Derivation of the Equation for Relativistic Mass

[nakurusil]

I'm not sure what you were hoping to accomplish by just reposting stuff that you've said before, but after some thinking I've come to the conclusion that your approach works too. I still think it's a very strange approach though. In my opinion, my approach makes a lot more sense.

Of course

You claim that u', u1 and u2 are all arbitrary,

Why don't you read the previous post (more carefully):

u' has been taken arbitrary (this makes both u_1 and u_2 arbitrary) , therefore one can easily make u'=V thus making u_2=0 and u_1=2V/(1+V^2/c^2)=v then (5) becomes:

m_1=m(v)=m(0)/\sqrt(1-v^2/c^2)=m_0/\sqrt(1-v^2/c^2)

but that V is fixed. If V is fixed, then (3) shows that once you choose a specific value of u2,

...but this is not what I am doing, why don't you read above?<rest snipped as irrelevant since it is based on a basic misunderstanding>

 

[Fredrik]

You're not choosing u2=0?! What you're saying in the last post is that you're choosing u'=0. Fine. You can do that, but that's equivalent to choosing u2=0. It's the same thing.

(2) and (3) is a system of equations with four "variables": u1, u2, u' and V. If you fix two of them, then the other two are completely determined by (2) and (3). You fixed V in your definition of S, so when you choose a specific value of any of the remaining three variables, the other two will be fixed as well. That's why choosing u'=V is the same thing as choosing u2=0.

 

[nakurusil]

You're not choosing u2=0?! What you're saying in the last post is that you're choosing u'=0. Fine. You can do that, but that's equivalent to choosing u2=0. It's the same thing.

(2) and (3) is a system of equations with four "variables": u1, u2, u' and V. If you fix two of them, then the other two are completely determined by (2) and (3). You fixed V in your definition of S, so when you choose a specific value of any of the remaining three variables, the other two will be fixed as well. That's why choosing u'=V is the same thing as choosing u2=0.

This is why math was invented, in order to make sure that there is no ambiguity.

u&#039; has been taken arbitrary (this makes both u_1 and u_2 arbitrary) , therefore one can easily make u&#039;=V thus making u_2=0 and u_1=2V/(1+V^2/c^2)=v then (5) becomes:

m_1=m(v)=m(0)/\sqrt(1-v^2/c^2)=m_0/\sqrt(1-v^2/c^2)
So, whenever you have problems you can read Tolman , pages 43,44 in the reference. If you still have problems with his solution, I suggest you take it up with the editor since Tolman is long dead.

 

[Fredrik]

I don't have problems with Tolman or his solution. I found two consistent interpretations of the partial solution you posted. One of them was based on stuff that you had said but later denied. I posted both interpretations in this thread.

I do have a problem with you though. A lot of the things you're saying are wrong. In this thread your statements aren't wrong in the absolute sense, but they are contradicting each other. Simply reposting the same thing again won't change that.

My part in this discussion ends here. <<mentor snip>>

 

[nakurusil]

Start with the identity (Rosser)
g(u)=g(V)g(u')(1+u'V/cc) (1)
Multiply both its sides with m (invariant rest mass) in order to obtain
mg(u)=g(V)mg(u')(1+u'V/cc). (2)
Find out names for
mg(u), mg(u') and mg(u')u'
probably in accordance with theirs physical dimensions. In order to avoid criticism on this Forum avoid the name relativistic mass for mg(u) and mg(u) using instead E=mc^2g(u) and E'=mc^2g(u') calling them in accordance with theirs physical dimensions relativistic energy in I and I' respectively using for p=Eu/c^2 and p'=E'u'/c^2 the names of relativistic energy. Consider a simple collision from I and I' in order to convince yourself that they lead to results in accordance with conservation of momentum and energy.
Is there more to say?
sine ira et studio

This seems to lead back to Tolman's derivation mentioned above. Same equations , you would need to add the momentum conservation in the collision as you mention.

