Dynamic Mass of Photon: E=mc^2 & hv

[cesiumfrog]

It's a bit more than that but rarely, if ever, do I see a person devle into the general cases of objects with mass including continuous media, stressed bodies, etc. [Griffiths and Owen]have a paper in Am. J. Phys. which illustrates some of the problems one encounters under certain situations.

Who here has studied the mass of continuous media?

I've studied the paper you're referring to (v.51 p.1120, 1983 - it's referred to in Griffith's Electrodynamics), and it is exactly what I was thinking of when I said:

alarm bells: you'd be surprised how many long standing paradoxes are actually coordinate transformation errors

Don't suppose you've studied the resolution in Phys. Rev. D 73, 104020 (2006)?

 

[pervect]

I'm surprised this thread has been going on this long, without any resolution.

I think what people need to do is to consult some textbooks on the topic. (Note that the problem doesn't really need a consideration of continuous media, though it wouldn't hurt to use one if people are up to it).

I think the following quote, from the dialog "Use and abuse of the concepts of mass" in "Spacetime physics" (Taylor & Wheeler) covers the main points.

Can a photon -- that has no mass - give mass to an absorber?

Yes. Light with energy E transfers mass m = E [ed note: in geometric units] (= E_conv/c^2 [ed note: in standard units]) to a heavy absorber. (Exercise 8.5).

Adding a photon to a hollow mirrored sphere will transfer both the energy and the momentum of the photon to the system, just as it will if the photon is absorbed by an absorber.

Adding many photons in random directions will increase the energy of the sphere, without increasing its momentum.

The only necessary formula is this. For an isolated system, the invariant mass of that system is \sqrt{E^2 - p^2} in geometric units \sqrt{\left( E^2 - \left( p c \right)^2} \right) / c^2 in standard units.

This invariant mass will, by definition, increase if one adds energy to the system in question (by heating it up, but adding photons to it, etc) without changing the momentum.

Note that Hurkyl gave this correct answer much earlier in the thread, but seemed to be ignored.

While there is more that could be said, (especially about non-isolated systems which can be very tricky), I hope (I'm probably too optimistic) that we can get some resolution on the simple textbook problem of the invariant mass of an isolated system in special relativity.

 

[nakurusil]

I

Adding a photon to a hollow mirrored sphere will transfer both the energy and the momentum of the photon to the system, just as it will if the photon is absorbed by an absorber.

Adding many photons in random directions will increase the energy of the sphere, without increasing its momentum.

The only necessary formula is this. For an isolated system, the invariant mass of that system is \sqrt{E^2 - p^2} in geometric units \sqrt{\left( E^2 - \left( p c \right)^2} \right) / c^2 in standard units.

The above is the well-known and a perfectly fine solution for the particular case of adding multiple photons whose random momenta cancel out. No question about it.
In unit values for c:

m=\sqrt{(\Sigma E)^2-<\Sigma p,\Sigma p>}

and since \Sigma p is arranged to be 0 while \Sigma E has increased due to the injection of photons in the box , it becomes obvious that the invariant mass of the system has increased.

But what about the case of the single photon? The momentum of the sole photon is non - zero , so the simple approach from Taylor and Wheeler no longer works. This is the problem we were discussing and it is a very interesting problem indeed. A different approach is needed, the one I was showing earlier, whereby the sole photon , due to the curved trajectory in the grvitational field is transfering a downward momentum to the vertical walls of the box at each collision, thus creating the effect of increased weight. The same approach can be generalized to all cases of adding photons whose momenta don't cancel out.

But wait a minute! Can't we use the fact that the added photon contributes "equally" with E=\sqrt{<p,p>} to both "sigmas" in the formula of the invariant mass of the system? Not really, since we know that due to the gravitational field , the photon impulse changes with time , so it will be hard to draw any conclusion from

m=\sqrt{(\Sigma E)^2-<\Sigma p,\Sigma p>}

since the dot product <\Sigma p,\Sigma p> has become variable, and, if you do all the calculations (not very difficult) one finds out that we can't determine what happened to the system's invariant mass (may have increased , may have decreased, depending on the direction of the photon momentum). So, IMHO I believe that we need to byte the bullet and use the approach that calculates the additional vertical forces.

