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A spacetime diagram is a graphical illustration of the properties of space and time in the
special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
. Spacetime diagrams allow a qualitative understanding of the corresponding phenomena like
time dilation In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
and
length contraction Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGera ...
without mathematical equations. The history of an object's location throughout all time traces out a line, referred to as the object's
world line The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...
, in a spacetime diagram. Points in spacetime diagrams represent a fixed position in space and time and are referred to as events. The most well-known class of spacetime diagrams are known as Minkowski diagrams, developed by
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
in 1908. Minkowski diagrams are two-dimensional graphs that depict events as happening in a
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
consisting of one space dimension and one time dimension. Unlike a regular distance-time graph, the distance is displayed on the horizontal axis and time on the vertical axis. Additionally, the time and space
units of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multi ...
are chosen in such a way that an object moving at the speed of light is depicted as following a 45° angle to the diagram's axes.


Introduction to kinetic diagrams


Position versus time graphs

In the study of 1-dimensional kinematics, position vs. time graphs (also called distance vs. time graphs, or p-t graphs) provide a useful means to describe motion. The specific features of the motion of objects are demonstrated by the shape and the slope of the lines. In Fig 1-1, the plotted object moves away from the origin at a uniform speed of 1.66 m/s for 6 seconds, halts for 5 seconds, then returns to the origin over a period of 7 seconds at a non-constant speed. At its most basic level, a spacetime diagram is merely a time vs position graph, with the directions of the axes in a usual p-t graph exchanged, that is, the vertical axis refers to temporal and the horizontal axis to spatial coordinate values. Especially when used in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
(SR), the temporal axes of a spacetime diagram are scaled with the speed of light , and thus are often labeled by This changes the dimension of the addressed physical quantity from <''Time''> to <''Length''>, in accordance with the dimension associated with the spatial axes, which are frequently labeled


Standard configuration of reference frames

To ease insight into how spacetime coordinates, measured by observers in different reference frames, compare with each other, it is useful to work with a simplified setup. With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. Setting the temporal component aside for the moment, two
Galilean reference frame In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
s (i.e. conventional 3-space frames), S and S′ (pronounced "S prime"), each with observers O and O′ at rest in their respective frames, but measuring the other as moving with speeds ±''v'' are said to be in ''standard configuration'', when: * The ''x'', ''y'', ''z'' axes of frame S are oriented parallel to the respective primed axes of frame S′. * Frame S′ moves in the ''x''-direction of frame S with a constant velocity ''v'' as measured in frame S. * The origins of frames S and S′ coincide for time ''t'' = 0 in frame S and ''t''′ = 0 in frame S′. This spatial setting is displayed in the Fig 1-2, in which the temporal coordinates are separately annotated as quantities ''t'' and ''t. In a further step of simplification it is often possible to consider just the direction of the observed motion and ignore the other two spatial components, allowing ''x'' and ''ct'' to be plotted in 2-dimensional spacetime diagrams, as introduced above.


Non-relativistic "spacetime diagrams"

The black axes labelled and on Fig 1-3 are the coordinate system of an observer, referred to as ''at rest'', and who is positioned at . This observer's 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 describes an object moving with constant speed to the right, such as a moving observer. This blue line labelled may be interpreted as the time axis for the second observer. Together with the axis, which is identical for both observers, it represents their coordinate system. Since the reference frames are in standard configuration, both observers agree on the location of the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * Origin (comics), ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * The Origin (Buffy comic), ''The Origin'' (Bu ...
of their coordinate systems. The axes for the moving observer are not
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to each other and the scale on their time axis is stretched. To determine the coordinates of a certain event, two lines, each parallel to one of 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 . Generally stated, all events on a line parallel to the axis happen simultaneously for both observers. There is only one universal time , modelling the existence of one common position axis. On the other hand, due to two different time axes the observers usually measure different coordinates for the same event. This graphical translation from and to and and vice versa is described mathematically by the so-called
Galilean transformation In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotat ...
.