 

[bernhard.rothenstein]

mass relativity

This seems to lead back to Tolman's derivation mentioned above. Same equations , you would need to add the momentum conservation in the collision as you mention.

Thanks. Please let me know in which way does the derivation I have presented leads back to Tolman as not involving conservation of momentum and energy?

 

[Fredrik]

Start with the identity (Rosser)
g(u)=g(V)g(u')(1+u'V/cc) (1)
Multiply both its sides with m (invariant rest mass) in order to obtain
mg(u)=g(V)mg(u')(1+u'V/cc). (2)
Find out names for
mg(u), mg(u') and mg(u')u'
probably in accordance with theirs physical dimensions. In order to avoid criticism on this Forum avoid the name relativistic mass for mg(u) and mg(u) using instead E=mc^2g(u) and E'=mc^2g(u') calling them in accordance with theirs physical dimensions relativistic energy in I and I' respectively using for p=Eu/c^2 and p'=E'u'/c^2 the names of relativistic energy. Consider a simple collision from I and I' in order to convince yourself that they lead to results in accordance with conservation of momentum and energy.
Is there more to say?
sine ira et studio

This looks interesting, but I don't understand what you're doing here. In particular, what is the definition of the function g, and how did you obtain identity (1)?

 

[bernhard.rothenstein]

mass relativity

This looks interesting, but I don't understand what you're doing here. In particular, what is the definition of the function g, and how did you obtain identity (1)?

thanks. g(u) and g(u') stand for the gamma factor in the inertial reference frames I and I. We obtain the relativistic identity by expressing g(u) as a function of u' via the addition law of relativistic velocities (see W.G.V Rosser "Classical electromagnetism via relativity" London Butterworth 1968) pp 165-173)
For more details please have a critical look at
arXiv.org > physics > physics/0605203

Date: Tue, 23 May 2006 22:28:50 GMT (259kb)
Relativistic dynamics without conservation laws

Subj-class: Physics Education

We show that relativistic dynamics can be approached without using conservation laws (conservation of momentum, of energy and of the centre of mass). Our approach avoids collisions that are not easy to teach without mnemonic aids. The derivations are based on the principle of relativity and on its direct consequence, the addition law of relativistic velocities.
Full-text: PDF only
I would highly appreciate your experience with it.

 

[nakurusil]

This looks interesting, but I don't understand what you're doing here. In particular, what is the definition of the function g, and how did you obtain identity (1)?

I showed you how (1) is obtained in my first post about Tolman's solution (the one that you keep trying to re-explain to me). <<mentor snip>> You get it from the fact that The Lorentz transforms satisfy the cndition:

L(u)*L(v)=L(w) where w=(u+v)/(1+uv/c^2))

where L(v)=gamma(v)*|1...-v|
......|-v/c^2...1|

Try it, you might even be able to calculate it all by yourself.

 

[nakurusil]

Thanks. Please let me know in which way does the derivation I have presented leads back to Tolman as not involving conservation of momentum and energy?

You follow the same steps.
You use a hypothetical collision experiment
You use the conservation of momentum equation as "seen" from two different frames (you call them I and I', right?).
This is the formalism employed by Tolman about 100 years ago.

 

[Fredrik]

g(u) and g(u') stand for the gamma factor in the inertial reference frames I and I. We obtain the relativistic identity by expressing g(u) as a function of u' via the addition law of relativistic velocities...

I haven't had time to look at this yet. Maybe tomorrow.

 

[bernhard.rothenstein]

mass relativity

You follow the same steps.
You use a hypothetical collision experiment
You use the conservation of momentum equation as "seen" from two different frames (you call them I and I', right?).
This is the formalism employed by Tolman about 100 years ago.

With all respect, I think that in the derivation I poposed "collision", "conservation laws" are not mentioned and so it has nothing in common with Tolman but it has with the 100 years old special relativity.

