Questions about the relativistic kinetic energy expressions

[Dale]

I still have some skepticism regarding the mass growing with speed and the energy going infinitely with v approaching c.

The concept of relativistic mass has been deprecated for decades now. I am not sure how it persists still. The energy increases without limit as v approaches c.

Do we have any experimental data?

Sure. Every high energy particle accelerator experiment ever conducted. The accelerators wouldn’t even work otherwise.

I know that the calculations demonstrating that a parameter rises to infinity are often a result of a poor model.

It is certainly possible that some future experiment will show a Lorentz violation, but so far there is none. This is despite a recent experimental renaissance on the topic.

 

[atyy]

I still have some skepticism regarding the mass growing with speed and the energy going infinitely with v approaching c. Do we have any experimental data? In turn, I know that the calculations demonstrating that a parameter rises to infinity are often a result of a poor model. Take for example contact mechanics, it has been shown that the stress at the edges of the roller of the rolling bearing increases infinitely. A paradox was resolved by understanding that with a stress increase the deformation transforms from elastic to plastic thus removing the artificial infinite stress.

Yes, there is experimental data. Nowadays, people prefer the term "energy" for what you are referring to as "mass".

 

[sweet springs]

I still have some skepticism regarding the mass growing with speed and the energy going infinitely with v approaching c. Do we have any experimental data?

"Mass growing with speed" is an old interpretation in the beginning of last century. Mass is constant.
"Energy going to infinity with v approoaching c" is right with relation
E^2=m^2c^4+p^2c^2
p=mc\frac{v}{\sqrt{1-v^2}}=mcu where v=V/c and instead of v
u:=\frac{v}{\sqrt{1-v^2}} is often used in SR as velocity instead of v.
V\rightarrow c, |u|\rightarrow +\infty, |p|\rightarrow +\infty, E\rightarrow +\infty

Peculiar case $m=0$ is experienced daily.
E=|p|c for photon. The relation
p=mcu does not work here due to m=0 and u=infinity.

 

[Ricardo]

Thanks, everybody for a quick response. I am getting rusty in general science and am trying to clean up. I am delighted with your confirmation that the mass does not grow with speed. Accordingly, should we consider to be obsolete the statement of Einstein that "...If a body [moving with velocity v] takes up an amount of energy Eo, then its inertial mass increases by the amount Eo/c^2; the inertial mass of a body is not constant, but varies according to the change in the energy of the body." Einstein talks here about the energy absorbed in the form of radiation. As a counterargument to this, I may invoke a situation when the mass is actually reduced when it absorbs the radiation." A good example of the situation is warming up of a substance in the microwave oven. The liquid in the substance will evaporate (while energy is absorbed) thus reducing the mass of the substance, which may or may not be in a state of rest. Instead of the microwave oven, one may think about energy transfer by any kind of electromagnetic waves. At the very list, the energy balance should be considered when talking about the mass increase via energy transfer.

 

[Dale]

Accordingly, should we consider to be obsolete the statement of Einstein that "...If a body [moving with velocity v] takes up an amount of energy Eo, then its inertial mass increases by the amount Eo/c^2; the inertial mass of a body is not constant, but varies according to the change in the energy of the body."

That statement is still valid. The mass increase referred to there is the invariant mass, not the relativistic mass.

 

[PeterDonis]

I may invoke a situation when the mass is actually reduced when it absorbs the radiation." A good example of the situation is warming up of a substance in the microwave oven. The liquid in the substance will evaporate (while energy is absorbed) thus reducing the mass of the substance

This is because the system is not closed; you are allowing material to escape by evaporation, and not counting the mass of the evaporated material in the mass of the system. All of the statements from relativity that you have been quoting assume that the system is closed; if the system is not closed, the analysis gets more complicated, and you need to get clear about the simpler case of a closed system first.

 

[Ricardo]

I see some caveats in the closed system as well.
1. If the system is completely closed, then, I guess, it wouldn't be able to absorb the external radiation.
2. If it still somehow absorbed the radiation and its mass increased, then it is not completely closed and thus its temperature will increase and it may start to emit thermal radiation losing its mass.
3. If, however, a mass of the body increased through irradiation, then its velocity should go down, which is a situation that contradicts to the main premise that the velocity wouldn't increase.

 

[PeterDonis]

If the system is completely closed, then, I guess, it wouldn't be able to absorb the external radiation.

That's correct. Either that, or you count the radiation as part of the system from the beginning, and add its energy in accordingly.

If it still somehow absorbed the radiation and its mass increased, then it is not completely closed and thus its temperature will increase and it may start to emit thermal radiation losing its mass.

Yes.

If, however, a mass of the body increased through irradiation, then its velocity should go down

Why?

which is a situation that contradicts to the main premise that the velocity wouldn't increase.

What main premise are you talking about?

 

[Dale]

If it still somehow absorbed the radiation and its mass increased, then it is not completely closed and thus its temperature will increase and it may start to emit thermal radiation losing its mass.

