I've seem sometimes people saying that although uniform motion on a straight line cannot be detected and hence it is not absolute, acceleration is indeed absolute in Classical Mechanics (I don't know yet how this turns out to work in GR, so for this question I'm explicitly talking about CM).
Now, thinking about it for a while I think I got the idea and I want to know if it is this way it works. Suppose we consider one observer inside a train. With the train at rest, a ball is attached to the ceiling by a string. For this observer there are then two forces to be considered: the weight of the ball and the tension on the string.
Of course in this situation, the string is fully stretched and the ball sits making angle $\theta_0 = 0$ with the $y$ axis. Now, suppose the train moves with constant acceleration $a$. In that case, the ball will raise and now will have a new angle $\theta$.
The observer, however, still sees just the two forces which were acting before. In that case, it should be possible to compute a new acceleration being added, in order to change the state of motion of the ball so that it now makes an angle $\theta$ with the vertical.
So the idea, is that performing experiments himself, inside the train it is possible to detect the value of the acceleration? In that sense, acceleration is not something that has to be measured with respect to some outside frame? Is that the idea of absolute acceleration?