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Now if the same body is in orbit around earth at same height, owing to its velocity its mass increases causing the PE at that height to be more than what it was when the body was stationary.

Where did the extra energy come from?

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- #1

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Now if the same body is in orbit around earth at same height, owing to its velocity its mass increases causing the PE at that height to be more than what it was when the body was stationary.

Where did the extra energy come from?

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Ibix

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Whatever you used to accelerate the mass.

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But the work done by that external agent would cause increase in kinetic energy.it is not doing any work against force of gravity(as they are perpendicular). So if its not doing work against gravity how is its work converted to gravitational PE?Whatever you used to accelerate the mass.

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Nugatory

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Your premise is mistaken - it's mass does not increase because of its velocity, at least not the way you're thinking, so the question does not arise. Confusion of this sort is why the the concept of relativistic mass is seldom used these days: https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/Now if the same body is in orbit around earth at same height, owing to its velocity its mass increases causing the PE at that height to be more than what it was when the body was stationary.

You can safely say that the total energy of the orbiting object is greater than the energy of the suspended object. The "extra" energy is kinetic energy, and as @Ibix says, it came from whatever you used to accelerate the object.

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Thank you. I understand nowYour premise is mistaken - it's mass does not increase because of its velocity, at least not the way you're thinking, so the question does not arise. Confusion of this sort is why the the concept of relativistic mass is seldom used these days: https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

You can safely say that the total energy of the orbiting object is greater than the energy of the suspended object. The "extra" energy is kinetic energy, and as @Ibix says, it came from whatever you used to accelerate the object.

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Now if the same body is in orbit around earth at same height, owing to its velocity its mass increases causing the PE at that height to be more than what it was when the body was stationary.

Where did the extra energy come from?

Suppose two bodies at different temperatures are held stationary at a height h above earth, having some PE

Now if the hotter body heats the cooler body, owing to the cooler body's increased heat energy its mass increases causing the PE at that height to be more than what it was when the body was cooler.

Where did the extra energy come from?

- #8

Nugatory

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The hotter body lost energy as it cooled.Now if the hotter body heats the cooler body, owing to the cooler body's increased heat energy its mass increases causing the PE at that height to be more than what it was when the body was cooler.

Where did the extra energy come from?

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Okay but there is still one problem: The hotter body lost potential energy as it cooled. Where did that energy go?The hotter body lost energy as it cooled.

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Okay I answer myself: It went to the body that gained potential energy.

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Special relativity by itself will not handle gravity. Getting gravity to work correctly along with relativity was why Einstein created General Relativity. This was not an easy, or short, task that can be fully explained in a post that's a few paragraphs long. It gets involved, but we can say a few things. The shortest reasonable answer is that, as I already said, we do need General relativity, not special relativity to answer your question, and that there is more to GR than plugging in some different values for mass into Newtonian formulae.

Now if the same body is in orbit around earth at same height, owing to its velocity its mass increases causing the PE at that height to be more than what it was when the body was stationary.

Where did the extra energy come from?

I could give you a brief rundown of some of the various sorts of masses that are defined in special and general relativity (I'm not sure I know all of the later but I could talk about the 'Big Three'). However, I'm not sure if that's interesting to you especially since it won't be sufficient to answer your question. None of these sorts of masses will allow you to plug them into Newton's formula, and get the answer you seek.

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It's not that easy. The hot body loses energy by emission of heat radiation and the cold body receives energy by absorption of this radiation. If the emitted and absorbed energy would be equal, the sum of the rest energies of both bodies would remain constant. But the potential energy of the system would change due to the changed mass of the bodies. The question remains, where the difference comes from or goes to and the answer is: The heat released from the hot body and the heat absorbed by the cold body are actually different due to red- or blueshift (depending on the mass ratio of the bodies). The difference comes from or goes to the potential energy of the system.The hotter body lost energy as it cooled.

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It's not that easy. The hot body loses energy by emission of heat radiation and the cold body receives energy by absorption of this radiation. If the emitted and absorbed energy would be equal, the sum of the rest energies of both bodies would remain constant. But the potential energy of the system would change due to the changed mass of the bodies. The question remains, where the difference comes from or goes to and the answer is: The heat released from the hot body and the heat absorbed by the cold body are actually different due to red- or blueshift (depending on the mass ratio of the bodies). The difference comes from or goes to the potential energy of the system.

Let's say object number two is a sphere around object number one. This way we get rid of the distracting redshift.

And let's put those two objects as far from all masses as possible. This makes the potential energy of the objects as large as possible. We usually do this when we do not want to consider general relativity, right? So maybe we can forget the general relativity here too?

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No, we don't. There is still a red- or blueshift in the gravitational field of object one.Let's say object number two is a sphere around object number one. This way we get rid of the distracting redshift.

