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italicus
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- TL;DR Summary
- A formula by R. Feynman about the radius excess of curvature
I think the best place to put this post is the section on special and general relativity. Reading Feynman’s lecture n.42 , volume II here linked :
https://www.feynmanlectures.caltech.edu/II_42.html
I’ve met the following formula 42.3 for the radius excess of curvature, that Feynman attributes to Einstein :
$$\text {Radius excess} = R{measured} – \sqrt {\frac {A} {4\pi} } = \frac {GM} {3c^2} $$
Feynman says the space is curved, according to Einstein. The first observation here is that spacetime is curved by matter, not only space. Anyway, let us go on, some expert will clarify this. I have understood that, if we take a sphere of matter small enough that density can be considered constant inside, and A is the surface area, then the measured radius is grater than ##\sqrt {\frac {A} {4\pi} }## , and that this difference is given by ##\frac {GM} {3c^2}## .
My question is : where does the last equality come from ? I have studied relativity on several books , Schutz, Rindler, D’Inverno...but haven’t found that formula. I know that in a curved geometry “r” is a coordinate and not a distance, but I’m perplexed. Perhaps there is something i haven’t well understood
Thank you for the attention.
https://www.feynmanlectures.caltech.edu/II_42.html
I’ve met the following formula 42.3 for the radius excess of curvature, that Feynman attributes to Einstein :
$$\text {Radius excess} = R{measured} – \sqrt {\frac {A} {4\pi} } = \frac {GM} {3c^2} $$
Feynman says the space is curved, according to Einstein. The first observation here is that spacetime is curved by matter, not only space. Anyway, let us go on, some expert will clarify this. I have understood that, if we take a sphere of matter small enough that density can be considered constant inside, and A is the surface area, then the measured radius is grater than ##\sqrt {\frac {A} {4\pi} }## , and that this difference is given by ##\frac {GM} {3c^2}## .
My question is : where does the last equality come from ? I have studied relativity on several books , Schutz, Rindler, D’Inverno...but haven’t found that formula. I know that in a curved geometry “r” is a coordinate and not a distance, but I’m perplexed. Perhaps there is something i haven’t well understood
Thank you for the attention.