 

[nakurusil]

With all respect, I think that in the derivation I poposed "collision", "conservation laws" are not mentioned and so it has nothing in common with Tolman but it has with the 100 years old special relativity.

Yes, I remember your paper now. We've talked about it.

 

[NanakiXIII]

It's been a while, I left this thread because of the now fortunately deleted bickering. However, having once again picked up this subject, no matter how useless, I've stumbled upon a problem. I refer to this post:

https://www.physicsforums.com/showpost.php?p=1227743&postcount=21

It states:

m(u_1)u_1+m(u_2)u_2 = (m(u_1)+m(u_2))V

However, since after the collision the speeds are no longer u_1 and u_1, how can you say that the relativistic mass of the "new" object is simply m(u_1)+m(u_2)?

 

[NanakiXIII]

Though he only had time for a quick glance, my physics teacher agreed that this looked incorrect.

He also suggested I should use the following equation to derive the equation for relativistic mass (I was too lazy to LaTeX, so here's a link):

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/imgrel/emcpc.gif

I'm not sure how to go about that, though. If anyone has any comment on why that which I think to be false is true or any suggestion on how to go about this using the referred equation, I'd much appreciate it.

 

[nakurusil]

It's been a while, I left this thread because of the now fortunately deleted bickering. However, having once again picked up this subject, no matter how useless, I've stumbled upon a problem. I refer to this post:

https://www.physicsforums.com/showpost.php?p=1227743&postcount=21

It states:

m(u_1)u_1+m(u_2)u_2 = (m(u_1)+m(u_2))V

However, since after the collision the speeds are no longer u_1 and u_1, how can you say that the relativistic mass of the "new" object is simply m(u_1)+m(u_2)?

Interesting, it appears that you found a weakness in Tolman's derivation. Since Tolman surmises that the two masses will have zero speed in S' (in order to move together with speed V wrt S after their collision) you would expect :

m(u_1)u_1+m(u_2)u_2 = (m(0)+m(0))V

However,I think that Tolman must have assumed that mass cannot vary non-continously (from m_1(u_1) to m_1(0)), thus justifying his use of :

m(u_1)u_1+m(u_2)u_2 = (m(u_1)+m(u_2))V

Granted, this is a very weak argument, so the best thing is to dump the blasted "relativistic mass" altogether, as I mentioned in the opening statement. The whole darned thing was introduced in order to reconcile the relativistic momentum/energy:

p=\gamma m(0)v (1)
E=\gamma m(0)c^2

with the Newtonian counterpart:p=mv (2)

So the best thing is to tell your teacher that your proof is you grouped together \gamma and proper mass m(0) into \gamma m(0) and you assigned that quantity to m

 

[NanakiXIII]

Interesting, it appears that you found a weakness in Tolman's derivation. Since Tolman surmises that the two masses will have zero speed in S' (in order to move together with speed V wrt S after their collision) you would expect :

m(u_1)u_1+m(u_2)u_2 = (m(0)+m(0))V

But since u_1 and u_2 are the speeds in S, why are you using the speed in S' after the collision?

However,I think that Tolman must have assumed that mass cannot vary non-continously (from m_1(u_1) to m_1(0)), thus justifying his use of :

Why should it vary non-continuously? For the relativistic mass to change non-continuously, wouldn't the velocity have to as well? I don't see why it would, it simply changes due to the collision, quite continuously, right?

 

[nakurusil]

But since u_1 and u_2 are the speeds in S, why are you using the speed in S' after the collision?

I see, correction:

m_1(u_1)u_1+m_2(u_2)u_2=(m_1(V)+m_2(V))V

Doesn't change anything, the derivation is still flawed.

Why should it vary non-continuously?

Because the formula above would imply that m_1(u_1) before collision becomes m_1(V) after collision and this would entail a discontinuity. This is why Tolman must be silently assuming that m_1 before after collision are exactly the same. In fact , he writes :


m_1u_1+m_2u_2=(m_1+m_2)V

So, Tolman must be assuming m_1(u_1) before and after the collision in his formula. Otherwise, the derivation falls apart.