 

[Hurkyl]

So you're also unconvinced by Pete's contrary proof?

Yep. It looks like an equivication fallacy to me: he's changed the meaning of both "momentum" and "energy", so that his argument is no longer relevant to the original question!

 

[pervect]

The above is the well-known and a perfectly fine solution for the particular case of adding multiple photons whose random momenta cancel out. Since (for c=1):

m=\sqrt{(\Sigma E)^2-(\Sigma p)^2} and since \Sigma p is arranged to be 0 while \Sigma E has increased due to the injection of photons in the box , it becomes obvious that the invariant mass of the system has increased.

But what about the case of the single photon? The momentum of the sole photon is non - zero , so the simple approach from Taylor and Wheeler no longer works. This is the problem we were discussing and it is a very interesting problem indeed. A different approach is needed, the one I was showing earlier, whereby the sole photon , due to the curved trajectory in the grvitational field is transfering a downward momentum to the vertical walls of the box at each collision, thus creating the effect of increased weight. The same approach can be generalized to all cases of adding photons whose momenta don't cancel out.

The case of a heavy absorber is discussed in Taylor & Wheeler. It's not terribly hard to show that in the limit of a large mass of the absorber (m_absorber >> E/c^2, where E is the energy of the photon) that the amount of momeuntum gained by the absorber from the photon is negligible.

Consider a 400nm photon, at the upper edge of the visible spectrum. It will have an energy of about 5e-19 joules, and a momentum of 1.6e-27 kg-m/sec

If it impacts a 1 gm absorber, the absorber will move at a velocity of 1.6e-24 m/s to conserve momentum. The kinetic energy due to its motion will be negligible (about 1e-51 joules). So most of the energy of the photon goes in heating up the absorber, i.e. virtually all of the 5e-19 joules gets turned into heat.

The conservation of energy and momentum then tells us the energy, E, and momentum, p of the absorber

E = (9e13 joules + 5e-19 joules)
pc = (5e-19 joules)

We can then compute the mass sqrt(E^2 - (pc)^2)/c^2, and verify using standard series approximations that it increases by essentially 5e-19/c^2 grams due to the absorption of the photon, just as Taylor & Wheeler state.

The effect of the (pc)^2 term on the mass is negligible

I think your observation that gravity causes the photons to travel in non-straight paths is interesting, but it doesn't affect the textbook answer.

Your observation is more of an illustration of how the system weighs more in a gravitational field. One does expect a heavier object to weigh more than a lighter object, and your analysis illustrates how that happens. But we don't even need to refer to a "gravitational field" to work the problem - in fact, it is easier if we do not, if we simply stick with the standard textbook definition of invariant mass.

 

[JustinLevy]

Hint: the photon does not add mass to the system. But it does add energy/momentum.

No, that is the very point we've been trying to show you is incorrect.

A box + photon will have more invarient mass than the same box without a photon.

The above is the well-known and a perfectly fine solution for the particular case of adding multiple photons whose random momenta cancel out. No question about it.
...
But what about the case of the single photon? The momentum of the sole photon is non - zero , so the simple approach from Taylor and Wheeler no longer works.

No, the invarient mass method really is that straight-forward. In the rest frame of the empty box we have the four momentum (c=1) of (E,0,0,0) where E is the rest energy of the empty box. Now add a photon, and we have (E,0,0,0)+(e,e,0,0) where e is the energy of the photon. (E+e)^2 - e^2 is clearly GREATER than E^2. The invarient mass increases.

The point, as stated before by me and others, is that adding photons to a system can contribute to the system's invarient mass. I am not really understanding your extreme reluctance to accept this.

Everytime I bring up the invarient mass of the system you accuse me of a "strawman arguement" even though the mass increase of the system was the very question that the thought experiment brought up. Am I really misunderstanding your statements that much? From your statements quoted above it really does appear to us that you are claiming the invarient mass does not increase. But if we really are misunderstanding you somehow, please start back a little further so we can see where the problem is arising.

