Minkowsky force and physical interpretation.

[Ribble]

Homework Statement

Compute the $\mu=0$ component of the Minkowski force law $K^\mu=q\eta_\nu F^{\mu\nu}.$ (Einstien summation convention applies.)

Homework Equations

$$\eta_\nu=\frac{1}{\sqrt{1-u^2/c^2}}(-c,u_x,u_y,u_z)$$
$F^{\mu\nu}$ is the field tensor where
$$F^{00}=0,F^{01}=\frac{E_x}{c},F^{02}=\frac{E_y}{c},F^{03}=\frac{E_x}{c}.$$

The Attempt at a Solution

$$K^0=q(\eta_0 F^{00} +\eta_1 F^{01} +\eta_2 F^{02} +\eta_3 F^{03}) = \frac {q \gamma}{c}(u_x E_x + u_y E_y +u_z E_z) = \frac {q \gamma}{c}(\bf{u}.\bf{E})$$


This all seems ok to me, but I have no idea what it actually means. What does $K^0$ physically represent and what does $\bf{u}.\bf{E}$ mean.

Thank you for your help.

 

[chrisk]

Review this website for an explanation of the Minkowski Force components and the dot product of u with E.

http://books.google.com/books?id=J4...DUQ6AEwBg#v=onepage&q=minkowski force&f=false

 

[jdwood983]

I think Chris's answer is okay, but not nearly sufficient.

The dot product is the magnitude of the projection of one vector onto the other. In this case, we are projecting the electric field vector onto the velocity vector (or vice versa). So the time component of the Minkowski 4-force is the magnitude of the electric field vector in direction of the velocity.

 

[vela]

Newton's second law

$$F^\mu = \frac{dp^\mu}{d\tau}$$

still holds in special relativity. Think about what the time component of the four-momentum is. Also, it might help to rewrite K⁰ slightly to get

$$K^0 = \frac{\gamma}{c}[\textbf{u}\cdot (q\textbf{E})]$$

to see what it physically means.

 

[paranormal]

I can provide a response to this content by explaining the physical interpretation of the Minkowski force law and its components.

The Minkowski force law, also known as the Lorentz force law, is a fundamental equation in electromagnetism that describes the force exerted on a charged particle moving through an electromagnetic field. It is named after the mathematician and physicist Hermann Minkowski, who first derived it in 1908.

The equation you have provided is a specific form of the Minkowski force law, where the subscript 0 or \mu=0 represents the time component and the subscript \nu represents the spatial components. This means that the equation is describing the force in the time direction, specifically in the direction of the particle's motion.

The term \mu=0 component of the Minkowski force law represents the force experienced by a charged particle due to an electric field in the direction of the particle's motion. This means that the force is acting to either accelerate the particle in the direction of motion or decelerate it, depending on the direction of the electric field.

The term \bf{u}.\bf{E} represents the dot product of the particle's velocity (represented by \bf{u}) and the electric field (represented by \bf{E}). This dot product represents the component of the electric field that is parallel to the particle's motion. This means that the force experienced by the particle is directly proportional to the component of the electric field in the direction of the particle's motion.

In summary, the \mu=0 component of the Minkowski force law represents the force experienced by a charged particle due to an electric field in the direction of its motion, and the term \bf{u}.\bf{E} represents the component of the electric field in that direction. This equation is essential in understanding the motion of charged particles in electromagnetic fields and has many applications in fields such as particle physics and astrophysics.