Conclusion: use the derivation based on relativistic momentum I gave you twice.

 

[NanakiXIII]

Because the formula above would imply that m_1(u_1) before collision becomes m_1(V) after collision and this would entail a discontinuity.

This I don't understand. Changing u_1 to V doesn't break continuity, does it? There is just an acceleration. The collision isn't an instantaneous event. So if the velocity changes continuously, why wouldn't the relativistic mass?

Conclusion: use the derivation based on relativistic momentum I gave you twice.

I'm probably overlooking it or something, but I'll risk looking stupid and ask: what derivation are you talking about?

 

[nakurusil]

This I don't understand. Changing u_1 to V doesn't break continuity, does it? There is just an acceleration. The collision isn't an instantaneous event. So if the velocity changes continuously, why wouldn't the relativistic mass?

You got things backwards, if m_1(u_1) before collision (LHS) is different from m_1(V) after collision (RHS), Tolman's derivation falls apart. So, the only out for his derivation is that m_1 before and after collision are the same. THOUGH the m_1 speed has jumped from u_1 to V

I'm probably overlooking it or something, but I'll risk looking stupid and ask: what derivation are you talking about?

https://www.physicsforums.com/showpost.php?p=1238273&postcount=46
Look at the bottom.

 

[country boy]

Try starting with:
ds^2 = (cdt)^2 - dx^2

Divide through by dt^2 and rearrange:
c^2 = v^2 + (ds/dt)^2

Multiply by m^2 to get p^2:
(E/c)^2 = p^2 + (m*ds/dt)^2

Make the last term a constant:
m*ds/dt = Eo/c

And arrive at:
mc = Eo*dt/ds

Which from the 2nd eqn above is:
mc^2 = Eo/[1-(v/c)^2] = E

The Lorentz transformations come from the invariance of ds. The mass behaves just so that the rest energy is invariant.

 

[bernhard.rothenstein]

a simple and convincing way to m=gm(0)

Throughout pages on the internet I've seen the following relationship between rest mass and relativistic mass:

m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}

However, I have been utterly unable to find any sort of derivation or explanation of this formula, other than such explanations as "nearing the speed of light, an object's mass nears infinity and thus.. etc.". Where did this equation come from? Does it somehow follow from the Lorentz transformations? Any helpful insights would be much appreciated.

I suggest to have a look at
Leo Karlov, "Paul Kard and Lorentz-free special relativity." Phys.Educ. 24,
165-168 (1969).
The derivation Kard proposes is based on a scenario which involves a body absorbin a photon, using conservation of momentum and mass, velocity dependent mass and the relation m=p/c between the mass m of the photon and its momentum p.

 

[rbj]

Throughout pages on the internet I've seen the following relationship between rest mass and relativistic mass:

m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}

However, I have been utterly unable to find any sort of derivation or explanation of this formula, other than such explanations as "nearing the speed of light, an object's mass nears infinity and thus.. etc.". Where did this equation come from? Does it somehow follow from the Lorentz transformations? Any helpful insights would be much appreciated.

you can do a thought experiment with two identical balls of identical mass moving toward each other at identical speeds (along the y-axis), striking each other in a perfectly elactic collision and bouncing back. let's call the top ball, "A" and the bottom ball, "B". now, if there is no "x" velocity, all this makes sense, the two balls having equal mass and equal speeds, then have equal momentum and ball A bounces back up (+y direction) with the same speed it had before and ball B bounces back down similarly.

now imagine that this same experiment is done but ball A is moving along the x-axis direction with a constant velocity of v. the y velocity is the same as before in the frame of reference of ball A. ball B is not moving along the x-axis direction but still has the previous y velocity and they collide at the origin. after the collision ball A is moving up, as before (but also to the right with velocity v) and ball B is moving down. now, for observers, one traveling with ball A and the other hanging around with ball B, we set this up so that both observers sense the y-axis velocity of the ball in their reference frame as the same as the other observer sees for their own ball.