But we don't even need to refer to a "gravitational field" to work the problem - in fact, it is easier if we do not, if we simply stick with the standard textbook definition of invariant mass.

True. Actually I kind of feel uncomfortable with the semi-Newtonian phrasing we've been using which is why I tried to move nakurusil's argument to considering measuring the inertial mass in free space instead.

Actually, going back to the hollowed mirror sphere example you brought up, the photons add an energy density inside, but also apply pressure on the walls of the sphere which would strain the sphere. Is there someway to show how this pressure/strain would add to the inertial mass of the sphere, or would that somehow be double counting the effect of the photons?

 

[country boy]

But we don't even need to refer to a "gravitational field" to work the problem - in fact, it is easier if we do not, if we simply stick with the standard textbook definition of invariant mass.

So we are not talking about "weighing" anymore? Of course the relationship between mass and energy can be presented in terms of inertial mass alone. But I was enjoying the gravitational aspect of the discussion.

 

[pmb_phy]

Yep. It looks like an equivication fallacy to me: he's changed the meaning of both "momentum" and "energy", so that his argument is no longer relevant to the original question!

Can you fill me in and explain what an "equivication fallacy" is and how I supposedly "changed" the meaning of both momentum and energy. I go by very strict rules which I try to adhere to at all times and that rule is to make sure that at least two SR/GR textbooks back me up as to what I'm saying. I also go to the journals too and see how they define things. I also try to read several articles on the subject to make sure that one author is unique among many.

Kind regards

Pete

ps - The term "equivocal" means "subject to two or more interpretations and usually used to mislead or confuse". If you are saying that I'm intentionally try to mislead people on purpose then I resent that remark and ask that you cease on that course of reasoning where I'm thought of a person who misleads. I've nevver done anything in my posting career which was ever intended to mislead. I would consider such a post dishonorable and I never say anything which which would lead one to question my honor.

 

[nakurusil]

No, the invarient mass method really is that straight-forward. In the rest frame of the empty box we have the four momentum (c=1) of (E,0,0,0) where E is the rest energy of the empty box. Now add a photon, and we have (E,0,0,0)+(e,e,0,0) where e is the energy of the photon. (E+e)^2 - e^2 is clearly GREATER than E^2. The invarient mass increases.

Correct, it is good to see that you finally understood the problem statement.
My point was (go back and read the post) that for this particular case the approach works.
For the general case, it doesn't, this was the point I was explaining to pervect. Here is why:

Assume that the box has a number of particles of non-vanishing resultant momentum P and energy E. Then, the invariant mass of the system is :

m=\sqrt{ E^2-<P,P>}

Now add the photon of energy e and arbitrary orientation momentum p

The invariant mass becomes :

m'=\sqrt{(E+e)^2-<P+p,P+p>}=\sqrt{E^2-P^2+2(Ee-<P,p>}

The term 2(Ee-<P,p>)=2(E\sqrt{<p,p>}-<P,p>) can be positive , zero or negative, depending on the relative orientation of P and p.

As such, m'>m, m'=m or ...m'<m !
Surprise, surprise, the photon doesn't always add to the invariant mass of the system!
The complete solution is further complicated by the fact that the photon momentum is not constant, as shown in my earlier post, it is time-varying. The photons will not describe straight lines, they will describe ever-descending parabolas bounded by the vertical walls. To calculate this part rigorously, you would need GR , please don't call my approach "semi -Newtonian", ok?
So, the exact contribution is a function of time:

2(E\sqrt{<p(t),p(t)>}-<P,p(t)>)

and this happens even for the very particular case you studied (P=0) :

2E\sqrt{<p(t),p(t)>}

Now, let's try another case. We will start with your simplified case (empty box) , i.e. P=0 and let's add a couple of photons with the arbitrary momenta p_1 and p_2. What happens to the invariant mass of the resulting system? I am quite sure that you can calculate it yourself after seeing the general solution.

