How to Draw a Spacetime Diagram

This interactive Minkowski diagram is based on the conventional setting of c = 1. Units along the axis may be interpreted as: t unit = second, then d unit = lightsecond, or alternatively, d unit = grand, t unit = iii.34E-9 south, etc. Every bit usual, the three spacial dimensions are represented by the single d axis. Alternatively, you may use c as a conversion factor and read the vertical axis in length units as ct in stead of c. Then both axes may exist read in meters.

The diagram is representing a model of 2 spacetime events, event A and result B. Ii observers in 2 inertial reference frames laissez passer each other in the origin. At that point is issue A - the green brawl. The user inputs the time and distance for result B - the red ball - and the relative velocity. The diagram will show fourth dimension dilation, the relativity of simultaneity and other furnishings of special relativity.

The diagram will testify the events equally measured by the two observers as designed by Einstein: Each observer has a set of synchronized clocks and previously established distance points. The diagram may also testify the observed times as calculated past taking into account the travel time of low-cal from the events to the observers.

One observer follows the worldline of t, the other follows a worldline t' which will appear when the button "Calculate" is pressed. Start input the relative velocity as a fraction of c, fourth dimension and distance for event B every bit measured by the observer post-obit worldline t.

Calculation of coordinates in the t' system are done by the inverse Lorentz transformations. The input values are as measured in the t coordinate system that is not moving. With c = 1 the simplified inverse Lorentz transformation is and .

For some situations where the goal is to find the coordinates in the t system, the simplified original Lorentz transformation is used. A reminder of those equations:

and .

The initial setting is to utilise the inverse Lorentz transformations. The choice of input and manner of calculation may be toggled with the push " Toggle input mode ".

The two events may also be shown in "real fourth dimension" by pressing the buttons " Play worldline t " or " Play worldline t' ". The two events volition then appear in space and time (given time unit of measurement = 1 second).

Beneath the animation is a more detailed tutorial and some examples and excercises.

Relative velocity: c	Event A :	d = 0	t = 0
	Event B :	d =	t =
		d'=	t'=
		Invariant interval:
		i2 = (ct)ii - dtwo = i'2 = (ct')two - d'2 =

Tutorial

The blitheness may be opened in a separate window by clicking this link.

Basic input

The user tin can input 3 variables, relative velocity and the distance value and fourth dimension value for event B. Values are entered in the fields as shown in a higher place.

The relative velocity is entered as a fraction of c, the speed of lite (299 792 458 m/s). Velocities >= 1 or <= -ane are not accustomed.

Effect A is always at t = t' = 0 and d = d' = 0.

When the push button Calculate is pressed the animation volition calculate the time and distance values for event B as measured past the observer in the worldline t'. These values are shown in the gray output fields.

The blitheness volition as well calculate the invariant spacetime interval (the Lorentz interval) squared.

The animation calculates and displays the slope and scale of the worldline t' depending on the relative velocity. The distance line d' associated with the worldline t' is as well displayed.

The button Toggle input style toggles the calculation mode of the animation and the respective input variables. In the initial setup the animation is using d and t as input and calculates with the inverse Lorentz transformations. Using the toggle push you may instead use d' and t' as input and summate with the original Lorentz transformations.

The button Articulate resets the display area, the user input and the calculated variables.

The graph

Event A is shown with a green ball. Event B is shown with a crimson ball. To observe the distance of outcome B in the t reference frame, follow the greyness line from the red ball and parallel to the t (ct) centrality and read off the d centrality. To observe the time of effect B, follow the greyness line from the scarlet brawl and parallel to the d axis and read off the t (ct) axis.

When the t' (ct') axis and the d' centrality are shown, the spacetime coordinates for event B are found in a similar mode. To detect the distance of event B in the t' reference frame, follow the blueish line parallel to the t' (ct') axis and read off the d' axis. To find the fourth dimension of event B in the t' reference frame, follow the bluish line parallel to the d' axis and read off the t' (ct') axis. Notice that the t' (ct) and d' axes are scaled co-ordinate to the changed Lorentz transformation in society to bear witness the correct fourth dimension and altitude values.

Real-time playback

The two Play buttons will activate playback on the distance-lines. The time volition progress equally shown past the clock, and the two events volition announced as predicted by the respective measurements co-ordinate to the ii worldlines. This display is most effective with sound activated as the two events are accompanied past a couple of chords. Event A appears at d = 0 and t = 0 and at d' = 0 and t' = 0.