Minkowski diagrams


Overview

The term Minkowski diagram refers to a specific form of spacetime diagram frequently used in special relativity. A Minkowski diagram is a two-dimensional graphical depiction of a portion of
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
, usually where space has been curtailed to a single dimension. The units of measurement in these diagrams are taken such that the
light cone In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
at an event consists of the lines of
slope In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
plus or minus one through that event. The horizontal lines correspond to the usual notion of ''simultaneous events'' for a stationary observer at the origin. A particular Minkowski diagram illustrates the result of a
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
. The Lorentz transformation relates two
inertial frames of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleratio ...
, where an
observer An observer is one who engages in observation or in watching an experiment. Observer may also refer to: Computer science and information theory * In information theory, any system which receives information from an object * State observer in co ...
stationary at the event makes a change of
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
along the -axis. As shown in Fig 2-1, the new time axis of the observer forms an angle with the previous time axis, with . In the new frame of reference the simultaneous events lie parallel to a line inclined by to the previous lines of simultaneity. This is the new -axis. Both the original set of axes and the primed set of axes have the property that they are orthogonal with respect to the
Minkowski inner product In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
or ''relativistic dot product''. Whatever the magnitude of , the line forms the universal bisector, as shown in Fig 2-2. One frequently encounters Minkowski diagrams where the time
units of measurement A unit of measurement is a definite magnitude of a quantity, defined and adopted by convention or by law, that is used as a standard for measurement of the same kind of quantity. Any other quantity of that kind can be expressed as a multi ...
are scaled by a factor of such that one unit of equals one unit of . Such a diagram may have units of * Approximately 30 centimetres length and
nanosecond A nanosecond (ns) is a unit of time in the International System of Units (SI) equal to one billionth of a second, that is, of a second, or 10 seconds. The term combines the SI prefix ''nano-'' indicating a 1 billionth submultiple of an SI unit ( ...
s *
Astronomical unit The astronomical unit (symbol: au, or or AU) is a unit of length, roughly the distance from Earth to the Sun and approximately equal to or 8.3 light-minutes. The actual distance from Earth to the Sun varies by about 3% as Earth orbits ...
s and intervals of about 8 minutes and 19 seconds (499 seconds) *
Light year A light-year, alternatively spelled light year, is a large unit of length used to express astronomical distances and is equivalent to about 9.46 trillion kilometers (), or 5.88 trillion miles ().One trillion here is taken to be 1012 ...
s and
year A year or annus is the orbital period of a planetary body, for example, the Earth, moving in its orbit around the Sun. Due to the Earth's axial tilt, the course of a year sees the passing of the seasons, marked by change in weather, the h ...
s *
Light-second The light-second is a unit of length useful in astronomy, telecommunications and relativistic physics. It is defined as the distance that light travels in free space in one second, and is equal to exactly . Just as the second forms the basis for ...
and second With that, light paths are represented by lines parallel to the bisector between the axes.


Mathematical details

The angle between the and axes will be identical with that between the time axes and . This follows from the second postulate of special relativity, which says that the speed of light is the same for all observers, regardless of their relative motion (see below). The angle is given by :\tan\alpha = \frac = \beta. The corresponding boost from and to and and vice versa is described mathematically by the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant velo ...
, which can be written : \begin ct' &= \gamma (ct - \beta x),\\ x' &= \gamma (x - vt) \\ \end where \gamma = \left(1 - \beta^2\right)^ is the
Lorentz factor The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
. By applying the Lorentz transformation, the spacetime axes obtained for a boosted frame will always correspond to
conjugate diameters In geometry, two diameters of a conic section are said to be conjugate if each chord (geometry), chord parallel (geometry), parallel to one diameter is bisection, bisected by the other diameter. For example, two diameters of a circle are conjugate ...
of a pair of
hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, ca ...
s. As illustrated in Fig 2-3, the boosted and unboosted spacetime axes will in general have unequal unit lengths. If is the unit length on the axes of and respectively, the unit length on the axes of and is: : U' = U\sqrt\frac\,. The -axis represents the worldline of a clock resting in , with representing the duration between two events happening on this worldline, also called the
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
between these events. Length upon the -axis represents the rest length or
proper length Proper length or rest length is the length of an object in the object's rest frame. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on ...
of a rod resting in . The same interpretation can also be applied to distance upon the - and -axes for clocks and rods resting in .


