Minkowski diagram

The Minkowski diagram was developed in 1908 by Hermann Minkowski and serves to illustrate the properties of space and time in special relativity theory. It allows a quantitative understanding of the related phenomena such as time dilation and length contraction without formulas.

The Minkowski diagram is a space -time diagram with only one spatial dimension. May be this case, a superposition of the coordinate systems for two against each other at a constant speed moving observer is presented, so that the spatial and time coordinates x and t, which uses an observer to the description of the events immediately that of the other x 'and t' read and vice versa. From this graphically -one correspondence of x and t to x 'and t' can be seen directly the consistency of many seemingly paradoxical statements of the theory of relativity. The invincibility of the speed of light reveals itself graphically as a result of the properties of space and time. The shape of the diagram follows immediately and without formulas from the postulates of special relativity theory and illustrates the close relationship between space and time, which results from the theory of relativity. An extension is the Penrose diagram, with which one can represent the global structure of more general, also curved space-times.

Basics

For the sake of presentability is omitted in the Minkowski diagrams on two of the three dimensions of space and only considers what is happening in a one-dimensional world. Unlike conventional in the path-time diagram, the path on the X- axis and the time on the y -axis is shown. Thus, the action on a horizontal path can immediately empathize with the chart, which also moves way with the passage of time from bottom to top through the diagram through. Each object in this way, such as an observer or a vehicle, describes in this way a line in the diagram, which is called its world line.

Each point in this diagram marks a specific point in space and time. Such a body is referred to as an event regardless of whether anything is happening at this time and in this place.

It proves to be advantageous on the time axis is not the time t directly, but apply the assigned size ct, where c = 299,792.458 km / s is the speed of light. Corresponds to one second in this manner, a portion of 299792.458 km on the ordinate. Since x = ct for a particle of light that passes through the origin to the right, its world line is a 45 ° inclined straight line in the diagram, if the same scale is used for both coordinate axes.

Path-time diagram in Newtonian physics

The diagram represents the coordinate system of an observer is that we want to denote for simplicity as the Dormant, and which is located at x = 0. The world line of the observer is therefore identical with the time axis. Each parallel to the said axis also correspond to a stationary object at a different location. The blue straight line, however, corresponds to an object moving with constant velocity to the right, for example, a moving observer.

This blue Just can now be interpreted as the time axis of the observer, which represents its coordinate system together with the identical for both observers space axis. This represents an agreement between the two observers, the point x = 0 and t = 0 with x '= 0 and t ' = 0 to denote. The coordinate system of the moving observer is obliquely. To read the coordinates of a point in this case, the two parallels are made to the axes through the event point and considered their point of intersection with the axes.

It is certainly the case of the event A in the diagram, so that the spatial coordinate, as expected, different values ​​can be determined because the moving observer = 0 has to be moved to the location of the event since t. On the other hand, takes place in the Newtonian physics an event from the point of view of both observers at the same time instead. The scale on the time axis of the moving observer is therefore stretched such that are the same height above the x-axis on both timelines same values ​​.

Generally find all events that are located on a line parallel to the path axis, simultaneous and indeed for both observers. There is only one universal time t = t ', which manifests itself in the existence of a common path axis. Analog is the existence of two different timelines in conjunction with the fact that both observers detect different spatial coordinates. This translation of the graphic coordinates of x and t in the x 'and t ', and vice versa takes place mathematically, the Galileo transformation.

Minkowski diagram in the special theory of relativity

Albert Einstein ( 1905) now discovered that the above description of the conditions, the reality does not reflect applicable. The higher the speeds involved, the stronger the error. Space and time are such that different rules apply for translating the coordinates between moving observers. Particular events occur, an observer which evaluates the same time, instead of one movable relative to the observer at different times. This was interpreted geometrically by Hermann Minkowski in an elegant way.

In Minkowski diagram corresponds to the relativity of simultaneity of the existence of different Wegachsen for the two observers. Each observer interpreted according to the above rule, all events on a straight line parallel to the distance axis than at the same time. The sequence of events from the point of view of a particular observer can thus be graphically by parallel displacement of such a straight line from bottom to top through the diagram through vividly understand.

In application of ct instead of t on the time axis, the angle α between the two Wegachsen proves to be identical with that between the two timelines. The cause of this orientation of the Wegachsen can interpret the principle of the constancy of the speed of light ( see below). The angle α is obtained from the relative velocity v to

The corresponding translation of the coordinates x and t in the x 'and t ', and vice versa takes place mathematically, the Lorentz transformation. The scaling of the axes is derived as follows: If U is the unit length on the ct- and x -axis, the unit length is given on the ct' and x' -axis with:

Sets the ct -axis is the world line of a stationary clock is in S, then U corresponds to the duration of the interval between two events that occur in this world line, which is called the proper time. The length of U on the x - axis corresponds to the resting length or proper length of a rest in S scale. The same relationships apply also for the distances U ' on the ct' and x' axis.

