# Calling pmb_phy

JesseM
Aer said:
should be "the mass of an object is the sum of all its constituents' rest masses".
No, not if you use the definition given by Tom Roberts, where the "rest mass" of a composite object is defined as its total energy (divided by c^2, presumably) in the center-of-mass frame. In this frame, most of the individual particles will have nonzero velocity, so their energy will be greater than just c^2 times their rest mass, it will be c^2 times their relativistic mass.

Here is another page (from mathpages.com, a pretty reliable internet resource) that says that the inertia of a composite object (its resistance to being accelerated) will be a function of its total energy, not just the energy of the rest mass of all the constituent particles:
Another derivation of mass-energy equivalence is based on consideration of a bound "swarm" of particles, buzzing around with some average velocity. If the swarm is heated (i.e., energy E is added) the particles move faster and thereby gain both longitudinal and transverse mass, so the inertia of the individual particles is anisotropic, but since they are all buzzing around in random directions, the net effect on the stationary swarm (bound together by some unspecified means) is that its resistance to acceleration is isotropic, and its "rest mass" has effectively been increased by E/c^2. Of course, such a composite object still consists of elementary particles with some irreducible rest mass, so even this picture doesn't imply complete mass-energy equivalence.
Do you have any sources to back up your claim that the inertia of a composite object is dependent only on the rest masses of its constituent particles? If not, why are you so confident about this?

Apparently you all need a little refresher course, I hope this helps.

The total energy of a particle is:

$$E = \gamma m c^2$$

where $\gamma$ is the Lorentz factor, m is the particle's rest mass and c is the speed of light.

We can also write:

$$E = E_0 + K$$

where K is the particle's kinetic energy and $E_0$ is the particle's rest energy. That is:

$$E_0 = m c^2$$

The relativistic kinetic energy is then easily seen to be:

$$K = (\gamma - 1) m c^2$$

which for $\gamma$ close to 1 (v << c) reduces to approximately

$$K = 1/2 m v^2$$

the usual Newtonian expression for kinetic energy.

JesseM said:
No, not if you use the definition given by Tom Roberts, where the "rest mass" of a composite object is defined as its total energy (divided by c^2, presumably) in the center-of-mass frame.
Very well, then his definition of "rest mass" is not the proper definition of "rest mass" :zzz:

JesseM said:
that says that the inertia of a composite object (its resistance to being accelerated) will be a function of its total energy, not just the energy of the rest mass of all the constituent particles:
The acceleration of an object is only properly measured in it's rest frame which implies the total energy is the rest energy.
:zzz:

JesseM
Aer said:
Apparently you all need a little refresher course, I hope this helps.

The total energy of a particle is:

$$E = \gamma m c^2$$

where $\gamma$ is the Lorentz factor, m is the particle's rest mass and c is the speed of light.
Uh, yes, and this is the same as E=Mc^2, where M is the relativistic mass which equals gamma*m. So the total energy of a collection of particles (again, ignoring potentials--assume the particles don't interact much) is equal to the sum of their relativistic masses times c^2. Thus, if you define the "rest mass" of a composite object as the total energy in its center-of-mass frame divided by c^2, then the rest mass of a composite object will be the sum of the relativistic masses of all the particles that make it up. That brings us to the issue of whether this is in fact the standard definition of "rest mass" for a composite object:
JesseM said:
No, not if you use the definition given by Tom Roberts, where the "rest mass" of a composite object is defined as its total energy (divided by c^2, presumably) in the center-of-mass frame.
Aer said:
Very well, then his definition of "rest mass" is not the proper definition of "rest mass"
What makes you so sure? Do you have any sources that tell us how "rest mass" should be defined for a composite object made up of many individual particles which are in motion relative to each other?

And aside from the issue of definitions, that mathpages.com page confirmed that the resistance to acceleration (inertia) of a composite object will be proportional to its total energy, so the inertia of a box filled with gas will increase as the gas is heated. Do you have any source that says otherwise? Have you actually done a calculation to see how a box filled with moving objects would react to external forces? If not, why are you so confident, when multiple sources say otherwise?

JesseM
Aer said:
The acceleration of an object is only properly measured in it's rest frame which implies the total energy is the rest energy.
:zzz:
Uh, but again, we're talking about a composite object. Even in the center-of-mass frame of a box filled with moving gas molecules, most of the individual molecules will not be at rest. Unless all the molecules are moving at the same speed and in the same direction (which would be a thermodynamic miracle) there is no frame where all the molecules are at rest. And in the center-of-mass frame, the total energy of a box of gas will be the sum of the relativistic masses of all the gas molecules (assuming the energy in the walls is negligible)--do you deny this?

