Energy in E=mc2

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True, I'm not asking about the frame of a certain particle properties.
It is hard to explain, I'm referring to the concept that the quantity is invariant regardless the state of motion.
Take momentum instead of energy, there is no invariant momentum. So it might sound somewhat confusing that there is an energy invariant when in SR what is conserved is the momentum-energy.
If we define invariant momentum like invariant mass at v=0 then we get that invariant momentum - the momentum at zero speed - is 0. I suppose that that is why there is no invariant momentum...
 
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If we define invariant momentum like invariant mass at v=0 then we get that invariant momentum - the momentum at zero speed - is 0. I suppose that that is why there is no invariant momentum...
So you are saying v=0 (therefore p=0) can be defined and agreed by all observers?
 
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So you are saying v=0 (therefore p=0) can be defined and agreed by all observers?
The properties of an object as determined with a reference system relative to which it is in rest are agreed by all observers - and yes, to be precise: zero speed relative to a certain object is agreed by all inertial reference systems.

Thus the group of reference systems relative to which the object has zero speed is called its "rest frame", and its energy relative to that "frame" is called its "rest energy" m0c2

As a matter of fact, an "inertial frame" is an imaginary or real system with rulers and clocks that are in rest relative to each other - according to that system as well as according to systems in uniform motion relative to it. If this was not agreed by all inertial systems, then no consistent transformations would be possible I'm afraid.
 
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Only if E and p are not conserved or E²/c² = m²·c² + p² is wrong.
I wrote "see #11": there is "missing" rest mass
- http://en.wikipedia.org/wiki/Binding_energy

As a matter of fact, rest mass diminishes with L/c2 as the first derivation already showed which I cited (post #12, just above your post!).

Nevertheless, while the extended equation includes radiation, it doesn't include potential energy - it's incomplete.
 
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As a matter of fact, rest mass diminishes with L/c2 as the first derivation already showed which I cited (post #12, just above your post!).
As a mater of fact energy diminishes with L. But that does not mean that energy is not conserved. The total energy of an isolated system is constant and so is the total rest mass of the system.
 

DrGreg

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The total energy of an isolated system is constant and so is the total rest mass of the system.
That depends what you mean by "total rest mass".
  • the sum of the individual rest masses
  • the invariant or system mass of the whole system
These are not the same. If you mean (b), you are right. If you mean (a), you are wrong.
 
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That depends what you mean by "total rest mass".
  • the sum of the individual rest masses
  • the invariant or system mass of the whole system
Of course I mean (b). (a) is just a number without any physical relevance because it can be changed by choosing different theoretical fragmentations of the same system into sub-systems and the properties of a real system must not depend on its theoretical description.
 

DrGreg

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That depends what you mean by "total rest mass".
  • the sum of the individual rest masses
  • the invariant or system mass of the whole system
Of course I mean (b). (a) is just a number without any physical relevance because it can be changed by choosing different theoretical fragmentations of the same system into sub-systems and the properties of a real system must not depend on its theoretical description.
Good. The problem is that most people equate the word "total" with "sum" and would go for option (a), which is why you've had people disagreeing with you in this thread.
 
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I find some ambiguity in the term rest(invariant) mass, because if rest is considered to be relative in SR, how can it be invariant? I mean if rest mass is the same in all reference frames, then all observers are agreeing about the rest frame, aren't they?

Unfortunately mass (like energy) is not a well defined quantity in GR (momentum-energy is), but it is well defined in SR so it should be possible to clarify the above within SR.
Ok, so Ithink the easiest way to clarify this is simply to acknowledge that in SR (unlike in reality) there are truly inertial frames, so rest point masses are unambiguously defined and invariant. And as previously commented by other posters there is nothing special with these rest frames within SR as any point mass can be chosen as the rest frame.
I wonder how this translates to more realistic scenarios where there are no truly inertial frames.
More specifically, shouldn't the equation E=mc2 even though derived from SR, really be more of a GR formula? I guess it can't be given that neither mass nor energy are well defined in GR.
But the situation is a bit paradoxical: invariant mass and energy are well defined within SR and therefore E=mc2 considered an SR formula, but at the same time in more physical terms we accept the equivalence of inertial and gravitational mass and therefore strictly speaking mass is outside the range of SR in as much as gravitation cannot be explained within SR.
 
