Understanding Velocity Transformation: A Crucial Factor in Particle Motion

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Discussion Overview

The discussion revolves around the transformation of velocity components in the context of special relativity, specifically focusing on the Y(Y') component of velocity when transitioning from one-dimensional to two-dimensional frameworks. Participants explore the implications of coordinate choices and the principles underlying these transformations.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that the transformation of the Y(Y') component of velocity is derived using the time transformation from a one-dimensional approach, questioning the rationale behind this choice.
  • Others argue that a more general choice of relative orientation and velocity does not change the underlying physics, although it complicates the computations.
  • A participant expresses confusion regarding the necessity of choosing the velocity along the X, X' axes rather than Y, Y', prompting further clarification requests.
  • One participant suggests that the simultaneous nature of events characterized by the same x coordinate may be a reason for the transformation approach, questioning if this should be explicitly mentioned.
  • Another participant emphasizes that Minkowski spacetime allows for flexibility in describing velocities, indicating that any direction or origin can be chosen, but acknowledges complications when velocities are not parallel.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and agreement regarding the transformation process and the implications of coordinate choices. The discussion remains unresolved, with multiple competing views on the necessity and implications of the chosen transformations.

Contextual Notes

Limitations include potential missing assumptions about the nature of simultaneity in different frames and the complexities introduced by non-parallel velocities, which are not fully explored in the discussion.

bernhard.rothenstein
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When textbooks derive the transformation of the Y(Y') component of the velocity of a particle, take into account that y=y' and use the time transformation depending only on the X(X') component of the parfticle's velocity and on the relative velocity as well. Please tell me why?
 
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A more general choice of the two coordinate frames' relative orientation and a more general relative velocity (rather than along a common X,X' axis) will not change the physics (what happens where and when). Time dilation and length contraction still appear. But the computations that show this will be more complicated.
 
Eli Botkin said:
A more general choice of the two coordinate frames' relative orientation and a more general relative velocity (rather than along a common X,X' axis) will not change the physics (what happens where and when). Time dilation and length contraction still appear. But the computations that show this will be more complicated.
Thanks. I have in mind only the case of the standard arrangement.
 
In that case I don't understand what your question requires. Certainly not why the velocity is chosen along X, X' rather than Y, Y'. Please explain.
 
Eli Botkin said:
In that case I don't understand what your question requires. Certainly not why the velocity is chosen along X, X' rather than Y, Y'. Please explain.
Consider the standard arrangement of the involved inertial reference frames. The derivation of the LT starts with the transformation of the x(x') space coordinates and with the derivation of the time transformation
t=g(t'+Vx'/cc). Extending the problem to two space dimensions we add y=y' and derive the transformation of the Y(Y') component of the speed using the time transformation derived in the one space dimension approach considering that the time transformation is not affected by that.
Thanks for discussing with me the problem.
 
bernhard.rothenstein said:
Consider the standard arrangement of the involved inertial reference frames. The derivation of the LT starts with the transformation of the x(x') space coordinates and with the derivation of the time transformation
t=g(t'+Vx'/cc). Extending the problem to two space dimensions we add y=y' and derive the transformation of the Y(Y') component of the speed using the time transformation derived in the one space dimension approach considering that the time transformation is not affected by that.
Thanks for discussing with me the problem.
Are you asking why y=y'?
 
Meir Achuz said:
Are you asking why y=y'?
No.
What I am asking for is:
Starting from an one space dimension approach, textbooks derive the transformation for the x(x') and t(t') space and time coordinate . Going to two space dimensions it is stated that y=y' (a direct consequence of the principle of relativity) and in order to derive the transformation of the Y(Y') component of the velocity the time transformation derived in the case of one space dimension is used. Is there an explanation for that?
Because all events characterized by the same x coordinate are simultaneous? If so, should that fact be mentioned?
Regards
 
I'm not sure I understand Bernhard's question either, but will try.
The velocity of the observer relative to light only occurs in one direction,
and is arbitrarily selected as x because this choice simplifies the transformation equations. If a random vector other than x was selected,
the observers velocity would have components in the y and z axes.
 
bernhard.rothenstein said:
No.
What I am asking for is:
Starting from an one space dimension approach, textbooks derive the transformation for the x(x') and t(t') space and time coordinate . Going to two space dimensions it is stated that y=y' (a direct consequence of the principle of relativity) and in order to derive the transformation of the Y(Y') component of the velocity the time transformation derived in the case of one space dimension is used. Is there an explanation for that?
Because all events characterized by the same x coordinate are simultaneous? If so, should that fact be mentioned?
Regards
Since Minkowski spacetime is an affine space it really does not matter how to describe parallel velocities. Any single direction or origin can be chosen.
In case of velocities that are not parallel the situation is more complicated and not even associative and commutative but one can still select any principle direction and origin.
 
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