# Energy, heat and speed

## Main Question or Discussion Point

My question is two-fold:

1). I understand that according to Einstein's theories that as one approaches the speed of light, e.g. interstellar flight in a spaceship, the mass of the moving object approaches infinity. E=mc^2 the increasing mass correspond to an increasing amount of energy. Is this energy in the form of potential energy, i.e. could be released by a collision with another object, or some other type of energy?

2). In reference to my first question, is there any relationship between the energy of a moving object and thermal energy? I know that c is the upper limit for speed of an object, but does mean it has the maximum amount of energy it can store as potential energy? Is there an upper bound for thermal energy? I have heard that thermal energy is the kinetic energy of particles in a system. I also have heard that all bodies emit thermal radiation in the form of black-body radiation, but I've never heard of any object emitting energy because of the speed its traveling at. Is there any correlation between the thermal energy of a hot object, e.g. a hot cloud of plasma, and the energy of a moving object,e.g. a fast moving comet?

I have no real knowledge of physics so feel free to correct me if I have missed anything or made absurd assumptions.

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To clarify the second question, is there correlation between the energy added by linear motion, e.g. a spaceship where all of its particles are moving with the same velocity, and heat added to a system causing the particles to move faster?

My question is two-fold:

1). I understand that according to Einstein's theories that as one approaches the speed of light, e.g. interstellar flight in a spaceship, the mass of the moving object approaches infinity. E=mc^2 the increasing mass correspond to an increasing amount of energy. Is this energy in the form of potential energy, i.e. could be released by a collision with another object, or some other type of energy?

2). In reference to my first question, is there any relationship between the energy of a moving object and thermal energy? I know that c is the upper limit for speed of an object, but does mean it has the maximum amount of energy it can store as potential energy? Is there an upper bound for thermal energy? I have heard that thermal energy is the kinetic energy of particles in a system. I also have heard that all bodies emit thermal radiation in the form of black-body radiation, but I've never heard of any object emitting energy because of the speed its traveling at. Is there any correlation between the thermal energy of a hot object, e.g. a hot cloud of plasma, and the energy of a moving object,e.g. a fast moving comet?

I have no real knowledge of physics so feel free to correct me if I have missed anything or made absurd assumptions.
Your question touches on the relationship of relativity and thermodynamics. The relationship is not clear. I have Tolman's book on relativity and thermodymanics that concludes the temperature of a moving body cools (T' = T/y). This is the classic interpretation but a number of experts since have claimed the relativistic temperature is covariant (T'=Ty) and yet others that claim temperature is invariant (T'=T). To that you can add experts that claim there is no simple relationship between temperature and relative velocity and recent paper that introduces a 4 inverse temperature. I have been comparing these papers on and off for a number of weeks to try and see how and why they differ and hope to start a thread on that soon.

I can comment on some of the issues you raise.

Yes, the energy would be released in some form in a collision.

An infinite amount of energy would be required to accelerate a massive body to the speed of light so that observation does not put any finite upper bound to the kinetic energy or temperature of a body.

Yes, all bodies that are not at absolute zero temperature (which is supposed to be impossible) emit radiation which is proportional (for an ideal black body) to their temperature. We can note that the peak frequency of the radiation from a black body is proportional to its temperature. For a body with relative motion we can also note that the frequency of the radiation from the body reduces by the Lorentz factor (after we have deducted the doppler shift due the the body comeing towards us or going away) hinting at the drop in relativistic temperature due to relative motion. Not all bodies are exact ideal black bodies but most cosmologists/ astronomers aproximate stars as black bodies when trying to figure out their temperaures.

While it is not absolutely clear if a body emits energy because of the speed it is traveling at, it is certain that particles emit radiation while accelerating. This is known as synchrotron radiation. This is regularly observed in synchroton accelerators and particles spiralling down towards a black hole in the acretion disk are thought to give off large quantities of radiation in the form of x-rays.

To clarify the second question, is there correlation between the energy added by linear motion, e.g. a spaceship where all of its particles are moving with the same velocity, and heat added to a system causing the particles to move faster?
The assumption that all the particles of a rocket are moving at the same velocity is an aproximation because a large object like a rocket can not be at absolute zero temperature. Thermal vibrations of atoms mean some of the atome will be be moving faster than the average velocity of the rocket and some will be moving slower. However, it is a reasonable aproximation at low temperatures. On the understanding of that aproximation, we used 10,000 Joules of energy to accelerate brick A to a given velocity and another 10,000 Joules of energy to heat brick B to a given temperature then the average kinetic energy of the atoms in both bricks will be the same. The only difference is that motion of individual atoms in the first brick are are all in the same direction (coherant), while the motion of the vibrating atoms of the second brick are in random directions (incoherant).

I hope at least some of that helps.

Yes, the explanation about relativity and temperature and the brick experiment did help, but also led me to some additional questions:

An infinite amount of energy would be required to accelerate a massive body to the speed of light so that observation does not put any finite upper bound to the kinetic energy or temperature of a body.
As far as the quote above, does this mean that there is no bound to the amount of thermal energy that can exist in a system? I thought there was a bound at the amount of energy/mass that can be in one place before a black hole is created?

The only difference is that motion of individual atoms in the first brick are are all in the same direction (coherant), while the motion of the vibrating atoms of the second brick are in random directions (incoherant).
Does one have to add more energy to an incoherent system than a coherent system to arrive at the same average speed of each particle? I would think that the collisions taking place in an incoherent system would cause it to radiate more energy as heat, and subsequently take longer for individual particles to speed up, than a coherent system. Therefore, would it take less time to accelerate the particles in a coherent system to a certain speed than heat up particles in an incoherent one? Do incoherent systems radiate more energy as heat than coherent systems? I've also heard the reason why a object would require an infinite amount of energy to accelerate to the speed of light was because the mass of increases as the speed of the object increases. Is it true that particles in an incoherent system become more massive as their thermal energy is increased?