History

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
discovered special relativity in 1905, with
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
providing his graphical representation in 1908. : Various English translations on Wikisource: Space and Time 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 In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative ra ...
t^2 - x^2 = 1 to show the locus of a unit of
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
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 Ludwik Silberstein (1872 – 1948) was a Polish-American physicist who helped make special relativity and general relativity staples of university coursework. His textbook '' The Theory of Relativity'' was published by Macmillan in 1914 with a ...
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 In geometry, two diameters of a conic section are said to be conjugate if each chord (geometry), chord parallel (geometry), parallel to one diameter is bisection, bisected by the other diameter. For example, two diameters of a circle are conjugate ...
. 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 Sir Edmund Taylor Whittaker (24 October 1873 – 24 March 1956) was a British mathematician, physicist, and historian of science. Whittaker was a leading mathematical scholar of the early 20th-century who contributed widely to applied mathema ...
has pointed out that the
principle of relativity In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. For example, in the framework of special relativity the Maxwell equations ha ...
is tantamount to the arbitrariness of what hyperbola radius is selected for
time Time is the continued sequence of existence and event (philosophy), events that occurs in an apparently irreversible process, irreversible succession from the past, through the present, into the future. It is a component quantity of various me ...
in the Minkowski diagram. In 1912
Gilbert N. Lewis Gilbert Newton Lewis (October 23 or October 25, 1875 – March 23, 1946) was an American physical chemist and a Dean of the College of Chemistry at University of California, Berkeley. Lewis was best known for his discovery of the covalent bond a ...
and Edwin B. Wilson applied the methods of
synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass ...
to develop the properties of the non-Euclidean plane that has Minkowski diagrams. 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.


Loedel diagrams

While a frame at rest in a Minkowski diagram has orthogonal spacetime axes, a frame moving relative to the rest frame in a Minkowski diagram has spacetime axes which form an acute angle. This asymmetry of Minkowski diagrams can be misleading, since
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
postulates that any two
inertial reference frames In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
must be physically equivalent. The Loedel diagram is an alternative spacetime diagram that makes the symmetry of inertial references frames much more manifest.


Formulation via median frame

Several authors showed that there is a frame of reference between the resting and moving ones where their symmetry would be apparent ("median frame"). In this frame, the two other frames are moving in opposite directions with equal speed. Using such coordinates makes the units of length and time the same for both axes. If and \gamma = \left(1 - \beta^2\right)^ are given between S and S^\prime, then these expressions are connected with the values in their median frame ''S''0 as follows: (Translation: The Lorentz–Einstein transformation and the universal time of Ed. Guillaume) : \begin &(1) & \beta &= \frac,\\ pt &(2) & \beta_ &= \frac. \end For instance, if between S and S^\prime, then by (2) they are moving in their median frame ''S''0 with approximately each in opposite directions. On the other hand, if in ''S''0, then by (1) the relative velocity between S and S^\prime in their own rest frames is . The construction of the axes of S and S^\prime is done in accordance with the ordinary method using with respect to the orthogonal axes of the median frame (Fig. 3-1). However, it turns out that when drawing such a symmetric diagram, it is possible to derive the diagram's relations even without mentioning the median frame and at all. Instead, the relative velocity between S and S^\prime can directly be used in the following construction, providing the same result: If is the angle between the axes of and (or between and ), and between the axes of and , it is given: : \begin \sin\varphi = \cos\theta &= \beta,\\ \cos\varphi = \sin\theta &= \frac,\\ \tan\varphi = \cot\theta &= \beta\gamma. \end Two methods of construction are obvious from Fig. 3-2: the -axis is drawn perpendicular to the -axis, the and -axes are added at angle ; and the ''x''′-axis is drawn at angle with respect to the -axis, the -axis is added perpendicular to the -axis and the -axis perpendicular to the -axis. In a Minkowski diagram, lengths on the page cannot be directly compared to each other, due to warping factor between the axes' unit lengths in a Minkowski diagram. In particular, if U and U^\prime are the unit lengths of the rest frame axes and moving frame axes, respectively, in a Minkowski diagram, then the two unit lengths are warped relative to each other via the formula: : U^\prime = U\sqrt\frac By contrast, in a symmetric Loedel diagram, both the S and S^\prime frame axes are warped by the same factor relative to the median frame and hence have identical unit lengths. This implies that, for a Loedel spacetime diagram, we can directly compare spacetime lengths between different frames as they appear on the page; no unit length scaling/conversion between frames is necessary due to the symmetric nature of the Loedel diagram.


