Minkowski Diagram - Minkowski Diagram in Special Relativity

Minkowski Diagram in Special Relativity

Albert Einstein discovered that the description above is not correct. Space and time have properties which lead to different rules for the translation of coordinates in case of moving observers. In particular, events which are estimated to happen simultaneously from the viewpoint of one observer, happen at different times for the other.

In the Minkowski diagram this relativity of simultaneity corresponds with the introduction of a separate path axis for the moving observer. Following the rule described above each observer interprets all events on a line parallel to his path axis as simultaneous. The sequence of events from the viewpoint of an observer can be illustrated graphically by shifting this line in the diagram from bottom to top.

If ct instead of t is assigned on the time axes, the angle α between both path axes will be identical with that between both time axes. This follows from the second postulate of the special relativity, saying that the speed of light is the same for all observers, regardless of their relative motion (see below). α is given by

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The corresponding translation from x and t to x' and t' and vice versa is described mathematically by the so called Lorentz transformation. Whatever space and time axes arise through such transformation, in a Minkowski diagram they correspond to conjugate diameters of a pair of hyperbolas.

For the graphical translation it has been taken into account that the scales on the inclined axes are different from the Newtonian case described above. To avoid this problem it is recommended that the whole diagram be deformed in such a way that the scales become identical for all axes, eliminating any need to stretch or compress either axis. This can be done by a compression in the direction of 45° or an expansion in the direction of 135° until the angle between the time axes becomes equal to the angle between the path axes. The angle β between both time and path axes is given by

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In this symmetrical representation (also referred to as Loedel diagram, named after the physicist Enrique Loedel Palumbo who first introduced this symmetrised Minkowski representation), the coordinate systems of both observers are equivalent, since both observers are traveling at the same speed in opposite directions, relative to some third point of view.

However, Loedel diagrams become more complicated than Minkowski diagrams for more than three observers and therefore lose their pedagogical appeal.