The point, as stated before by me and others, is that adding photons to a system can contribute to the system's invarient mass.

I will make this point one last time:

1. It is a bad idea to talk about relativistic mass.
2. It is an even worse idea to talk about the relativistic mass of photons
3. It is a bad idea to use thought experiments in the style "photon in the box", which give variable results depending on initial and final conditions, depending on momentum directions in order to prove that "photons can contribute to the invariant mass of a system". Because sometimes they don't add any mass and other times they even subtract, thus making the whole issue muddled.
4. It is a good idea to say that photons add to the overall energy of the system and (vectorially) to the overall momentum of a system. We should leave it to that.

"Justin",

I understand that we are having a pedagogical dispute, you can continue teaching your way, I will continue teaching my way. I have expunged "relativistic mass" and "photon contribution the the invariant mass of a system " from my course notes and I am very happy with the results.

 

[pmb_phy]

I will make this point one last time:

1. It is a bad idea to talk about relativistic mass.
2. It is an even worse idea to talk about the relativistic mass of photons.

May I ask why? The reason it is used goes far beyond what you'll find in your intro to SR/GR classes.

Pete

 

[nakurusil]

My I ask why? The reason it is used goes far beyond what you'll find in your intro to SR/GR classes.

Pete

Because relativistic mass is not necessary in teaching relativity.
The worst thing it does, is that it brings about the notion "photons do not have rest mass but they have relativistic mass" resulting into never ending discussions that lead nowhere.

 

[nakurusil]

The case of a heavy absorber is discussed in Taylor & Wheeler. It's not terribly hard to show that in the limit of a large mass of the absorber (m_absorber >> E/c^2, where E is the energy of the photon) that the amount of momeuntum gained by the absorber from the photon is negligible.

Consider a 400nm photon, at the upper edge of the visible spectrum. It will have an energy of about 5e-19 joules, and a momentum of 1.6e-27 kg-m/sec

If it impacts a 1 gm absorber, the absorber will move at a velocity of 1.6e-24 m/s to conserve momentum. The kinetic energy due to its motion will be negligible (about 1e-51 joules). So most of the energy of the photon goes in heating up the absorber, i.e. virtually all of the 5e-19 joules gets turned into heat.

The conservation of energy and momentum then tells us the energy, E, and momentum, p of the absorber

E = (9e13 joules + 5e-19 joules)
pc = (5e-19 joules)

We can then compute the mass sqrt(E^2 - (pc)^2)/c^2, and verify using standard series approximations that it increases by essentially 5e-19/c^2 grams due to the absorption of the photon, just as Taylor & Wheeler state.

The effect of the (pc)^2 term on the mass is negligible

Yes, we are in total agreement on this.

I think your observation that gravity causes the photons to travel in non-straight paths is interesting, but it doesn't affect the textbook answer.

Your observation is more of an illustration of how the system weighs more in a gravitational field. One does expect a heavier object to weigh more than a lighter object, and your analysis illustrates how that happens. But we don't even need to refer to a "gravitational field" to work the problem - in fact, it is easier if we do not, if we simply stick with the standard textbook definition of invariant mass.

Actually, the situation is more complicated (and more interesting) , please see my answer to JustinL. I believe that we need to go thru the slightly more complicated calculations if we want an exact quantitative result.

 

[pmb_phy]

Because relativistic mass is not necessary in teaching relativity.
The worst thing it does, is that it brings about the notion "photons do not have rest mass but they have relativistic mass" resulting into never ending discussions that lead nowhere.

What tools are necessary in teaching is what may help the student in his work. Relativistic mass provides a good tool for that given all the applications it can provide and ways of looking at a problem that is difficult to look at in another way. For example; Suppose someone asked you what the center of mass of a system of particles is. You may have a hard time doing that so you might just do away with the question altogether and say that the center of mass is meaningless and has no use whereas the center of energy does have meaning. That would be a very weak arguement. But other physicists (e.g. Rindler) had no difficulty with this problem. They simply use the regular formula replacing rest mass with inertial mass. That is but one example. Now suppose that there is a particle in a static gravitational field. The particle will have constant energy. But it won't be given by E = mc² (were m is inertial mass). m would be P^0 whereas energy would be P_0. The list goes on.

