Spotting Errors in Transverse & Longitudinal Accelerations

[greswd]

Let's say you know all 3 cartesian components of a particle's velocity and all 3 for it's acceleration.

You can split the acceleration vector into two vectors, one parallel (longitudinal) to the velocity vector and one perpendicular (transverse) to the velocity vector.

Then, I found the x-components of both acceleration vectors in terms of the 6 variables listed at the start. That would be the component of each acceleration vector parallel to the x-axis. I used the dot product to derive it.

As you can see below, I multiplied the transverse-x acceleration component by gamma and the longitudinal-x acceleration component by gamma cube. These are the formulas for the longitudinal and transverse masses.

I can get a formula for the x-component of the Force.

In order to get the second formula, I differentiated the x-component of the 3-momentum with respect to time.
I expected both equations to be equivalent, but try as I might, I can't make them equal.Can you help me spot the error(s) in my formulas?

 

[PeterDonis]

The Wikipedia page on 4-acceleration might be helpful:

https://en.wikipedia.org/wiki/Four-acceleration

 

[greswd]

Sorry, it didn't help me at all.

Anyway, I've discovered my careless mistake. It is supposed to be multiplied, not divided, by gamma cube for the second equation.