Reading Griffiths, he states that the Lorentz Transform is useful for describing where an 'event' occurs in a different inertial frame. What about describing the motion of a particle in this moving frame if I know its motion in my frame?(adsbygoogle = window.adsbygoogle || []).push({});

Really, I'm looking at pickup ions in the solar wind. A hydrogen atom is at rest in the solar wind when it is ionized. If we consider the solar wind to be a plasma moving at Vsw in the positive x direction and in the presence of a perpendicular magnetic field in the z direction, then we know that in the solar wind's frame the ion will undergo gyromotion and trace a circle,

[tex]

x(t) = -\frac{V_{sw}}{\omega} Sin(\omega t), y(t) = V_{sw} \frac{Cos(\omega t)}{\omega}.

[/tex]

If we then do Lorentz transform back to the ion's original rest frame, we can see that it will still be undergoing the same simple harmonic oscillation in y, but will have

[tex]

x'(t) = -\gamma V_{sw}(\frac{Sin(\omega t)}{\omega} -t).

[/tex]

This seems to work, as this reproduces the inverted "U" shape that I was told to expect, and (assuming Vsw<<c), we get Vx(t) = 2Vsw at the top of this motion, which seems to be correct based on what google can tell me.

The only thing that is making me worried about this is that Griffiths seems pretty adamant that this is a transformation of an 'event.' Can we merely say that the first 'event' is that the ion is at x=0, t=0, and the second event is the ion at x=ds, t=dt?

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# Can you use the Lorentz transform for a function of time?

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