Energy of a photon in different frames.

[Uniqueuponhim]

Just a quick question here, that I've been having a bit of trouble figuring out:
How do you calculate the energy of a photon in a frame if you know the energy of the photon in another frame as well as the relative velocities of the two frames?

Thanks a lot.

 

[dextercioby]

How do you relate the 4-momenta of the photons in the two frames...?

Daniel.

 

[Uniqueuponhim]

There is only one photon, being observed in two different reference frames.

 

[pmb_phy]

Just a quick question here, that I've been having a bit of trouble figuring out:
How do you calculate the energy of a photon in a frame if you know the energy of the photon in another frame as well as the relative velocities of the two frames?

Thanks a lot.

Write doen the 4-momentum in the first frame and use the Loretz transformation to boost to the new frame. Let P' be the original 4-momenutum in the oiginal frame. Then P = (m'c, p') where the primed quantities represent the quantities in the old frame, i.e. the old photon mass and the onld photon momentum. Now transform to the new frame you'' get P = (mc, p). Suppose you're standing on the S<sub>x axis. Then all we need to is find P⁰ so we calculate P^u = L^uvP^v or P⁰ = L⁰vP<sup>v.

P0 = L0v[/suo]Pv

This will give you mc = P0 where m = E/c2 and E = hf. The rest should be easy for you to do since all there is left is calulation and then solve for f.

Pete

ps - my HTML is all messed but I'm too tired to fix it. Tommoror perhaps.</sup></sub>

 

[robphy]

Doppler Effect.

 

[dextercioby]

It's actually Doppler-Fizeau effect, though, correctly, it should bear only Armand Hyppolyte Fizeau's name, as he discovered it.

Daniel.

 

[Trilairian]

Just a quick question here, that I've been having a bit of trouble figuring out:
How do you calculate the energy of a photon in a frame if you know the energy of the photon in another frame as well as the relative velocities of the two frames?

Thanks a lot.

Since energy is proportional to frequency you can use the relativistic doppler shift equation. However that doesn't tell you much about what physics is taking place. Since a photon has zero mass you can't ligitimately use p^{\mu } = (\gamma mc, p) for the generalization as was suggested. Instead for generalization you use p^{\mu } = (\hbar \omega, p^{x}, p^{y}, p^{z}) where p^{i} = \hbar k^{i}. For any particle with or without mass v_{g} =d\omega /dk. For photons the relation reduces to v_{g} = c = v_{p} = \omega /k

 

[pmb_phy]

Trilairian; You've made an erroneous assumption above. Nobody was using the expression

p^{\mu} = (\gamma m c, p)

other than yourself. You failed to notice the context in which the quantity "m" was being used you therefore assumed that m was the photon's proper mass, which is indeed zero. Here "m" is not being used to represent proper mass. m = photon's inertial mass = p/v where here v = c (aka m = relatvistic mass). Had I had the case to use proper mass then it would have read m₀. It is always wise consider the context a quantity like "m" is being used before you attempt to make a correction in a post such as mine. It was obvious from context what my "m" meant, unless you have a limited knowledge of SR (E.g. see Introducing Einstein's Relativity by D'Inverno, Basic Relativity by Richard A. Mould, Special Relativity by A.P. French, or the Feynman Lectures)

Especially in relativity since some people use m₀ for proper mass and some use m.

The "m" used in my expression for the 4-momentum was not the photon's proper mass. It was the inertial mass m = p/v = p/c = E/c² = hf/c² since E = pc and E = hf. The h would have dropped out in the end of all the calculations and we'd have the relativistic expression for the doppler frequency.

The expression you used, i.e.

m = \gamma m_0

(gamma * proper mass) for P⁰/c is valid if and only if m₀ is not zero which is not the case here. In SR the relation

m = \gamma m_0

is derived on the assumption that the particle is a tardyon (a particle which always moves at v < c) which is not the case here. Since in this case it is zero another expression is required and that's m = E/c² where E = hf and therefore m = hf/c².

Pete

 

[Pyro]

Trilairian = You've made an erroneous assumption above. Nobody was using the expression

p^{\mu} = (\gamma m c, p)

as you suggest. You didn't note the context in which the quantity "m" was being used in and you therefore assumed that m was the photon's proper mass which is indeed zero.

