Most of us learned Euclid's geometry at school and we were -and still are- convinced that it was true. But is it?
Geometry is based on some concepts such as "plane", "point" and "straight line" from which we deduce some ideas and certain propositions (Axioms) that we accept as true. All other propositions are deduced from these "true" axioms and, therefore, they are also proven to be true. This reasoning is logically correct as long as the axioms are true but as there is no proof of it, we must simply admit that they are true. For instance we cannot say if it is true that one straight line goes through two points. All we can say is that the Euclidian geometry deals with "straight lines", each of which is determined by two points located on it. This does not really fit with the concept of "truth" as we know it. However for all practical everyday uses, these axioms and the propositions derived from them can be considered as being "true".
Within the concept of the Euclidian geometry we can also admit the following proposition: the distance between two points on a solid body is the same if the body is at rest or is moving. On this basis we can obtain the distance between two fixed points, A and B, on a rigid body by measuring it. Let assume that a rod S is our standard measure or unit of length. Let us draw a straight line between A and B then starting from A we can step by step find how many time we have to move the rod S to cover the distance between A and B. This number will be the measurement of the distance AB using the rod S as the unit of measurement.
The position where any event takes place or an object is located is based on the specification of the point by reference to a rigid body with which the event or object is located. To facilitate the description of a given position, scientists invented what is known as a Cartesian set of co-ordinates. In practise the system consists of three plane surfaces perpendicular to each other and rigidly attached to a rigid body such as the earth. In this system the position of an event or an object will be given by the lengths of the three perpendiculars or co-ordinates x, y, z resulting from the projection of the position of the event on the three planes.
In classical mechanics space and time are absolute. But is it really true?
Let us assume that a traveller in a train going to a constant speed drops
an apple through his carriage's open window. He will see it falling on a
straight line but, to someone standing still on the ground, the apple trajectory
is parabolic. What is the true trajectory of the apple, a straight line
or a parabola? It fact the trajectory depends on the rigid body of reference
or, in other words, to the chosen coordinates of reference fixed to the
train or to the earth in this example. Moreover we must describe how the
apple is changing its position with time. In classical mechanics both the
traveller and the man on the ground have the same time but, if we take into
consideration the finite velocity of light, we will see later on that this
is not exactly true.