Symmetrical Minkowski diagram

To avoid the scaling of the different axes, without change to the assignment of x and t X 'and T', the diagram can be deformed as follows: The axes are compressed in the direction of 45 ° or stretched in the direction 135 ° to the situation that x is perpendicular to ct ' and x' perpendicular to ct.

This corresponds to a Minkowski diagram from the perspective of a third inertial frame in which the other two move with the same speed in the opposite direction ( " Central System"). If and are between two inertial systems and given, then they are as follows connected to the corresponding variables in the agent system:

For example, if given between S and S ', then they move according to (2 ) in their central system with approximately ± 0.268 c in opposite directions. Or if is given, according to (1 ) is the relative velocity between S and S ' in their own quiet systems given with 0.8 c.

It appears, however, that the construction of related symmetric Minkowski diagrams neither the agent system have yet to be listed. The same result is namely also obtained when directly the relative velocity between S and S ' is used in the following construction: The ct' - axis is drawn perpendicular to the x- axis, just like the ct- to the x' axis. When the angle between the axis ct' and CT ( and between the x and x 'axis ), and between the x - axis and ct' is then obtained:

In addition, it follows that the parallel projections of vector its contravariant components (x, t, x ', t' ) correspond to the orthogonal and its covariant components (see picture right).

• Max Born ( 1920) used in his book " The Theory of Relativity Einstein's" different Minkowski diagrams with two perpendicularly propagating light beams as axis cross. To illustrate the symmetry of length contraction and time dilation, he added, the axes of two systems S and S 'where the x - axis approximately perpendicular to the ct' - axis and the x' axis was approximately perpendicular to the ct axis.
• Dmitry Mirimanoff ( 1921) discovered the existence of " means systems " that are always relative to two inertial frames moving relative to each other can be found. However, he showed no graphical interpretation of this relationship.
• Paul Gruner ( 1921), together with Josef Sauter, developed symmetric diagrams in a systematic way. It relativistic effects such as length contraction and time dilation have been derived, as well as the relationship between contravariant and covariant components. Gruner extended this method in other works (1922-1924), and appreciated the performance Mirimanoffs.
• The construction of such symmetric graphs was repeatedly rediscovered later. For example, Enrique Loedel Palumbo published starting in 1948 with several works in Spanish, in which he developed this method. In 1955 she was again rediscovered by Henri Amar. In some textbooks such diagrams are therefore referred to as " Loedel Diagrams".

Time dilation

The time dilation says that a clock which displays their own time and moves relative to an observer, from whose point of view seems to run slower, and therefore also the time in this system itself This fact can be read directly from the adjacent Minkowski diagram. The observer is moving within the space-time from the origin O in the direction A and the clock of O in the direction B. All events that interprets this observer at A than at the same time, lying on the parallels to his path axis, ie the line through A and B. Because OB < OA, however, is on the moving relative to him clock a smaller time passed than on the clock that leads the observer with it.

A second observer, who has moved with a clock from O to B is, however, claim that the other clock was in this moment only in C, and it was, therefore, the run slower. These different interpretations of what is happening at the same time in another place, is the cause for this seemingly paradoxical situation. Given the principle of relativity is the question of who assessed the situation correctly, in principle unanswerable and therefore pointless.

Length contraction

The length contraction says that a length scale of a certain length at rest, moves relative to an observer, from whose point of view appears shortened, and thus the space in this system itself, the observer is moving again on the ct- axis. The world lines of the two end points of a moving relative to him scale move along the ct' - axis and parallel to it by A and B. For the observer, the scale ranges at time t = 0 only from O to A. For a along the ct ' axis comoving second observer, for the rest of the scale, it has at the moment t ' = 0, the resting length OB. So it appears to the first observer shortened < OB due to OA.

The comoving observers will argue that the first observer and therefore have not recorded start and end points at O and A at the same time, so that he had identified an incorrect length due to its intermittent movement. About the same reasoning determines the second observer for the length of a scale whose endpoints move along the ct- axis and parallel to it by C and D, a length contraction of OD to OC. The seemingly paradoxical situation that for each the standards of the other appear shortened due in turn to the relativity of simultaneity, as the Minkowski diagram.

With all these considerations, it was assumed that the observer into account in their statements that they know speed of light. That is, they do not indicate what they see directly, but rather what they think real basis of the signal propagation time and the determined physical distance from them to the seen events for.

Principle of the constancy of the speed of light

The more important of the two postulates of special relativity is the principle of the constancy of the speed of light. It states that the vacuum speed of light, in each inertial system has the same value c, independently of the speed of the light transmitter or the light receiver. All observers measure the speed of light, so come, regardless of their own state of motion, the same result. This statement seems paradoxical at first, but results graphically immediately from the Minkowski diagram. It also explains the result of the Michelson -Morley experiment, which ensured before the discovery of relativity theory for amazement.

