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TobyC
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We had a physics lesson today on nuclear binding energy, where we looked at the mass of two protons and two neutrons independently and compared it with the mass of a helium nuclei, and were told how the discrepancy related to the release of energy during nuclear fusion.
The first question in the textbook then asked us to calculate how much mass a 10kg block gains when it is raised by 2 meters above the ground, using E=mc^2, I guess it was intended as a way of introducing us to the concept of binding energy in a way that was familiar to us.
The first problem I have is that I've read while learning about electromagnetism that the potential energy of a system is not stored with the particles, but is stored in the field, so the block in this example would not gain mass at all, but rather the field would. If the book was correct then as a particle falls its increase in kinetic energy would always be directly countered by a decrease in its potential energy (so that it's mass doesn't change) and then what would stop things exceeding the speed of light?
The second thing that occurred to me is that its mass probably does change, not directly because of its change in energy, but because of the relativistic effects of a gravitational field, in the same way the mass of an object increases as it approaches the speed of light. I know the concept of mass becomes a bit weirder once you get into general relativity and start dealing with non-inertial coordinate systems, but I thought that it if you take the weak field limit where you can treat space-time as having a minkowski metric with a tensor field over the top representing the pertubation, then you could come up with a meaningful definition of mass (and when I'm talking about mass here I'm not referring to gravitation, but inertial mass).
I calculated the three vector momentum for a particle in a weak field by differentiating position with respect to proper time and multiplying by the rest mass, and I got a result which was the product of the particle's velocity and some other complicated function of the field, the rest mass, and the velocity, and I defined this function to be the mass. I thought this should work because if an experiment was conducted somewhere in the field then it would follow the normal Newtonian laws of physics if you interpreted this function to be the mass.
So basically what I came up with is that the mass of an object decreases if you raise its position in a gravitational field, and its dependence on velocity is different to standard special relativity. My expression for the 'mass' of a particle in a gravitational field is this:
[tex]\frac{m}{\sqrt{1-v^{2}}} - m\phi - m(\phi)(v^{2})[/tex]
Where the units are chosen so that the speed of light equals 1, and m is the rest mass, [tex]\phi[/tex] is the gravitational potential, and v is the speed of the particle.
The physical significance of this expression is the mass the particle would appear to have in mechanical processes. For example, if this particle was at a great height and viewed from the ground, when the motion is analysed this is the mass the particle would appear to have.
I tried to check this result online when I derived it, but I couldn't find a reference to it anywhere, so I decided to post here to ask whether I've done anything wrong or overlooked anything, I don't really know a great deal about general relativity. Does the mass of an object really appear to decrease in this way when you raise it in a gravitational field?
The final question I have isn't really related to relativity, but goes back to the original subject of the lesson. If potential energy is stored in the field rather than in the particles, why is a proton more massive on its own than when it is combined in a nucleus?
The first question in the textbook then asked us to calculate how much mass a 10kg block gains when it is raised by 2 meters above the ground, using E=mc^2, I guess it was intended as a way of introducing us to the concept of binding energy in a way that was familiar to us.
The first problem I have is that I've read while learning about electromagnetism that the potential energy of a system is not stored with the particles, but is stored in the field, so the block in this example would not gain mass at all, but rather the field would. If the book was correct then as a particle falls its increase in kinetic energy would always be directly countered by a decrease in its potential energy (so that it's mass doesn't change) and then what would stop things exceeding the speed of light?
The second thing that occurred to me is that its mass probably does change, not directly because of its change in energy, but because of the relativistic effects of a gravitational field, in the same way the mass of an object increases as it approaches the speed of light. I know the concept of mass becomes a bit weirder once you get into general relativity and start dealing with non-inertial coordinate systems, but I thought that it if you take the weak field limit where you can treat space-time as having a minkowski metric with a tensor field over the top representing the pertubation, then you could come up with a meaningful definition of mass (and when I'm talking about mass here I'm not referring to gravitation, but inertial mass).
I calculated the three vector momentum for a particle in a weak field by differentiating position with respect to proper time and multiplying by the rest mass, and I got a result which was the product of the particle's velocity and some other complicated function of the field, the rest mass, and the velocity, and I defined this function to be the mass. I thought this should work because if an experiment was conducted somewhere in the field then it would follow the normal Newtonian laws of physics if you interpreted this function to be the mass.
So basically what I came up with is that the mass of an object decreases if you raise its position in a gravitational field, and its dependence on velocity is different to standard special relativity. My expression for the 'mass' of a particle in a gravitational field is this:
[tex]\frac{m}{\sqrt{1-v^{2}}} - m\phi - m(\phi)(v^{2})[/tex]
Where the units are chosen so that the speed of light equals 1, and m is the rest mass, [tex]\phi[/tex] is the gravitational potential, and v is the speed of the particle.
The physical significance of this expression is the mass the particle would appear to have in mechanical processes. For example, if this particle was at a great height and viewed from the ground, when the motion is analysed this is the mass the particle would appear to have.
I tried to check this result online when I derived it, but I couldn't find a reference to it anywhere, so I decided to post here to ask whether I've done anything wrong or overlooked anything, I don't really know a great deal about general relativity. Does the mass of an object really appear to decrease in this way when you raise it in a gravitational field?
The final question I have isn't really related to relativity, but goes back to the original subject of the lesson. If potential energy is stored in the field rather than in the particles, why is a proton more massive on its own than when it is combined in a nucleus?