Transformation equations presented in a different way

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In summary, the conversation discusses the use of transformation equations for a particle moving at different speeds relative to two inertial reference frames. The use of gamma factors and rest mass and energy are mentioned, as well as the preference for using proper physical quantities in equations. The use of rapidities and spacetime diagrams are also suggested for a more intuitive understanding. Overall, there is a focus on presenting the equations in a transparent and time-saving manner.
  • #1
bernhard.rothenstein
991
1
Consider a particle that moves with speed u relative to the inertial reference frame I and with speed u' relative to the inertial reference frame I'. Let g(u), g(u') and g(V) be the orresponding gamma factors (V the relative speed of I and I'). m(0) stands for its rest mass, E(0) for its rest energy, p and E for its momentum and energy measured by observers from I. It is obvious that
p=g(u')g(V)m(0)(V+u')=g(u')g(V')E(0)(V+u')cc
E=g(u')g(V)E(0)(1+Vu'/cc)
I consider that such a presentation presents some (pedagogical) advantages showing clearly what observers from the I frame measure in the case when u'=0 and when u' and V are both equal to zero.
Even if I know that the concept of relativistic mass is persona non grata on the Forum I would also suggest for the relativistic mass
m=g(u')g(V)E(0)(1+Vu'/cc)=g(u')g(V)m(0)(1+Vu'/cc).
The oppinion of those who teach or learn special relativity theory is highly appreciated of course in the spirit of
sine ira et studio
 
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  • #2
Never use "it is obvious" in pedagogy. That is like the old mathematics professor joke.
 
  • #3
bernhard.rothenstein said:
E=g(u')g(V)E(0)(1+Vu'/cc)
Don't you think that [tex]E=\gamma(V)*E(0)[/tex] is much cleaner?
 
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  • #4
bernhard.rothenstein said:
Even if I know that the concept of relativistic mass is persona non grata on the Forum [...]
As an aside, persona non grata means 'an unwelcome person' and is not usually used in reference to a concept. Absit invidia :wink:
 
  • #5
transformation equation

Meir Achuz said:
Never use "it is obvious" in pedagogy. That is like the old mathematics professor joke.
Thanks. I think that there is a big difference between posting on the Forum, where the participants can easily transform the usual transformations in those I propose and presenting them in all its steps.
I know very much joks with teachers of physics and mathematics. Which of them do you mean?
 
  • #6
transformation equations

nakurusil said:
Don't you think that [tex]E=\gamma(V)*E(0)[/tex] is much cleaner?
Thanks. Yes it is but I think that the Forum could offer a simple equation editor.
 
  • #7
bernhard.rothenstein said:
Thanks. Yes it is but I think that the Forum could offer a simple equation editor.

The forum does! LaTex is very easy to learn, and it is easy to use on this forum too; simply put [tex] [ /tex] tags (without the space) around the equations.
 
  • #8
latina ginta est Regina

Hootenanny said:
As an aside, persona non grata means 'an unwelcome person' and is not usually used in reference to a concept. Absit invidia :wink:
Thanks. My first language is close to Latin. I thought that physicists are able to extrapolate from persona non grata to relativistic mass which is there non grata. I end with
absit invidia which is shorter and more adequate then sine ira et studio I used so far.
 
  • #9
bernhard.rothenstein said:
m=g(u')g(V)E(0)(1+Vu'/cc)=g(u')g(V)m(0)(1+Vu'/cc).

[tex]\gamma(u')\gamma(V)(1+Vu'/c^2)=\gamma(u)[/tex] , so the above reduces the the much cleaner, well known :

[tex]m(u)=\gamma(u)*m(0)[/tex]
 
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  • #10
The identity (rearranged in a minor way)
[tex]\gamma(u')\gamma(V)(1+u'V/cc)=\gamma(u)[/tex]
is more recognizable in terms of rapidities:
[tex]
\begin{align*}
\gamma(u')\gamma(V)(1+u'V/cc)
&=
\cosh{\theta'}\cosh{\phi}(1+c\tanh{\theta'}\ c\tanh{\phi}/cc)\\
&=
\cosh{\theta'}\cosh{\phi}+\sinh{\theta'}\sinh{\phi}\\
&=
\cosh{(\theta'+\phi)}\\
&=
\cosh{\theta}\\
&=
\gamma(u)
\end{align*}
[/tex]
where u[itex]=c\tanh{\theta}[/itex] is the spatial-velocity obtained by "spatial-velocity-composition of u' and V".

