Potential Energy Explained: Does It Contradict Conservation of Mass/Energy?

[learningphysics]

No.
It makes no difference.
No.

The above assumes that "rest mass" referres to the mass of a closed system as measured in the zero momentum frame.

The following may help (what you call rest mass others call "invariant mass". The usage is not really consistent in the literature.)
http://www.geocities.com/physics_world/sr/invariant_mass.htm

This is a perfect worked example of what this thread is about
http://www.geocities.com/physics_world/sr/nuclear_energy.htm

Pete

Thanks for your help Pete. Your website is great.

It seems like I never really "get" special relativity. Every time I think I've understood, there seems to be something I've missed.

What I've learned now... Invariant mass of a system is not necessarily the sum of the invariant masses of all the particles within the system.

Energy of a system is equal to the sum of the energies of the constituent particles... Momentum of a system is equal to the sum of the momentums of the constituent particles... is this right?

Is this right: If energy is added to a system... and the total momentum of the system remains the same (in whatever frame you're using), then the invariant mass of the system increases... whether or not the energy went into increasing the kinetic energy of the constituent particles... or the potential energy of the constituent particles.

What is still bothering me is the weight mechanism (the force of gravity)... I'm unsure as to why the "weight" of a system would be dependent of the invariant mass of the system (as opposed to the sum of the invariant masses of the constituent particles)... It was no problem before when I thought that invariant mass was additive... but it isn't clear now. Can you elaborate? Thanks.

 

[pervect]

I think that you need some familiarity with GR and the stress-energy tensor to really understand the weight issue (or gravity itself, for that matter - you need Einstien's field equations before you can start dealing with gravity seriously). Steve Carlip, for instance, has a paper on the issue of gravitational mass of a system (which is trickier than it first appears BTW), but it definitely takes more than simple SR to understand the paper. The abstract, though, at least clearly states that kinetic energy does contribute to the gravitational mass of a system of particles - so you may have to just take this on faith a bit, unless and until you get the needed background.

Kinetic Energy
and the Equivalence Principle
S. Carlip∗
Department of Physics
University of California
Davis, CA 95616
USA
Abstract

According to the general theory of relativity, kinetic energy contributes
to gravitational mass. Surprisingly, the observational evidence for this
prediction does not seem to be discussed in the literature. I reanalyze
existing experimental data to test the equivalence principle for the
kinetic energy of atomic electrons, and show that fairly strong limits
on possible violations can be obtained. I discuss the relationship
of this result to the occasional claim that “light falls with twice the
acceleration of ordinary matter.”

 

[pmb_phy]

I think that you need some familiarity with GR and the stress-energy tensor to really understand the weight issue (or gravity itself, for that matter - you need Einstien's field equations before you can start dealing with gravity seriously).

"really" understand? If one understands it without GR does that mean neccesarily that he "really" doesn't understand it? The first derivation I ever did to determine whether a moving body weighed more I did before I learned GR to any great extent. All one requires is the equivalence principle. From that one can extrapolate to a more complex object such as a 2-d gas in a 2-d box. If one recalls Einstein's 1905 SR paper then in it Einstein states that to measure the transverse mass of a particle one can use a balance. Since the transverse mass = relativistic mass one obtains the weight of the particle. Its always best to try to figure these things out before one puts pen to paper. I've rarely made any progress in anything I've ever done by starting with the math.

Pete

 

[pmb_phy]

Thanks for your help Pete. Your website is great.

Thank you.

It seems like I never really "get" special relativity. Every time I think I've understood, there seems to be something I've missed.

Join the club.

What I've learned now... Invariant mass of a system is not necessarily the sum of the invariant masses of all the particles within the system.

That is correct.

Energy of a system is equal to the sum of the energies of the constituent particles... Momentum of a system is equal to the sum of the momentums of the constituent particles... is this right?

Sure. As long as you're talking about non-interacting particles. You don't even need to employ energy conservation. This fact can be derived without ever employing the principle of the conservation of energy. Using the theory of the conservation of mass makes it trivial.

Is this right: If energy is added to a system... and the total momentum of the system remains the same (in whatever frame you're using), then the invariant mass of the system increases...

Brigtht boy!

For a derivation of what you just stated see
http://www.geocities.com/physics_world/sr/mass_energy_equiv.htm

What is still bothering me is the weight mechanism (the force of gravity)... I'm unsure as to why the "weight" of a system would be dependent of the invariant mass of the system (as opposed to the sum of the invariant masses of the constituent particles)... It was no problem before when I thought that invariant mass was additive... but it isn't clear now. Can you elaborate? Thanks.

See -
http://www.geocities.com/physics_world/sr/weight_moving_sr.htm
http://www.geocities.com/physics_world/gr/weight_move.htm

Pete

 

[El Hombre Invisible]

Pete - I now realize that it made no sense to address my post to you. We seem to be in complete agreement, bar the point about the rest mass of the system where I temporarily went insane - you are of course 100% correct.

 

[pmb_phy]

Pete - I now realize that it made no sense to address my post to you. We seem to be in complete agreement, bar the point about the rest mass of the system where I temporarily went insane - you are of course 100% correct.

I will remind you of this when it becomes my turn and my head explodes and messes up the place.

Pete

 

[pmb_phy]

What is still bothering me is the weight mechanism (the force of gravity)... I'm unsure as to why the "weight" of a system would be dependent of the invariant mass of the system (as opposed to the sum of the invariant masses of the constituent particles)... It was no problem before when I thought that invariant mass was additive... but it isn't clear now. Can you elaborate? Thanks.

I too thought about this. I knew that E = mc^2. But what I wanted to know was Why? People rearely ask themselves this question since they seem to assume that there can be no answer to "Why?" questions. I'm not one of those people. Please see

On the concept of mass in relativity, Peter M. Brown

I placed this online today at
http://www.geocities.com/physics_world/mass_paper.pdf

See page 52 section - XI Why does E = mc²

I addressed the mechanis for 3 different kinkds of energy as I recall. This should be the best answer you're going to get on the interent.

Pete

 

[learningphysics]

I too thought about this. I knew that E = mc^2. But what I wanted to know was Why? People rearely ask themselves this question since they seem to assume that there can be no answer to "Why?" questions. I'm not one of those people. Please see

On the concept of mass in relativity, Peter M. Brown

I placed this online today at
http://www.geocities.com/physics_world/mass_paper.pdf

See page 52 section - XI Why does E = mc²

I addressed the mechanis for 3 different kinkds of energy as I recall. This should be the best answer you're going to get on the interent.

Pete

Hi Peter. The paper was very useful. I'm wondering... mass in a gravitational field... If we have two masses in a gravitational field at rest... then we take the two masses further apart and keep them at rest. The way I understand it, the total mass has increased now. Is the mechanism involved in increasing the mass analogous to that of the electrical force? What about the other fundamental forces?

The mechanism that creates mass associated with electrostatic potential energy... it almost seems like the mass of E/c^2 comes out coincidentally. On page 55 you write "Utilizing electrodynamics Boyer has shown that... Comparison with proper energy gives us the expected result"... Could the result have been otherwise?


Is there any book that you can recommend on Special Relativity that will help give a good solid foundation (hopefully give enough to prepare you to learn about GR) ? More rigorous the better.