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The Minkowski diagram was developed in 1908 by Herman Minkowski and provides an illustration of the properties of space and time in the Special Theory Of Relativity . It 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 System s 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 BASICS For simplification in Minkowski diagrams usually only events in a one dimensional world are considered. Different to usual path-time diagrams the path 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, moving from bottom to top as time passes. In this way each object on this path, 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 Event irrespectively if there happens anything at all. For convenience, not the time ''t'' itself is assigned on the time axis but the corresponding quantity ''ct'', where ''c'' =299,792 km/s is the speed of light. This way one Second corresponds with a distance of 299,792 km on the ordinate. Due to ''x''=''ct'' for a Photon passing 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 identical. PATH-TIME DIAGRAM IN NEWTONIAN PHYSICS 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 und ''t''=0 also with ''x 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 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 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). ''α'' is given by :. The corresponding translation from ''x'' and ''t'' to ''x For the graphical translation it has been taken into account that the scales on the inclined axes is different from the Newtonian case described above. To avoid this problem it is recommended to deform the whole diagram in such a way that the scales becomes identical for all axes, whereas the assignment between the coordinates of the observers remains unchanged. This can be done by a compression in the direction of 45° or an expansion in the direction of 135° until the situation for which the angle bisector of the time axes becomes perpendicular to the angle bisector of the path axes. The angle ''β'' between both time respectively path axes for this situation is given by :. In this symmetrical representation both observers respectively coordinate systems are equivalent. No difference can be seen between the moving and the non moving system, and according to the theory of relativity there is not any. For this symmetrical representation the identity of the axes scales follows from the Principle Of Relativity , the first postulate of the special relativity. It says that the physical laws are identical for all observers moving with constant velocity i. e. without Acceleration . Therefore all these so called Inertial Reference Frame s are undistinguishable. In case of different axes scales the system with smaller scale could be distinguished from that with larger scale. TIME DILATION 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 A second observer having moved together with the clock from O to B will argue that the other clock has reached only C until this moment and therefore this clock runs slower. The reason for this apparently paradox statements is the different determination of the events happening synchroneously at different locations. Due to the principle of relativity the question who is right, has no answer and does not make sense. LENGTH CONTRACTION Relativistic length contraction means that the length of an object moving relative to an observer is decreased and finally also the space itself is contracted in this system. The observer is assumed again to move along the ''ct''-axis. The world lines of the endpoints of an object moving relative to him are assumed to move along the ''ct The second observer will argue that the first observer has evaluated the endpoints of the object at O and A respectively and therefore at different times, leading to a wrong result due to his motion in the meantime. If the second observer investigates the length of another object with endpoints moving along the ''ct''-axis and a parallel line passing C and D respectively he concludes the same way this object to be contracted from OD to OC. Each observer estimates objects moving with the other observer to be contracted. This apparently paradox situation is again a consequence of the relativity of simultaneity as demonstrated by the analysis via Minkowski diagram. For all these considerations it was assumed, that both observers take into account the speed of light and their distance to all events they see in order to determine the times at which these events happen actually from their point of view. CONSTANCY OF THE SPEED OF LIGHT The more important of the two postulates of special relativity is the constancy of the speed of light. It says that any observer in an inertial reference frame measuring the speed of light relative to himself obtains the same value regardless of his own motion and that of the light source. This statement seems to be paradox, but it follows immediately from the Minkowski diagram. It explains also the result of the Michelson–Morley Experiment which was considered to be a mystery before the theory of relativity was discovered. For world lines of photons passing the origin in different directions ''x''=''ct'' and ''x''=−''ct'' holds. That means any position on such a world line corresponds with steps on ''x''- and ''ct''-axis of equal absolute value. From the rule for reading off coordinates in coordinate system with tilded axes follows that the two world lines are the angle bisectors of the ''x''- and ''ct''-axis. The Minkowski diagram shows, that they are angle bisectors of the ''x'-'' and ''ct'-''axis as well. That means both observers measure the same speed ''c'' for both photons. In principle further coordinate systems corresponding to observers with