now, because of time-dilation, the y velocity of ball A, as observed by an observer hanging around with ball B must be slower than the velocity that the "moving" observer measures for ball A by a factor of:

\sqrt{1 - \frac{v^2}{c^2}}

but for the y-axis momentums to be the same, then the mass of ball A, as observed by the "stationary" observer m must be increased by the same factor (from what the "moving" observer sees as the mass of ball A m_0) or

\frac{m}{m_0} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}

and that is where the increased inertial mass comes from.

now a legitimate question would be "Why would the y-axis momentums of the two balls have to be the same? Why can't the masses of the two balls remain the same resulting in a decreased y-axis momentum for ball A (as observed by the "stationary" observer) and less than the y-momentum of ball B?"

the answer to that is then, after the collision, both balls would tend to be moving upward since ball B had more y-axis momentum than A from the "stationary" observer's POV. so what's wrong with that? well, instead of hanging out with the ball B observer, now let's hang out with the ball A observer who, "moving" at a constant velocity has just as legitimate perspective as does observer B. so from observer A's POV, he is stationary and it is ball B moving to the left at velocity v. so, if the y-axis momentums were not equal in magnitude, from observer B's perspective, they would be be moving up together after the collision, but from oberver A's perspective (which is just as legit as B's) they would be moving down after the collision. that is contradictory, so they must have the same y-axis momentum, whether you are hanging out with ball A or with ball B.

but since observer A sees ball B as having less y-axis velocity than ball A (due to time-dilation) observer A must see ball B as having larger mass so that the y-axis momentum of ball A is the same as the y-axis momentum of ball B. likewize since observer B sees ball A as having less y-axis velocity than ball B (due to time-dilation) observer B must see ball A as having larger mass so that the y-axis momentum of ball B is the same as the y-axis momentum of ball A.

 

[nakurusil]

I found a very nice treatment here. It seems devoid of the problem that plagues Tolman's solution. In addition, it gives a very nice derivation of relativistic momentum/energy from base principles.

 

[NanakiXIII]

Thanks, nakurusil, that site is quite perfect for me. However, I've stumbled upon one problem. I don't understand how they get from

http://upload.wikimedia.org/math/9/1/9/919ed04c37014cd574c512ec1333f658.png

to

http://upload.wikimedia.org/math/1/2/b/12b6bc3d9ac1a3bde8d9fa85d6afb8ed.png

There seems to be part of one of the interlying equations missing, and I just generally don't see how the did it. I tried getting there myself, but to no avail. Could anyone explain these steps in some detail?rjb, your way of handling the problem seems similar to the one on the site nakurusil posted and helped me understand it somewhat better, thanks.

 

[nakurusil]

Thanks, nakurusil, that site is quite perfect for me. However, I've stumbled upon one problem. I don't understand how they get from

http://upload.wikimedia.org/math/9/1/9/919ed04c37014cd574c512ec1333f658.png

to

http://upload.wikimedia.org/math/1/2/b/12b6bc3d9ac1a3bde8d9fa85d6afb8ed.png

There seems to be part of one of the interlying equations missing, and I just generally don't see how the did it. I tried getting there myself, but to no avail. Could anyone explain these steps in some detail?

Hmm, I looked, it requires solving a simple equation degree 2 in v followed by an easy series of substitutions. Try again.

 

[Sunset]

Hi!

Throughout pages on the internet I've seen the following relationship between rest mass and relativistic mass:

m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}

However, I have been utterly unable to find any sort of derivation or explanation of this formula, other than such explanations as "nearing the speed of light, an object's mass nears infinity and thus.. etc.". Where did this equation come from? Does it somehow follow from the Lorentz transformations? Any helpful insights would be much appreciated.