Write doen(sic) the 4-momentum in the first frame and use the Loretz(sic) transformation to boost to the new frame. Let P' be the original 4-momenutum in the oiginal frame. Then P = (m'c, p')

You are so confused it is funny. Trilairian did not assume that that m was the photon's proper mass. He assumed m was the ridiculous "relativistic mass" (i.e. \gamma m). Are you now suggesting that your relativisitic mass is not always defined as \gamma m? I assume you are going to propose some kind of conditionality.

It is not. It is m = photon's inertial mass = p/v where here v = c (aka m = relatvistic mass). Always considder the context a quantity like "m" is being used in before you attempt to correct something like this.

m = \frac{p}{v} = \gamma m_0

 

[pmb_phy]

You are so confused it is funny.

I'm known for having a great sense of humor but I doubt that's how you meant that to be read. So while you're here please kill the poor attitude. This is a moderated board.
I'm sorry to inform you, pyro, that it is you who's confused. I said that the only person who was using the expression
P = (\gamma m c, [bp)
was Trilairian. When he saw my "m" in my 4-momenta, i.e. in my expression (mc, p) he must have thought it was proper mass since he stated "Since a photon has zero mass you can't ligitimately use p^{\mu } = (\gamma mc, p) for the generalization as was suggested. He tossed in the gamma himself thus changing the context from "m" being inertial mass to "m" being proper mass.
This follows since for my expression (mc, p) to be true then m must be the particle's inertial mass (what you enjoy referring to as "relativistic mass" in your condescending manner). And that is exactly what I meant. I'm well known on this and every forum for using m to mean inertial mass. But since you're new here you wouldn't know that yet. It may be that Trilairian doesn't know the difference but that remains to be seen if and when he chooses to respond.
Many people use "m" to mean proper mass. I do not. Trilairian does. Regardless, what a quantity is can be deduced from its usage or its context.

Trilairian did not assume that that m was the photon's proper mass.

You're being too vauge. What "m" are you referring to? The one when he read my 4-momentum and later wrote "...for the generalization as was suggested."? It was he who claimed that my "m" was zero and thus assumed that my "m" was proper mass when it was really inertial mass (aka relativistic mass)

He assumed m was the ridiculous "relativistic mass" (i.e. \gamma m).

If you think that relativistic mass is "ridiculous" then you don't know SR as well as you'd like to think you do.

Are you now suggesting that your relativisitic mass ...

Mine? Since when did relativistic mass become mine? I wasn't even alive when the quantitiy was defined.

..is not always defined as \gamma m?

If "m" is the proper mass of the particle then \gamma mThe so-called "relativistic mass" of a particle is the quantity "m" such that the quantity "mv" is a conserved quantity. If the particle is a luxon (I.e. a particle which always moves at v < c. I made an error above on terminology regarding "luxon" which has now been corrected). In such a case the particle's (relativistic) mass is a constant. Otherwise its a function of speed. One only need look it up to see this is the case. I know of no SR text which uses relativistic mass that says anything different than I just told you.

then the particle moves at I assume you are going to propose some kind of conditionality.
m = \frac{p}{v} = \gamma m_0

Well at least you're sort of on the right track. You can think of inertial mass as the ratio of momentum to speed. See footnote in French's text on bottom on page 16 "By inertial mass we mean the ratio of the linear momentum to speed"or for a complete coverage see D'Inverno's text section 4.5 Photons on pages 49->50. See Eq. (4.27) if nothing else.
And it wouldn't hurt you to kill the poor attitude.

 

[pmb_phy]

... the stupidity of your own definitions.[/quotes]Its comments like this which are proof that pyro has very little familiarity of the literature on this subject.

The definition is stated first by Richard C. Tolman. It is also repeated here

The Classical and Relativistic Concepts of Mass, Erik Eriksen, Kjell Voyenli, Foundations of Physics, Vol. 6, No. 1, 1976

which is online at
http://www.geocities.com/pmb_phy/ref/eriksen_voyenli_1976.pdf

Its also implicit in all derivations (E.g. Rindler, Feynman, Mould, French etc) of the (relativistic) mass of the particle and is stated quite clearly in French's text as was explained to they whiny pyro. And this includes the mass of a photon. And pyro claims to be familiar with this literature I bet?

