Some formulas in special relativity and the twin paradox
Victor Orsini

In his 1905 paper, Einstein follows the same line of reasoning that Lorentz had followed. Given the constancy of the speed of light, length contraction in the direction of motion relative to a static observer, is evident. The delay in time is not evident and needs to be derived from longitudinal contraction. The delay is a function of the contraction.

Consider two systems of reference, one static and other in motion relative to the first. Let’s call c the constant speed of light which is the same for the two systems.

In the static system, it is satisfied

c = lr / tr

where lr is the distance traveled by light in the rest frame and tr the time the light takes to travel it. tr is calculated as tr = lr / c.

In the system in relative motion also holds that

c = lm / tm

where lm is the distance traveled and tm is the time delay. tm is calculated as tm = lm / c.

The length contraction in the moving system can be written as the difference between the length in the moving system minus the length in the static system, so

contraction = lr - lm

Similarly, the time delay can be defined as the difference in time the light takes to travel lr in the system at rest minus the time it takes to travel lm in the moving system

delay = tr - tm

Substituting lr by tr / c and tm by lm / c we obtain

delay = tr - tm = lr / c - lm / c = (lr - lm) / c = contraction / c

Taking the first and the last term we have the equality

delay = contraction / c

It is clear that the delay turns zero when contraction becomes zero.
Einstein himself states that, when movement stops, the contraction in the direction of motion becomes zero. This demonstration implies that the delay turns zero when movement stops together with the contraction .

QED

This shows that the clock does not return late because the delay stops when motion stops. Equally, the traveling twin does not return younger, otherwise he will be a plane figure.

PD: Explanation

The expression obtained for the delay, is an obvious and direct result of the constancy of the speed of light that can be expressed as:

c = contraction / delay = lr / tr = lm / tm

This truth may be obscured if we take a bar of fixed length (that does not grow over time) and it is compared with a watch going indefinitely . This is because they compare different phenomena.

Be clear that, if the contraction of a millimeter goes away when movement stops, also the contraction of a meter or a billion of meters disappears when movement stops. Equally, if contraction of a millisecond becomes zero, the contraction of an hour or of a thousand million years also disappears.

For the constancy of the speed of light to be maintained, it is necessary that the delay disappear when ever contraction disappears.