Time dilation - what is the time on the Earth?

[PeroK]

So they both have to agree on times when the particular event happened, for S it was $t$ and for S' it was $t'$, if I use the Lorentz transformation for $t$ or $t'$, I have to get the same relation between these two times. And each stationary observer thinks that in the moving frame of reference time passes more slowly.

You have a special case where the two events both occur at $x = 0$. That means that throwing the ball was an unnecessary detail. You really just took two times at the spatial origin of $S$. And have essentially an example of time dilation.

The Lorentz Transformation (LT) can transform any two events. In general, as I think has been mentioned before in this thread, the LT encapsulates time dilation, length contraction and relativity of simultaneity.

Ultimately, the Lorentz Transformation is a pair of invertible equations and you can't boil them down to "each observer thinks time passes more slowly".

 

[Sagittarius A-Star]

So if I understand it well, if I use $\Delta t = \gamma (\Delta t' + v\Delta x'/c^2)$, that is when I take datas from S' into S. And when I use $\Delta t' = \gamma (\Delta t - v\Delta x/c^2)$, I take (the same?) datas from S into S'.

As I wrote, these are different data:

Please be aware, that the following deltas refer to a different pair of events.

When I calculate time $t'$ from the first equation, it is the acual time that passed in the rocket.

Yes, it is the proper time passed in the rocket while the coordinate-time interval $\Delta t$ with reference to the rest-frame of the earth. Time-dilation is in this case the ratio between the proper time of a moving clock and the coordinate-time of the inertial reference frame.

And when I want to calculate time $t$ from the second equation, I cannot suddenly say that this time really passed on Earth. According to the man in the rocket it passed, that I understand, but why cannot I say it is the real time that passed on Earth? In the first case I can say that it is the real time in the rocket.

That's wrong. The situation is reciprocal. To understand time-dilation, you need to understand the definition of a standard inertial 4D-coordinate system:

The basic principle of clock synchronization is to ensure that the coordinate description of physics is as symmetric as the physics itself. For example, bullets shot off by the same gun at any point and in any direction should always have the same coordinate velocity dr/dt .
...
We should, strictly speaking, differentiate between an inertial frame and an inertial coordinate system, although in sloppy practice one usually calls both IFs. An inertial frame is simply an infinite set of point particles sitting still in space relative to each other. For stability they could be connected by a lattice of rigid rods, but free-floating particles are preferable, since keeping constant distances from each other is also a criterion of the non-rotation of the frame. A standard inertial coordinate system is any set of Cartesian x,y,z axes laid over such an inertial frame, plus synchronized clocks sitting on all the particles, as described above. Standard coordinates always use identical units, say centimeters and seconds.

Source:
http://www.scholarpedia.org/article...nematics#Galilean_and_Lorentz_transformations

 

[Herman Trivilino]

According to the man in the rocket it passed, that I understand, but why cannot I say it is the real time that passed on Earth? In the first case I can say that it is the real time in the rocket.

Because in this context there is no such thing as a real time. Your question is prompted by a misunderstanding of the way time passes.