Where does the ISCO inequality L ≥ √3 c Rsch come from?

quasarLie · Jan 6, 2018

Hello,
Here's an interesting question inspired by a homework probem (not mine), we know that circular orbit (for scjwarzchild black hole) exist only if L ≥ sqrt3 c Rsch=Lisco . Where does this inequality come from? do you have a lecture which can help me to understand?
Thanks

mfb · Jan 6, 2018

Circular orbit or stable circular orbit?

To find the closest stable orbit you can look at the effective potential. In Newtonian mechanics this always has a minimum, in GR you get a 1/r³ term (if I remember correctly) that ruins the minimum below this critical L. For lower L you have a potential that goes down all the way to the black hole.

m4r35n357 · Jan 7, 2018

quasarLie said:

Hello,
Here's an interesting question inspired by a homework probem (not mine), we know that circular orbit (for scjwarzchild black hole) exist only if L ≥ sqrt3 c Rsch=Lisco . Where does this inequality come from? do you have a lecture which can help me to understand?
Thanks

It is covered by equations 7.29 to 7.55 in Carrol's online notes.

Where does the ISCO inequality L ≥ √3 c Rsch come from?

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