Relativistic dynamics as a semantic problem

[bernhard.rothenstein]

Consider the following equations
m/(1-u2/c2)1/2=m(1+u’V/c2)/(1-u’2/c2)1/2(1-V2/c2)1/2 (1)
mu/(1-u2/c2)1/2=m(u’+V)/(1-u’2/c2)1/2(1-V2/c2)1/2 (2)
where m represents the Newtonian mass of a particle that moves with speed u relative to the inertial reference frame I and with velocity u’ relative to the inertial reference frame I’ in the positive direction of the OX(O’X’) axes, I’ moving with velocity V relative to I, all velocities showing in the positive direction of the overlapped axes.
An exercised eye recognizes in (1) the transformation equation for mass whereas in (2) the transformation for momentum. Physicists, known as starters or developers of trends (godfathers?), combine the physical quantities that appear in (1) and (2) presenting them as
M=(M’+p’V/c2)/(1-V2/c2)1/2
P=(P’+M’V)/(1-V2/c2)1/2
being obliged to find out names for M,M’,P and P’. From that point the problem is no more then semantics. What names would you prefer? Has the name we choose some importance as long as we know that
M=m/(1-u2/c2)1/2
M’=m/(1-u’2/c2)1/2 P=mu/(1-u2/c2)1/2
P’=mu’(1-u’2/c2)1/2.
Is it compulsory to find out names for them?
I get accustomed with harsh answers!

 

[robphy]

It hurts to read these non-$$\LaTeX$$ed equations.

 

[bernhard.rothenstein]

latex?

It hurts to read these non-$$\LaTeX$$ed equations.

Sorry! Could the forum offer a simple equations editor? Knowing your experience in reading equations, I think you could see what is behind the equations and what is my problem.
Thanks

 

[tim_lou]

latexed:

Consider the following equations
$$\frac{m}{\sqrt{1-u^2/c^2}}=\frac{m(1+u'V/c^2)}{\sqrt{1-u'^2/c^2}\sqrt{1-V^2/c^2) }}$$(1)
$$\frac{\mu}{\sqrt{1-u^2/c^2}}=\frac{m(u'+V)}{\sqrt{1-u'^2/c^2}\sqrt{1-V^2/c^2}}$$(2)

where m represents the Newtonian mass of a particle that moves with speed u relative to the inertial reference frame I and with velocity u’ relative to the inertial reference frame I’ in the positive direction of the OX(O’X’) axes, I’ moving with velocity V relative to I, all velocities showing in the positive direction of the overlapped axes.
An exercised eye recognizes in (1) the transformation equation for mass whereas in (2) the transformation for momentum. Physicists, known as starters or developers of trends (godfathers?), combine the physical quantities that appear in (1) and (2) presenting them as
$$M=\frac{M'+P'V/c^2}{\sqrt{1-V^2/c^2}}$$
$$P=\frac{P'+M'V}{\sqrt{1-V^2/c^2}}$$

being obliged to find out names for M,M’,P and P’. From that point the problem is no more then semantics. What names would you prefer? Has the name we choose some importance as long as we know that
$$M=\frac{m}{\sqrt{1-u^2/c^2}}$$
$$M'=\frac{m}{\sqrt{1-u'^2/c^2}}$$
$$P=\frac{\mu}{\sqrt{(1-u^2/c^2)}}$$
$$P'=\frac{\mu'}{\sqrt{1-u'^2/c^2}}$$
Is it compulsory to find out names for them?
I get accustomed with harsh answers!

click to see the latex codes.

edit: misunderstood ^2 as _2

as for the reply, honestly, I don't know what you did and I hate the the square roots and messes that are generally involved with special relativity equations... I much prefer gamma and beta and matrix... make lives much easier.

 

[tim_lou]

Ok, how stupid of me... i see, basically, you use Lorentz transformation on the four vector
$$\left< Mc, \vec{P} \right>$$

I think M is just relativistic mass and P is just relativistic momentum (relative to a particular frame of reference of course).

 

[bernhard.rothenstein]

Ok, how stupid of me... i see, basically, you use Lorentz transformation on the four vector
$$\left< Mc, \vec{P} \right>$$

I think M is just relativistic mass and P is just relativistic momentum (relative to a particular frame of reference of course).

That is your contribution as a godfather and I aggree with you. Others will multiply M with cc avoiding the concept of relativistic mass using instead of it the concept of energy. Pure semantics I think

 

[bernhard.rothenstein]

an improved version

It hurts to read these non-$$\LaTeX$$ed equations.

That is an improved version of my thread

Many contributors to the forum mention that the concept of relativistic mass is harmful. Some of them agree with the fact that the concept can be used in special relativity if we apply it correctly.
I tell you a special relativity story. Consider that we are in the year 1905 when Bucherer, Kaufmann and others probably as well, have discovered that in the case of an electron that moves on a circular trajectory in an uniform magnetic field we obtain results in accordance with experiment if we do not use the concept of Newtonian mass m(0) using instead
m(0)g(u) (1)
u representing the electron’s speed. We do not give a name to the physical quantity defines by (1). Equation (1) can be considered as a well tested experimental fact.
A relativist who knows Einstein’s special relativity will consider Bucherer’s experiment from an inertial reference frame I and from an inertial reference frame I’, the electron moving with velocity u relative to I, with velocity u’ relative to I’, I’ moving at its turn with velocity V relative to I all in the positive direction of the OX axis. The involved velocities are related by
u=(u’+V)/(1+u’V/cc) (2)
Equation (1) becomes in I
m(0)g(u) (3)
whereas in I’ it becomes
m(0)g(u’). (4)
Combining (2),(3) and (4) we obtain
m(0)g(u)=m(0)g(u’)g(V)(1+Vu’/cc) (5)
m(0)ug(u)=m(0)g(u’)u’g(V)(1+V/u’) (6)
m(0)ccg(u)=m(0)g(u’)ccg(V)(1+Vu’/cc) (7)
m(0)uccg(u)=m(0)g(u’)u’ccg(V)(1+V/u’) (8)
An experimented eye will detect that we have derived the transformation equations for the physical quantities with which we operate in relativistic dynamics without using conservation laws. As a godfather he will try to make shorter the equations he has derived above introducing the notations
M(u)=m(0)g(u) (9)
M’(u)=m(0)g(u’) (10)
P(u)=M(u)u (11)
P’(u)=M’(u’)u’ (12)
E=M(u)cc (13)
E’=M’(u’)cc. (14)
Considering a collision between two particles from I and I’ and using the equations derived above we conclude that we obtain results in accordance with the conservation laws. We can find names for (9)-(14) in a free choice.
As a moral we can ask: Does the name we have given to the new physical quantities influence the validity of the obtained results?