I have Tolman's book on relativity and thermodymanics that concludes the temperature of a moving body cools (T' = T/y).This is the classic interpretation but a number of experts since have claimed the relativistic temperature is covariant (T'=Ty) and yet others that claim temperature is invariant (T'=T). To that you can add experts that claim there is no simple relationship between temperature and relative velocity and recent paper that introduces a 4 inverse temperature. I have been comparing these papers on and off for a number of weeks to try and see how and why they differ and hope to start a thread on that soon.
I though as an object approached the speed of light, its mass increased, not its temperature. I would think though that adding energy to a system in the form of speeding it up would make it hotter? Why would it lose energy as heat? Would this cooling reduce the objects speed in any way, or make the object less massive? I assume by the last sentence of this quote that the relation between relativity and thermodynamics is still not completely understood, so if these questions are unanswerable at this time I understand. Thank you.

............... does this mean that there is no bound to the amount of thermal energy that can exist in a system? I thought there was a bound at the amount of energy/mass that can be in one place before a black hole is created?
You are assuming that the particles in the sytem stay within the same volume ("in one place") as it is heated. A star with just over ten solar masses has enough gravitational energy to form a black hole. While the star still has nuclear fuel to burn, the thermal energy of its gas particles is enough to resist it collapsing to a black hole. If we somehow added even more energy to the star it would expand, increasing its volume and reducing its density preventing it becomeing a black hole. When the star runs out of nuclear fuel, it collapses rapidly, releasing a great deal of gravitational potential energy which creates a supernova explosion ejecting matter all over the place but enough mass remains to form a black hole.

.............. Does one have to add more energy to an incoherent system than a coherent system to arrive at the same average speed of each particle? I would think that the collisions taking place in an incoherent system would cause it to radiate more energy as heat, and subsequently take longer for individual particles to speed up, than a coherent system.Therefore, would it take less time to accelerate the particles in a coherent system to a certain speed than heat up particles in an incoherent one?
In the brick example you would have to insulate the brick that you are heating up to make it a fair comparison.

Do incoherent systems radiate more energy as heat than coherent systems?
Yes

I've also heard the reason why a object would require an infinite amount of energy to accelerate to the speed of light was because the mass of increases as the speed of the object increases.
This is a difficult subject. Classically mass increases with relative velocity. The formal modern view is that rest mass is invariant and we do not discuss relativistic mass. One reason for this is that an object moveing at a very high relative speed to the observer might be considered to have enough mass within a given volume to become a black hole. To an observer comoving with the object, it is not a black hole. An object that is a black hole to one one observer while not a black hole to another observer is clearly a contadiction. Einstein said there is no clear definition of mass. Like I said, it is not an easy subject.

Consider a particle fired horizontally from a tower at 0.8c relative to an observer on the tower. Say that observer measures 10 seconds for that particle to fall to the ground. (It is a very long flat planet :P) To an observer comoving with the particle it takes 6 seconds for the particle to fall to the ground. Assuming the the height of the tower and mass of the planet are such that the difference in gravitational time dilation between the top of the tower and the ground is negligable then why the difference?

Is it true that particles in an incoherent system become more massive as their thermal energy is increased?
Once again this is difficult to answer, if we can not talk about relativistic mass. What we can note, is that if we have a very hot brick and a very cold brick that it will take more energy to accelerate the the hot brick to a given velocity than the cold brick. The hot brick is harder to accelerate. Some people interpret that as the hot brick having more inertial mass (more resistance to acceleration) than the cold brick.

I though as an object approached the speed of light, its mass increased, not its temperature.
As I said before, the formal aproach is that its mass does not increase, but it does resist acceleration more. I think the more formal version is that the momentum energy of the system increases. As for the temperature, the experts are not agreed on whether it increases or decreases or something in between.

I would think though that adding energy to a system in the form of speeding it up would make it hotter?
Well, at least some of the energy has to spent on speeding up the system and increasing the overall (coherent) momentum of the system). This usually involves ejecting mass in the opposite direction so that momentum of the universe is conserved. The difficulty comes when analysing the motions of individual particles in a moving system in deciding how much of the motion is coherent and how much is incoherent. I have been looking at it for a while and it is not easy, and papers on the subject disagree.

Why would it lose energy as heat? Would this cooling reduce the objects speed in any way, or make the object less massive?
See all the above.

I assume by the last sentence of this quote that the relation between relativity and thermodynamics is still not completely understood, so if these questions are unanswerable at this time I understand. Thank you.
Here are some papers on relativistic thermodynamics if you want to sift through them:

P' and T' are the transformed pressure and temperature.

P' = Py^2 , T' = Ty http://arxiv.org/PS_cache/arxiv/pdf/0712/0712.3793v2.pdf

P' = ? , T' = ? http://arxiv.org/PS_cache/physics/pdf/0303/0303091v3.pdf

P' = ? , T' = ? http://arxiv.org/PS_cache/gr-qc/pdf/9803/9803007v2.pdf

P' = P , T' = Ty http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.2639v1.pdf

P' = P , T= T/y http://arxiv.org/PS_cache/physics/pdf/0505/0505004v2.pdf

P' = P, T= T/y is my interpretation of the last paper but they do not spell it out. Tolman's book on relativity and thermodynamics also concludes P' = P, T= T/y but the book is rather old.

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Thank you for the help.