Spacetime diagram

The Minkowski diagram, also known as a spacetime diagram, was developed in 1908 by Hermann 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 term Minkowski diagram is used in both a generic and particular sense. In general, a Minkowski diagram is a graphic depiction of a portion of Minkowski space, often where space has been curtailed to a single dimension. These two-dimensional diagrams portray worldlines as curves in a plane that correspond to motion along the spatial axis. The vertical axis is usually temporal, and the units of measurement are taken such that the light cone at an event consists of the lines of slope plus or minus one through that event.[1]

A particular Minkowski diagram illustrates the result of a Lorentz transformation. The origin corresponds to an event where a change of velocity takes place. The new worldline forms an angle α with the vertical, with α < π/4. The Lorentz transformation that moves the vertical to α also moves the horizontal by α. The horizontal corresponds to the usual notion of simultaneous events, for a stationary observer at the origin. After the Lorentz transformation the new simultaneous events lie on the α-inclined line. Whatever the magnitude of α, the line t = x forms the universal[2] bisector.

In Minkowski’s 1908 paper there were three diagrams, first to illustrate the Lorentz transformation, then the partition of the plane by the light-cone, and finally illustration of worldlines. The first diagram used a branch of the unit hyperbola \scriptstyle t^2 - x^2 = 1 to show the locus of a unit of proper time depending on velocity, thus illustrating time dilation. The second diagram showed the conjugate hyperbola to calibrate space, where a similar stretching leaves the impression of Fitzgerald contraction. In 1914 Ludwik Silberstein[3] included a diagram of "Minkowski’s representation of the Lorentz transformation". This diagram included the unit hyperbola, its conjugate, and a pair of conjugate diameters. Since the 1960s a version of this more complete configuration has been referred to as The Minkowski Diagram, and used as a standard illustration of the transformation geometry of special relativity. E. T. Whittaker has pointed out that the Principle of relativity is tantamount to the arbitrariness of what hyperbola radius is selected for time in the Minkowski diagram. In 1912 Gilbert N. Lewis and Edwin B. Wilson applied the methods of synthetic geometry to develop the properties of the non-Euclidean plane that has Minkowski diagrams.

Basics

For simplification in Minkowski diagrams, usually only events in a one dimensional world are considered. Unlike common 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

The black axes labelled x and ct on the adjoining diagram are the coordinate system of an observer which we will refer to as 'at rest', and who is positioned at x=0. 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, such as a moving observer.

This blue line labelled ct' may be interpreted as the time axis for the second observer. Together with the path axis (labeled x, which is identical for both observers) it represents his coordinate system. Both observers agree on the location of the origin of their coordinate systems. The axes for the moving observer are not perpendicular to each other and the scale on his time axis is stretched. To determine the coordinates of a certain event, two lines parallel to the two axes must be constructed passing through the event, and their intersections with the axes read off.

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 stated, all events on a line parallel to the path axis (x 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

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

\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. Whatever space and time axes arise through such transformation, in a Minkowski diagram they correspond to conjugate diameters of a pair of hyperbolas.

While the rest frame has space and time axes at right angles, the moving frame has primed axes which form an acute angle. Since the frames are meant to be equivalent, the asymmetry may be disturbing. When Einstein visited Argentina in 1925, he was asked by Enrique Loedel Palumbo if there might be a frame of reference between the resting and moving ones where their symmetry would be apparent. Such a frame of reference is a Loedel frame.

Time dilation

Relativistic time dilation means that a clock moving relative to an observer is observed to run slower. In fact, time itself in the frame of the moving clock is observed to run slower. This can be read immediately from the adjoining Minkowski diagram. The observer whose reference frame is given by the black axes is assumed to move from the origin O towards A. The moving clock has the reference frame given by the blue axes and moves from O to B. For the black observer all events happening simultaneously with the event at A are located on a straight line parallel to its space axis. This line passes through A and B, so A and B are simultaneous from the reference frame of the observer with black axes. However, the clock that is moving relative to the black observer marks off time along the blue time axis. This is represented by the distance from O to B. Therefore, the observer at A with the black axes notices his or her clock as reading the distance from O to A while he or she observes the clock moving relative him or her to read the distance from O to B. Due to the distance from O to B being smaller than the distance from O to A, he or she concludes that the time passed on the clock moving relative to him or her is smaller than that passed on his own clock.

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 synchronously at different locations. Due to the principle of relativity the question of "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'-axis and the parallel line passing through A and B. 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

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 through C and D 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 paradoxical 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 actual times at which these events happen from their point of view.

Constancy of the speed of light

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 vacuum 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 paradoxical, 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 undetectable 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.


Further coordinate systems corresponding to observers with arbitrary velocities can be added to 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, 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 time distance greater than zero for all observers. 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

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 was 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 spacetime, 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, but depends on the structure of spacetime.

Eponym

When Taylor and Wheeler composed Spacetime Physics (1966), they did not use the term "Minkowski diagram" for their spacetime geometry. Instead they included an acknowledgement of Minkowski’s contribution to philosophy by the totality of his innovation of 1908.[4]

As an eponym, the term Minkowski diagram is subject to Stigler’s law of eponymy, namely that Minkowski is wrongly designated as originator. The earlier works of Alexander Macfarlane contain algebra and diagrams that correspond well with the Minkowski diagram. See for instance the plate of figures in Proceedings of the Royal Society in Edinburgh for 1900. Macfarlane was building on what one sees in William Kingdon Clifford’s Elements of Dynamic (1878), page 90.

When abstracted to a line drawing, then any figure showing conjugate hyperbolas, with a selection of conjugate diameters, falls into this category. Students making drawings to accompany the exercises in George Salmon’s A Treatise on Conic Sections (1900) at pages 165–71 (on conjugate diameters) will be making Minkowski diagrams.

See also

References

  • Hermann Minkowski (1908) "Raum und Zeit", (German ).
    • Various English translations on : Space and Time.
  • Anthony French (1968) Special Relativity, pages 82 & 83, New York: W W Norton & Company.
  • E.N. Glass (1975) "Lorentz boosts and Minkowski diagrams" American Journal of Physics 43:1013,4.
  • N. David Mermin (1968) Space and Time in Special Relativity, Chapter 17 Minkowski diagrams: The Geometry of Spacetime, pages 155–99 McGraw-Hill.
  • W.G.V. Rosser (1964) An Introduction to the Theory of Relativity, page 256, Figure 6.4, London: Butterworths.
  • Edwin F. Taylor and John Archibald Wheeler (1963) Spacetime Physics, pages 27 to 38, New York: W. H. Freeman and Company, Second edition (1992).
  • (see page 10 of e-link)

External links

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