JesseM said:
Uh, yes, and this is the same as E=Mc^2, where M is the relativistic mass which equals gamma*m.
!! There is clearly no getting through to you. The concept of relativistic mass is not physical - it only exists in frames other than the frame of the actual object. You have an infinite number of "relativistic masses" according to your definition. What makes you think "relativistic mass" is any type of measure of "actual mass" (i.e. the weight an object would feel in a gravitational potential)?

JesseM said:
What makes you so sure? Do you have any sources that tell us how "rest mass" should be defined for a composite object made up of many individual particles which are in motion relative to each other?
I already did! It is the sum of all the constituents own rest masses.

If I have objects in my car moving at .9999999999999999999999999999999999999999999999c bouncing all over the place, what is the mass of my car?

JesseM
Aer said:
!! There is clearly no getting through to you. The concept of relativistic mass is not physical - it only exists in frames other than the frame of the actual object.
It's just as physical as energy--in fact it is simply the energy divided by c^2. If you prefer, we can ignore the concept of "relativistic mass" altogether and just talk about the total energy of a composite object in its center-of-mass frame.
Aer said:
You have an infinite number of "relativistic masses" according to your definition.
No, because the definition specifies that you're looking at things in a particular frame, the center-of-mass frame of the composite object.
Aer said:
What makes you think "relativistic mass" is any type of measure of "actual mass" (i.e. the weight an object would feel in a gravitational potential)?
Again, forget relativistic mass and just talk about energy. The reason I think it's the total energy rather than the sum of all the rest masses that determines weight is because I have several sources written by experts which say that it's the total energy that determines resistance to acceleration (inertia). What makes you think that the inertia of a composite object is proportional only to the sum of the rest masses of the particles that make it up rather than proportional to the total energy of the particles that make it up, when several sources written by experts say otherwise?

JesseM said:
If you prefer, we can ignore the concept of "relativistic mass" altogether and just talk about the total energy of a composite object in its center-of-mass frame.
This is stupid, then we would just be talking about adding up the masses and kinetic energy to get the total energy.

JesseM
Jesse said:
What makes you so sure? Do you have any sources that tell us how "rest mass" should be defined for a composite object made up of many individual particles which are in motion relative to each other?
Aer said:
I already did! It is the sum of all the constituents own rest masses.
I didn't ask you to just repeat the assertion, I asked if you had any sources that back up your assertion. Show me a source that specifically addresses the issue of composite objects made up of sub-objects in motion relative to one another. If you don't have a source, you're just guessing, you don't really know if the physics community would agree with your definition.

JesseM
Aer said:
This is stupid, then we would just be talking about adding up the masses and kinetic energy to get the total energy.
Sure, and this would be equal to the sum of the relativistic masses. What's the problem here? Why do you doubt that the total energy is the thing that determines the resistance to acceleration of the composite object?

JesseM said:
Again, forget relativistic mass and just talk about energy. The reason I think it's the total energy rather than the sum of all the rest masses that determines weight is because I have several sources written by experts which say that it's the total energy that determines resistance to acceleration (inertia).
How about you link to these sources instead of just saying they exist. If this were true, then the theory as it is now defined is wrong. I'd don't claim that the theory is neccessarily right - but lets be clear, are we talking about the theory or experiments?

One experimental proof should suffice. Someone's personal opinion is not experimental proof.

JesseM
Aer said:
If I have objects in my car moving at .9999999999999999999999999999999999999999999999c bouncing all over the place, what is the mass of my car?
That depends on how the physics community chooses to define the mass of a composite object. I have no reason to doubt that Tom Roberts and learningphysics are giving the standard definition, and you haven't provided any sources that indicate otherwise. But ignoring the issue of definitions, I'm confident that the same force will not accelerate your car as quickly as if the objects in your car were moving slower (in the center-of-mass frame of the car), ie the inertia of the car will be different, since two expert sources have said this is true.

JesseM said:
I'm confident that the same force will not accelerate your car as quickly as if the objects in your car were moving slower (in the center-of-mass frame of the car), ie the inertia of the car will be different, since two expert sources have said this is true.
I'd be more than happy to see these sources. This would do nothing but undermine the foundations of SR and probably neccessitate modifications to GR.

