- #1
MarkovMarakov
- 33
- 1
I would be very grateful if someone would kindly explain this generalization of the Lorentz force law to the special relativity domain. I am not entirely sure if what I have jotted down is exactly as the speaker intended to convey. But here is what I have got. Please bear with me.
>Classically, the Lorentz force law is [tex]m\frac{d^2x}{dt^2}=q(E+v\times B)[/tex]. We want to reformulate this relativistically. Noticing that [itex]\frac{dt}{d\tau}\approx 1[/itex], we could postulate that the relativistic version of the equation has a factor of [itex]\frac{dt}{d\tau}[/itex] multiplying E. But the force now depends on velocity linearly classically, so the only possibility hat is consistent with both the classical limit and SR is [tex]m\frac{d^2x^\mu}{d\tau^2}=qF^\mu{}_\nu \frac{dx^\nu}{d\tau}[/tex]
Here are some of the things I don't understand:
1. What does it mean to reformulate it relativistically? As far as I understand, it means we want an equation that holds at high velocities but reduces to the Lorentz equation in the Newtonian limit. OK. But what does that mean mathematically? How does one go about generalizing equations to fit SR?
2. Why does [itex]\frac{dt}{d\tau}\approx 1[/itex] in the classical case suggest multiplying [itex]\frac{dt}{d\tau}[/itex] in front of [itex]E[/itex]?
3. How do they come about with [tex]m\frac{d^2x^\mu}{d\tau^2}=qF^\mu{}_\nu \frac{dx^\nu}{d\tau}[/tex]? I suppose the linear-dependence of the force on velocity in the classical limit suggests the RHS has to be linear in [itex]dx\over d\tau[/itex]. But what is so SR about it? How is it consistent with/accommodate SR?
Thank you very much!
>Classically, the Lorentz force law is [tex]m\frac{d^2x}{dt^2}=q(E+v\times B)[/tex]. We want to reformulate this relativistically. Noticing that [itex]\frac{dt}{d\tau}\approx 1[/itex], we could postulate that the relativistic version of the equation has a factor of [itex]\frac{dt}{d\tau}[/itex] multiplying E. But the force now depends on velocity linearly classically, so the only possibility hat is consistent with both the classical limit and SR is [tex]m\frac{d^2x^\mu}{d\tau^2}=qF^\mu{}_\nu \frac{dx^\nu}{d\tau}[/tex]
Here are some of the things I don't understand:
1. What does it mean to reformulate it relativistically? As far as I understand, it means we want an equation that holds at high velocities but reduces to the Lorentz equation in the Newtonian limit. OK. But what does that mean mathematically? How does one go about generalizing equations to fit SR?
2. Why does [itex]\frac{dt}{d\tau}\approx 1[/itex] in the classical case suggest multiplying [itex]\frac{dt}{d\tau}[/itex] in front of [itex]E[/itex]?
3. How do they come about with [tex]m\frac{d^2x^\mu}{d\tau^2}=qF^\mu{}_\nu \frac{dx^\nu}{d\tau}[/tex]? I suppose the linear-dependence of the force on velocity in the classical limit suggests the RHS has to be linear in [itex]dx\over d\tau[/itex]. But what is so SR about it? How is it consistent with/accommodate SR?
Thank you very much!