So pyro - I've been informed that you've already been warned 8 times. You have 7 more to go and then g'bye!

 

[JesseM]

I am sorry to have to inform you, but a photon has neither rest mass nor inertial mass.

How do you define "inertial mass"? A mirrored box filled with photons would be slightly more resistant to acceleration than an empty but otherwise identical box, no?

 

[JesseM]

The quote specifically said a photon does not have inertial or gravitational mass. The effects of a photon as predicted by General Relativity is based on spacetime curvature and a photon following the said curvature. Not some type of mass relation as Newtonian gravity relates.

Your source was a usenet post, hardly authoritative. I'd like to see a definition of what you mean by "inertial mass", because "inertia" is usually defined in terms of resistance to acceleration, and as I said before a box filled with photons will be more resistant to acceleration that an empty but otherwise identical box.

It's true that you can't use Newtonian formulas to predict the path of photons, but I don't see where pmb_phy ever said you could. Also, note that if you take the weight of a box filled with photons and subtract the weight of the box when empty, the extra weight will be equal to the total energy of the photons divided by c^2. You can see this based on the equivalence principle, and the fact that the inertia of a bound system is proportional to the system's total energy (in its rest frame) divided by c^2.

 

[pmb_phy]

Your source was a usenet post, hardly authoritative. I'd like to see a definition of what you mean by "inertial mass", because "inertia" is usually defined in terms of resistance to acceleration, and as I said before a box filled with photons will be more resistant to acceleration that an empty but otherwise identical box.

That's easily shown. "Inertial mass" means that quantity which resists changes in momentum. As I explained to pyro (who wasn't able to follow) was that inertial mass (what he calls 'relativistic mass') is defined as the m such that mv is conserved.

Contrary to pyro I can back everything I say up either by straight forward calculation or a reference to the physics literature.

It's true that you can't use Newtonian formulas to predict the path of photons, but I don't see where pmb_phy ever said you could.

Nope. Since when did Newton come into the picture? A particle follows the path of a parabola in Newtonian gravity in a uniform g-field regardless of its mass. As the mass goes to zero in a uniform g-field the spatial portion of the geodesic approaches that of a semi-circle.

Jesse - Please note that when pyro demonstrated that he was one of those people who can't disagree with someone without flaming then I plonked him right away (i.e. he's on my ignore list) so I won't be able to understand what you're replying to if its from pyro unless you quote him.

To be 100% precise for a definition of mass (aka inertial mass) please see - http://www.geocities.com/physics_world/sr/inertial_mass.htm

Also, note that if you take the weight of a box filled with photons and subtract the weight of the box when empty, the extra weight will be equal to the total energy of the photons divided by c^2. You can see this based on the equivalence principle, and the fact that the inertia of a bound system is proportional to the system's total energy (in its rest frame) divided by c^2.

I believe I put all the derivations in this paper

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

However from the part of pyro's post that you quoted its quite evident that pyro doesn't have a clue as to what he's talking about. Its a fact of nature that a photon has mass but zero proper mass. When it is said that a photon has "mass" it means that it has inertial properties, active gravitational properties and passive gravitational properties. All of which are well known facts in GR.

Remember what I said about quotring the references? Then recall this. From The Feynman Lectures Vol -I page 7-11, section entitled Gravitation and Relativity

One feature of this new law is quite easy to understand is this: In Einstein relativity theory, anything which has energy has mass -- mass in the sense that it is attracted gravitationaly. Even light, which has energy, has a "mass". When a light beam, which has energy in it, comes past the sun there is attraction on it by the sun.

That's a quote. Here's a calculation

http://www.geocities.com/physics_world/gr/grav_light.htm

This was first done by Tolman et al but is so easy that anyone who knows GR can do it as a matter of course.

Pete

 

[pmb_phy]

How do you define "inertial mass"? A mirrored box filled with photons would be slightly more resistant to acceleration than an empty but otherwise identical box, no?

The relation E₀ = m₀c² means that an increase in the proper energy of the bnody (aka "rest energy") wil give yield to an increase in proper mass. This also says that since the proper inertial mass of a body is proportional to the proper passive gravitational mass of the body then that too will increase with an increase in an increase in rest energy.

