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

[barriboy]

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?

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,
$$x(t) = -\frac{V_{sw}}{\omega} Sin(\omega t), y(t) = V_{sw} \frac{Cos(\omega t)}{\omega}.$$
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
$$x'(t) = -\gamma V_{sw}(\frac{Sin(\omega t)}{\omega} -t).$$
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?

 

[bcrowell]

It's very difficult to read your math. You'll be more likely to get replies to your posts if you mark them up using LaTeX. Here's how to do that: https://www.physicsforums.com/help/latexhelp/ .

If you really want to transform into a different frame, then you also probably want to eliminate t in favor of t'.

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?

It's hard to know what Griffiths means without some more context. You have a world-line that is composed of events. That world-line can be described using (t,x,y) coordinates. The Lorentz transformation tells you what the description would be in terms of (t',x',y') coordinates.

Re the title question, "Can you use the Lorentz transform for a function of time?," the answer to this is basically that it depends on what kind of function of time it is. Different functions transform in different ways. Most people would not actually think of a world-line in terms of spatial coordinates as a function of time. Time is just another coordinate. It would be more common to think of it as a 4-vector (t,x,y,z) that is a function of some arbitrary parameter. Then that function would transform according to the Lorentz transformation, since that's the definition of what a 4-vector is.

 

[barriboy]

Roger dodger, made the math prettier.

 

[HomogenousCow]

Just replace t with some parameter and write the whole thing as a three-vector. Boost the three vector and rearrange for x(t) and y(t).

 

[vanhees71]

You mean four-vector, I guess?

 

[HomogenousCow]

You mean four-vector, I guess?

The third spatial dimension appears to be constant and he's boosting in a direction orthogonal to it. When I wrote three-vector I meant something like (t,x,y).

 

[Dale]

A three vector is usually the spatial part of a four vector. I have never seen anyone refer to an vector with one time and two space components as a three vector.

 

[HomogenousCow]

A three vector is usually the spatial part of a four vector. I have never seen anyone refer to an vector with one time and two space components as a three vector.

My bad.