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If you insist calling a particles energy it's mass, then sure :D I usually call mass mass and energy energy, so there's no confusion.
What "relativistic mass"? The "missing mass" is in the binding energy!

I definitely agree with clamtrox here. Mixing up those two, especially when someone isn't familiar with the concept, is causing nothing but confusion.

Zz.
Are you are saying that the rest mass of an atom should exclude the binding energy (since it is energy and not mass)? Should we also exclude the binding energy of the gluon field that holds its constituent quarks together? Or should we arbitrarily consider that gluon field to really be a gluon particle so that we can include its mass (which turns out to be the lion's share of the quark mass)?.

What is rest mass? As Naty1 pointed out, rest mass includes numerous energy components, both kinetic and potential. 1Kg of uranium consists of the rest mass of quarks, qluon field energy, atomic binding energy, thermal kinetic energy etc.

All of these mass-energy components make up its 1Kg rest mass.

Now it seems odd to me that if I now set this complex 1Kg package of mass-energy in motion, that I am obligated to record the increase in energy in Joules, not Kg, because it happens to be motion.

And, yet, if I place this moving 1Kg into a larger, stationary, container, its kinetic energy is now allowed to be converted to Kg and counts toward rest mass of the larger container.

I know that teaching relativistic mass has gone out of favor, and for some valid reasons, but I think that there is an unintended consequence.

Students should not get the impression that mass and energy are fundamentally different things or that e=mc^2 is a rule that tells how much of one can be converted to the other.

Energy can neither be created nor destroyed, and energy, in all of its forms, has mass.
Mass also can neither be created nor destroyed, and in all of its forms, has energy.
Mass and energy are two names for the same thing, and neither one is changed nor transformed into the other.
Rather than mass being changed into energy, the view of special relativity is that rest mass has been changed to a more mobile form of mass, but remains mass. In the transformation process, neither the amount of mass nor the amount of energy changes, since both are properties which are connected to each other via a simple constant.
 
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Good. The problem is that most people equate the word "total" with "sum" and would go for option (a), which is why you've had people disagreeing with you in this thread.
Exactly. Thanks for figuring that one out. :smile:
 
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Now it seems odd to me that if I now set this complex 1Kg package of mass-energy in motion, that I am obligated to record the increase in energy in Joules, not Kg, because it happens to be motion.
It's just a convention. Mass as given by definition 2 and lex 1 to 3 of Newton's Principia includes all kinds of energy if Lorentz transformation applies. But this definition of mass is outdated. Only rest mass is used in relativity today.

And, yet, if I place this moving 1Kg into a larger, stationary, container, its kinetic energy is now allowed to be converted to Kg and counts toward rest mass of the larger container.
No it isn't allowed - not generally. It depends on the momentum of the system. If you place a photon into an empty container the mass of the container would be zero. If you add another photon with the same energy but opposite momentum than the mass of the container is equivalent to the total energy of both photons. The velocity of the container does not matter.

Students should not get the impression that mass and energy are fundamentally different things or that e=mc^2 is a rule that tells how much of one can be converted to the other.
Of course not.

Energy can neither be created nor destroyed
That's true.

and energy, in all of its forms, has mass.
It would be true for the classical definition of mass (aka as relativistic mass) but with the current meaning of mass (rest mass) it is wrong.

Mass also can neither be created nor destroyed
That's true but you have to keep in mind that rest mass is not additive (see discussion above).

and in all of its forms, has energy.
That's true.

Mass and energy are two names for the same thing
See above. It depends on the definition of mass.

Rather than mass being changed into energy, the view of special relativity is that rest mass has been changed to a more mobile form of mass, but remains mass. In the transformation process, neither the amount of mass nor the amount of energy changes, since both are properties which are connected to each other via a simple constant.
It's easier: If a system at rest looses energy than it looses an equivalent amount of mass too and visa versa. Maybe I missed a corresponding original source but the "conversion of mass into energy" seems to be a concept of popular science only.
 
