- #1

kreil

Gold Member

- 668

- 67

I'm having trouble getting all of the relativistic dynamics information straight in my head. In particular, I'm having trouble with reference frames. It seems that when I am trying to solve problems, even simple ones, I am struggling with what each observer in the different frames should be seeing.

For example, lets say in the rest frame O an observer sees a particle moving with velocity V. The particle has a four velocity given by (note: I use c=1 units everywhere)

[tex] \bar u = \frac{d \vec x}{d \tau} = (\gamma, \gamma \vec V)[/tex]

and the four force on the particle would be

[tex] \bar f = \gamma (\vec F \cdot \vec V, \vec F)[/tex]

Now let's say an observer in the frame O', moving with velocity v along the x-axis of O, observes the particle. From the velocity law, he sees the particle moving at speed:

[tex] V^{x'} = \frac{V^x-v}{1-vV^x}[/tex]

where [itex]V^x[/itex] is the observed velocity in O.

If it is relevant, I encountered these thought problems while trying to find the four force and four velocity components for a particle moving in the Lorentz Force Field (Hartle problem 5.21),

[tex] \vec F = \frac{d \vec p}{dt} = q(\vec E + \vec V \times \vec B)[/tex]

which is invariant between inertial frames.

I realize that this is all quite basic stuff math wise, but the thought process that leads to the correct calculations is giving me trouble, and it is crucial I get it solidified into my head correctly.

For example, lets say in the rest frame O an observer sees a particle moving with velocity V. The particle has a four velocity given by (note: I use c=1 units everywhere)

[tex] \bar u = \frac{d \vec x}{d \tau} = (\gamma, \gamma \vec V)[/tex]

and the four force on the particle would be

[tex] \bar f = \gamma (\vec F \cdot \vec V, \vec F)[/tex]

Now let's say an observer in the frame O', moving with velocity v along the x-axis of O, observes the particle. From the velocity law, he sees the particle moving at speed:

[tex] V^{x'} = \frac{V^x-v}{1-vV^x}[/tex]

where [itex]V^x[/itex] is the observed velocity in O.

**Question:**Is it true that the particle's four velocity and four force are transformed to use this [itex]V^{x'}[/itex] ? Why or Why not?**Question:**Is the velocity in the Lorentz Factor [itex]\gamma[/itex] the velocity of the particle in the rest frame V, or the relative velocity between the frames v?**Question:**Is it correct to say that for any problem where a particle's velocity is being observed there are 3 reference frames to worry about: the rest frame O, the moving frame O', and the frame of the particle?If it is relevant, I encountered these thought problems while trying to find the four force and four velocity components for a particle moving in the Lorentz Force Field (Hartle problem 5.21),

[tex] \vec F = \frac{d \vec p}{dt} = q(\vec E + \vec V \times \vec B)[/tex]

which is invariant between inertial frames.

I realize that this is all quite basic stuff math wise, but the thought process that leads to the correct calculations is giving me trouble, and it is crucial I get it solidified into my head correctly.

Last edited: