# Energy of a photon in different frames.

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.

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JesseM
Trilairian said:
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.
Trilairian said:
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?
JesseM said:
So are you disagreeing with both Einstein and Carlip
Trilairian said:
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.
Trilairian said:
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.
Trilairian said:
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.
JesseM said:
With this clarification, do you still disagree that inertia is proportional to total energy divided by c^2?
Trilairian said:
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:
JesseM said:
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?
Trilairian said:
No.

JesseM said:
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 refering 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 sence 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
Trilairian said:
No I wasn't. You were the one being sloppy by not qualifying that you were refering 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:
JesseM said:
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?
JesseM said:
And what's your basis for saying you understand relativity better than Steve Carlip?
Trilairian said:
I understand it better than anyone sence Noether.
Yup, definitely sensing a troll here.

JesseM said:
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 refering 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

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