Gauss’s Law only says that about spherically symmetric mass distributions and quite a few of the situations here are not that.
False, it applies to any closed surface. Spherical symmetry is just the simplest problem where the Gauss law can be usefully applied.
(Otherwise the three body problem would be trivial to solve!)
The three-body problem has no relation to the Gauss law, except insofar as it is equivalent to the inverse square law for point masses in flat space.
Or how about some of the comments here that involve accelerating massive bodies to near light-speed very rapidly. Sure this is fine in Newtonian mechanics, but in GR the acceleration of a large body creates gravitational waves
This is true in most cases, but not in spherical symmetry (there are no spherically symmetric gravitational waves). That’s why I could write “an amount of matter equivalent to her mass has to travel inwards out of nowhere and coalesce into a person” and not worry about energy losses to gravitational radiation. Though there would be, of course, a lot of losses to EM radiation if the matter in question consisted of charged particles.
False, it applies to any closed surface. Spherical symmetry is just the simplest problem where the Gauss law can be usefully applied.
Oh I absolutely don’t deny that. What I do assert is that you can’t ignore what happens on the inside of a closed surface if you don’t know that the mass is spherically symmetric and you want to calculate the force at a specific force. If this is not what you are saying, then this is a miscommunication not a disagreement. What I took you to mean when you made comments like “This still conserves mass, since black holes have mass. It also obeys the Gauss Law” is to mean not just that Gauss’s law holds, but that it’s integral form can still be used to calculate the force on the Earth in a trivial matter depending only upon the mass inside the closed surface and hence then the force on the Earth would stay the same as if the sun had not split in two.
I hope that we both agree that Gauss’s law could not be used in such a manner (as it only gives the integral of flux over the surface and so without symmetry this force is not constant over the surface). It believe that you would also agree that splitting the sun in two and sending them in opposite directions as described would result (in the case of Newtonian gravity) in a decrease in the gravitational force the Earth experienced. Given this, I am not sure how to charitably interpret the comment I quoted. Could you elaborate on what you meant if you didn’t mean what I thought you meant? I will at this point certainly admit and apologize for the fact that I interpreted your comments uncharitably since it sounded like you were making a common beginner’s error.
My three-body comment meant this: if we could use Gauss’s law to say that the force from inside of an enclosed area is directly proportional to the mass inside, then we would be able to draw a region in space around two of the three bodies and calculate the force on the third. This force would then depend only upon the total mass which is constant and would point to the center of the enclosed area. So then we’d be able to solve for that one body’s motion independent of the other two. This is patently absurd and not possible. I chose to give it as an example of how such a naive application of Gauss’ law would give obviously incorrect results but the illusion of transparency grabbed me and I failed to make myself clear.
This is true in most cases, but not in spherical symmetry (there are no spherically symmetric gravitational waves).
That comment was not directed at your original post but at some of the comments on splitting the sun in two and shooting them apart. Sorry for the confusion!
What I took you to mean when you made comments like “This still conserves mass, since black holes have mass. It also obeys the Gauss Law” is to mean not just that Gauss’s law holds, but that it’s integral form can still be used to calculate the force on the Earth in a trivial matter depending only upon the mass inside the closed surface and hence then the force on the Earth would stay the same as if the sun had not split in two.
This would be a novice mistake I have been correcting countless times as a tutor and TA. The Gauss law holds even when there is no symmetry, but it is much less useful to calculating electric or gravitational field at a given point. It is, however, can be profitably used to argue other points, like the one I made.
I will at this point certainly admit and apologize for the fact that I interpreted your comments uncharitably since it sounded like you were making a common beginner’s error.
No need for an apology, I’m glad we cleared that up.
if we could use Gauss’s law to say that the force from inside of an enclosed area is directly proportional to the mass inside, then we would be able to draw a region in space around two of the three bodies and calculate the force on the third.
I understand what you mean now, given what you said previously and I certainly agree that this would make no sense.
As an expert in the area:
False, it applies to any closed surface. Spherical symmetry is just the simplest problem where the Gauss law can be usefully applied.
The three-body problem has no relation to the Gauss law, except insofar as it is equivalent to the inverse square law for point masses in flat space.
This is true in most cases, but not in spherical symmetry (there are no spherically symmetric gravitational waves). That’s why I could write “an amount of matter equivalent to her mass has to travel inwards out of nowhere and coalesce into a person” and not worry about energy losses to gravitational radiation. Though there would be, of course, a lot of losses to EM radiation if the matter in question consisted of charged particles.
Hope this makes sense.
Oh I absolutely don’t deny that. What I do assert is that you can’t ignore what happens on the inside of a closed surface if you don’t know that the mass is spherically symmetric and you want to calculate the force at a specific force. If this is not what you are saying, then this is a miscommunication not a disagreement. What I took you to mean when you made comments like “This still conserves mass, since black holes have mass. It also obeys the Gauss Law” is to mean not just that Gauss’s law holds, but that it’s integral form can still be used to calculate the force on the Earth in a trivial matter depending only upon the mass inside the closed surface and hence then the force on the Earth would stay the same as if the sun had not split in two.
I hope that we both agree that Gauss’s law could not be used in such a manner (as it only gives the integral of flux over the surface and so without symmetry this force is not constant over the surface). It believe that you would also agree that splitting the sun in two and sending them in opposite directions as described would result (in the case of Newtonian gravity) in a decrease in the gravitational force the Earth experienced. Given this, I am not sure how to charitably interpret the comment I quoted. Could you elaborate on what you meant if you didn’t mean what I thought you meant? I will at this point certainly admit and apologize for the fact that I interpreted your comments uncharitably since it sounded like you were making a common beginner’s error.
My three-body comment meant this: if we could use Gauss’s law to say that the force from inside of an enclosed area is directly proportional to the mass inside, then we would be able to draw a region in space around two of the three bodies and calculate the force on the third. This force would then depend only upon the total mass which is constant and would point to the center of the enclosed area. So then we’d be able to solve for that one body’s motion independent of the other two. This is patently absurd and not possible. I chose to give it as an example of how such a naive application of Gauss’ law would give obviously incorrect results but the illusion of transparency grabbed me and I failed to make myself clear.
That comment was not directed at your original post but at some of the comments on splitting the sun in two and shooting them apart. Sorry for the confusion!
This would be a novice mistake I have been correcting countless times as a tutor and TA. The Gauss law holds even when there is no symmetry, but it is much less useful to calculating electric or gravitational field at a given point. It is, however, can be profitably used to argue other points, like the one I made.
No need for an apology, I’m glad we cleared that up.
I understand what you mean now, given what you said previously and I certainly agree that this would make no sense.
Inferential distance is a b****.