pmb

The kinetic momentum [math]\pi[/math] can just be written [math]\frac{\partial \mathcal{L}}{\partial \dot{\phi}}[/math]. This is the Canonical momentum to [math]\phi[/math]. The ordinary momentum term is just freely switched for [math]\pi[/math] even though they don't exactly describe the the same things. For instance, [math]\sum_i p_i(q) \rightarrow \int \pi(x) dx[/math]. I don't call it kinetic momentum, I tend to call it a field momentum.

I've seen that term used in Jackson's text where it's also used to refer to ordinary momentum.

 

I found an article on this topic. See

Momentum conservation and the vector potential of moving charges, Grant R. Fowles, Am. J. Phys.. 48(8), Sept. 1980

The specific problem of the interaction between two freely moving charges has been treated in this journal in an excellant artilce by Breitenberger. His analysis is a simplified version of an earlier for formulation by Darwin in which it is demonstrated that conservation of linear momentum pi]does[/i] hold when, instead of the ordinary particle momentum mv + eA where A is the vector potential.

The author explains that eA can be identified with momentum of the joint electromagnetic field of the moving chages.

If it is in an email, I should doubt it is in breach of copyright.

Especially since its only a smal portion. I'll e-mail the publisher and author and ask for their permission.

Energy = Force * Length-->(Force is vector)

Energy is not defined that way. That expression merely the work done on a system. Work is not a form of energy. It merely has the same dimensions of energy.

Though I understand mass seems to be a buzz topic ..

A response is definitely required here:

 

"buzz topic"? No. I just happen to have some expertise on the topic since its my chosen area of expertise. So when the subject comes up I find it quite easy to respond to them. The responses I post can be found in most SR/GR texts as well as the Feynman Lectures. In fact every single thing I have ever posted can be found in the modern physics literature, without exception. And in all cases I'm quite clear that the proper mass of a photon is zero.

 

Some examples from the physics literature are in order:

 

From Relativity: Special, General and Cosmological, Wolfgang Rindler, Oxford Univ., Press, (2001), page 120

According to Einstein, a photon with frequency f has energy hf /c^2, and thus (as he only came to realize several years later) a finite mass and a finite momentum hf/c.

 

From Introducing Einstein's Relativity, Ray D'Inverno, Oxford Univ. Press, (1992), page 50

Finally, using the energy-mass relationship E = mc^2, we find that the relativistic mass of a photon is non-zero and given by

 

m = p/c.

 

Combining these results with Planck's hypothesis, we obtain the following formulae for the energy E, relativistic mass m, and linear momentum p of the photons:

 

E = hf m = hf/c2 p = hf/c

I choose to use the term "inertial mass" instead of "relativistic mass."

 

Mind you that these are the texts that many university professors are using to teach relativity.

 

..., please be advised that you already have a topic on this and that hijacking other threads with off-topic discussion won't be permitted.

A post literally cannot be considered to be hijacking a thread if the response is precisely the response to what the OP asks, which it was.

 

Please address the topic in the OP, or move your discussion to somewhere else.

If one were to read my post carefully they'd see that I responded only to the first post and exactly on the topic the OP asked. A question was asked. I answered it. It's as simple as that. I certaintly can't be blammed for the fact that the subject keeps coming up in several of the physics forums. Especially when people keep responding as do I.

 

Note that my responses are essentially the same as found in the Physics FAQ, i.e, at

http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.html

 

I respond the same way because I had influence on the latest version.

It might help you to take a trip to the library:-)

I'm currently reading "101 Quantum Questions" by Kenneth W. Ford. Question 74 is "What is the uncertainty principle?" and question 75 is "How does the uncertainty principle relate to the wave nature of matter?". The author's explainations are aimed at readers who are intelligent and curious, and also willing to devote some time and effort in pondering his explainations.

Do you have a scanner? If so then can you scan question 74 into a file and e-mail it to me? Thanks.

...(snipped erroneous claims)...

I recommend once again that you simply admit your mistakes and move on with life. Even if you just admit them to yourself. You've got to let this bone go.

 

I'll explain it yet once more. In relativity when one sees "p = mv" it is understood that the p here is mechanical momentum. It's universally accepted to be that. That expression is the definition of what the p is. Canonical momentum has a different symbol and a different definition and is made explicit what the symbols mean. In QM the 'p' is always assumed to be canonical momentum. We know that due to the context. One never needs to say that p = mv is the mechanical momentum since when p = mv appears its understood. That's universally true with the exception of QM but even there its made clear. All QM texts will state that p is canonical momentum.

 

So accept your mistake and move on. It's a small mistake which you're making into a mountain. I can see how you resist admitting your mistakes though. Try it. You might find it refreshing.

Hello pmb, I read your work, it is a good read for anyone who is wanting to learn this stuff... I perhaps only have one suggestion - it's a bit of a quibble.

