Can someone explain the 4-force to me.

[ozone]

I understand that we can write the 4-force in terms of the vector force based on velocity, and the energy of the system in general.

What I don't understand is what this gains for us, and why we use it. Also correct me if I am incorrect in my definition above.. I don't have my book infront of me.

 

[Muphrid]

Knowing dp/d\tau = f isn't useful?

 

[ozone]

no that's very useful. I don't understand though why we have a vector we call the energy-momentum four-vector,

defined as
p = (E, p)

Where the vector p is just our ordinary momentum.

Also we have a 4-vector force defined as
f = (\gamma F \cdot V, \gamma \text{ }F)

Obviously both terms contain standard 3-vector, but I'm not sure what physical good the fourth piece is.

This is out of James Hartle's Intro to relativity.

 

[Muphrid]

I'm not sure I understand what "fourth" piece you're referring to.

At any rate, I find the formulas for converting between ordinary force and four-force (and the same for acceleration) to be very annoying and tedious. I personally try to avoid dealing in ordinary force and acceleration precisely for this reason. It's just much easier to deal only with four-vectors or only with three-vectors and not constantly convert back and forth between them.

 

[ozone]

The fourth piece (the time piece) is just confusing to me.. I understand having momentum in terms of velocity but my book doesn't really explain why/what the 4th piece of the vector is.

Also I solved a relativistic energy conservation problem (in the lab frame) and I am pretty sure I only needed to use the 3-vector but I want to make sure that I did that correctly.

 

[Muphrid]

The time component of the four-force is basically the power the force applies to the object in question.