Sure. The system would gain mass as it absorbed radiation and lose mass as it emits radiation. That is the point.

If, however, a mass of the body increased through irradiation, then its velocity should go down, which is a situation that contradicts to the main premise that the velocity wouldn't increase.

You should work out the math on this. It doesn’t work the way you think.

 

[Ricardo]

Here I am trying to put one of the former questions somewhat differently.

If we all agree that the energy that is transferred to a substance by electromagnetic radiation converted to heat (for example, E = σTs^4, where σ is the Stefan-Boltzmann constant), then
>>>>. Where does the energy come from to increase the mass of the substance?<<<

 

[PeterDonis]

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

From the energy in the electromagnetic radiation that you said was converted to heat. Converting to heat means raising the temperature of the substance; that increases its rest mass.

 

[Dale]

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

The energy came from whatever the source of the EM radiation was.

You may be thinking of mass as a form of energy and energy that is converted into mass is not available to be in other forms. That is incorrect: $m^2 c^2=E^2/c^2-p^2$ holds at all times regardless of the form of the energy.

 

[sweet springs]

In SR mass is defined as energy measured in IFR called center-of-mass, COM, system where momentum of the system is zero. There mass is conserved before and after the interaction,so as for photon absorption case,

Mass of (Material + photons) = Mass of heated material

As another peculiar case, in electron and positron pair annhilation,

Mass of electron and positron = Mass of two $\gamma$ photons

with electromagnetic interaction and kinetic energy of electron and positron neglected.
Though photon has no mass, system of a photon and material or a group of photons has mass in general.

 

[Dale]

Mass of electron and positron = Mass of two γγ\gamma photons
with electromagnetic interaction and kinetic energy of electron and positron neglected.

The KE is not neglected. It contributes to the energy of the photons.

 

[sweet springs]

You are right. I mean I do not write them down in my formula for brevity in expectation that they are much smaller than $m_e$. So for more detail,

mass of (electron, positron, their kinetic energy and their electromagnetic interaction)=mass of two $\gamma$ photons

Moreover, for absorption of photon case, it is usually so and convenient for us that photons around material have no momentum in total. If not, material is moving more or less in COM so its kinetic energy should be also involved in mass of the system.

 

[Ricardo]

Thanks to everybody for a quick response. Your answers, some of which I expected, are definitely within the SRT orthodoxy.

To 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

To sweet springs:
Regarding your statement: Mass of (Material + photons) = Mass of heated material
Is this generalization truly justifiable? One can easily envision a situation when the mass of the heated material has remained heated even when the process of irradiation (associated with photons' interaction with the material) has ceased-- for a whatever reason- and the photons are no longer "stuck" within the material.

 

[PeterDonis]

Your answers, some of which I expected, are definitely within the SRT orthodoxy.

Your use of the word "orthodoxy" here is not a good sign. We are using SR to make predictions about the scenarios you describe, yes. That's because, for scenarios like the one you describe, SR has massive experimental confirmation. There is no "orthodoxy" about it.

You need to decide what you are here for. If you are here to learn what SR, which has massive experimental confirmation, as I said just now, says about the scenario you describe, that's fine; but then you need to stop using words like "orthodoxy" and recognize that, if you don't quite understand something we're saying or it doesn't quite make sense to you, that's not a sign that SR might be wrong; it's a sign that you don't understand it well enough yet.

If you are here to push your own personal viewpoint, then this thread will be closed; that's not what PF is here for.

One can easily envision a situation when the mass of the heated material has remained heated even when the process of irradiation (associated with photons' interaction with the material) has ceased-- for a whatever reason- and the photons are no longer "stuck" within the material

What photons are no longer "stuck" within the material? When photons are absorbed by a material and heat it up, those photons don't exist any more. They don't somehow stick around and escape from being "stuck" when the irradiation stops. Their energy is absorbed by the material and stays there.

 

[PeterDonis]

here is, perhaps, a typo in your expression

No, there isn't a typo in @Dale's expression; it is correct. Your expression is the one that's wrong; if you want $m^2 c^4$ on the LHS, then you would multiply @Dale's expression by $c^2$ to obtain

$$
m^2 c^4 = E^2 - p^2 c^2
$$

 

[Ricardo]

I am sorry for the misunderstanding. My word "orthodoxy" is a positive sign in a context that your response helps me to refresh the knowledge that I received decades ago. I am quite appreciative for your responses. With my questions and your answers, I am trying to get some additional/advanced knowledge, which, I thought, I could apply in some of my professional fields, associated, particularly, with lubrication.

Now, this.
It bothers me that loss of heat/energy has not been considered as a natural outcome of the energy gain. As we agreed, even if the system closed, the radiation energy may penetrate the material and heat it. But by the same token, a portion of the gained energy will escape. A natural heat balance.

 

[Ricardo]

To PeterDonis.

Yes. It is my typo. Thank you.

 

[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.