Pete

 

[Trilairian]

You didn’t “explain” anything to Pyro. In fact he was the one who explained why you were wrong. Since your own slanted papers are in fact just plain wrong referencing them proves nothing. You don’t even know what mass is, inertial gravitational etc-all the same thing, i.e. equivalent, and invariant which I can prove easily if anyone is intereseted. Gravitational mass passive or active are Newtonian gravitation concepts. In relativity even massless entities such as electromagnetic plane waves gravitate and deviate from constant motion in the presence of gravitation. In relativity gravitation is expressed by Einstein’s field equations, not the Poisson equation that yields Newtonian gravitation. In relativity the source term is the stress-energy tensor, not the mass density as in that Poisson equation. In relativity all particles massive or massless follow geodesics under the influence of gravitation alone, not the paths of motion as forced about by Newtonian gravitation. The photon is massless. The mass m, the proportionality between the conserved momentum and the four-vector velocity p = mU, is invariant. It is also the m in the relation to rest energy given by E_{0} = mc^{2}. In short you need to catch up with the real world of physics or at least stop misrepresenting it.

 

[JesseM]

You don’t even know what mass is, inertial gravitational etc-all the same thing, i.e. equivalent, and invariant which I can prove easily if anyone is intereseted.

Can you address my question to Pyro above, since he has not replied? I'd like to know what mathematical definition you are using for "inertial mass" in relativity, since the only types of mass I have seen defined are rest mass and relativistic mass. Do you agree that the inertia of a bound system in its rest frame is proportional to its total energy in that frame (including the kinetic energy of its constituents) divided by c^2? Do you agree that according to the equivalence principle, inertial mass and gravitational mass must be equal? (ie if the bound system is sitting on a scale being accelerated at 9.8 m/s^2 in deep space, the reading on the scale will be the same as if it is placed on a scale on earth)

 

[Trilairian]

Can you address my question to Pyro above, since he has not replied? I'd like to know what mathematical definition you are using for "inertial mass" in relativity, since the only types of mass I have seen defined are rest mass and relativistic mass.

Both are bad terminology. Mass is rest energy. Inertial mass means the measure of resistence something has to change in motion. In Newtonian physics this means that it is the m in f = ma, and in relativistic physics this means it is the m in four-vector force equals mass times four-vector acceleration
F = mA. This mass m is invariant. It does not change with speed, no not in relativity.

Do you agree that the inertia of a bound system in its rest frame is proportional to its total energy in that frame (including the kinetic energy of its constituents) divided by c^2?

No.

Do you agree that according to the equivalence principle, inertial mass and gravitational mass must be equal?

Yes, although gravitational mass is a purely Newtonian quantity and is not the source term in the more general case of general relativity. Instead, the stress energy tensor is. The m which is invariant and is the inertial mass is also the m_{g} called gravitational mass which only is defined in the case of the Newtonian applicability of |g| = Gm_{g}/r^{2}.

 

[JesseM]

Do you agree that the inertia of a bound system in its rest frame is proportional to its total energy in that frame (including the kinetic energy of its constituents) divided by c^2?

No.

Then your understanding of inertia in relativity is incomplete. This issue was discussed at length on this thread--for example, someone posted a quote from this page, where Einstein explains that all forms of energy contribute to an object's resistence to motion (inertia) in the same way, so that a hot iron is harder to accelerate than a cold one (the only difference being the average kinetic energy of the atoms that make it up):

In his 1938 book, The Evolution of Physics, 1 Einstein writes:

Energy, at any rate kinetic energy, resists motion in the same way as ponderable masses. Is this also true of all kinds of energy?

The theory of [special] relativity deduces, from its fundamental assumption, a clear and convincing answer to this question, an answer again of a quantitative character: all energy resists change of motion; all energy behaves like matter; a piece of iron weighs more when red-hot than when cool; radiation traveling through space and emitted from the sun contains energy and therefore has mass, the sun and all radiating stars lose mass by emitting radiation. This conclusion, quite general in character, is an important achievement of the theory of relativity and fits all facts upon which it has been tested.