Observation times

The Minkowski diagram shows the events as measured according to the reference frames. This should not be dislocated with observations of the events. Unless consequence B lies directly on the worldline of one of the observers, ascertainment time volition be unlike from measured time. To show the ascertainment times it is necessary to calculate the travelling fourth dimension for the light from the effect to the observers. The blitheness can show the light path and the observation times.

When the Show/Hide push button is toggled on, the graph shows the light path from event B to the ii observers at t' and t every bit a yellow line. Since light moves at v = c = ±1 in this diagram, these lines are e'er at ±45^o to the t and d axes. The observation times are marked on the two worldlines and shown in a connected text field. To the right is an example.

The ascertainment info may exist removed from the graphic past toggling the Show/hide push button.

Examples and exercises

i) Gradient and stretch of worldline for the moving observer.

Input t = d = 0, ie., both events are at the origin at time 0. Input relative velocity 0.3 c. Notice the slope of worldline t' and the approximate altitude betwixt the fourth dimension and distance markers. Alter the velocity to 0.v c and recalculate. Notice that the angle between the t and the t' axes has increased and that the distance betwixt the markers has increased.

Excercise 1a: Try other velocities like 0.7 c and 0.8 c.

Excercise 1b: At present study the d' axis when different velocities are chosen. What is the relationship betwixt the angle between t and t' and the angle between d and d'?

2) Alter point of view

If observer t' moves at velocity 5 in relation to observer t, one may modify the point of view and say that observer t moves at velocity -v in relation to observer t'. The two velocities are equal in magnitude and opposite in direction. If nosotros alter the indicate of view of the motion, the locations of events A and B will be exchanged as well. What was d and t for effect B is at present d and t for issue A and vice versa.

An example: Input v = 0.half dozen c, d = 5, t = 6. Notice that d' = 1.75 and t' = 3.75. At present switch point of view by inputting 5 = -0.6 c, d = 1.75 and t = 3.75. Notice that now d' = 5, t' = 6. In other words, the alter in point of view keeps the same coordinates for the 2 events.

Excersise 2a: Input v = 0.25 c, d = 6, t = iv. Switch bespeak of view. What are the new input values?

Excercise 2b: Input v = 0.eight c, d = -1, t = 2.5. Switch bespeak of view. What are the new input values?

Excercise 2c: Input five = -0.6 c, d = -v.v, t = 5. Switch indicate of view. What are the new input values?

iii) Time dilation and the slowing of clocks.

Toggle the input manner so that the d' and t' fields are active. Input v = 0.half-dozen c, d' = 0 and t' = four. This indicates that both events are on the worldline of the moving observer. Suppose both events represents a tick of a clock that ticks every second. From the moving observer's indicate of view, the clock shows that 4 seconds has passed. Detect that the nonmoving observer measures that five seconds has passed. The nonmoving observer will conclude that the moving clock is tiresome.

Now turn the situation around. Toggle the input way so that the d and t fields are active. Input v = 0.6 c, d = 0 and t = -4 (negative four). The clock is now on the worldline of the observer that is at rest. Since the distance is the same, 0, for the observer we are following in the corresponding situations, it must hither have taken the moving observer some time to reach this point in the reference frame of the observer at rest. And so the time is input as negative this fourth dimension. The nonmoving clock shows that iv seconds has passed. Simply according to the moving observer, 5 seconds has passed. The moving observer volition correctly conclude that the nonmoving clock is boring. Conclusion: Both observers volition conclude that the other clock is slow (by a cistron of 4/v = 0.8). Annotation that in the first situation, the observer t measures that the distance in fact is iii, and the moving observer measures the distance as 0. In the 2nd situation, the observer t measures the distance equally 0, and the moving observer the distance as three.

Excercise 3a: Repeat this experiment with five = 0.eight c. What are the d and t values for the nonmoving observer be when the moving observer's clock shows t' = 1.v? What is the charge per unit of time dilation each observer will calculate for the other clock?

Excercise 3b: Echo the experiment with v = 0.5 c. What are the d and t values for the nonmoving observer when the moving clock shows t' = three? What is the rate of fourth dimension dilation?

Excercise 3c: After observing the slowing of clocks for 3 unlike velocities, 0.5 c, 0.6 c and 0.viii c, what will you conclude regarding the relationship between the relative velocity and the time dilation?

iv) Simultaneity

Let v = 0 and input d = 4, t = 0. The two events A and B are simultaneous in both reference frames. Now change the velocity to v = 0.iv c. Notice that t' changes to -i.7457. This ways that in the reference frame of the moving observer event B is measured to happen about 2 seconds earlier event A, they are non simultaneous. Just in the nonmoving reference frame they are however happening at the same time!