JesseM
Aer said:
The sources I'm talking about are the ones I already linked too--the FAQ written by a physicist at Virginia Tech, and the mathpages.com page. I can look for more if you like.
Aer said:
If this were true, then the theory as it is now defined is wrong.
Why? Perhaps it is just your understanding of the theory that is wrong.

JesseM said:
The sources I'm talking about are the ones I already linked too--the FAQ written by a physicist at Virginia Tech, and the mathpages.com page.

JesseM said:
I can look for more if you like. Why? Perhaps it is just your understanding of the theory that is wrong.
No, my understanding is just fine.

You might want to look to experimental proof before you go believing anything. I am not saying I believe any of this - only that it is what the theory says. To my knowledge, there has been no experiment to confirm that rest masses are the true "rest mass" of any object (composite or otherwise).

Before you go on posting any more drivel, please read my post explaining energy, mass, et al completely: read here

And just an FYI, I compiled this from an expert source, it is not just my own personal understanding.

JesseM
Aer said:
No you haven't. Your only comment about the FAQ was the incorrect statement that the author didn't contradict the questioner, and you didn't say anything about the mathpages.com page at all.
Aer said:
No, my understanding is just fine.
Well, would you care to explain the basis for your statement "then the theory as it is now defined is wrong"? What obvious flaw do you see in the idea that resistance to acceleration is proportional to total energy in the center-of-mass frame?

JesseM said:
No you haven't. Your only comment about the FAQ was the incorrect statement that the author didn't contradict the questioner, and you didn't say anything about the mathpages.com page at all.
And then he proceeds to answer the question as if Einstein actually said that.

JesseM said:
Well, would you care to explain the basis for your statement "then the theory as it is now defined is wrong"? What obvious flaw do you see in the idea that resistance to acceleration is proportional to total energy in the center-of-mass frame?
Did you read my post completely?

Anyway, I said it would undermine the foundations. Acceleration is measured in the objects rest frame. You want to say that relative velocity in a gravitational potential increases the objects inertia - which means the acceleration would have to be measured in the rest frame of the gravitational potential. But this is kind of meaningless as it assumes the gravitational potential has a rest frame, perhaps it is the rest frame of the massive body creating the potential - OK, sounds acceptable. All of this seems to be pointing to a local ether around massive bodies. SR prohibits local ethers - in fact, it assumes they don't exist.

JesseM
Aer said:
And then he proceeds to answer the question as if Einstein actually said that.
No he doesn't, the first sentence he writes is "Actually, here's the way it should be said: energy and mass are related." Saying "actually, here's the way it should be said" indicates that the way the questioner said it was incorrect.

You still haven't addressed the mathpages.com page, either.
Aer said:
Did you read my post completely?
Yes, I'm already familiar with that stuff.
Aer said:
Anyway, I said it would undermine the foundations. Acceleration is measured in the objects rest frame.
But for a composite object, there is no frame in which every part of it is at rest. The best you can do is the center-of-mass frame.
Aer said:
You want to say that relative velocity in a gravitational potential increases the objects inertia - which means the acceleration would have to be measured in the rest frame of the gravitational potential.
What are you talking about? I never said anything about looking at the frame of the "gravitational potential" (presumably you mean the frame of the source of this potential?), I said that according to the sources I mentioned, inertia is a function of the total energy in the object's own center-of-mass frame. This would be just as true for an object in empty space being accelerated by some non-gravitational force (or even accelerated by a collision rather than a constant force) as it would be for an object in a gravitational field.

EnumaElish
Homework Helper
Aer said:
The concept of relativistic mass [...] only exists in frames other than the frame of the actual object. You have an infinite number of "relativistic masses" according to your definition.
As a general statement I am with you here. One cannot open up this box (relativistic mass) and then deny this consequence, as far as I am able to follow the subject.

JesseM said:
You still haven't addressed the mathpages.com page, either.
Yes I did

JesseM said:
Yes, I'm already familiar with that stuff.
Apparently you don't understand it.

JesseM said:
But for a composite object, there is no frame in which every part of it is at rest. The best you can do is the center-of-mass frame.
You are assuming an objects inertia will increase with an increase in energy content. What is your basis for this? Inertia only increases with mass, and mass is only defined as "rest mass" in physics. "Relativistic mass" is on the fringe edge of physics, in fact, it is not even mentioned in any of my physics text books.

*and was never brought up in any physics lecture.