In addition, in terms of rapidities, one can immediately transcribe the calculation into a spacetime diagram, which provides a hopefully more intuitive interpretation of what is happening physically [and mathematically].

So, it's not clear to me if anything is gained in the proposed formula, except maybe for a particular type of problem.
 
Last edited:
  • #11
transformations

robphy said:
The identity (rearranged in a minor way)
[tex]\gamma(u')\gamma(V)(1+u'V/cc)=\gamma(u)[/tex]
is more recognizable in terms of rapidities:
[tex]
\begin{align*}
\gamma(u')\gamma(V)(1+u'V/cc)
&=
\cosh{\theta'}\cosh{\phi}(1+c\tanh{\theta'}\ c\tanh{\phi}/cc)\\
&=
\cosh{\theta'}\cosh{\phi}+\sinh{\theta'}\sinh{\phi}\\
&=
\cosh{(\theta'+\phi)}\\
&=
\cosh{\theta}\\
&=
\gamma(u)
\end{align*}
[/tex]
where u[itex]=c\tanh{\theta}[/itex] is the spatial-velocity obtained by "spatial-velocity-composition of u' and V".

In addition, in terms of rapidities, one can immediately transcribe the calculation into a spacetime diagram, which provides a hopefully more intuitive interpretation of what is happening physically [and mathematically].

So, it's not clear to me if anything is gained in the proposed formula, except maybe for a particular type of problem.

Thanks. My intention is to present the transformation equations in such a way that theirs right sides contain only a proper physical quantity and velocities reducing the long discussions related to the concept of relativistic mass. That is the direction in which I hope our discussions will evolve.
Reading my lines please take into account that English is not my first language.
 
  • #12
robphy said:
The identity (rearranged in a minor way)
[tex]\gamma(u')\gamma(V)(1+u'V/cc)=\gamma(u)[/tex]
is more recognizable in terms of rapidities:
[tex]
\begin{align*}
\gamma(u')\gamma(V)(1+u'V/cc)
&=
\cosh{\theta'}\cosh{\phi}(1+c\tanh{\theta'}\ c\tanh{\phi}/cc)\\
&=
\cosh{\theta'}\cosh{\phi}+\sinh{\theta'}\sinh{\phi}\\
&=
\cosh{(\theta'+\phi)}\\
&=
\cosh{\theta}\\
&=
\gamma(u)
\end{align*}
[/tex]
where u[itex]=c\tanh{\theta}[/itex] is the spatial-velocity obtained by "spatial-velocity-composition of u' and V".

In addition, in terms of rapidities, one can immediately transcribe the calculation into a spacetime diagram, which provides a hopefully more intuitive interpretation of what is happening physically [and mathematically].

So, it's not clear to me if anything is gained in the proposed formula, except maybe for a particular type of problem.
Thank you for having brought the formula to a more transparent shape. Consider the concept of proper mass m(0) and multiply both sides of with it. It leads to[tex]m(0)gamma(u0=m(0)gamma(u')gamma(V)(1+u'V/cc[/tex]An exercised eye will recognise in the left side of the equation the expression of the relativistic mass in I in the left side its expresion as a function of phyhsical quantities measured in I. Do you consider that such a presentation is time saving, transparent and convincing for the fact that conservation laws are not compulsory in the derivation. I do not convinced that the equation will appear correctly in myh message.
 

1. What are transformation equations?

Transformation equations are mathematical expressions that describe the relationship between two coordinate systems. They are used to convert coordinates from one system to another.

2. Why do we need to present transformation equations in a different way?

Presenting transformation equations in a different way can help us better understand the underlying concepts and make them easier to apply in practical situations. It can also provide alternative methods for solving problems.

3. What are some common ways to present transformation equations in a different way?

Some common ways to present transformation equations in a different way include using matrices, vectors, and geometric representations. Other methods may involve using different coordinate systems or changing the order of operations.

4. How can transforming equations in a different way improve our understanding?

By presenting transformation equations in a different way, we can gain a deeper insight into the underlying principles and connections between different coordinate systems. This can help us to better understand the applications of these equations in various fields such as physics, engineering, and mathematics.

5. Are there any challenges to presenting transformation equations in a different way?

Yes, there can be challenges in presenting transformation equations in a different way. Some methods may be more complex and require a strong understanding of mathematical concepts. It may also take more time and effort to learn and apply these alternative methods.

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