Relativistic Limit of Tardyon Momentum and Mass

[bernhard.rothenstein]

Consider the following transformation equations for the momentum and mass of a tardyon
p=gp'(1+V/u')
m=gm'(1+Vu'/cc)
Question: What happens if we take in both sides the limit for u' going to c.
Has that some physical meaning? Thanks for the answer.
soft words and hard arguments

 

[Meir Achuz]

Consider the following transformation equations for the momentum and mass of a tardyon
p=gp'(1+V/u')
m=gm'(1+Vu'/cc)
Question: What happens if we take in both sides the limit for u' going to c.
Has that some physical meaning? Thanks for the answer.
soft words and hard arguments

Thank you. That is a good demonstration why using the 4-vector (E,p)
is better than u, which has complicated transformation properties.
You also have shown an example of how using m instead of E/c^2 can lead to confusion.
In terms of E and p, taking u' to c is trivial. (Nothing happens.)

 

[bernhard.rothenstein]

Thank you. That is a good demonstration why using the 4-vector (E,p)
is better than u, which has complicated transformation properties.
You also have shown an example of how using m instead of E/c^2 can lead to confusion.
In terms of E and p, taking u' to c is trivial. (Nothing happens.)

Consider the following transformation equations for the momentum and mass of a tardyon
p(u)=gp'(u')(1+V/u') (1)
m(u)=gm'(u')(1+Vu'/cc) (2)
Question: What happens if we take in both sides the limit for u' going to c.
Has that some physical meaning? Thanks for the answer.
soft words and hard arguments

Thanks for your participation on my thread. As you can see I have made some minor changes in the presentation of the transformation equations where u and u' are the OX(O'X') components of the speed of the same tardyon.
a. Using (1) and (2) we should not use the transformation of speeds. It was used deriving them.
b. Taking the mentioned limit we obtain from a mathematical point of view
p(c)=gp'(c)(1+V/c) (1)
m(c)gm'(c)(1+V/c) (2)
or
E(c)gE'(c)(1+V/c) (3)
and from a physical point of view we have obtained the transformation equations for the momentum and the energy of the photon.
Has that some physics behind it? Some thing happens! soft words and hard arguments

 

[Meir Achuz]

"Has that some physics behind it?"
I think it is just elementary algebra.

 

[bernhard.rothenstein]

simple mathematics or some physics as well

"Has that some physics behind it?"
I think it is just elementary algebra.

Thanks. Consider the addition law of parallel velocities
u=[u'+V]/[1+u'V/cc]
and take there the limit for u=u'=c in order to obtain c=c.
Consider the relativistic Doppler shift formula in the acoustic wave and take there the limit when the propagation velocity becomes equal to c in order to obtain the Doppler shift formula in the electromagnetic wave.
Is that simple mathematics or it reflects some physical facts?
Is it correct to postulate:
If you have in your hands formulas which account for the behaviour of a tardyon take there the limit u=u'=c in order to obtain the corresponding equations that account for the behaviour of the photon.
soft words and hard arguments

 

[Mentz114]

If you have in your hands formulas which account for the behaviour of a tardyon take there the limit u=u'=c in order to obtain the corresponding equations that account for the behaviour of the photon.

The physical content of such formulae would imply that photons can be created by accelerating matter. This is not true. Photons are born photons and are not transformed matter.

 

[bernhard.rothenstein]

photon tardyon

The physical content of such formulae would imply that photons can be created by accelerating matter. This is not true. Photons are born photons and are not transformed matter.

Thanks. It is a fact that formulas for tardyons become formulas for photons. Is there an explanation for that?
There are mnemonic rules in physics. Could be my statement such a rule?

 

[Mentz114]

What equations do is certainly of interest to mathematicians. We have no evidence of matter being accelerated to photons. The idea does not hold water because speed is relative.

 

[MeJennifer]

A particle that has mass can never reach the speed of light not even in the limit. In fact there is no limit.

Convert the speed to rapidity and you can clearly see that there is no limit.

 

[Mentz114]

Hi MJ,
if a rocket with enough fuel accelerates away from me at 1g, then after a certain time won't it reach c relative to me ? If it shone a laser back at me I would see it shift down the energy spectrum until it blinked out. I suppose for the signal to truly reach zero would take an infinite time on my clock, which bears you out.

