B Questions about the relativistic kinetic energy expressions

[PeterDonis]

It bothers me that loss of heat/energy has not been considered as a natural outcome of the energy gain.

Why do you think it has not been considered? The fact that once an object heats up it will give off heat is commonplace.

As we agreed, even if the system closed, the radiation energy may penetrate the material and heat it

No, we did not agree to this. If radiation is penetrating the material and heating it, the material, as a system, is not a closed system. If you consider the source of the radiation, plus the radiation, plus the material being heated by the radiation, all together, that might be a closed system (but you would also, as you note, need to consider the heat given off by the material once it was heated).

 

[Ricardo]

Well, I guess, I unsuccessfully used your former line below.

"That's correct. Either that, or you count the radiation as part of the system from the beginning, and add its energy in accordingly."
............
In reference to your recent line,
"Why do you think it has not been considered? The fact that once an object heats up it will give off heat is commonplace."

What I am trying to say, is that I don't see in the formulas displayed here or in other sources the accounting of this process. For example, in E. "Relativity" (which may be not the best source) the reference is given only to the absorbed energy. It is possible that under this term the net absorbed energy is considered.

 

[Ibix]

What I am trying to say, is that I don't see in the formulas displayed here or in other sources the accounting of this process. For example, in E. "Relativity" (which may be not the best source) the reference is given only to the absorbed energy. It is possible that under this term the net absorbed energy is considered.

If you put a body on a scale and heat it, its mass will increase from $m$ to $m'=m+H/c^2$, where $H$ is the input energy (less anything lost to the environment during heating). This is because its rest energy has changed from $E$ to $E'=E+H$ and, for a stationary body, $mc^2=E$. It will then sit there and cool, with its mass decreasing back to $m$ as it does so. If you absorb the radiated energy in some other body, that body's mass will increase - although presumably it'll radiate at a similar rate to what it absorbs, so the increase will be tiny.

You may like to consider the size of $H/c^2$ compared to $m$ for reasonable values of $H$ and $m$.

 

[PeterDonis]

What I am trying to say, is that I don't see in the formulas displayed here or in other sources the accounting of this process.

That's because doing so makes the analysis a lot more complicated. Most real systems are constantly exchanging energy with lots of other systems.

For example, in E. "Relativity" (which may be not the best source)

As books for the lay person go, it's a pretty good one; but it's not the same as a textbook or peer-reviewed paper. If you really want to learn science, you need to look at textbooks and peer-reviewed papers.

the reference is given only to the absorbed energy. It is possible that under this term the net absorbed energy is considered.

No, Einstein is considering a highly idealized situation, where an object is absorbing some energy and that's it. This makes the analysis simple enough to explain in a book for the lay person. But of course such a situation is not realistic.

 

[Dale]

There is, perhaps, a typo in your expression
m^2c^2=E^2/c^2−p^2

I guess it should be
m^2c^4=E^2/c^2−p^2

Check your units. The units in your expression are not correct

My expression is correct. It is the norm of the four momentum. Are you familiar with four-vectors? If not then I would strongly recommend that you study them. Your impressions seem very haphazard and disorganized, and four-vectors are a good way to bring organization and understanding, particularly if you have a strong grounding in using standard 3D vectors in your previous work (as I am guessing that you do)

 

[Dale]

What I am trying to say, is that I don't see in the formulas displayed here or in other sources the accounting of this process. For example, in E. "Relativity" (which may be not the best source) the reference is given only to the absorbed energy. It is possible that under this term the net absorbed energy is considered.

E covers any energy, energy transferred in or out, energy changing forms, whatever.

Here is a thread that came to mind about energy being emitted. It is well understood and discussed. In fact, the usual derivation of E= mc^2 uses mass loss due to emitted energy. https://www.physicsforums.com/threa...and-relativistic-beaming.783514/#post-4928786

 

[Ricardo]

Dale,

It was my mistake. I almost immediately recognized it almost 10 hr ago and posted as:

"To PeterDonis.

Yes. It is my typo. Thank you."

Yours and your colleagues' advice is tremendously helpful to refresh for me all this information and make my thoughts in the field of SRT more organized.

Thank you very much.

 

[Herman Trivilino]

>>>>. Where does the energy come from to increase the mass of the substance?<<<

It comes from the electromagnetic energy.

Think of it this way. Electromagnetic radiation can cause the atoms in a piece of material to move faster. The increased kinetic energy associated with that increased motion makes a contribution to the mass of that piece of material. If an amount of energy $E$ is transferred, the mass increases by $E/c^2$.

There is nothing special about the electromagnetic radiation. It's just what Einstein chose to use as his example.

If you have a box of mass $M$, and inside the box is a ball of much much smaller mass $m$ moving with speed $v$ relative to the box, then the mass of the system is $M+\frac{m}{\sqrt{1-(v/c)^2}}$. The energy of the ball's motion contributes to the mass of the system.

In most cases this increase in mass is way too small to notice, but there are some situations where it is large enough to notice. Everyone notices it when it's large enough to be noticed, but most people don't realize it's there when it's too small to be noticed.

 

[Herman Trivilino]

It bothers me that loss of heat/energy has not been considered as a natural outcome of the energy gain.

Why is that? The thought experiment involves the absorption of radiation. Everything else is indeed ignored. Why wouldn't it be?
Suppose you place a cold pan in a hot oven. Heat energy will be transferred to the pan, increasing its mass. There is no loss of energy that's a "natural outcome". And even if there were, it would constitute a loss of mass.

When an object gains rest energy, it gains mass. When an object loses rest energy, it loses mass. Just because someone gave you an example of the former unaccompanied by an example of the latter doesn't make the former any less valid. If it still bothers you, have the object emit some radiation, and you will see that its mass decreases.