What is the concept of four velocity and four force in special relativity?

[granpa]

if the magnitude of the four velocity vector is always c then it can be rewritten in polar form with 3 numbers giving the direction that it points and a fourth number giving the magnitude. since the fourth number is constant it adds nothing to our understanding so it seems like it should be dropped.

also what is four force? does force have a component in the time direction? does an electric field have a component in the time direction?

I just accepted these things before but I'm seeing now that I don't really understand them at all.

 

[JustinLevy]

also what is four force? does force have a component in the time direction? does an electric field have a component in the time direction?

If we write events in an inertial frame as (ct,x,y,z)
then using that same convention for ordering, the four momentum is (E/c, p_x, p_y, p_z).

The force four vector, is d P / d tau (where tau is the proper time of the object the force is applied to). So, this should make it clear what the time like component of the four force is ... it relates to the rate at which energy is being transferred to the object (where as the other components relate to the rate at which momentum is being transferred to the object).

The electric field is actually a component of a tensor, and does not transform like components of a four-vector. However, the "scalar potential" and "vector potential" together form a four-vector.

I hope that helped some.

 

[Chestermiller]

The four velocity vector has 4 components for any given inertial frame of reference. Three of these components are in the 3 spatial directions of that frame, and the 4th is in the (inaccessible) time direction. In their rest frames of reference, all objects have a four velocity vector pointing exclusively in the time direction of that rest frame. The spatial components in an object's rest frame of reference are all zero. The magnitude of time component in the rest frame is c.

The four force is equal to the rest mass times the four acceleration, where the four acceleration is equal to the derivative of the 4 velocity with respect to proper time. Because the four velocity has constant magnitude, the four acceleration (and four force) are always perpendicular to the four velocity. Therefore, in a body's rest frame of reference, the four force has no component in the time direction, and only the 3 spatial components are non-zero. However, in any other inertial frame of reference, all four components are non-zero.

I've been writing up some notes on spectial relativity, and the second chapter of my notes is devoted to providing a better mechanistic understanding of the fundamental geometric structure of 4D space-time, and how frames of reference are traveling through 4D space-time. I've been very unhappy with how special relativity is taught in the various texts that I've worked with, and feel that it is much easier to understand than the way it is presented. If you are interested in seeing what i have written so far, please send me your email address at chestermiller@mindspring.com, and I will email you a copy. This is a work in progress, but I'm sure you will find the second chapter very illuminating.