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[..] Mass and energy are two names for the same thing, and neither one is changed nor transformed into the other. [..]
I would say that Einstein's formulation* in his first paper on this is very accurate, and therefore I disagree with the first of your two statements.

*"The mass of a body is a measure of its energy-content"
It's just a convention. [...] If you place a photon into an empty container the mass of the container would be zero. [..]
Yes, it's just a convention and for consistency with length and time I prefer the "outdated" one; that's a matter of taste. But I think that the example doesn't work: a photon is measured as moving at v=c whatever box you choose it to be in, using any standard reference system. The system mass of such a (mass-less) container includes the relativistic mass of the light and is thus not zero (the common example is a box with mirrors; the mass doesn't switch to zero when the photon happens to be in-between the walls). However, this could be another free choice of definition. :rolleyes:
 
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I prefer the "outdated" one
Me too but to avoid confusions I accustomed myself to indicate when I use the word mass in this (currently) unusual sense. If I talk about mass without such an explanation than I mean rest mass (e.g. in my statement about photons in a virtual box).
 
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Me too but to avoid confusions I accustomed myself to indicate when I use the word mass in this (currently) unusual sense. If I talk about mass without such an explanation than I mean rest mass (e.g. in my statement about photons in a virtual box).
Yes, that's how I understood your statement. The relativistic mass of a box in rest equals the rest mass of a box in rest. BTW I found an account of a real box with photon:
http://www.abc.net.au/science/articles/2007/03/15/1872353.htm
 
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The relativistic mass of a box in rest equals the rest mass of a box in rest.
That depends what you mean by "in rest". If you mean that the walls of the box are stationary than you are wrong. If you mean that the box has no momentum than you are right but a virtual box containing one photon has momentum in every frame of reference.
 

stevendaryl

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I wonder how this translates to more realistic scenarios where there are no truly inertial frames.
More specifically, shouldn't the equation E=mc2 even though derived from SR, really be more of a GR formula? I guess it can't be given that neither mass nor energy are well defined in GR.
The SR equation E2 = p2c2 + m2c4 has an analogy in GR:

gαβ Pα Pβ = m2 c2

where Pα is the 4-momentum, which is defined in GR, as well as SR.
 
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That depends what you mean by "in rest". If you mean that the walls of the box are stationary than you are wrong. If you mean that the box has no momentum than you are right but a virtual box containing one photon has momentum in every frame of reference.
It seems that we are actually disagreeing on this. :uhh:
As a photon has no rest mass (it's an impossibility!), I assumed that you meant the mass of a system in which the box is in rest - which is also the time-averaged rest mass of the system if the box has perfect mirrors. Compare:
- http://www.weburbia.com/physics/light_mass.html
- http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html (note: I was slightly involved with that)

It's conceptually similar to a box with loss-less bouncing tennis balls inside: the time-averaged "rest mass" of the system is the sum of E/c2 of the parts.
 
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I assumed that you meant the mass of a system in which the box is in rest
See above: What do you mean with "in rest"? I mean "with no momentum".
 
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See above: What do you mean with "in rest"? I mean "with no momentum".
A virtual mass-less container cannot have momentum at any condition, incl. "not in rest"; and a photon cannot have "no momentum" and cannot be "in rest". In such an example "in rest" can only refer to the displacement of the container itself; expressed in Lorentz coordinates its walls have that dx/dt=0.
 
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A virtual mass-less container cannot have momentum
Do you really want to talk about the mass of a mass-less container? This discussion makes sense for the container including its content only and that's what I am talking about.
 
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Do you really want to talk about the mass of a mass-less container? This discussion makes sense for the container including its content only and that's what I am talking about.
It appears that you did mean a system with a photon in rest which is impossible in SR, as I already mentioned in post #44. As we didn't advance from there and it has little to do with the topic, I'll leave it at that.
 
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It appears that you did mean a system with a photon
No, I don't. I am talking about a virtual box containing one or more photons. As I told you in #42 a box containing a single photon has momentum in every frame of reference (due to the momentum of the photon). Thus it has no rest frame and therefore no rest mass. But it has relativistic mass (due to the energy of the photon).
 

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