What is the meaning of transverse acceleration due to work in special relativity? If we derive the momentum in Special Relativity we get :
$$
\dfrac{\mathrm{d}\boldsymbol{p}}{\mathrm{d}t} = \dfrac{\mathrm{d}m \gamma \boldsymbol{v}}{\mathrm{d}t} = m\dot{\gamma}\boldsymbol{v}+m\gamma\dot{\boldsymbol{v}} = \boldsymbol{f}
$$
It's the relation between the 3-force and 3-acceleration in SR we can find on Wikipedia on other textbook. We can see that if it exist a "work" done by a force and the motion is not in the direction of that force (velocity is not longitudinal to the force) then, the "object" feel an acceleration in the transverse direction.
For example the object is a bullet with :
$$
\boldsymbol{v} = (v_x,v_y) \quad \boldsymbol{f}=(f_x,0)
$$
$\boldsymbol{f}$ can be due to an uniform electric field or gravity. The work is not 0, and the velocity in $y$ direction is not 0 too. Even if there is no force in the $y$ direction we have :
$$
\dfrac{\dot{\gamma}v_y}{\gamma}=\color{red}{-}\dot{v_y}
$$
And so, the acceleration is not parallel to the force like in Newtonian/Classical physics, where $\gamma$ does not exist. Here, the change of energy due to the force along $x$ imply an acceleration in $y$ direction.
I read several text where authors spoke about longitudinal and transverse mass, but it seems it's an old concept which is don't use anymore.
This phenomenon can be expressed with simple word ? I don't find an explanation in my own brain apart "The derivation chain rule"...
 A: If a positively charged particle A moves to the north, and a negatively charged particle B also moves to the north, and if both particles enter an electric field that points straight down in our frame, then the system consisting of the two particles starts gaining more rest-mass, and losing northwards speed. The system's northwards momentum stays constant. 
In other words the moving system scoops up mass-energy that is originally not moving , and that causes the slowdown. 
The system's kinetic energy decreases and the system's internal kinetic energy increases. So "inelastic collision" is a good term to describe this phenomenon.
So both particles "scatter".
A: "Increase of speed in the transverse direction causes a decrease of speed in the longitudinal direction" means the same as: "Motion causes time dilation". That can be also expressed as "The faster a clock moves, the slower the clock-hand of that clock moves".
If we have a particle moving to the north-east direction, then we can say the east-wards component of the motion is a clock's clock hand moving, and the north-wards component of the motion is the whole clock moving, or the other way around.
I'll add something about transverse and longitudinal masses:
After a car has braked down from 0.99c to 0.9 c, its brakes have tons of mass-energy which is still moving quite fast which means that the brakes have huge amount of momentum. This is one way how a relativistic car can have a large longitudinal inertia.
If the driver of said car locks the brakes at 0.99 c, then the car starts losing mass-energy quite fast, in this case the mass energy and its momentum go to the road, which means that even a slippery road feels a large force when a relativistic object is sliding on it.  
After a car has braked down from 0.99c to 0.9 c, its brakes have tons of mass-energy - now that extra mass has an effect on steering forces. These steering forces change the velocity of the transverse mass. The car has the same mass-energy and the same transverse inertia before and after the braking. 