Let me ask you this - How would you define the mass density of an ideal gas?

Pete

 

[nakurusil]

Let me ask you this - How would you define the mass density of an ideal gas?

Pete

Here is an http://www.owlnet.rice.edu/~jigarb/density.htm as good as any.
This is getting interesting, let's see where it leads.

 

[Hurkyl]

Can you fill me in and explain what an "equivication fallacy" is and how I supposedly "changed" the meaning of both momentum and energy.

You switched from ordinary momentum to the generalized momentum that includes a stress term, and likewise, you added a stress term to the energy that was being considered.

AFAIK, nobody has ever argued that E/c²=m if you mix in a few new energy terms and pass to some notion of generalized relativistic mass.


As an aside, it may be possible that generalized relativistic mass becomes a useful notion -- if you agree, then it would be nice to see a precise definition and an example of it actually being useful. For 3-vectors, p = mv can almost never be true for generalized momentum (and velocity might even become difficult to define, depending on the situation), so I'm highly skeptical that generalized relativistic mass is useful.

ps - The term "equivocal" means "subject to two or more interpretations and usually used to mislead or confuse". If you are saying that I'm intentionally try to mislead people

I didn't mean to imply it was intentional: I simply meant I thought it was happening.

 

[pmb_phy]

You switched from ordinary momentum to the generalized momentum that includes a stress term, and likewise, you added a stress term to the energy that was being considered.

That is not a change in definition. I was explaining what the general formula for momentum was and that reduces to what you call the "ordinary" momentum for a single particle. There was no switching going on. I was filling in where I believed tjere was a hole.

AFAIK, nobody has ever argued that E/c²=m if you mix in a few new energy terms and pass to some notion generalized relativistic mass.

Have you ever seen anyone every discuss such a situation using the most general form of mass there is at all?

As an aside, it may be possible that generalized relativistic mass becomes a useful notion -- if you agree, then it would be nice to see a precise definition and an example of it actually being useful.

What I call useful I'm sure you will have another idea of what is useful. I call "useful" that which gives the correct answer for any legitimate question in relativity. Because nobody uses it in practive today cannot be taken as any form of proof that it won't be used 300 years from now. But I did give an example of a stressed rod many times in the past. To see that derivation please take a look at the web page I created specifically for this purpose.

http://www.geocities.com/physics_world/sr/inertial_energy_vs_mass.htm

There is also an article in the Am. J. Phys. Called "The inertia of stress" which you might want to look at if you have access to this journal.

For 3-vectors, p = mv can almost never be true for generalized momentum (and velocity might even become difficult to define, depending on the situation), so I'm highly skeptical that generalized relativistic mass is useful.

Skeptical? You seemed to be saying earlier in this thread that you were 100% sure that generalized relativistic mass is useful. And there is no reason to keep referring to it as "generalized relativistic mass" since the term has a ring to it as if it was special in some sense whereas the cases like a free partilce is actually a special case.

I didn't mean to imply it was intentional: I simply meant I thought it was happening.

The term you used has that meaning. I recommend that you just say what you mean instead of using glossy language.

Best wishes

Pete

 

[rbj]

The worst thing it does, is that it brings about the notion "photons do not have rest mass but they have relativistic mass" resulting into never ending discussions that lead nowhere.

you say that, but you don't support it.

 

[pmb_phy]

you say that, but you don't support it.

I myself don;t like getting into these discussions on mass so I think it is best for me to agree to disagree with those opposed to inertial mass (aka relativistic mass) and whomever wishes to talk to me about it can PM me.

Kind regards all

Pete

 

[country boy]

I'm afraid this thread has frayed.

 

[pervect]

My $.02 - now that we've hopefully resolved the basic issues we can talk about some of the more advanced issues.

Inertial mass is actually a second rank tensor, as Hurkyl points out.