Classical physics introduced two substances: matter and energy. The first had weight, but the second was weightless. In classical physics we had two conservation laws: one for matter, the other for energy.7

Also posted was a link to this paper, which shows some experimental evidence that the kinetic energy of the parts of a bound system contribute to its inertial and gravitational mass. And potential energy certainly has to be taken into account in calculating inertial or gravitational mass too--as this page which was linked to earlier in that thread points out, the mass of a deuteron is less than the sum of the masses of the proton and neutron which make it up, because the potential energy of a proton and neutron bound into a deuteron is lower than the potential energy of the same proton and neutron kept a large distance apart.

 

[Trilairian]

Then your understanding of inertia in relativity is incomplete.

No it isn't.

...where Einstein explains that all forms of energy contribute to an object's resistence to motion ...

prior to the formulation of the modern version of the law, the four-vector equation I gave - so it is your understanding that isn't complete. I understand the dynamics much better than you and the inertia is not proportional to the energy as you assert. In fact the kind of inertia you are referring to, the proportionality between ordinary force and coordinate acceleration even has a directional dependence. You're so far off in assuming that it is the energy that it isn't even funny.

 

[JesseM]

No it isn't. prior to the formulation of the modern version of the law, the four-vector equation I gave - so it is your understanding that isn't complete.

So you're saying that the modern formulation is not just a more elegant way of describing findings in special relativity which were already known, but that it actually overturned ideas about special relativity which were held by Einstein? I doubt that very much...in any case, the more modern paper on the contribution of kinetic energy to gravitational mass which I mentioned, written by physicist Steve Carlip, said much the same thing as that Einstein quote:

The principle of equivalence—the exact equality of inertial and gravitational mass—is a cornerstone of general relativity, and experimental tests of the universality of free fall provide a large set of data that must be explained by any theory of gravitation. But the implication that energy contributes to gravitational mass can be rather counterintuitive. Students are often willing to accept the idea that potential energy has weight—after all, potential energy is a rather mysterious quantity to begin with—but many balk at the application to kinetic energy. Can it really be true that a hot brick weighs more than a cold brick?

So are you disagreeing with both Einstein and Carlip that a hot brick weighs more than a cold brick, and that the difference is proportional to the difference in kinetic energy divided by c^2?

In fact the kind of inertia you are referring to, the proportionality between ordinary force and coordinate acceleration even has a directional dependence.

I mentioned earlier that I was only talking about resistance to acceleration in the object's rest frame--the directional dependence only comes in when you are talking about the resistance to acceleration of an object that is already moving at some nonzero velocity v in your frame. With this clarification, do you still disagree that inertia is proportional to total energy divided by c^2?

 

[pervect]

No it isn't. prior to the formulation of the modern version of the law, the four-vector equation I gave - so it is your understanding that isn't complete. I understand the dynamics much better than you and the inertia is not proportional to the energy as you assert. In fact the kind of inertia you are referring to, the proportionality between ordinary force and coordinate acceleration even has a directional dependence. You're so far off in assuming that it is the energy that it isn't even funny.

Let's see if we agree about some basics.

Suppose we have a system occupying some volume V with a stress-energy tensor

T_{uv}

in a static Minkowskian space-time with a metric

|g_{ab}| = \delta^a{}_b+\epsilon

where \epsilon is small (<<1).

What is the formula for the mass (invariant mass) of the entire system in terms of it's stress energy tensor?

 

[pmb_phy]

You didn’t “explain” anything to Pyro.

Why you use the term "you" in a multiparticipant thread then you need to clarify whom it is you're referring to.

In fact he was the one who explained why you were wrong.

He's as wrong as you were in that post.

Since your own slanted papers..

To say I made a "slanted" statement or wrote a "slanted" paper means to present with a special interest. Since the concept of mass in relativity is a subject I do enjoy why on God's green Earth would I write anything else?? Unless you also have your own special meaning for the word "slanted" too?

Everything I've ever written is simply hard core special relativity. Had your education been a bit broader then you'd know that fact. In fact there's nothing I've ever posted on the internet here that can't be found in the mos prestigious relativity textbooks such as

Relativity; Special, General and Cosmological, Wolfgang Rindler, Oxford University Press, (2001)

Introducing Einstein's Relativity Ray D'Inverno, Oxford University Press, (1992)

Basic Relativity, Richard A. Mould, Springer University Press, (1994)

Gravity from the ground up, Bernard Schutz, Cambridge University Press, (2003)

Concepts of Mass in Contemporary Physics and Philosophy, Max Jammer, Princeton University Press, (2000)

And if you had Gravitation by Misner, Thorne and Wheeler (MTW) and you knew what you were reading then I'd point out where in that text they use the term in the exact same way that I do.

