     # Minkowski diagram deals with space and time.        Minkowski diagram for the translation of the space and time coordinates x and t of a first observer into those of a second observer (blue) moving relative to the first one with 40% of the speed of light c. This particular Minkowski diagram is of a special type - called a Loedel diagram - in which the scales of all four axes are identical.The Minkowski diagram was developed in 1908 by Herman Minkowski. The Minkowski diagram provides an illustration of the properties of space and time in the special theory of relativity. The Minkowski diagram allows a quantitative understanding of the corresponding phenomena like time dilation and length contraction without mathematical equations. The Minkowski diagram is a space-time diagram with usually only one space dimension. It is a superposition of the coordinate systems for two observers moving relative to each other with constant velocity. Its main purpose is to allow for the space and time coordinates x and t used by one observer to read off immediately the corresponding x' and t' used by the other and vice versa. From this one-to-one correspondence between the coordinates the absence of contradictions in many apparently paradox statements of the theory of relativity becomes obvious. Also the role of the speed of light as a non conquerable limit results graphically from the properties of space and time. The shape of the diagram follows immediately and without any calculation from the postulates of special relativity, and demonstrates the close relationship between space and time discovered with the theory of relativity.The Basics of the Minkowski diagram. Choosing ct instead of t on the time axis the world line of a photon becomes a straight line with a slope of 45°.For simplification in Minkowski diagrams, usually only events in a one dimensional world are considered. Different from usual distance-time diagrams the distance will be displayed on the x-axis (abscissa) and the time on the y-axis (ordinate). In this manner the events happening on a horizontal path in reality can be transferred easily to a horizontal line in the diagram. Objects plotted on the diagram can be thought of as moving from bottom to top as time passes. In this way each object, like an observer or a vehicle, follows in the diagram a certain curve which is called its world line. Each point in the diagram represents a certain position in space and time. Such a position is called an event whether or not anything happens at that position. For convenience, the (vertical) time axis represents, not t, but the corresponding quantity ct, where c =299,792,458 m/s is the speed of light. In this way, one second on the ordinate corresponds to a distance of 299,792,458 m on the abscissa. Due to x=ct for a photon passing through the origin to the right, its world line is a straight line with a slope of 45°, if the scales on both axes are chosen to be identical.Path-time diagram in Newtonian physics of the Minkowski diagram. In Newtonian physics for both observers the event at A is assigned to the same point in time.The adjoining diagram shows the coordinate system of an observer which we will refer as at rest, and who is positioned at x=0. Obviously his world line is identical with the time axis. Each parallel line to this axis would correspond also to an object at rest but at another position. The blue line however describes an object moving with constant speed v to the right like a moving observer for instance. This blue line can be interpreted as the time axis for this observer. Together with the path axis, which is identical for both observers, it represents his coordinate system. This corresponds with the agreement between both observers to denote the position x=0 and t=0 also with x'=0 and t'=0. The axes for the moving observer are not perpendicular to each other and the scale on its time axis is stretched. To read off coordinates of a certain point, both parallel lines to the axes passing the event have to be constructed and the intersections with the axes to be considered. Determining position and time of the event A as an example in the diagram leads to the same time for both observers as expected. Only for the position different values result, because the moving observer has approached the position of the event A since t=0. Generally all events on a line parallel to the path axis happen simultaneously for both observers. There is only one universal time t=t' which corresponds with the existence of only one common path axis. On the other hand due to two different time axes the observers usually measure different path coordinates for the same event. This graphical translation from x and t to x' and t' and vice versa is described mathematically by the so called Galilean transformation.Minkowski diagram in special relativity. In the theory of relativity both observers assign the event at A to different times.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 a between both path axes results to 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). is given by The corresponding translation from x and t to x' and t' and vice versa is described mathematically by the so called Lorentz transformation. Symmetrical representation with lines of simultaneity for both observers.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 ß between both time and path axes is given by 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 travelling 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.Minkowski diagram time dilation. Time dilation: Both observers consider the clock of the other as running slower.Relativistic time dilation means that a clock moving relative to an observer is running slower and finally also the time itself in this system. This can be read immediately from the adjoining Minkowski diagram. The observer at A is assumed to move from the origin O towards A and the clock from O to B. For this observer at A all events happening simultaneously in this moment are located on a straight line parallel to its path axis passing A and B. Due to OB Universe - Galaxies and Stars: Links and Contacts  the web this site   