Relativistic mass is usually understood to be a scalar quantity, another name for energy (so it is not the same as inertial mass.) Pete's usage in this area is IMO non-standard, and causes a lot of confusion.

Invariant mass is also a scalar - it is the invariant length of the energy-momentum 4-vector for a point particle - or the invariant length of the energy momentum 4-vector for a system with a finite volume.

Interestingly enough, the total energy-momentum of a non-isolated system with finite volume does not transform as a 4-vector. This is behind some of what Pete is (IMO) trying to say.

If one reads Taylor & Wheeler, "Spacetime physics" closely, for instance, one will see that they are always careful to say that the mass of an isolated system is a Lorentz invariant, not that the mass of any arbitrary system is Lorentz invariant.

How does one deal with relativistic systems with a finite volume? Via the stress-energy tensor, which always transforms properly (i.e. covariantly) as a rank 2 tensor.

Given a particular frame of reference, the total energy in a given volume can be expressed as the integral of T_00, and the components of the momentum can be expressed as integrals of T_0i.

Given these volume integrals for the total energy E and mass p in some volume V in special relativity, the mass contained within a volume can be (and as far as I can tell from a close reading of the textbooks) is defined in special relativity as sqrt(E^2 - (pc)^2) / c^2, in spite of the fact that the above quantity is not always Lorentz invariant.

(I haven't seen any textbook specifically say that the mass of a non-isolated system is defined in this manner - rather, the above formula is offered as a general defintion of mass, and the comment is made additionally that the above quantity is an invariant for isolated systems. I believe it is correct to say that the quantity is still defined for non-isolated systems, but is not invariant).

In other words, in spite of the name, the "invariant mass" of a system is actually an invariant only if the system has zero volume, or if the system is isolated.

For one source for this in the literature, see http://arxiv.org/abs/physics/0505004 or the peer-reviewed

http://www.springerlink.com/content/534j31t61675w010/

by the same author.

 

[JustinLevy]

Correct, it is good to see that you finally understood the problem statement.
My point was (go back and read the post) that for this particular case the approach works.

No, you previously very clearly stated that the mass did not increase. In suddenly changing your opinion now you are trying to make it sound like you always claimed it did increase. And on top of that, condescendingly implying that I was the one claiming the invarient mass didn't increase and you proved me wrong... a complete switch-a-roo. It is because of attitude like this that it was easy to tell you were a sockpuppet of the banned clj4.

Everyone makes mistakes, but unless we can admit to ourselves that we've made a mistake, we can't learn from them. I hope you can take this with you in the future and I wish you well.

While the discussion is now ended, I would still like to help you learn the results here as you are still misunderstanding some pieces.

As such, m'>m, m'=m or ... m'<m !
Surprise, surprise, the photon doesn't always add to the invariant mass of the system!

That is incorrect. The simple solution given before is general. You can translate to any inertial frame and the photon will still be there. So translate to the rest frame of the box, and calculate the result of adding the photon in that frame. The result as shown previously is that the invarient mass increases. Always.

3. It is a bad idea to use thought experiments in the style "photon in the box", which give variable results depending on initial and final conditions, depending on momentum directions in order to prove that "photons can contribute to the invariant mass of a system". Because sometimes they don't add any mass and other times they even subtract, thus making the whole issue muddled.

No, add a photon to a closed system and it will always increase the invarient mass. Please, please take the time to think this through so you can gain something from this discussion.

I have expunged "relativistic mass" and "photon contribution the the invariant mass of a system " from my course notes and I am very happy with the results.

I somehow doubt such course notes exist. Assuming you are indeed clj4, who admitting he is Adrian Sfarti when I mentioned I believed Sfarti's papers would get rejected from inclusion in last year's Grossmann Meeting on General Relativity published conference proceedings, then the only record I could find of you ever teaching was as a guest lecturer in a CS course.

If I am wrong, feel free to email me (bj0umow02@sneakemail.com) and I appologize in advance.