The entire paper needs to be rewritten. I initially was addressing this debate on what mass is in nodern physics. I've changed my mind. I hate debates so I'll take another tact in the next version

 

''Notice how the energy moves back and forth from system to system yet the total energy of the universe remains constant. ''

 

I don't think current mainstream cosmology believes that the universe conserves energy any more.

I am familiar with that. It brings the entire meaning of energy into question. Energ has to be conserved for it to have meaning. I know that energy (outside of gravity) is conserved. With gravity in Netonian gravity, its conserved.

 

The T^00 component of the stress-energy-momentum (SEM) tensor is energy density. It's no different that the energies we've been talking about. E.g. in many cases its simply potential energy, em engery, kinetic energy, etc. None of which is a physical thing.

 

The one funny thing about T^00 is that it assigns a location for energy. Normally one doesn't assign a location to, say, kinetic energy. But the SEM tensor does this. A very odd thing for classical mechanics.

Thank you for your reply pmb. I will follow your link and have a look at it.

You're most welcome. Here is a more detailed treatment on inertial mass

http://home.comcast.net/~peter.m.brown/sr/inertial_mass.htm

 

There is more here too

http://home.comcast.net/~peter.m.brown/sr/invariant_mass.htm

 

That page describes proper mass (sometimes called invariant mass) of macroscopic systems

It might help you to take a trip to the library:-)

Nah. I already know what uncertainty is. I learned what it was when I studied quantum mechanics in graduate shool. I don't want to say more than I already have so that I don't interfer with the responses.

Pour me a cup of energy, then. While you're at it, I'll have a cup of length as well.

How would you suggest that one go about pouring a cup of dark energy?

Can I read your work later, btw, I need to go do something the now. Thanks.

Sure.

Hmmm....

 

 

... I disagree with energy not being a physical ''thing''.... The rest I agree with. Physical is not restrained to mass.

What led you to believe that energy is a physical thing? I'm sure that you've seen all the derivations of the various forms of energy and how nothing about them was physical. E.g. what do you think is physical about, say, potential energy. There is always a constant added to the potential which is chosen for convenience. What's physical about such an arbitrary constant?

 

In case you missed my post on the definition of energy you can take a gander at it at

http://home.comcast.net/~peter.m.brown/mech/what_is_energy.htm

 

This is a page in my own website. What do you think? Do you have any comments on it? Something I might add?

Dear Fellow Physics Lovers,

 

I'm looking to put together a blog discussing Heisenberg's Uncertainty Principle and the some of its ramifications. The intent of the piece will be to educate people who have an interest in it, but limited knowledge of it. So that I can pitch what I say at the right level I need a better handle on what people think uncertainty is. I'd appreciate it if you would take a couple of minutes to write what uncertainty means to you in the context of quantum mechanics.

 

I'll wait a few days before I post in this thread again. At that time please let me know if you'd like my input on the definition. If you don't wish to wait then feel free to PM me. Thanks.

Can I ask what it is meant to be then? thank you.

I think that it appears anywhere a greek symbol appears inline.

Can you friends help me in understanding relativity???

Sure. I've seen many people ask this question over the years. I didn't like sending people off to read books since people came here to learn, knowing full well they could have gone to a book store. So what I did was gradually over the years, as the topics came up, I created a web page for each topic and placed it on my website. I don't know your math background. There's a lot that canbe learned using siple algebra. Some elementary differential and integral calculus is required in some places. The part of the site which discusses special relativity is here

 

http://home.comcast.net/~peter.m.brown/sr/sr.htm

 

Please let me know if you have any questions. I make no claims of having made a perfect website either.

Hi there

 

My question is: "If light can a particle, does it have mass. If so how can something that has mass travel at the speed of light?"

A complete treatment can be found in the physis FAQ at - http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.html

 

It really depends on what one means by the term "mass". Particle physicists and quantum physicists tend to use the term to mean "proper mass" aka "rest mass." Others use the term to mean inertial mass. When it comes to more general systems the inertial mass is more meaningful. proper mass can't be applied to open and extened systems such as a body which is radiating energy, thus having a variable mass. Inertial mass is defined as the p in p = mv. For a particle traveling at the speed of light this relationship still holds. We now have v = c so that p = mc. For photons E = pc which implies that m = E/c² = hf/c².

 

I see cosmologists and GRists use the term to mean inertial mass as well as active and gravitational mass.

I dont believe photons have matter.But if they could exist at a rest state,perhaps they might.

Consider a gas of massless photons for which the photons have random directions and momenta. Calculate the invariant mass m of this system and you'll get a non-zero value whose magnitude can be found by using the relation

 

P*P = (mc)²

 

P = total 4-momenta of system.

So, are you saying in your paper, its not meant to be the operator, because it was an error when it was error in the translation?

Yes.

If gravity is described as a force between two masses, then how exactly can light be effected by it? From my understanding, from what I have been taught, gravity would effect the photons of light which have no mass. Please correct me if I am wrong. Any responses would be much appreciated!

When you say "light has no mass" it means that the proper mass of a photon is zero. While the proper mass of a photon is zero it still has what is called passive gravitational mass which means it has the property for which particles are affected by a gravitational field. This is due to its inertial mass which is found by p = mv = mc. Use E = pc or p = E/c and substitue into p = mc and solve for m to get m = E/c².