You don’t even know what mass is, ..

Now that has to be the most ignorant I've seen made on this board for the longest time. Comments such as this are designed to ruffle feathers as anyone knows. So why do you, a newbie, comment ever placed on this board. Its intuitively obvious even to the most casual observer that your claim is Total nonsense.

Most people here who know me also know that this is one of my favorite topic and the topic I'm most educated in. In fact David Morin (The guy who wrote this text with this section http://www.courses.fas.harvard.edu/~phys16/Textbook/ch11.pdf) was happy that I pointed out an error regarding this point. He's a nice guy so if you e-mail him please don't be as rude to him as you have been with me.

It is apparent that you've taken the position that the term "mass" should never be taken to mean anything other than "proper mass", probably because you're physics prof told you tha'ts the way it was or that your texts said that "relativistic mass" was old-fashioned or some other hogwash.

That you're most likely not even familiar with the fact the fact that people use "m = mass" differently than you do. Otherwise you wouldn't be so insulting as you're being here. Or perhaps you've never really put much thought into the whole "mass debate" found in the physics literature and it is that which is what is most likely responsible for you not knowing that the "m" in my post regarding the photon's 4-momentum (mc, p) means "inertial mass" (aka "relativistic mass") and not proper mass is the source of your mistake to begin with.

[qupte]...inertial gravitational etc-all the same thing, i.e. equivalent, and invariant which I can prove easily if anyone is intereseted.[/quote]Whomever is interested has already been shown the proof, by me of course.

Inertial mass (I.e. relativistic mass) = passive gravitational mass = active gravitational mass - All of which are not invariant. Simply crack open MTW and learn the fact yourself. See page 404 and especially EQ. (17.1) where the authors write

Mass is the source of gravity. The density of mass-energy as measured by any observer with 4-velocity u is
(17.1) \rhou*T*u

In short you need to catch up with the real world of physics or at least stop misrepresenting it.

Get a life hack. Or come back to reality.

Sorry mr. flamer. I don't waste my time with ignorant first year relativity students who think they know it all. If you choose to learn the subject correctly then ask the good Dr. Reilly Atkinson. If you learn the subject to the point where Reilly tells me that you;re no longer a waste of time then I'll take you off my block list. Until then you can keep flaming people until the moderator tosses you and your side kick pyro (who may be the same person) and get lost.

Note - Folks, it is a habit that some insecure people form that they will come to a new place to post and find that their odd-ball ideas are not being agreed with then they create a phantom poster who they'll use to support the ideas of the nutcase who created the phantom. They're easy to spot if they're used enough.

Pete

 

[quantumdude]

Let's tone it down here. If you feel that you are being personally attacked or if you witness a personal attack, rather than respond in kind please use the "Report Bad Post" feature. Moderators can't be everywhere at once, and we rely on reported posts to do our jobs here.

Carry on.

 

[Trilairian]

So you're saying that the modern formulation is not just a more elegant way of describing findings in special relativity which were already known, but that it actually overturned ideas about special relativity which were held by Einstein?

Yes, Einstein overturned some of his ideas in relativity and the definition of mass was one of them.

I doubt that very much

So?

...in any case, the more modern paper on the contribution of kinetic energy to gravitational mass which I mentioned,…

Any paper talking about relativistic corrections to Newtonian physics is not talking about doing purely relativistic physics. The source term in purely relativistic physics is the stress energy tensor, not mass.

So are you disagreeing with both Einstein and Carlip

No I am not on either case, not that it would matter. For one I do understand relativity better than both you and Steve combined, but for another he goes with the mass as invariant definition, not the relativistic mass definition adamantly and if you don’t believe me ask him! He has argued against pmb on this point as well as others endlessly in google groups. At least look up his position before attributing him with one.

I mentioned earlier that I was only talking about resistance to acceleration in the object's rest frame

Fine then heating it up added to its invariant mass. So?

With this clarification, do you still disagree that inertia is proportional to total energy divided by c^2?