Also, I appologize about my comments on Sfarti's papers as there was no way for me to know you were him at the time. Actually, it is usually rare for presented material to be withheld from conference proceedings, so they probably will go through fine. It is too time consuming to adequately peer-review conference proceedings, but if they do make comments on your papers, I hope you take them to heart and stop believing that you are defending the mainstream view against a sea of "crackpots" even though many people have taken time to help point out how your arguements actually conflict with current mainstream theory. People aren't "out to get you", they are trying to help you learn.

I wish you good luck in your endeavors and hope you never lose the thirst for learning.

==================================

Returning to the original topic, there was a question I had that got buried earlier:

Going back to the hollowed mirror sphere example pervect brought up, the photons add an energy density inside, but also apply pressure on the walls of the sphere which would strain the sphere. Is there someway to show how this pressure/strain would add to the inertial mass of the sphere, or would that somehow be double counting the effect of the photons?

 

[Hurkyl]

What I call useful I'm sure you will have another idea of what is useful. I call "useful" that which gives the correct answer for any legitimate question in relativity.

Requiring one to write "rest mass" instead of "mass" has the benefit of added precision and the drawback of being more cumbersome. I define this requirement to be useful iff the benefit outweighs the drawbacks.

When I talk about the "usefulness" of relativistic mass in this context, I mean in the above sense: does the notion of relativistic mass have sufficient utility to justify encumbering the notion of rest mass?

The term you used has that meaning. I recommend that you just say what you mean instead of using glossy language.

Er, I did. Equivocation is one of the standard logical fallacies.

 

[pervect]

Returning to the original topic, there was a question I had that got buried earlier:

Going back to the hollowed mirror sphere example pervect brought up, the photons add an energy density inside, but also apply pressure on the walls of the sphere which would strain the sphere. Is there someway to show how this pressure/strain would add to the inertial mass of the sphere, or would that somehow be double counting the effect of the photons?

I talk about this in an arrticle I wrote for the wikipedia. (For those who care about such things, I need to point out that this should only be considered to be wikipedian reviewed and not peer reviewed).

http://en.wikipedia.org/wiki/Mass_i...simple_examples_of_mass_in_general_relativity

If you consider the simplest case of an isolated sphere, the tension terms in the walls of the sphere are exactly counterbalanced by the pressure terms in the interior of the sphere, and they make no net contribution to the Komar mass of the sphere + photons.

So if you have an empty sphere, and add photons to it, the mass of the system of sphere + photons increases by E/c^2, where E is the energy of the photons you added to the sphere.

I don't think I go into all the gory details, see for instance http://www.efunda.com/formulae/solid_mechanics/mat_mechanics/pressure_vessel.cfm for how to compute the stresses in a sphere.

Also note that one particular type of mass from general (and not special) relativity is being used in this calculation, a type of mass known as Komar mass.

GR does not have a single, general definition of mass, but has several different definitions that apply under different circumstances.

The Komar mass formula is one of the simplest, and applies to any static system, such as our mirrored sphere. (Actually, with enough care, the Komar formula can also be applied to stationary systems, like rotating spheres or Kerr black holes, but some of the details get a bit more complex).

It should be noted here that we are using the Komar formula, and that for the isolated sphere it gives the same answer for mass as the special relativistic formula.

The simpler special relativistic formula could actually be used here - note that for the isolated system of sphere + photons it gives the same answer. Note also that the SR formula does NOT have any pressure terms - there are only momentum and energy terms in the SR formula.

It greatly simplifies things to consider the mass only of an isolated system - you can get numbers for the mass of a non-isolated system, but you should realize that they are coordinate dependent.

 

[cesiumfrog]

That is not a change in definition. I was explaining what the general formula for momentum was and that reduces to what you call the "ordinary" momentum for a single particle. [..] There is also an article in the Am. J. Phys. Called "The inertia of stress" [..] And there is no reason to keep referring to it as "generalized relativistic mass"

Regardless of whether Pete's "generalisation" is reasonable, it is distinct from the usual concept of relativistic mass. (And why haste to sacrifice the niceties of that mass concept, except to use invariant mass instead?)

