Spacetime diagram

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 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 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.

Basics

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

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 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

$\tan(\alpha )={\frac {v}{c}}$.

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 β between both time and path axes is given by

$\sin(\beta )={\frac {v}{c}}$.

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.

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<OA he concludes that the time passed on the clock moving relative to him is smaller than that passed on his own clock since they were together at O.

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 these apparently paradoxical 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

Length contraction: Both observers consider objects moving with the other observer as being shorter.

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'-axis and the parallel line passing A and B respectively. For this observer the endpoints of the object at t=0 are O and A. For a second observer moving together with the object, so that for him the object is at rest, it has the length OB at t'=0. Due to OA<OB the object is contracted for the first observer.

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

For the speed of a photon passing A both observers measure the same value even though they move relative to each other.

Another postulate 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 differential equation yielding this, and the Minkowski diagram agrees. It explains also the result of the Michelson–Morley experiment which was considered to be a mystery before the theory of relativity was discovered, when photons were thought to be waves through an undetectible medium.

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 tilted 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.

Minkowski diagram for 3 coordiante systems. For the speeds relative to the system in black v' = 0.4c and v" = 0.8c holds.

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

Past and future relative to the origin. For the grey areas a corresponding temporal classification is not possible.

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, then 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 cones with apexes touching each other at the origin. They are called light cones.

The speed of light as a limit

Sending a message at superluminal speed from O via A to B into the past. Both observers consider the temporal order of the pairs of events O and A as well as A and B different.

Following the same argument, all straight lines passing through the origin and which are more nearly horizontal 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 by a light-signal, nor by any object or signal moving with less than the speed of light. Such pairs of events 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 space coordinate 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 an object moving faster than light, say from O to A in the adjoining diagram, would imply that, for any observer watching the object moving from O to A, there can be found another observer (moving at less than the speed of light with respect to the first) for whom the object moves from A to O. The question of which observer is right has no unique answer, and therefore makes no physical sense. Any such moving object or signal would violate the principle of causality.

Also, any general technical means of sending signals faster than light would permit information to be sent into the originator's own past. In the diagram, an observer at O in the x-ct-system sends a message moving faster than light to A. At A it is received by another observer, moving so as to be in the x'-ct'-system, who sends it back, again faster than light by the same technology, 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. Indeed, if it were be possible to accelerate an observer to the speed of light, the space 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 objects such as technologically imperfect space ships. The prohibition of faster-than-light motion actually has nothing in particular to do with electromagnetic waves or light (applying also to eg gravitational waves), but depends on the structure of spacetime and our notion of causality, which such motions appear to violate.

The relation between space and time

Rotation of an orthogonal coordinate system in usual space. It is formally related to a shear of space and time coordinates in relativity.

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'-axis and also the x'-axis to the ct-axis. For this reason the arrangement of the four axes is identical with that of two usual orthogonal coordinate systems rotated against each other by the angle β with subsequent exchange of the time axes. This results in a shear of the axes instead a rotation. This exchange of two axes and all other differences between space and time are finally a consequence of a simple algebraic sign in a fundamental equation of relativity. It is the equation which connects space and time by defining the metric in spacetime playing the role of a certain kind of distance between events.

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 x₀ to x₃ where x₀=ct. This simplifies all equations considerably, and the speed of light in these units becomes simply a dimensionless number c=1.

See also

• Minkowski space
• Penrose diagram

References

• Rindler, Wolfgang (2001). Relativity: Special, General and Cosmological. Oxford University Press. ISBN 0-19-850836-0.