Lorentz Boost & Galileo Speed: Exploring Relationship

In summary: Bell's theorem has nothing to do with SRT. It's a theorem about the (non-)existence of a local hidden-variable theory explaining all the predictions of QT, and it has been falsified by experiment. That's all there is to say about it.In summary, the conversation discusses the relationship between the Lorentz boost and Galilean speed, and introduces the concept of rapidity and the hyperbolic tangent function. There is also a mention of Bell's theorem and its implications for Lorentz transformation, which is clarified as having no contradiction with relativistic quantum field theory.
  • #1
jk22
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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##.
 
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  • #2
Have you heard of the term “rapidity” or the hyperbolic tangent function ##\tanh (\alpha)##? You might want to look into those.
 
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  • #3
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##.
 
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  • #4
What Vanhees71 says: how would this Lie-group look like? And: why are you interested in the first place?
 
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  • #5
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.
 
  • #6
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.
 
  • #7
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.
 
  • #8
jk22 said:
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.
 
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  • #9
jk22 said:
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##.
 
  • #10
jk22 said:
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##
jk22 said:
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).
 
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  • #11
jk22 said:
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.
 
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1. What is a Lorentz Boost?

A Lorentz Boost is a mathematical transformation that relates the observations of space and time between two reference frames moving at a constant velocity relative to each other. It is a fundamental concept in the theory of relativity.

2. How does a Lorentz Boost differ from a Galileo Boost?

A Lorentz Boost takes into account the effects of special relativity, such as time dilation and length contraction, while a Galileo Boost does not. This means that a Lorentz Boost is more accurate at high speeds, while a Galileo Boost is a good approximation at low speeds.

3. What is the relationship between Lorentz Boost and Galileo Speed?

The relationship between Lorentz Boost and Galileo Speed is that they both describe the motion of objects in different reference frames. Lorentz Boost takes into account the effects of special relativity, while Galileo Speed does not. However, at low speeds, the two concepts are equivalent.

4. How does Lorentz Boost affect the perception of time and space?

Lorentz Boost affects the perception of time and space by showing that they are not absolute, but rather relative to the observer's reference frame. Time can appear to pass slower and distances can appear to be shorter depending on the relative velocity between two reference frames.

5. Can Lorentz Boost be applied to everyday situations?

Yes, Lorentz Boost can be applied to everyday situations, especially in modern technology such as GPS systems. These systems use the principles of special relativity, including Lorentz Boost, to accurately determine the position and time for navigation purposes.

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