That article considers some material that is painted with electric charge. The article basically presumes the total relativistic mass of the system to be (naturally) the relativistic mass of the material (m_r) plus the mass-energy required to pull the electric charge distribution onto place. The total energy/momentum result by multiplying that with c^2 or with its velocity.

Then the article claims* to demonstrate obtaining the same results by calculating the momentum as "m_rv + stress momentum + field momentum (E x B)" and the energy as "m_r c^2 + stress energy + field energy (E^2 + B^2)". (In this sense the article concludes that there is inertia in stress, as in mass and in classical EM fields).

Now, Pete seems to be asserting that (when we consider individual components of this system) the stress term should be associated with the intervening material (rather than with the external charge distribution applying the forces.. which is presumably ascribed only the field's energy/momentum). It isn't shown whether or not the intervening material actually behaves as having increased inertia physically.

*The technicalities seem a bit odd, I'll look into the references a little - the whole stress thing seems like a hack to avoid a paradox which other authors claim never existed..

 

[pmb_phy]

*The technicalities seem a bit odd, I'll look into the references a little - the whole stress thing seems like a hack to avoid a paradox which other authors claim never existed..

Modern authors over the last 40 years seemed to have left this part of relativity out of there texts for the most part. The only one that comes to mind is Rindlers 1982 intro to SR text. It appears that new students wouldn't touch a book that was written before they were born, supposeldy they believe that they can't learn anything from them that they can from a newer text. But such texts by, say, Moller are a gem of a textbook. I myself haven't even gotten to read Moller but that's due to a lack of access and a lack of $.

Pete

 

[robphy]

It appears that new students wouldn't touch a book that was written before they were born, supposeldy they believe that they can't learn anything from them that they can from a newer text. But such texts by, say, Moller are a gem of a textbook. I myself haven't even gotten to read Moller but that's due to a lack of access and a lack of $.

Pete

Apparently, this is Moller's text:
http://www.archive.org/details/theoryofrelativi029229mbp

(Off main topic:
Recently, I've been interested in old pre-1925 relativity books, especially those pre-GR books. It's interesting and inspiring to see the physical, mathematical, philosophical, and pedagogical approaches taken to understand relativity back then. And it certainly is possible that some idea or technique which has not been continued in the modern textbooks can be useful for pedagogy or even cutting edge research.

One of the most interesting are the works of A.A. Robb, starting with his 1911 book "Optical geometry of Motion" http://www.archive.org/details/opticalgeometryo00robbrich
which (I think) was the first to use the word "rapidity" (for the Minkowskian analogue of angle) and has the foundations of the Bondi k-calculus and the beautiful but not well-known formula for the interval between a local event and a distant one in terms of three clock readings from a radar experiment [as featured in texts by Synge, Geroch, and Burke]. Some of his other books: http://www.archive.org/search.php?query=robb AND (geometry OR relativity) suggest that he was one of the first to emphasize the causal order... in fact, recovering practically all of the structure of Minkowski spacetime from the causal order... in a methodical but tortuous way.
)

More from archive.org: http://www.archive.org/search.php?query=subject:Relativity. Enjoy.

 

[pmb_phy]

Recently, I've been interested in old pre-1925 relativity books, especially those pre-GR books. It's interesting and inspiring to see the physical, mathematical, philosophical, and pedagogical approaches taken to understand relativity back then. And it certainly is possible that some idea or technique which has not been continued in the modern textbooks can be useful for pedagogy or even cutting edge research.

Thanks for that insightful opinion Rob!

There was some comments before about a gas of massless photons which had mass itself and the weight of a box of such a gas. There was an article published on this topic

The mass of a gas of massless photons, H. Kolbenstvedt, Am. J. Phys. 63(1), January 1995

I have this one and am in the process of placing my journal files on CD. Since this one was already scanned and on disk I thought I'd post it. Its a very interesting read!

http://www.geocities.com/physics_world/Kolbenstvedt_1995.pdf

Best wishes and enjoy

Pete