I Is the Lorentz Boost Speed Nontrivially Related to Galilean Speed?

[jk22]

TL;DR Summary

boost

What about if the speed parameter in a Lorentz boost were in fact related nontrivially to a Galilean speed ?

More formally $L(v_L)=G(v)\circ F$ where L is a Lorentz boost with Lorentz speed $v_L$, G is a Galileo transformation with speed $v$ and $F$ is still an unknown linear transformation that has to fulfill the previous matrix equation, which by solving should lead to a relationship $v_L=g(v)$ that possibly could have the property $v_L=g(v\rightarrow\infty)\rightarrow c$.

 

[Pencilvester]

Have you heard of the term “rapidity” or the hyperbolic tangent function $\tanh (\alpha)$? You might want to look into those.

 

[vanhees71]

How should this ever work? The Galilei group is not a subgroup of the Poincare group. How should your composition then make sense?

In a way the "natural" parameter for a Lorentz-boost along a direction $\vec{n}$ is the rapidity $\alpha$ (I use the notation of the previous post). With it a boost in the $tx$-Minkowski plane reads
$$x'=\begin{pmatrix} c t' \\ x' \end{pmatrix} = \begin{pmatrix} \cosh \alpha & -\sinh \alpha \\ -\sinh \alpha & \cosh \alpha \end{pmatrix} \begin{pmatrix} ct \\ x \end{pmatrix}=\hat{\Lambda}(\alpha) x.$$
You can easily show by setting $x'=0$ that the velocity of the frame $\Sigma'$ against $\Sigma$ is
$$v=c \tanh \alpha.$$
The rapidity is "natural" in the sense that for boosts in one direction you have
$$\hat{\Lambda}(\alpha_1) \hat{\Lambda}(\alpha_2)=\hat{\Lambda}(\alpha_1+\alpha_2).$$
From this you very simply get the addition law for velocities in one direction:
$$v''=c \tanh(\alpha_1+\alpha_2)=c \frac{\sinh(\alpha_1+\alpha_2)}{\cosh(\alpha_1+\alpha_2)}=c \frac{\sinh \alpha_1 \cosh \alpha_2 + \sinh \alpha_2\cosh \alpha_1}{\cosh \alpha_1 \cosh \alpha_2 + \sinh \alpha_1 \sinh \alpha_2}=\frac{v+v'}{1+v v'/c}.$$
In the last step I devided numerator and denominator by $\cosh \alpha_1 \cosh \alpha_2$ and used $\tanh \alpha_1=v/c$ and $\tanh \alpha_2=v'/c$.

 

[haushofer]

What Vanhees71 says: how would this Lie-group look like? And: why are you interested in the first place?

 

[jk22]

I have no knowledge on Lie group I just want to solve this equation system. But since it's now years from my last courses or physics books I make lot of mistakes.

I'm just wondering what Bell's theorem implies for Lorentz transform then the conclusion of his theorem on wikipedia says the theory explaining quantum covariances could not be Lorentz invariant. But I didn't find his work on this, the Lorentz transformation.

 

[Ibix]

Your matrix is trivial to work out. Using t and x as the zeroth and first coordinates, it's$$\mathbf F=\left(\begin{array}{cc}\gamma&-v\gamma\\0&1/\gamma\end{array}\right)$$and $\mathbf\Lambda=\mathbf G.\mathbf F$. But as others have pointed out, all you've done is decomposed a coordinate transform that reflects the symmetry of spacetime into two stages , each of which doesn't.

 

[jk22]

F should be a function of $v$ and $v_L$, is the $\gamma$ a function of $v_L$ ?

Anyhow my goal was to find $\gamma(v_L(v))$ but I got lost in calculations.

 

[Pencilvester]

F should be a function of $v$ and $v_L$, is the $\gamma$ a function of $v_L$ ?

Anyhow my goal was to find $\gamma(v_L(v))$ but I got lost in calculations.

Before you start looking for equations that, with your current apparent knowledge, you likely wouldn’t know how to interpret, you should probably read any introductory text on special relativity. They will cover the gamma factor, $\gamma$, early on.

 

[Dale]

What about if the speed parameter in a Lorentz boost were in fact related nontrivially to a Galilean speed ?

More formally $L(v_L)=G(v)\circ F$ where L is a Lorentz boost with Lorentz speed $v_L$, G is a Galileo transformation with speed $v$ and $F$ is still an unknown linear transformation that has to fulfill the previous matrix equation, which by solving should lead to a relationship $v_L=g(v)$ that possibly could have the property $v_L=g(v\rightarrow\infty)\rightarrow c$.

I don’t understand what is the difference between $v_L$ and $v$.

 

[Ibix]

F should be a function of $v$ and $v_L$, is the $\gamma$ a function of $v_L$ ?

I hadn't realized you were using different velocities for your Galilean and Lorentz transforms. In that case $\mathbf F$ is $$\pmatrix{\gamma&-v_L\gamma\cr \left(v-v_L\right)\gamma&\left(1-v_Lv\right)\gamma\cr }$$where $\gamma$ is indeed a function of $v_L$

Anyhow my goal was to find $\gamma(v_L(v))$ but I got lost in calculations.

Assuming what you mean is that you want to regard $\mathbf F$ as a Lorentz transform, I don't see how you think you are going to do this. $\mathbf F$ isn't even symmetric (because the Galilean transform isn't but the Lorentz transform is).

 

[vanhees71]

I have no knowledge on Lie group I just want to solve this equation system. But since it's now years from my last courses or physics books I make lot of mistakes.

I'm just wondering what Bell's theorem implies for Lorentz transform then the conclusion of his theorem on wikipedia says the theory explaining quantum covariances could not be Lorentz invariant. But I didn't find his work on this, the Lorentz transformation.

Which Wikipedia article are you referring to? You have to be careful with Wikipedia. Though it's a great resource to get a first rough information about some topic, it's not a reliable source for research.

Everything concerning QT and SRT is well-understood in terms of relativistic QFT, and there's no contradiction between SRT and QFT whatsoever.