History

*
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German physicist and mathematician who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics and supervised the work of a ...
(1920) drew Minkowski diagrams by placing the -axis almost perpendicular to the -axis, as well as the -axis to the -axis, in order to demonstrate length contraction and time dilation in the symmetric case of two rods and two clocks moving in opposite direction. * Dmitry Mirimanoff (1921) showed that there is always a median frame with respect to two relatively moving frames, and derived the relations between them from the Lorentz transformation. However, he didn't give a graphical representation in a diagram. * Symmetric diagrams were systematically developed by
Paul Gruner Paul may refer to: * Paul (given name), a given name (includes a list of people with that name) *Paul (surname), a list of people People Christianity *Paul the Apostle (AD c.5–c.64/65), also known as Saul of Tarsus or Saint Paul, early Chri ...
in collaboration with Josef Sauter in two papers in 1921. Relativistic effects such as length contraction and time dilation and some relations to covariant and contravariant vectors were demonstrated by them. (Translation: Elementary geometric representation of the formulas of the special theory of relativity) (translation: An elementary geometrical representation of the transformation formulas of the special theory of relativity) Gruner extended this method in subsequent papers (1922–1924), and gave credit to Mirimanoff's treatment as well. (translation: Graphical representation of the four-dimensional space-time universe) * The construction of symmetric Minkowski diagrams was later independently rediscovered by several authors. For instance, starting in 1948, Enrique Loedel Palumbo published a series of papers in Spanish language, presenting the details of such an approach. In 1955, Henri Amar also published a paper presenting such relations, and gave credit to Loedel in a subsequent paper in 1957. Some authors of
textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions. Schoolbooks are textbook ...
s use symmetric Minkowski diagrams, denoting as ''Loedel diagrams''.


Relativistic phenomena in diagrams


Time dilation

Relativistic time dilation refers to the fact that a clock (indicating its
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval ...
in its rest frame) that moves relative to an observer is observed to run slower. The situation is depicted in the symmetric Loedel diagrams of Fig 4-1. Note that we can compare spacetime lengths on page directly with each other, due to the symmetric nature of the Loedel diagram. In Fig 4-2, 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 their clock as reading the distance from O to A while they observe 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, they conclude that the time passed on the clock moving relative to them is smaller than that passed on their own clock. A second observer, having moved together with the clock from O to B, will argue that the black axis clock has only reached C and therefore 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 refers to the fact that a ruler (indicating its
proper length Proper length or rest length is the length of an object in the object's rest frame. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on ...
in its rest frame) that moves relative to an observer is observed to contract/shorten. The situation is depicted in symmetric Loedel diagrams in Fig 4-3. Note that we can compare spacetime lengths on page directly with each other, due to the symmetric nature of the Loedel diagram. In Fig 4-4, the observer is assumed again to move along the -axis. The world lines of the endpoints of an object moving relative to him are assumed to move along the -axis and the parallel line passing through A and B. For this observer the endpoints of the object at 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 proper length OB at . Due to . 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 -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 themself 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 The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
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 and holds. That means any position on such a world line corresponds with steps on - and -axes 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 - and -axes. As shown in Fig 4-5, the Minkowski diagram illustrates them as being angle bisectors of the - and -axes as well. That means both observers measure the same speed 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 x 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-and-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 in Fig 4-6. 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 any 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
cone A cone is a three-dimensional geometric shape that tapers smoothly from a flat base (frequently, though not necessarily, circular) to a point called the apex or vertex. A cone is formed by a set of line segments, half-lines, or lines con ...
s with apexes touching each other at the origin. They are called
light cones In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
.


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. Given an object moving faster than light, say from O to A in Fig 4-7, then for any observer watching the object moving from O to A, another observer can be found (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 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 system, who sends it back, again 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 were possible to accelerate an observer to the speed of light, their space and time axes would coincide with their angle bisector. The coordinate system would collapse, in concordance with the fact that due to
time dilation In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
, time would effectively stop passing for them. 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, therefore, has nothing in particular to do with electromagnetic waves or light, but comes as a consequence of the structure of spacetime.


