6. Some formula.
The separation of events in spacetime
In ordinary 3D, flat, Euclidean space, the distance from the origin s of a point P with coordinates (x,y,z) is obtained from the formula
which in two dimensions reduces to the well-known Pythagorean theorem
In four-dimensional space-time, a point or event is identified by four coordinates (ct,x,y,,z); to the spatial coordinates is added time, multiplied by the speed of light so that the dimensions are homogeneous.
The 4D space-time of special relativity is “flat,” but the distance between two events is not the same as a space in which Euclidean geometry applies. Its geometry is quite different from that of ordinary three-dimensional space.
Given two neighboring events in spacetime with coordinates are:
the separation between events, the same for all observers, is given by the formula:
separation between time-type events
separation of space-type events
separation between light-type events
1. Time-type separation: positive time interval between the two events, the events are connected by a signal moving at a speed less than the speed of light, it is possible to go from one event to the other by following a trajectory at a speed less than the speed of light.
2. Space type separation: positive space interval between the two events, the events are not connected by any signal moving at a speed less than or equal to that of light, it is not possible to go from one event to the other following a trajectory at a speed less than or equal to that of light.
3. Light-type separation: zero interval of space and time between the two events, the events are connected by a signal moving at the speed of light, it is possible to go from one event to the other by following a trajectory at exactly the speed of light.
Only events with time-type separation can be ordered in time, while events with space-type separation have no definite temporal order between them.
Own time
The formula
where:
- is an interval of proper time, i.e., time measured by an observer integral to an event, i.e., moving along with it
is the time interval measured by a reference observer, at rest with respect to the event.
are the spatial coordinate ranges measured by the reference observer.
When the observer integral to the event moves at a constant speed relative to the reference observer,
where is the Lorentz factor, we thus derived the formula 
the proper time of an event is always less than or equal to the time measured by an outside observer, a phenomenon known as special relativity time dilation
Time dilation and length contraction
are intervals of time and length measured by an observer considered stationary
are intervals of time and space measured by an observer in an inertial reference system in uniform relative motion with velocity v relative to that of the stationary observer
Time dilation: 
Contraction of lengths:
A1 Lorentz and Galileo transformations
Note: this paragraph can be skipped on a first reading
Siano:
- the coordinates of an inertial reference system
the coordinates in another inertial reference system, in uniform rectilinear motion relative to the first, with velocity v, in the direction of the x-axis
The coordinate transformation rules between the two observers are Galileo transformations in classical mechanics, Lorentz transformations in special relativity
Galileo transformations
time is the same for the two observers; it is absolute.
Lorentz transformations
time and space vary and are related to each other in the formulas. These transformations leave the formula for the separation of events in spacetime and Maxwell’s equations of electromagnetism unchanged.
Classic limit
Galileo’s transformations are the limit of relativistic Lorentz transformations when
in fact, we rewrite the Lorentz transformations using the ratio
For it can be placed in the Lorentz transformation as an excellent approximation
by obtaining Galileo’s transformations
in which time is absolute and does not depend on the space traveled and the relative speed of the observers.