The light clock : just an illusion ?

Two observers, one in the S station and the other on board to a super fast train S’ that is going on with v speed, want to measure the duration of a physical phenomenon (that is temporal separation between two events) obviously each of them from his reference system. They use a light clock made up by two plane mirrors put at a known vertical distance d; a light beam that moves all along the axis of the mirrors reflects alternatively on them and the time required to go and return on the same mirror is the period of the clock.
The period of the clock (measured by an observer in quiet respect to the clock) is T0 = 2d/c.
The interval of time T0 is the temporal separation between two events : the event departure and the event arrival of the light beam on the lower mirror.
For an observer in quiet the two events have zero spatial separation. The temporal separation of two events with zero spatial separation is said "own time".
Since that both in S and in S’ there are two equal light clocks, the two observers measure the same temporal interval each of them in his reference system.
But what happens if the observer in S station tries to make a measure of time with light clock that stands in S’ train ? For the observer in S, the clock moves on with v speed, so light covers from departure to return, a zig zag line with lenght 2L greater then 2d.











Since that light has always c speed in every inertial reference system the T’ period of the clock on move (for S) is : T' = 2 L / c > T0 . So the T’ period of clock on move is greater then the one of the standing clock.
Let we see that this thing is not true for the passenger on board of the train that can, for the same reason, say that the clock in the station is slower!
In this statement, that can seems paradoxical, there is all the meaning of Special Relativity principle : the physical laws are the same for all inertial reference systems, that is that each of the two observers says that clock in motion slows down.
All of this can summarize, with simple geometrical considerations, with the formula :
(1) : delta tau = delta t x (1 – ( v/c)^2)^0,5, being delta tau = T0 and delta t = T'.
known as formula for the time dilatation in the clocks in motion.
From the (1), considering this time the light beam directed all along the motion, we get out, with simple passages, the
(2) : l = lambda x (1- (v/c)^2) ^0,5
said Lorentz - Fitzgerald's lenghts contractions formula that expresses the symmetric concept of the light clock lenght contraction l, measured from the station all along the direction of motion, respect to the effective length λ. The lenghts contraction is perfectly symmetrical to the time dilatation.
The (1) and (2) are the Special Relativity fundamental formulas.

Marius, that is a curious type, asked more time to "experts" of the sector (not without some malice) if time dilatation and lenghts contractions were real or they were only “special effects”.
The answer is been : “in physics is real what is measurable, so time dilatations and lenghts contraction are real.”

But let's see what the last statement entails.
Without gravitational fields space time isn’t curved (it is the Minkowsky’s plane space – time) ; in it can be chosen infinite inertial reference systems and between them are valid Special Relativity and Lorentz transformations. A general curved space – time has a much important property that connects it, as we can say, to the most familiar euclideus plane space. As far as it could be curved, it’s always possible to consider a little portion of it where it's practically plane. We can better understand this concept considering earthly surface. It's a two dimensional space (variety) curved where we can define curvilinear coordinates as latitude and longitude. In great scale we can’t take away the curvature of earthly surface and the effects are well visible to all. Instead for a mason building an house earthly surface is plane and he doesn’t mind of the problem. In every curved space – time it's always possible to choose a reference system respect on which space – time is locally plane and inertial (Minkowski’s space – time).
To do it is sufficient to imagine a body that falls down free in a gravitational field. Respect to this body, for a limited time, the other free bodies falling down with it look to satisfy the inertia law. The ones standing keep standing, the ones in uniform motion get on in this way. Respect to this reference system, falling down for a short time, space – time is the plane Minkowski’s one of SR. The astronauts have experience of this when they are parked on earthly stationary orbits (In effect it is the same thing as they really would fall down freely). In their space craft they test zero gravity. Here on the Earth is possible to verify this for a short time when, for example, an airplane gets an air vacuum or in some games at lunapark.
The fact that space – time is curved by masses that create gravitational field is a concept outside commune experience. In a curved space – time are invalid the rules and the properties of euclideus geometry, that is the geometry of our daily life.
To clarify better this concept let's consider an inertial reference system K and a not inertial reference system K’ in uniform rotation respect to K. Let's consider also a circumference tied with K.













Respect to K the fraction between the circumference and its diameter is π. Respect to K’, that rotates in a clockwise direction, the circumference is seen rotating in the opposite way. Every little segment of the circumference is seen from K’ moving on with v speed. For some instant every little segment by which is made up the circumference is seen contracting respect to K’ in accord with the Lorentz contraction law for which the fraction between circumference and its diameter is, respect to K’, unequal of π (diameter doesn’t be subjected to the Lorentz contraction for the reason that it doesn’t move respect to K’ in the sense of its length).
This simple example shows that space, respect to an accelerated reference system, is not plane but is curved, for the reason that are not valid the rules of euclideus geometry and so, remembering the precedent statement : “in physics is real what is measurable", we could say that time dilatation and lenght contraction are real (or, that is the same thing, absolute) and that since a gravitational field is equivalent to an accelerated reference system, space – time is curved by a gravitational field.
This is the fundamental principle of General Relativity : weight force is due to the curvature induced by masses in the space – time.
At this point Marius, that always wondered what could be the real nature of gravity force, could have had satisfied, but fortunately things didn’t go in this way.

Stefano Gusman