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No, we don't. There is still a red- or blueshift in the gravitational field of object one.

Oh. Well then object one is two thin and light papers, one paper attached to the inside of the object #2, other one on the outside of the object #2. (Object #2 was a sphere, if anyone has forgotten)

Now I hope redshift is eliminated. I mean on the average redshift should be zero.

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Nugatory

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That's not right. One way to see this is to analyze the problem with the zero point of the gravitational potential set at the height of the two-body system, instead of at infinity or the surface of the earth as we usually do. Now the potential energy of the system is zero and remains zero no matter what happens to the masses of the individual objects. Thus, conservation of energy requires that whatever is lost by one body is gained by the other; there's nowhere else for it to go.It's not that easy. The hot body loses energy by emission of heat radiation and the cold body receives energy by absorption of this radiation. If the emitted and absorbed energy would be equal, the sum of the rest energies of both bodies would remain constant. But the potential energy of the system would change due to the changed mass of the bodies. The question remains, where the difference comes from or goes to and the answer is: The heat released from the hot body and the heat absorbed by the cold body are actually different due to red- or blueshift (depending on the mass ratio of the bodies). The difference comes from or goes to the potential energy of the system.

You are right that if the two bodies are exchanging heat through thermal radiation there will be a (tiny) gravitational redshift/blueshift if their masses are different. But in this case clocks on the surface of each object are also running at different rates because gravitational time dilation is at work. The "uphill" object will receive low intensity red-shifted light for a longer time, so ends up receiving the same amount of energy as was emitted by the "downhill" object.

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That depends on the specific conditions. If yes, than the change of the potential energy should also be zero.I mean on the average redshift should be zero.

What is "the height of the two-body system"?One way to see this is to analyze the problem with the zero point of the gravitational potential set at the height of the two-body system, instead of at infinity or the surface of the earth as we usually do.

Edit:

After reading your post again I have an idea what you mean. Are you talking about the potential energy of the two-body system in the gravitational field of a third body only? That is indeed constant, but that's not the full story. There is also a potential energy due to the gravitational field of the two bodies itself and this part of the total potential energy is changing.

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https://www.physicsforums.com/threads/potential-energy-and-annihilation.936094/

And here in this thread a poster has a problem with appearing potential energy.

So maybe the solution to the disappearing potential energy there is the appearing potential energy here, and vice versa.

(Seems obvious to me that it it so)

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Yes, that's what I mean. The potential energy of the system is increased and the energy of the radiation is decreased by redshift (or the other way around with blueshift). The total energy remains constant.So maybe the solution to the disappearing potential energy there is the appearing potential energy here, and vice versa.

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Do you mean this redshift:Yes, that's what I mean. The potential energy of the system is increased and the energy of the radiation is decreased by redshift (or the other way around with blueshift). The total energy remains constant.

Light climbs away from the annihilation point and redshifts, because light is climbing away from other light. The amount of redshift is maybe a few microjoules.

Or this:

Matter-antimatter fuel is lifted from the ground to a position high above the earth. The lifting causes the light of annihilation of the fuel to have more energy, maybe a few megajoules more.

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That's not right. One way to see this is to analyze the problem with the zero point of the gravitational potential set at the height of the two-body system, instead of at infinity or the surface of the earth as we usually do. Now the potential energy of the system is zero and remains zero no matter what happens to the masses of the individual objects. Thus, conservation of energy requires that whatever is lost by one body is gained by the other; there's nowhere else for it to go.

You are right that if the two bodies are exchanging heat through thermal radiation there will be a (tiny) gravitational redshift/blueshift if their masses are different. But in this case clocks on the surface of each object are also running at different rates because gravitational time dilation is at work. The "uphill" object will receive low intensity red-shifted light for a longer time, so ends up receiving the same amount of energy as was emitted by the "downhill" object.

No. Anything that climbs loses energy. When we ask where the lost energy goes, the answer is: "it goes to the system". I have seen that answer many times here.

If two things climb away from each other, both things lose energy. The system gains potential energy.

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How can two objects bothNo. Anything that climbs loses energy. When we ask where the lost energy goes, the answer is: "it goes to the system". I have seen that answer many times here.

If two things climb away from each other, both things lose energy. The system gains potential energy.

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When an object moves from lower potential to higher potential, it 'climbs', except when something else 'lifts' the object. To 'climb' is to 'self-lift'.How can two objects bothclimbaway from each other unless they are on opposite sides of the larger body? Also I'm pretty sure in Nugatory's example the height of the objects with respect to the designated origin is not changing.

When space contains two objects, there is a potential difference between any two points of the space, except when the two points happen to be on the same equipotential surface.

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Earth and Moon climb away from each other without a larger body between them. During this process the kinetic energies of both objects decrease and the common potential energy increases.How can two objects bothclimbaway from each other unless they are on opposite sides of the larger body?