Gravitational wave emission from electrons

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Summary:

An electron would take ~10^110 seconds to collapse into the nucleus due to gravitational wave emission, assuming gravity is not quantized.

Main Question or Discussion Point

Ok, so I've been on a kick trying to really understand why QM and GR are incompatible. I think I get that GR cant be realistically converted into a quantum field because it creates some infinite series that you can't use the normal tricks you would for other QM fields. Hard block, ok got it.

So then what if gravity just isn't quantized? The best argument I was able to find for why it should be quantized was that an electron orbiting the nucleus would eventually lose energy from gravitational waves and fall into the nucleus. This is the same argument that led to a lot of QM in the first place.

So I managed to do the calculation of how long it would take an electron to collapse in on the nucleus based on nothing but gravitational wave emission. The result I got is ~10110 seconds. If this calculation is right, it just wouldn't have ever happened yet. That number is insane.

Equation for calculating the time to collapse from here: https://en.wikipedia.org/wiki/Gravitational_wave#Binaries
Shortened url for the wolfram alpha link to calculation here: https://bit.ly/2KUmmBa
I used the bohr radius for the distance between nucleus and electron.
 

Answers and Replies

Vanadium 50
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First, even classically, your calculation is wrong. The time variation of the quadrupole moment is zero, so no radiation.

Second about "The result I got is ~10110 seconds. If this calculation is right, it just wouldn't have ever happened yet. That number is insane. " Why is the stability of atoms "insane"? Atoms are stable.
 
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Of course atoms are stable, I didn't mean to suggest that they aren't.

When I was trying to learn why it seems that there are a fair number of people that start with the assumption that gravity is quantized, one of the arguments that seemed the most solid was that if it wasn't, atoms wouldn't be stable. I wondered about that for a bit, because gravity is so weak that it might just be that the instability is on such a long timeline, that it might not have ever happened. When I ran the calculation I found a number that is huge, that's what I meant by insane. It could have be 1040 seconds and the result would be the same, atoms are stable. But it's so much bigger than that.

Thank you for pointing out my calculation was wrong, I marked this as beginner for the reason that I really am a beginner in GR. I think I've begun to have decent grasp of SR, only scratched the surface on the maths of GR. I literally don't even know what "time variation of the quadrupole moment" means. I'm going to have to do some more learning.
 
PeroK
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So then what if gravity just isn't quantized? The best argument I was able to find for why it should be quantized was that an electron orbiting the nucleus would eventually lose energy from gravitational waves and fall into the nucleus. This is the same argument that led to a lot of QM in the first place.

I used the bohr radius for the distance between nucleus and electron.
In the ground state of hydrogen the electron isn't "orbiting" the nucleus in any classical sense. A measurement of the angular momentum of the electron will always return a value of zero. So, however you want to describe this state, it's not going round and round the nucleus.

Moreover, the electron cannot lose energy. It is in the lowest energy state possible.

An electron in some excited states (higher energy levels) has non-zero angular momentum. But, even so, it can only release energy in well-defined quanta that would take it to a specific lower energy level. It can't "spiral in".

I would encourage you to learn SR properly and try to grasp GR if you can. But, if you want to analyse the incompatibility between GR and QM, you are going to need some serious study of QM first. QM is not just classical physics with a bit of quantisation thrown in.
 
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there are a fair number of people that start with the assumption that gravity is quantized, one of the arguments that seemed the most solid was that if it wasn't, atoms wouldn't be stable.
Where have you heard this? Mostly I’m curious.
 
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Where have you heard this? Mostly I’m curious.
I'll be honest, the first place I heard this argument was in this video at the 2:30 mark:



I also came across it here while googling to find answers: https://physics.stackexchange.com/questions/275619/are-gravitational-atoms-stable-without-quantization

And since I know those aren't really decent sources, I just went looking for papers that have talked about this. I found a (very) old paper trying to apply electromagnetic theories to gravity, and it appears that the author was probably thinking about the same issue (though they don't mention it directly) when writing it: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC522457/pdf/pnas01017-0027.pdf

To be clear, Vanadium50 and PeroK have made it very clear that I've made some assumptions that are just wrong. Just posting so you know where I got the idea :)
 
PeroK
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I'll be honest, the first place I heard this argument was in this video at the 2:30 mark:



I also came across it here while googling to find answers: https://physics.stackexchange.com/questions/275619/are-gravitational-atoms-stable-without-quantization

And since I know those aren't really decent sources, I just went looking for papers that have talked about this. I found a (very) old paper trying to apply electromagnetic theories to gravity, and it appears that the author was probably thinking about the same issue (though they don't mention it directly) when writing it: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC522457/pdf/pnas01017-0027.pdf

To be clear, Vanadium50 and PeroK have made it very clear that I've made some assumptions that are just wrong. Just posting so you know where I got the idea :)
It depends what you want to learn. Looking at a random paper by Weyl from 1929 may be interesting but it won't lead to a systematic understanding.

The stack exchange post is misguided, I believe. A gravitational atom would satisfy the Schrodinger equation for a suitable gravitational potential.

All very interesting, but perhaps deflects you from a systematic study of the subject of QM.

What level of maths can you handle?
 
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