A Some questions about the derivation steps in the Gravitational deflection of light section in Schutz

MathematicalPhysicist
Science Advisor
Gold Member
Messages
4,662
Reaction score
372
TL;DR Summary
My question is referring to some derivation from pages 293-294 of Schutz's second edition (2009) of A First Course in GR.
In the screenshots below there are the equations (11.49) and (11.53).

I don't understand how did he derive equation (11.53) from Eq.(11.49)?
From (11.49) I get: ##d\phi/dy= d\phi/du du/dy = (1/b^2-u^2+2Mu^3)^{-1/2}(1+2My)##.
It seems he neglected the ##2Mu^3## since ##Mu\ll 1##, so ##y\approx u##, but how do we get the added term of ##O(M^2u^2)##?

schutz2.pngschutz1.png
schutz1.png
schutz2.png
 
Physics news on Phys.org
I would have thought\begin{align*}
\dfrac{d\phi}{dy} = \dfrac{d\phi}{du} \left( \dfrac{dy}{du} \right)^{-1} &= \dfrac{( 1 - 2Mu)^{-1} }{\left( b^{-2} - u^2(1-2Mu) \right)^{1/2}}
\end{align*}then because ##y^2 = u^2(1-Mu)^2 \sim u^2(1-2Mu)## and also ##(1-2Mu)^{-1} \sim 1+2Mu + O(M^2u^2) \sim 1 + 2My + O(M^2u^2)## you get ##\mathrm{11.53}##...
 
  • Like
Likes Ibix and MathematicalPhysicist
ergospherical said:
I would have thought\begin{align*}
\dfrac{d\phi}{dy} = \dfrac{d\phi}{du} \left( \dfrac{dy}{du} \right)^{-1} &= \dfrac{( 1 - 2Mu)^{-1} }{\left( b^{-2} - u^2(1-2Mu) \right)^{1/2}}
\end{align*}then because ##y^2 = u^2(1-Mu)^2 \sim u^2(1-2Mu)## and also ##(1-2Mu)^{-1} \sim 1+2Mu + O(M^2u^2) \sim 1 + 2My + O(M^2u^2)## you get ##\mathrm{11.53}##...
It seems I got it wrong. Thanks for correcting me.
 
Thread 'Can this experiment break Lorentz symmetry?'
1. The Big Idea: According to Einstein’s relativity, all motion is relative. You can’t tell if you’re moving at a constant velocity without looking outside. But what if there is a universal “rest frame” (like the old idea of the “ether”)? This experiment tries to find out by looking for tiny, directional differences in how objects move inside a sealed box. 2. How It Works: The Two-Stage Process Imagine a perfectly isolated spacecraft (our lab) moving through space at some unknown speed V...
Does the speed of light change in a gravitational field depending on whether the direction of travel is parallel to the field, or perpendicular to the field? And is it the same in both directions at each orientation? This question could be answered experimentally to some degree of accuracy. Experiment design: Place two identical clocks A and B on the circumference of a wheel at opposite ends of the diameter of length L. The wheel is positioned upright, i.e., perpendicular to the ground...
According to the General Theory of Relativity, time does not pass on a black hole, which means that processes they don't work either. As the object becomes heavier, the speed of matter falling on it for an observer on Earth will first increase, and then slow down, due to the effect of time dilation. And then it will stop altogether. As a result, we will not get a black hole, since the critical mass will not be reached. Although the object will continue to attract matter, it will not be a...
Back
Top