No, I always said mass is rest frame energy. I never said otherwise. You didn’t qualify that you meant the rest frame which made the statement wrong.

 

[Trilairian]

This is how modern relativistic dynamics is formulated.
The equation of motion for general relativity is four-vector force equals mass times four-vector acceleration
F^{\mu } = mA^{\mu }
I have already shown in another thread that in special relativity four-vector acceleration A is related to coordinate acceleration a and coordinate velocity v by
A^{\mu } = \gamma ^{2}(a^{\mu } + \gamma ^{2}v^{\mu }(\mathbf{v}\cdot \mathbf{a}/c^{2}))
(Im choosing \mathbf{v}\cdot \mathbf{a} to represent the ordinary three component dot product of coordinate velocity and coordinate acceleration and using v and a to represent those even when I choose to give them a fourth element as indicated by the greek index.)
In special relativity four-vector force is the derivative of four-vector momentum with respect to proper time:
F^{\mu } = dp^{\mu }/d\tau
From time dilation this can be written
F^{\mu } = \gamma dp^{\mu }/dt
But the coordinate time derivative of momentum is the ordinary force so
F^{\mu } = \gamma f^{\mu }
Now using the results in the first equation one arrives at
f^{\mu } = \gamma m a^{\mu } + \gamma ^{3}mv^{\mu }(\mathbf{v}\cdot \mathbf{a}/c^{2}))
This last equation is the correct dynamics equation for special relativity expressed in terms of mass which does not change with speed m and coordinate velocity and coordinate acceleration yielding the ordinary force.

 

[JesseM]

Yes, Einstein overturned some of his ideas in relativity and the definition of mass was one of them.

That doesn't count as "overturning an idea about relativity" in a true physical sense, it would just be changing some terminology for pedagogical reasons.

Any paper talking about relativistic corrections to Newtonian physics is not talking about doing purely relativistic physics. The source term in purely relativistic physics is the stress energy tensor, not mass.

I never said anything about gravitational mass being a term in the equations of GR (and as far as I know neither did pmb_phy), when I refer to gravitational mass I'm just talking about a physical measurement like the reading on a scale when an object is placed on it. Do you agree that GR predicts the increase in the weight of a box when you fill it with some photons will be equal to the total energy of the photons in the box's rest frame divided by c^2?

So are you disagreeing with both Einstein and Carlip

No I am not on either case, not that it would matter. For one I do understand relativity better than both you and Steve combined

You probably do understand it better than me, seeing as I have only undergraduate-level knowledge of it, but nevertheless you have been either thinking or speaking sloppily on this thread when you disagreed with my statements about inertial and gravitational mass being proportional to total energy. And what's your basis for saying you understand relativity better than Steve Carlip? He's a physics professor at UC Davis and frequently discusses issues relating to relativity online, from what I've seen he seems extremely knowledgeable.

but for another he goes with the mass as invariant definition, not the relativistic mass definition adamantly and if you don’t believe me ask him! He has argued against pmb on this point as well as others endlessly in google groups.

It's true that most physicists prefer to use the term "mass" to refer only to rest mass, but I think they would all agree this is an aesthetic choice, that using "relativistic mass" is just an alternate way of analyzing certain problems that, if used properly, won't lead to any different conclusions. And in any case, take this up with pmb_phy, because I am not trying to defend the use of relativistic mass here, I'm just saying that the inertia of a compound object in its rest frame is proportional to the sum of the rest masses, kinetic energies and potential energies of all its constituent parts.

Fine then heating it up added to its invariant mass. So?

So you were wrong to object to my statement that "Einstein explains that all forms of energy contribute to an object's resistence to motion" by saying that it was made "prior to the formulation of the modern version of the law, the four-vector equation I gave". Einstein's statement is correct, as you just agreed.

With this clarification, do you still disagree that inertia is proportional to total energy divided by c^2?

No, I always said mass is rest frame energy. I never said otherwise. You didn’t qualify that you meant the rest frame which made the statement wrong.

Yes I did, you should have read more carefully:

Do you agree that the inertia of a bound system in its rest frame is proportional to its total energy in that frame (including the kinetic energy of its constituents) divided by c^2?

No.