There is no derivation of this formula because it's simply the abreviation for the expression \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} which you CALL
m. Mass of a particle m_0 is a parameter of your theory , which doesn't change under Lorentz-transformation. The factor \gamma only appears in formulas - by accident - in the combination with m_0. That's why people speak of m_0 as "rest-mass", but the relation above is nothing but introducing an abreviation.
m_0 is a Lorentz-scalar, m is not m_0 &#039; i.e. it is NOT the mass measured by a moving guy. There exists only a act of measurement for particles at rest! You can't put a moving particle on a balance. That what you call mass is always for particles at rest - the prefix rest is unnecessary!

Best regards Martin

 

[NanakiXIII]

Hmm, I looked, it requires solving a simple equation degree 2 in v followed by an easy series of substitutions. Try again.

Simple to you, perhaps. Either I am to use techniques I'm not familiar with, or I'm just not seeing it. The variable v is in there three times, I've been trying but I can't seem to get it to one side the the equal sign.

By "degree 2 in v", you mean I should square the equation? I looked into that but still see no obvious solution.

 

[rbj]

There is no derivation of this formula because it's simply the abreviation for the expression \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} which you CALL m.

okay, so why don't we just CALL m this:

m = \frac{m_0}{\sqrt{1-\frac{c^2}{v^2}}}

or this

m = m_0 \sqrt{1-\frac{c^2}{v^2}}

or this

m = m_0 \frac{v^2}{c^2}

?

there's a lot of formulae (that are dimensionally correct) that we could pull out of our butt and say it's the inertial mass so that when you multiply it by v, it becomes momentum. why not use those? why is this one

p = \frac{m_0 v}{\sqrt{1-\frac{v^2}{c^2}}}

the correct one for momentum as perceived by an observer that observes the mass m_0 whizzing by at velocity v?

Mass of a particle m_0 is a parameter of your theory , which doesn't change under Lorentz-transformation. The factor \gamma only appears in formulas - by accident - in the combination with m_0.

yeah, right.

That's why people speak of m_0 as "rest-mass", but the relation above is nothing but introducing an abreviation.
m_0 is a Lorentz-scalar, m is not m_0 &#039; i.e. it is NOT the mass measured by a moving guy. There exists only a act of measurement for particles at rest! You can't put a moving particle on a balance.

but you can see how it behaves in a collision with a particle that you can possibly put on a scale.

That what you call mass is always for particles at rest - the prefix rest is unnecessary!

that's an opinion, not an undisputed fact.

i know this is controversial, but i think more confusion happens when speaking of photons ("massless" vs. m = (h \nu)/c^2) and such. even the new proposed definition of the kilogram uses the term "at rest" and if they adopt it, i'll bet those two words survive in the final definition.

 

[Sunset]

Hi rbj,

why is this one

p = \frac{m_0 v}{\sqrt{1-\frac{v^2}{c^2}}}

the correct one for momentum as perceived by an observer that observes the mass m_0 whizzing by at velocity v.

The relativistic momentum p_r = \frac{m_0 v}{\sqrt{1-\frac{v^2}{c^2}}} is defined in that way, because you want to have momentum-conservation.
Let me explain:
When you step from classical mechanics (Newtons axioms) to relativistic mechanics you need a new axiom-system. You have to define p_r in such a way that your relativistic theory is covariant and reproduces classical mechanics for low v. There might be indeed other choices for p_r which fullfill those two conditions - its simply the question if they describe reality.
p_r = \frac{m_0 v}{\sqrt{1-\frac{v^2}{c^2}}}
is the easiest choice which garantees conservation of p_r (e.g. Nolting, a german text-book, discusses this). Indeed it doesn't mean there are other possible choices, then you could describe physics with this p_r &#039; , too! The thing is you have a NEW physical quantity, which you should call "relativistic momentum".

Martin