arbitrary velocities can be added in this Minkowski diagram. For all these systems both photon world lines represent the angle bisectors of the axes. The more the relative speed approaches the speed of light the more the axes approach the corresponding angle bisector. The path axis is always more flat and the time axis more steep than the photon world lines. The scales on both axes are always identical, but usually different from those of the other coordinate systems. SPEED OF LIGHT AND CAUSALITY Straight lines passing the origin which are steeper than both photon world lines correspond with objects moving more slowly than the speed of light. If this applies to an object than it applies from the viewpoint of all observers because the world lines of these photons are the angle bisectors for any inertial reference frame. Therefore any point above the origin and between the world lines of both photons can be reached with a speed smaller than that of the light and can have a cause-effect-relationship with the origin. This area is the absolute future, because any event there happens later compared to the event represented by the origin regardless of the observer, which is obvious graphically from the Minkowski diagram. Following the same argument the range below the origin and between the photon world lines is the absolute past relative to the origin. Any event there belongs definitely to the past and can be the cause of an effect at the origin. The relationship between of such pairs of event is called ''timelike'', because they have a finite time distance different from zero for all observers. On the other hand a straight line connecting these two events is always the time axis of a possible observer for whom they happen at the same place. Two events which can be connected just with the speed of light are called ''lightlike''. In principle a further dimension of space can be added to the Minkowski diagram leading to a three-dimensional representation. In this case the ranges of future and past become Cone s with apexes touching each other at the origin. They are called Light Cones . THE SPEED OF LIGHT AS A LIMIT Following the same argument all straight lines passing the origin more flat than the photon world lines would correspond to objects or signals moving Faster Than Light regardless of the speed of the observer. Therefore no event outside the light cones can be reached from the origin even with the speed of light. Such pairs of event are called ''spacelike'' because they have a finite spatial distance different from zero for all observers. On the other hand a straight line connecting such events is always the path axis of a possible observer for whom they happen at the same time. By a slight variation of the velocity of this coordinate system in both directions it is always possible to find two inertial reference frames whose observers estimate the chronological order of these events to be different. Therefore moving faster than light would imply that for any observer estimating the corresponding object moving from X to Y there can be found another one for whom it moves from Y to X. The question who is right has no answer and gives no sense. The principle of causality would be violated. Furthermore with signals moving faster than light information could be sent in the own past. In the adjoining diagram an observer in the ''x''-''ct''-system sends a message moving faster than light from O to A. At A it is received by an observer in the ''x'-ct'-''system who sends it back faster than light arriving at B. But B is in the past relative to O. The absurdity of this process becomes obvious when both observers subsequently confirm that they received no message at all but all messages were directed towards the other observer as can be seen graphically in the Minkowski diagram. Furthermore if it would be possible to accelerate an observer until the speed of light the path and time axes would coincide with their angle bisector. The coordinate system would collapse. These considerations show that the speed of light as a limit is a consequence of the properties of space and time and not of the properties of the objects like an imperfect space ship. THE RELATION BETWEEN SPACE AND TIME In the basic equations of relativity space and time appear formally equivalent to a large extent. For this reason they can be unified to a four dimensional Spacetime . This close relationship between space and time manifests oneself also in the Minkowski diagram. The known equivalence of the three space dimensions expresses itself especially through the possibility of rotations. Therefore the three dimensions are not predefined but can be chosen arbitrarily by definition of the coordinate system. On the other hand in Newtonian physics space and time appear strictly separated. However in the theory of relativity relative motions are closely related with rotations of coordinate systems with space and time axes in spacetime: The angle between both space axes and both time axes in the symmetrical representation of a Minkowski diagram are identical. Therefore the ''x''-axis is perpendicular to the ''ct For that reason the main role of the speed of light as a fundamental Physical Constant is primarily to establish the close relationship between space and time. The fact that photons move with this speed is considered to be merely a consequence of this close relationship. Therefore in relativity calculations usually are not performed with coordinates ''x'', ''y'', ''z'' and ''t'' but with ''x1'' to ''x4'' where ''x4''=''ct''. This simplifies all equations considerably, and the speed of light in this units becomes simply a dimensionless number ''c''=1. SEE ALSO REFERENCE |
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