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the (537), and (139), time (69), relativity (63), with (61), #diagram (58), are (53), minkowski (52), spacetime (51), frame (50), for (49), that (48), light (47), #observer (47), axes (47), axis (42), line (40), from (39), diagrams (38), moving (38), world (37), fig (37), special (34), this (33), speed (30), reference (30), edit (27), frames (26), two (25), object (25), other (24), event (24), events (24), observers (24), space (23), length (22), position (22), lines (22), both (22), can (21), relative (21), loedel (20), between (20), such (19), which (19), origin (19), lorentz (18), theory (18), same (18), textstyle (18), their (18), transformation (17), different (17), all (16), has (16), each (16), than (16), beta (16), relativistic (15), representation (15), velocity (15), inertial (15), they (15), due (14), any (14), angle (14), proper (13), unit (13), non (12), page (12), einstein (12), physics (12), general (12), dilation (12), isbn (12), coordinate (12), also (12), distance 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ribes 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 x 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 of their coordinate systems the axes for the moving observer are not perpendicular 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 t 0 generally stated all events on a line parallel to the x axis happen simultaneously for both observers there is only one universal time t t 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 x and t to x and t and vice versa is described mathematically by the so called galilean transformation minkowski diagrams edit overview edit fig 2 1 in the theory of relativity each observer assigns the event at a to a different time and location fig 2 2 minkowski diagram for various speeds of the primed frame which is moving relative to the unprimed frame the dashed lines represent the light cone of a flash of light at the origin 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 usually where space has been curtailed to a single dimension the units of measurement in these diagrams are taken such that the light cone at an event consists of the lines of slope plus or minus one through that event 3 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 the lorentz transformation relates two inertial frames of reference where an observer stationary at the event 0 0 makes a change of velocity along the x axis as shown in fig 2 1 the new time axis of the observer forms an angle α with the previous time axis with α π 4 in the new frame of reference the simultaneous events lie parallel to a line inclined by α to the previous lines of simultaneity this is the new x 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 or relativistic dot product the original position on your time line ct is perpendicular to position a the original position on your mutual timeline x where t is zero this timeline where timelines come together are positioned then on the same timeline even when there are 2 different positions the 2 positions are on the 45 degree event line on the original position of a hence position a and position a on the event line and t 0 relocate a back to position a whatever the magnitude of α the line ct x forms the universal bisector as shown in fig 2 2 one frequently encounters minkowski diagrams where the time units of measurement are scaled by a factor of c such that one unit of x equals one unit of t such a diagram may have units of approximately 30 centimetres length and nanoseconds astronomical units and intervals of about 8 minutes and 19 seconds 499 seconds light years and years light second and second with that light paths are represented by lines parallel to the bisector between the axes mathematical details edit fig 2 3 different scales on the axes the angle α between the x and x axes will be identical with that between the time axes ct and ct 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 α is given by 4 tan α v c β displaystyle tan alpha frac v c beta the corresponding boost from x and t to x and t and vice versa is described mathematically by the lorentz transformation which can be written c t γ c t β x x γ x β c t displaystyle begin aligned ct gamma ct beta x x gamma x beta ct end aligned where γ 1 β 2 1 2 textstyle gamma left 1 beta 2 right frac 1 2 is the lorentz factor by applying the lorentz transformation the spacetime axes obtained for a boosted frame will always correspond to conjugate diameters of a pair of conjugate hyperbolas as illustrated in fig 2 3 the boosted and unboosted spacetime axes will in general have unequal unit lengths if u is the unit length on the axes of ct and x respectively the unit length on the axes of ct and x is 5 u u 1 β 2 1 β 2 displaystyle u u sqrt frac 1 beta 2 1 beta 2 the ct axis represents the worldline of a clock resting in s with u representing the duration between two events happening on this worldline also called the proper time between these events length u upon the x axis represents the rest length or proper