 

[Gib Z]

Enough Fuel? Remember as we get closer to the speed of light, our mass increases making it harder to accelerate. In this endless process of gaining more mass and accelerating, you will never actually reach c.

 

[Meir Achuz]

Thanks. Consider the addition law of parallel velocities
u=[u'+V]/[1+u'V/cc]
and take there the limit for u=u'=c in order to obtain c=c.
...
Is that simple mathematics or it reflects some physical facts?
Is it correct to postulate:
If you have in your hands formulas which account for the behaviour of a tardyon take there the limit u=u'=c in order to obtain the corresponding equations that account for the behaviour of the photon.
soft words and hard arguments

The answer to this is yes, but that is trivial algebra and not a hard argument.
But, in terms of E and p, it is still simpler.

"Consider the relativistic Doppler shift formula in the acoustic wave and take there the limit when the propagation velocity becomes equal to c in order to obtain the Doppler shift formula in the electromagnetic wave."
What do you mean by "the relativistic Doppler shift formula in the acoustic wave"?

 

[Mentz114]

Enough Fuel? Remember as we get closer to the speed of light, our mass increases making it harder to accelerate. In this endless process of gaining more mass and accelerating, you will never actually reach c.

No.

'...as you approach c' is nonsense if YOU DON'T SAY WHAT FRAME IS MEASURING THE VELOCITY.

Do you think there's an absolute reference frame ? If you were on the spaceship would you detect an increase in your own mass ?

 

[bernhard.rothenstein]

The answer to this is yes, but that is trivial algebra and not a hard argument.
But, in terms of E and p, it is still simpler.

"Consider the relativistic Doppler shift formula in the acoustic wave and take there the limit when the propagation velocity becomes equal to c in order to obtain the Doppler shift formula in the electromagnetic wave."
What do you mean by "the relativistic Doppler shift formula in the acoustic wave"?

Thanks for the first "yes" answer from you.
IMHO Doppler shift in acoustic wave is
f=gf'(1+V/u') (1)
whereas in the electromagnetic wave it is
f=gf'*1+V/c) (2)
and again (1) becomes (2) for u=u'=c.
Yes or not?
The problem is not " trivial argument" but "why"?
Regards

 

[Mentz114]

Any comparison between the acoustic wave and the EM wave is naive and unfruitful. Sound is a longitudinal wave in a compressible medium and nothing like EM radiation.

 

[bernhard.rothenstein]

acoustic and electromagnetic wave

Any comparison between the acoustic wave and the EM wave is naive and unfruitful. Sound is a longitudinal wave in a compressible medium and nothing like EM radiation.

I was asked about acoustic and electromagnetic waves. I think your statement is to radical. Have a look please at
C.Moller The Theory of Relativity, Clarendon Press Oxford 1972 chapter 2.9in order to see how acoustic and electgromagnetic waves could be related and how the comparison between them could lead to de Broglie.
What I mentioned is the coincidence between the equations which describe the Doppler shift in the two waves for u=u'=c.
soft words and hard arguments

 

[Mentz114]

Yes, my reply is too radical. They are both waves after all, and have that much in common. I do not have access to the book you mention so no comment on that.

 

[robphy]

Apparently, Moller is available here at
http://www.archive.org/details/theoryofrelativi029229mbp

 

[Mentz114]

Moller (1952) is available. I'll have a look at it when I get the time, because I enjoy 'old' physics texts.

Thanks for the reference.

 

[bernhard.rothenstein]

electronic books

Apparently, Moller is available here at
http://www.archive.org/details/theoryofrelativi029229mbp

Please give us a list of relativity books available on the internet.
Thanks

 

[robphy]

Please give us a list of relativity books available on the internet.
Thanks

You can search archive.org's "Texts" for the keyword "relativity".

 

[Meir Achuz]

Thanks for the first "yes" answer from you.
IMHO Doppler shift in acoustic wave is
f=gf'(1+V/u') (1)
whereas in the electromagnetic wave it is
f=gf'*1+V/c) (2)
and again (1) becomes (2) for u=u'=c.
Yes or not?
The problem is not " trivial argument" but "why"?
Regards

Would you believe I don't know what IMHO means?
Your acoustic Doppler is for motion of the observer with a fixed source.
The relativistic Doppler is for the relative velocity.
For a moving source, your derivation gives the wrong answer.
But why bother, when the RDS follows so easily from the LT acting on k^\mu.