For the world lines of two light particles that pass through the origin in different directions x = ct and x = -ct, ie each path point correspond in amount equal sections on the x and ct- axis. From the rule for the reading of coordinates in an oblique coordinate system results in the fact that these world lines are the two bisectors of the x and ct- axis. The Minkowski -diagram takes now that they are the bisectors of the x 'and ct' axis simultaneously. That is, both observers determine the magnitude of the velocity of these two photons same value c.

In principle, can be used in this Minkowski diagram further coordinate systems to observers at any speed add. In all these coordinate systems, the world lines of light particles are the bisectors of the coordinate axes. The more approximate the relative speeds of the speed of light, the more the coordinate axes at least one of the participating systems cling to the bisector. The Wegachsen are always flatter than this bisector and the time axes always steeper. The scales on the respective distance and time axes are always the same, but differ in general from those of the other coordinate systems.

Speed ​​of light and causality

All straight line through the origin, are steeper than the two world lines of light particles correspond to objects that move slower than the speed of light. Since the world lines of the particles of light are the same for all observers, this statement is independent of the observer. From the origin of each point can be achieved above and between the world lines of the two photons with sub- light speed, so any event that may stand there with the origin in a cause - effect relationship. This area is known as absolute future, as each local event independent of the observer is later than the event which marked the origin of which one can easily convince graphical way.

Analogously, the area below the origin and between the world lines of the two light particles, the absolute past relative to the origin. Each event there may be a cause of action at the origin and is clearly in the past.

The ratio of two event points that can stand in a cause -effect relationship in this manner is referred to as time-like, as they have a finite time interval for all observers. In contrast, the link is always represents the time axis of a coordinate system possible to take place for the observer, the two events are therefore in the same place. Let's just combine the speed of light, two events, as they are called light-like.

In principle, the Minkowski diagram suggests a further dimension of space to add, so that a three-dimensional representation. In this case, the areas of past and future into cones whose tips are touching at the origin are. They are referred to as the light cone.

Speed ​​of light as a limit

Likewise, any straight line through the origin correspond to running flatter than the two world lines of light particles, objects or signals that move faster than light, with the above argument again independent of the observer. Thus, among all the events outside the light cone and the speed of light is no contact be made ​​at the origin itself. The ratio of two such event points is referred to as space-like, because they have a finite distance for all observers. In contrast, the link always constitute the distance axis of a possible coordinate system, taking place for the observer the two events at the same time. By a slight variation of the velocity of this coordinate system in both directions can therefore be always find two coordinate systems whose observers judge the temporal order of these two events differently.

Starting from the postulate of constant light speed faster than light would, therefore, mean that for each observer, for the would move such an object from X to Y, another could find, for it would move from Y to X, again without the question of who describes the situation correctly, would make sense. The principle of causality would be violated thereby.

In addition, it follows from the theory of relativity that could send information into his own past with superluminal signals. So send in to the diagram of the observer in the x -ct system a message faster than light from O to A. At point A it is received by an observer in the x'- ct' system, which in turn sends back a response signal faster than light, so that it arrives at B and therefore in the past by O. The absurdity of the operation is thus clear that both observers would then claim to have received the answer to their message before the send.

The incompatibility of relativity theory and the possibility of an observer at the speed of light or even going to accelerate beyond, is also expressed in the fact that at the speed of light its time and distance axis would coincide with the bisector so that the coordinate system would collapse as such.

These considerations show graphically using the Minkowski diagram that the invincibility of the speed of light is a consequence of the relativistic structure of space and time and not a property of things, such as a merely imperfect spaceship.

The relationship of space and time

Space and time appear formally broadly equivalent to the basic equations of the theory of relativity together and leave unite to form a four-dimensional space-time therefore. This close relationship of time and space is reflected in the Minkowski diagram.

The well-known equivalence of the three dimensions of space manifests itself in particular in the ability to rotate in space. Thus, the three dimensions are not fixed, but on the definition of a coordinate system freely chosen. Space and time appear, however, strictly separated in Newtonian physics. In the special theory of relativity, however, prove to relative movements as closely related to rotations of coordinate systems with space and time axes in space-time: Since the angle between the two space - and the two timelines in the symmetric representation is equal to the x - axis is perpendicular to ct' the axis as well as the x 'axis of the CT -axis. The arrangement of the four axes is thus identical to the two ordinary rectangular coordinate systems are rotated only by an angle φ to each other, followed by reversal of the time axis. This results in a shear of the axes results instead of a rotation. This interchange of two axes, as well as all the differences between space and time can be ultimately to a single sign in the equation attributed that links space and time, by defining the metric of spacetime.

For this reason, the importance of the speed of light as a fundamental constant of nature of physics in the first place is to make this connection between space and time. The fact that photons move at this speed is to be regarded rather as a consequence of this close relationship. In relativity theory, it is therefore also common, instead of the coordinates x, y, z and t with x1 to x4 anticipated, in which x4 = ct. All formulas are simplified so much, and for the speed of light results in these units is a dimensionless number c = 1