# Relativistic dynamics without conservation laws

1. Jan 7, 2007

### bernhard.rothenstein

Globalization makes that we can become aware of what other physicists, located at different remote places, have achieved.
Kard1,2 is the author of the derivations of the fundamental equations of relativistic dynamics without using conservation laws. He starts by defining the momentum of a particle in I as
p=mu (1)
and as
p’=m’u’ (2)
in I’, the particle moving in the positive direction of the OX(O’X’) axes.
Combining (1) and (2) and taking into account the addition law of relativistic velocities he obtains
p/m=(p’/m’)(1+V/u’)/(1+Vu’/c2) (3)
Equation (3) suggests considering
p=g(V)p’(1+V/u’)=g(V)(p’+m’V)
and
m=g(V)m’(1+Vu’/c2)=g(V)(m’+Vp’/c2). (4)
Isotropy of space requires that g(V)=g(-V) the inverse of (4) being
m’=g(V)(m-Vp/c2). (5)
Combining (4) and (5) we obtain
g(V)=(1-V2/c2)-1/2 (6)
and so
p=(p’+Vm’)/(1-V2/c2)1/2 (7)
m=(m’+Vp’/c2)/(1-V2/c2)1/2 (8)
and the way is paved to all the transformation equations we encounter in relativistic dynamics.
Considering a collision from I and I’ and using the transformation equations derived above (and others we could derive from them) we see that they lead to results in accordance with conservation of momentum and mass (energy).
Is the derivation circular?
IMHO no!
Is it time saving?
IMHO yes!
1Leo Karlov, “Paul Kard and Lorentz-free special relativity,” Phys.Educ. 24 165 (1989)
2Paul Kard, “Foundation of the concepts of relativistic mass and energy,” EESTI NSV Teaduste Akademia Toimetised 25 Koide Fuusika Matematika 1976 No.1 75-77 (in Russian)

2. Jan 7, 2007

### nakurusil

Of course it is circular.
The composition of speed is a direct consequence of the Lorentz transform. The derivation of Lorentz transform, as you well know, already contains the proper derivation of g(V)=1/sqrt(1-(V/c)^2). So (6) is circular.

3. Jan 7, 2007

### bernhard.rothenstein

relativistic dynamics

Thanks. As far as I know, the addition law of relativistic velocities can be derived without using the LT. I could offer a long list of papers, which confirm that fact, published by authors I respect. The problem of what we deive from what in SR is an open and important problem.

4. Jan 7, 2007

### nakurusil

This only means that the derivation that you presented is getting longer than the original derivation because it now requires the papers that allegedly derive speed composition without resorting to LT.
In addition to this, chances are that these new papers that you are quoting contain other circular arguments.

5. Jan 7, 2007

### bernhard.rothenstein

relativistic dynamics

IMHO in Kard's derivation the fact that the addition law is derived using LT or not is of any importance.

6. Jan 7, 2007

### nakurusil

I understand , like in every other discussion on this subject: the fact that LT already contain the correct derivation of gama=1/squrt(1-(v/c)^2) from base principles and that Kard "rederives" it from some made up stuff, is of no importance and presents no problems with circular reasoning.
Out of curiosity, as an instructor, don't you think that gama needs to be introduced in the relativistic kinematics, way before it is derived (Kurd style) in the dynamic section? Wouldn't this make more sense?

If you want to present "alternative derivations", this is fine but be aware that a textbook must be self consistent and devoid of circular reasoning. So, youy cannot make a coherent textbook from all these "alternative derivations" , sorry.

Last edited: Jan 7, 2007
7. Jan 8, 2007

### bernhard.rothenstein

Out of curiosity, as an instructor, don't you think that gama needs to be introduced in the relativistic kinematics, way before it is derived (Kurd style) in the dynamic section? Wouldn't this make more sense.

Thanks for your answers. I consider you a potential referee of the book I prepare. As an instructor what would you say concerning titles we find in the literature of the subject:
A derivation of of the LT from Newton's first law and the homogeneity of time.
Derivation of the LT from the Maxwell equations
A derivation of LT based on frequency standards
From E=mcc to the LT via the law of addition of relativistic velocities
Special theory of relativity through the Doppler Effect
Derivation of the LT from gedanken experiments (time dilation and length contraction)
and the list is open. Are all of them circular?

8. Jan 8, 2007

### nakurusil

I can't judge by the titles only. Judging by your previous posts, i would say that they are circular, i.e. they use the conclusion in the demonstration.
Now, if you really wanted a "four line demonstration" for how the momentum transforms from frame to frame , here is one from base principles

A particle moving with speed $$u$$ wrt fram S has the momentum:

$$p=\gamma(u)*m_0*u$$ [1]

From a frame S' moving with speed $$V$$ wrt S , the momentum is :

$$p'=\gamma(w)*m_0*w$$ [2]

where :

$$w=\frac{u+V}{1+\frac{uv}{c^2}}$$ [3]

Substiture [3] into [2] and you obtain:

$$p'=\gamma(V)(p+m(u)V)$$ [4]

i.e. exactly the formula [7] in your post.

End

Last edited: Jan 8, 2007