Newton's shell theorem with negative mass
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Newton's shell theorem with negative mass
As Newton proved in his shell theorem, "If the body is a spherically symmetric shell (i.e., a hollow ball), no net gravitational force is exerted by the shell on any object inside, regardless of the object's location within the shell." So if we make a big hole in the centre of the Earth and assume it's perfectly spherical and with equally distributed mass, you won't be able to walk on the inside of the hole. But what if the object inside had negative mass? (Let's assume for the sake of this argument that it exists and it repels gravitationally positive mass.) I'm inclined to think it would also experience no gravitational force, but I kinda hope I'm wrong because that would make my life a bit easier.
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Re: Newton's shell theorem with negative mass
Changing the sign would make no difference in this case. The theorem still holds.
Re: Newton's shell theorem with negative mass
Or to put it another way, the mathematics is all done without considering whether the mass is positive or negative, and since the final answer doesn't depend on the mass it must still work.
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Re: Newton's shell theorem with negative mass
One of the most important properties of gravity, and one which was not properly understood until the dawn of general relativity was the fact that all bodies are affected equally by gravity in the sense that they all experience the same acceleration.
In Newtonian gravity, this is just an unexplained quirk arrising from the fact that gravitational "charge" is proportional to the inertial mass with no good justification as to why this need be the case. Once you realise that gravity is not a force per se but rather arises due to the underlying curvature of space, it becomes a lot clearer why all bodies moving through that same curvature should be affected equally.
There are a few subtleties: for one, the relativistic value for the deflection of light differs from the "Newtonian" one (only really derived in the 20th century but using the idea that light must also accelerate at the same rate as any other body) by a factor of 2. But as long as you stay away from 0 mass, gravity doesn't care what your mass is so a ve mass body inside the shell will not experience a force any more than a +ve mass one.
In Newtonian gravity, this is just an unexplained quirk arrising from the fact that gravitational "charge" is proportional to the inertial mass with no good justification as to why this need be the case. Once you realise that gravity is not a force per se but rather arises due to the underlying curvature of space, it becomes a lot clearer why all bodies moving through that same curvature should be affected equally.
There are a few subtleties: for one, the relativistic value for the deflection of light differs from the "Newtonian" one (only really derived in the 20th century but using the idea that light must also accelerate at the same rate as any other body) by a factor of 2. But as long as you stay away from 0 mass, gravity doesn't care what your mass is so a ve mass body inside the shell will not experience a force any more than a +ve mass one.
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Re: Newton's shell theorem with negative mass
eSOANEM wrote:One of the most important properties of gravity, and one which was not properly understood until the dawn of general relativity was the fact that all bodies are affected equally by gravity in the sense that they all experience the same acceleration.
Not super relevant to Matt_Rethyu's question, but this doesn't hold for negative mass objects, right? If I drop a 10 kg object and a 1 kg object from the same height they will hit the ground at the same time (ignoring air resistance), but an object that weighs 1 kg will never hit the ground at all.
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Re: Newton's shell theorem with negative mass
No, the 1kg object will fall down.
The difference is that actually the 10kg object hits the earth faster than the 1kg object since the earth falls towards the 10kg object faster. Likewise, while the 1kg object is falling down, the earth is falling in the same direction, (so away from the 1kg object), so the 1kg object will hit the ground after the other two.
The difference is that actually the 10kg object hits the earth faster than the 1kg object since the earth falls towards the 10kg object faster. Likewise, while the 1kg object is falling down, the earth is falling in the same direction, (so away from the 1kg object), so the 1kg object will hit the ground after the other two.
Re: Newton's shell theorem with negative mass
Nicias has the right of it (although the effect is tiny because the earth is so many times heavier than the other bodies).
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Re: Newton's shell theorem with negative mass
Nicias wrote:No, the 1kg object will fall down.
The difference is that actually the 10kg object hits the earth faster than the 1kg object since the earth falls towards the 10kg object faster. Likewise, while the 1kg object is falling down, the earth is falling in the same direction, (so away from the 1kg object), so the 1kg object will hit the ground after the other two.
Oh, that is super interesting, so what happens if you have a 1kg object and a 1kg object, does the positive mass object fall away from the negative mass object at the same speed the negative mass object falls towards the positive mass object? That seems like it violates conservation of energy/momentum.
Edit: And one wiki search later, it appears that this is indeed what happens, but it does not violate conservation laws and it is one of the things that makes negative mass so weird. So, yeah, I learned things today
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Re: Newton's shell theorem with negative mass
Nicias wrote:Likewise, while the 1kg object is falling down, the earth is falling in the same direction, (so away from the 1kg object), so the 1kg object will hit the ground after the other two.
First I thought "what the hell are you on about?", but it's even correct in newtonian gravity: the force is repelling both masses (negative times positive is negative), but a negative mass accelerates in the opposite direction of the force (vector divided by a negative scalar is an opposing vector), so in the end it ascends down to the Earth.
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peregrine_crow wrote:Oh, that is super interesting, so what happens if you have a 1kg object and a 1kg object, does the positive mass object fall away from the negative mass object at the same speed the negative mass object falls towards the positive mass object? That seems like it violates conservation of energy/momentum.
How would it violate anything? As a closed system it's in perfect equilibrium.

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Re: Newton's shell theorem with negative mass
Flumble wrote:How would it violate anything? As a closed system it's in perfect equilibrium.
That appears to be the case, yes, so can we now do the magnet attached to a pole attached to a train thing and have it actually work (in deep space at least)?
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Re: Newton's shell theorem with negative mass
I actually did my doctoral dissertation on negative mass, but mostly on cosmological scales.
A matched pair of positive and negative mass particles would indeed accelerate forever. This doesn't violate conservation of energy, since the negative mass has negative kinetic energy.
Generalizing, a gas with negative mass and positive mass particles would have particles of both masses have their average particle speeds increase without bound. The temperature would not increase since a negative mass gas has negative temperature. Since we do not observe this, we can be confident that there are no negative mass particles in the universe. (I'm not super confident about the exact details of the gas situation. It's been years since I've looked at this stuff.)
A matched pair of positive and negative mass particles would indeed accelerate forever. This doesn't violate conservation of energy, since the negative mass has negative kinetic energy.
Generalizing, a gas with negative mass and positive mass particles would have particles of both masses have their average particle speeds increase without bound. The temperature would not increase since a negative mass gas has negative temperature. Since we do not observe this, we can be confident that there are no negative mass particles in the universe. (I'm not super confident about the exact details of the gas situation. It's been years since I've looked at this stuff.)
Re: Newton's shell theorem with negative mass
Nicias wrote:I actually did my doctoral dissertation on negative mass, but mostly on cosmological scales.
A matched pair of positive and negative mass particles would indeed accelerate forever. This doesn't violate conservation of energy, since the negative mass has negative kinetic energy.
Generalizing, a gas with negative mass and positive mass particles would have particles of both masses have their average particle speeds increase without bound. The temperature would not increase since a negative mass gas has negative temperature. Since we do not observe this, we can be confident that there are no negative mass particles in the universe. (I'm not super confident about the exact details of the gas situation. It's been years since I've looked at this stuff.)
If you attached a 1kg mass to a 1kg mass, would you not essentially have a massless object? Would its inertial mass also be zero, meaning it had to travel at the speed of light? (edit: assuming for the moment the ideal case were you could make it exactly zero)
Re: Newton's shell theorem with negative mass
It's a compound object though so it's not quite that simple. Both objects have +ve mass squared so are restricted to timelike paths (speeds below c) even though the total mass could be zero.
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