 

In Einstein's first derivation of gravitational deflection he treated it like a wave rather than a particle. Given that the velocity of light depends on the gravitational potential Einstein showed that the wave can be deflected by a gravitating body. I think he got the wrong answer since he didn't take spatial curvature into account.

What you say is correct, but the AMOUNT that light bends is consistent with the predictions of Einstein's theory of GR, and not with Newton's theory. Newton's theory predicts an amount of bending of light from stars by our sun that is only half the amount that is experimentally observed.

The additional delfection obtained using relativity is due to the spatial curvature around the sun. Without taking the curvature of space into account you're left with only the gravitational acceleration. It can be said that half the delfection is due to gravitational acceleration and half due to spatial curvature. In Einstein's first prediction he didn't take spatial curvature into account and ended up with half the correct value. When he finished his GR theory Einstein recalculated the deflection, now taking spatial curvature into account, and got the correct value.

Can you define [math]\Box[/math] this more clearly, you say Minkowski defined it as the mechanical mass, but I have never seen this before - I believe this symbol is more generally used to define the d'Alembertian.

Yes, it's the d'Alembertian operator. If/when you see that in my paper its because there was an error translating it from an MS Word file to a PDF file.

You continue repeating the same mistakes that I reported in post #33 above, except that you have now corrected your expression for P_jafter that I noticed above that your previous expression for P_j was wrong if by m you mean "inertial mass".

The mistakes are all yours, and that includes your confusion of canonical momentum with mechanical momentum. Nobody else in this forum would make such a mistake. Simply start a thread and pose the question and they'll enlighten you

 

What is mass?

The inertial mass m of a body is that property which resists changes in momentum. The inertial mass of a body is defined as p = mv where v = 3-velocity and 3 = 3-momentum = mechanical momentum.

 

The active gravitational mass of a body is that which as as the source of gravity.

 

The passive gravitational mass of a body is that quantity on which gravity acts.

 

Thanks for the link. I'll take a peak.

So... you want someone here, to show how one would find the mass density of an electromagnetic field...

No. I didn't want anything. I was merely making a challenge. I think I decided not to post the result in open forum. If you'd like to see the answer please send me a PM and I'll send it to you.

 

Thanks for the link. When I get a printer I'll prinit it out and read it. I suspect that is wrong though. I can't concentrate reading involved documents online. Sitting too long causes pain and the pain distracts me.

 

By the way, you gave the value of the energy density for a magnetic field, not the mass density. Please take note that if the answer was just mass density = energy density/c[sup2[/sup] then I'd never have asked. That would be trivial and not worth discussing about.

 

The answer is also in this paper that I wrote - http://arxiv.org/abs/0709.0687

What is energy, exactly?

 

In the 19th century and before, according to Newtonian physics, energy was defined by matter whereby a force applied to it, times the distance traveled, defined the resultant energy absorbed. F x D = E.

That is the relationship for work. It is not a definition of energy. When you do work on a system it doesn't equal the energy of the system. It merely changes the energy of the system. But it can in no way be considered a form of energy.

 

In the 20th century matter was proposed to have an energy equivalent whereby Energy accordingly is equal to mass times the speed of light squared: E = MC² .

In Einstein's derivation of E = mc² in 1905, the mass/matter wasn't provced to have energy. The E in that expression is the amount energy emitted by a body at rest du to, say, a change in the electric structure of the body being perturbed so as t decreased the amount of EM energy in the system and thus emitting energy of equal amounts in opposite directions.

 

The m is the amount of mass that the body decreased. Einstein didn't prove that energy had mass. He started out using that as assumed. Later on the same thing held true. Never in the 20th century did this change. There was an application in elementary particle physics wherein the form of the mass changed, i.e. the creation of two photons from a matter/anti-matter annihilation with energy/inertial mass and momentum being conserved in the process

 

 

In the 19th century EM radiation was the wave motion of the aether, in the 20th century Einstein proposed the idea of pure energy with no carrier particle since an aether was proposed to be no longer needed.

Einstein never proposed that something could have pure energy. That's just a myth.

 

Are the present theories of energy correct?

There are no "theories" if energy. There is only a property of energy which din part defines energy and that's that the total energy of a closed system is a constant.

 

Most also believe that energy conservation in a closed system must be maintained.

All educated physicists know that to be true.

 

If any part of the present theory of energy could be wrong it seemingly could only be Einstein's formulation of mass-energy conversion, or his idea of pure energy which would be invalid if EM waves were comprised of a carrier particle of some kind.

Mass cannot be converted to energy and energy cannot be converted to mass. That fact was known from the start and an article to remind the scientific world to that fact was published. The article is A Relativistic Misconception, C. Roland Eddy, Science, September 1946

Absolutely not! There is no difference between F=ma and F=dp/dt for constant mass systems.

I take it that you're talking about Newtonian systems?