length of a rod resting in s the same interpretation can also be applied to distance u upon the ct and x axes for clocks and rods resting in s history edit light cone and hyperbolas in minkowski 1908 albert einstein announced his theory of special relativity in 1905 6 with hermann minkowski providing his graphical representation in 1908 7 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 7 the first diagram used a branch of the unit hyperbola t 2 x 2 1 textstyle 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 8 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 9 10 self published source 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 11 loedel diagrams edit 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 postulates that any two inertial reference frames 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 edit fig 3 1 view in the median frame fig 3 2 symmetric diagram several authors showed that there is a frame of reference between the resting and moving ones where their symmetry would be apparent median frame 12 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 β v c and γ 1 β 2 1 2 textstyle gamma left 1 beta 2 right frac 1 2 are given between s displaystyle s and s displaystyle s prime then these expressions are connected with the values in their median frame s 0 as follows 12 13 1 β 2 β 0 1 β 0 2 2 β 0 γ 1 β γ displaystyle begin aligned 1 beta frac 2 beta _ 0 1 beta _ 0 2 3pt 2 beta _ 0 frac gamma 1 beta gamma end aligned for instance if β 0 5 between s textstyle s and s textstyle s prime then by 2 they are moving in their median frame s 0 with approximately 0 268 c each in opposite directions on the other hand if β 0 0 5 in s 0 then by 1 the relative velocity between s displaystyle s and s displaystyle s prime in their own rest frames is 0 8 c the construction of the axes of s textstyle s and s textstyle s prime is done in accordance with the ordinary method using tan α β 0 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 β 0 at all instead the relative velocity β v c between s textstyle s and s textstyle s prime can directly be used in the following construction providing the same result 14 if φ is the angle between the axes of ct and ct or between x and x and θ between the axes of x and ct it is given 14 15 16 17 sin φ cos θ β cos φ sin θ 1 γ tan φ cot θ β γ displaystyle begin aligned sin varphi cos theta beta cos varphi sin theta frac 1 gamma tan varphi cot theta beta gamma end aligned two methods of construction are obvious from fig 3 2 the x axis is drawn perpendicular to the ct axis the x and ct axes are added at angle φ and the x axis is drawn at angle θ with respect to the ct axis the x axis is added perpendicular to the ct axis and the ct axis perpendicular to the x 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 textstyle u and u textstyle 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 u 1 β 2 1 β 2 displaystyle u prime u sqrt frac 1 beta 2 1 beta 2 by contrast in a symmetric loedel diagram both the s textstyle s and s textstyle 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 edit max born 1920 drew minkowski diagrams by placing the ct axis almost perpendicular to the x axis as well as the ct axis to the x axis in order to demonstrate length contraction and time dilation in the symmetric case of two rods and two clocks moving in opposite direction 18 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 did not give a graphical representation in a diagram 12 symmetric diagrams were systematically developed by paul gruner 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 15 16 gruner extended this method in subsequent papers 1922 1924 and gave credit to mirimanoff s treatment as well 19 20 21 22 23 24 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 25 26 in 1955 henri amar also published a paper presenting such relations and gave credit to loedel in a subsequent paper in 1957 27 28 some authors of textbooks use symmetric minkowski diagrams denoting as loedel diagrams 14 17 relativistic phenomena in diagrams edit time dilation edit fig 4 1 relativistic time dilation as depicted in two loedel spacetime diagrams both observers consider the clock of the other as running slower fig 4 2 relativistic time dilation as depicted in a single loedel spacetime diagram both observers consider the clock of the other as running slower relativistic time dilation refers to the fact that a clock indicating its proper time 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 edit fig 4 3 relativistic length contraction as depicted in two loedel spacetime diagrams both observers consider objects moving with the other observer as being shorter fig 4 4 relativistic length contraction as depicted in a single loedel spacetime diagram both observers consider objects moving with the other observer as being shorter relativistic length contraction refers to the fact that a ruler indicating its proper length in its rest frame that moves relative to an observer is observed to contract shorten the situation is depicted in symmetric loedel diagr...