Accelerating observers

It is often, incorrectly, asserted that special relativity cannot handle accelerating particles or accelerating reference frames. In reality, accelerating particles present no difficulty at all in special relativity. On the other hand, accelerating ''frames'' do require some special treatment, However, as long as one is dealing with flat, Minkowskian spacetime, special relativity can handle the situation. It is only in the presence of gravitation that general relativity is required. An accelerating particle's 4-vector acceleration is the derivative with respect to proper time of its 4-velocity. This is not a difficult situation to handle. Accelerating frames require that one understand the concept of a ''momentarily comoving reference frame'' (MCRF), which is to say, a frame traveling at the same instantaneous velocity of a particle at any given instant. Consider the animation in Fig 5-1. The curved line represents the world line of a particle that undergoes continuous acceleration, including complete changes of direction in the positive and negative x-directions. The red axes are the axes of the MCRF for each point along the particle's trajectory. The coordinates of events in the unprimed (stationary) frame can be related to their coordinates in any momentarily co-moving primed frame using the Lorentz transformations. Fig 5-2 illustrates the changing views of spacetime along the world line of a rapidly accelerating particle. The ct' axis (not drawn) is vertical, while the x' axis (not drawn) is horizontal. The dashed line is the spacetime trajectory ("world line") of the particle. The balls are placed at regular intervals of proper time along the world line. The solid diagonal lines are the light cones for the observer's current event, and they intersect at that event. The small dots are other arbitrary events in the spacetime. The slope of the world line (deviation from being vertical) is the velocity of the particle on that section of the world line. Bends in the world line represent particle acceleration. As the particle accelerates, its view of spacetime changes. These changes in view are governed by the Lorentz transformations. Also note that: * the balls on the world line before/after future/past accelerations are more spaced out due to time dilation. * events which were simultaneous before an acceleration (horizontally spaced events) are at different times afterwards due to the relativity of simultaneity, * events pass through the light cone lines due to the progression of proper time, but not due to the change of views caused by the accelerations, and * the world line always remains within the future and past light cones of the current event. If one imagines each event to be the flashing of a light, then the events that are within the past light cone of the observer are the events visible to the observer. The slope of the world line (deviation from being vertical) gives the velocity relative to the observer.


Case of non-inertial reference frames

The photon world lines are determined using the metric with d \tau = 0. The
light cones In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...
are deformed according to the position. In an inertial reference frame a free particle has a straight world line. In a
non-inertial reference frame A non-inertial reference frame is a frame of reference that undergoes acceleration with respect to an inertial frame. An accelerometer at rest in a non-inertial frame will, in general, detect a non-zero acceleration. While the laws of motion are ...
the world line of a free particle is curved. Let's take the example of the fall of an object dropped without initial velocity from a rocket. The rocket has a uniformly accelerated motion with respect to an inertial reference frame. As can be seen from Fig 6-2 of a Minkowski diagram in a non-inertial reference frame, the object once dropped, gains speed, reaches a maximum, and then sees its speed decrease and asymptotically cancel on the
horizon The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether i ...
where its proper time freezes at t_\text. The velocity is measured by an observer at rest in the accelerated rocket.


See also

*
Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the iner ...
*
Penrose diagram In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity. It is an ext ...
*
Rapidity In relativity, rapidity is commonly used as a measure for relativistic velocity. Mathematically, rapidity can be defined as the hyperbolic angle that differentiates two frames of reference in relative motion, each frame being associated with d ...


References

*
Anthony French Anthony Philip French (November 19, 1920 – February 3, 2017) was a British professor of physics at the Massachusetts Institute of Technology. He was born in Brighton, England. French was a graduate of Cambridge University, receiving his ...
(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 The ''American Journal of Physics'' is a monthly, peer-reviewed scientific journal published by the American Association of Physics Teachers and the American Institute of Physics. The editor-in-chief is Beth Parks of Colgate University."Current F ...
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 McGraw Hill is an American educational publishing company and one of the "big three" educational publishers that publishes educational content, software, and services for pre-K through postgraduate education. The company also publishes refere ...
. * * W.G.V. Rosser (1964) ''An Introduction to the Theory of Relativity'', page 256, Figure 6.4, London:
Butterworths LexisNexis is a part of the RELX corporation that sells data analytics products and various databases that are accessed through online portals, including portals for computer-assisted legal research (CALR), newspaper search, and consumer inform ...
. *
Edwin F. Taylor Edwin F. Taylor is an American physicist known for his contributions to the teaching of physics. Taylor was editor of the American Journal of Physics, and is author of several introductory books to physics. In 1998 he was awarded the Oersted Meda ...
and
John Archibald Wheeler John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in ...
(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

{{Relativity Special relativity Geometry Diagrams