\frac{dx^\alpha}{dt}\frac{d}{dt}(\frac{dt}{d\tau})####+\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}\frac{dt}{d\tau}=0####\Rightarrow\frac{d^2x^\alpha}{dt^2}\frac{dt}{d\tau}+0+\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}\frac{dt}{d\tau}=0\Rightarrow\frac{d^2x^\alpha}{dt^2}+\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}####=0####\Rightarrow\frac{1}{c^2}\frac{d^2x^\alpha}{dt^2}+\frac{1}{c^2}\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}=0####\quad##So where is the ##1+O(\varepsilon)##?
 
  • Like
Likes Dale
OK, it's actually a lot simpler than what you've written in post #7.

From (10.43) we have ##d\tau = dt + O(\varepsilon^2)##. That means you can replace all the ##d\tau##'s in the derivatives by ##dt##. (Think of those derivatives as ratios of differentials.)

The trick they don't seem to explain is that ##d^2 x^\alpha/dt^2## is ##O(\varepsilon)##, as is ##\Gamma^a_{~bc}##, which is shown in (10.45) as an ##\varepsilon## term plus ##O(\varepsilon^2)##. In the geodesic equation after (10.46), they divided throughout by 1 factor of ##\varepsilon##.

(Is that explanation sufficient? I'll agree they probably could have been clearer.)
 
  • Informative
  • Like
Likes Dale and vanhees71
strangerep said:
OK, it's actually a lot simpler than what you've written in post #7.

From (10.43) we have ##d\tau = dt + O(\varepsilon^2)##. That means you can replace all the ##d\tau##'s in the derivatives by ##dt##. (Think of those derivatives as ratios of differentials.)

The trick they don't seem to explain is that ##d^2 x^\alpha/dt^2## is ##O(\varepsilon)##, as is ##\Gamma^a_{~bc}##, which is shown in (10.45) as an ##\varepsilon## term plus ##O(\varepsilon^2)##. In the geodesic equation after (10.46), they divided throughout by 1 factor of ##\varepsilon##

(Is that explanation sufficient? I'll agree they probably could have been clearer.)
Why is "##d^2 x^\alpha/dt^2## is ##O(\varepsilon)##"related with this question?
Do you mean it is ##\frac{\varepsilon}{c^2}\frac{d^2x^\alpha}{dt^2}+\frac{\varepsilon}{c^2}\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}(1+O(\varepsilon))=0##before" they divided throughout by 1 factor of ##\varepsilon##"?Thank you!
 
  • #10
GR191511 said:
Why is "##d^2 x^\alpha/dt^2## is ##O(\varepsilon)##"related with this question?
Do you mean it is ##\frac{\varepsilon}{c^2}\frac{d^2x^\alpha}{dt^2}+\frac{\varepsilon}{c^2}\Gamma^\alpha{}_{bc}\frac{dx^b}{dt}\frac{dx^c}{dt}(1+O(\varepsilon))=0##before" they divided throughout by 1 factor of ##\varepsilon##"?Thank you!
That's the general idea. But... (sigh)... now you've got me wanting to rewrite that stuff properly without skipping steps. Probably not today though.
 
  • Like
Likes GR191511
Back
Top