 

[Trilairian]

Do you agree that GR predicts the increase in the weight of a box when you fill it with some photons will be equal to the total energy of the photons in the box's rest frame divided by c^2?

Yes you've increased the invariant mass of the box and what it contains by the amount that you increased the center of momentum frame energy. For a particle mass is rest energy, for a system its center of momentum frame energy.

You probably do understand it better than me, seeing as I have only undergraduate-level knowledge of it, but nevertheless you have been either thinking or speaking sloppily on this thread when you disagreed with my statements about inertial and gravitational mass being proportional to total energy.

No I wasn't. You were the one being sloppy by not qualifying that you were referring to the rest frame energy.

And what's your basis for saying you understand relativity better than Steve Carlip?

I understand it better than anyone sense Noether.

So you were wrong to object to my statement that "Einstein explains that all forms of energy contribute to an object's resistence to motion" ...

No I wasn't.

Yes I did, you should have read more carefully:

No you didn't. You should read and write more carefully.

 

[JesseM]

No I wasn't. You were the one being sloppy by not qualifying that you were referring to the rest frame energy.

Are you just being a troll now? Please explain why the part in bold does not count as a qualification that I was referring to the rest frame energy:

Do you agree that the inertia of a bound system in its rest frame is proportional to its total energy in that frame (including the kinetic energy of its constituents) divided by c^2?

And what's your basis for saying you understand relativity better than Steve Carlip?

I understand it better than anyone sense Noether.

Yup, definitely sensing a troll here.

 

[pmb_phy]

It's true that most physicists prefer to use the term "mass" to refer only to rest mass, but I think they would all agree this is an aesthetic choice, that using "relativistic mass" is just an alternate way of analyzing certain problems that, if used properly, won't lead to any different conclusions. And in any case, take this up with pmb_phy, because I am not trying to defend the use of relativistic mass here,

He doesn't dare to take it up with me. All he's done so far is make claims and insult me. I believe I know who this person is and if I'm correct that's about all he can do. He rarely backs up anything he says and when he does and his errors are pointed out to him then he'll ignore him and simply repeat that he made no such error. He's like a parror in that sense. And if I'm correct then note that the person at sci.physics.relativity that I'm referring too's knowledge is horrificly terrible. He has the worst understanding of relativity than anyone I've ever seen.

Its amazing the course this thread has taken due simply to the fact that somone didn't understand how I use the term "m" and then gets nasty when I explain it! Yipes!

Its true that many people who work in particle physics use the term "mass" to refer to "proper mass." Since that's all they study (i.e. research) then they have no need to keep adding the qualifier "rest" or "proper" to "mass" just as they have no reason to add "proper" to the term lifetime when they are speaking about the "proper lifetime" of a particle. And as everybody knows the proper lifetime is invariant while the lifetime is observer dependant.

Then there are cosmologists who use "mass" to refer to (relativistic) mass. It is not their place as a group to study the intrinsic properties of matter but to study the large scale matter and objects in the universe and how they interact with other matter.

In neither case do they actually study relativity. To them its a tool. The are particle physicists or cosmologists, they are not relativists, at least not when they are working in those areas as most do for much of their time.

For some scientists its a matter of the audience to whom they're writing (audience - not education). If the author was writing a textbook on GR then they might use "m" and simply refer to it as "mass" but they duely note it somewhere in the text and usually at the beginning. However that same person may just as well write a text on special relativity (or a text which is to a large extent SR) and then use "m" to refer to (relativistic) mass.

However particle physicists and cosmologists study matter in the very small or matter in the very large. Neither of them study matter on scales inbetween. Ever hear of someone talking about the relativistic properties of a capacitor? Rarely. I only know Rindler and Denur to do things like this.

Also it is quite misleading to say that a tensor is the source of something. That's just plain silly. Take EM as an example; the source of an EM field is charge and when those charges move the current produces a magnetic field. But it is still the charge itself that is creating all EM fields, either directly or indirectly. The mathematical quantity which is ueful in relativity in the equations is the 4-current = (charge density, current density).

JesseM - As I always say "Life is too short." Find someone more pleasant to discuss physics with. You can always tell the totally arrogant ones - They never allow you to PM them. Someone here is famous for that.

I'm outta here until the place is cleared of the troublemakers by Tom. You can reach me in PM since I'll be checking in there.

Pete