Change the relative velocity to v = -0.4. Now event B is measured to happen almost 2 seconds subsequently upshot A according to the moving observer. To conclude: Events that are simultaneous to 1 observer are generally not simultaneous to another observer moving relative to the outset one.

Repeat this experiment as above, with the two different five-values. Each time, click the buttons Play worldline t and Play worldline t' respectively and follow the animation of the events as measured past the two observers. Notice when and where the two events announced in both cases.

Another example. Input v = 0.half dozen c, d = iii.75 and t = two.25. Notice that t' = 0. Result B is now simultaneous to event A in the reference frame of the moving observer, but not in the nonmoving reference frame. Hither event B happens two.25 seconds later on event A. Notice that event B is placed on the centrality d'. All points on this centrality are simultaneous to the origin in the moving reference frame. Likewise: All points on any line that is parallell to d' are equidistant in time from the origin in this reference frame.

Utilize the "Play worldline"-buttons likewise for this experiment and for the two exercises that follow!

Excercise 4a: Two observers are moving at 0.8 c relative to each other. For ane observer two events A and B are simultaneous and separated by iii units of space. How will the two events exist measured by the other observer?

Excercise 4b: At a relative velocity of 0.8 c, what will the nonmoving observer mensurate if the moving observer measures result B to be simultaneous with effect A and at the distance of 1.v? (Hint: Use the input mode for the original Lorentz transformations.)

5) Temporal guild

Nether some weather condition the temporal order of events may be any of these: Before, simultaneous or afterwards. An example: Let outcome B be located at d = 4, t = ii. Showtime, attempt five = 0.2 c. Both observers now concur that event A happens earlier event B. Modify the velocity to v = 0.5 c. Now the ii events are simultaneous co-ordinate to the moving observer. Finally set up five = 0.viii c. This time the moving observer will measure event B to happen earlier consequence A. All three possibilities of temporal order are nowadays, only depending on the velocity of the observer. Use the Play worldline-buttons for all three scenarios to confirm the numerical and graphical information.

Excercise 5: Set v = 0.6, d = 5 and t = i. Then find a velocity where the two events are simultaneous for the moving observer and a velocity where effect A happens before event B for the moving observer.

Examples 4 & 5 demonstrate conspicuously that there is no such thing equally accented simultaneity.

6) The invariant interval

In special relativity there is a quantity that is invariant (not irresolute) for all inertial reference frames. In other words, given some event B ane can summate an interval which is the aforementioned regardless of the relative velocity of the two observers. The equation for this interval (besides chosen the Lorentz interval or the invariant spacetime interval) is given by i² = (ct)^two - d² . The animation calculates the square of the invariant interval .

Case: Fix v = 0.3 c, d = 3, t = 4. Notice that iⁱⁱ = i'² = 7. Try changing the velocity, e.yard set 5 = 0.5 c. Try any other value for 5. Try a negative value for v.

Excercise 6: Detect the invariant interval for d = v.v, t = two.v. Find the invariant interval for d = 2, t = 6.

[In three spacial dimensions the Lorentz interval is given by the equation i² = (ct)² - (x² + y^two + z²).
(In some texts the sign is switched between the fourth dimension and space dimensions).]

vii) Timelike and spacelike intervals

In the animation the spacetime diagram is divided by the two lightlines into 4 equal sectors. 2 of these are labelled "Timelike intervals", two "Spacelike intervals". Choose whatsoever value for v and plug in these value-pairs for d and t: d = 5 t = iii, d = 6 t = -1, d = -5 t = four, d = -half dozen t = -3. Notice that all instances of result B are in the sectors with spacelike intervals. Notice also that for all these instances the spacetime interval squared is negative. In all such cases the spacial relation between events A and B cannot exist reversed. With spacelike intervals the spacial separation dominates, d^two > (ct)ⁱⁱ , and at that place can be no causual relationship between the two events. Another way to view this: The interval is such that in that location is not enough fourth dimension for low-cal to laissez passer from one event to the other.

At present choose whatsoever value for c and plug in these value-pairs for d and t: d = 1 t = 2, d = 3 t = five, d = -1 t = half-dozen, d = four t = -v, d = -3 t = -6. Detect that all instances of event B are in the sectors with timelike intervals. Notice also that for all these instances the spacetime interval squared is positive. In all such cases the sequence in time between events A and B cannot be reversed. With timelike intervals the separation in time dominates, (ct)² > dⁱⁱ , and there is no stock-still directional human relationship between the ii events: left and right tin can be exchanged depending on the motion of the ovserver.

There is a third type of interval to consider. Let (ct)^two = d² which is to say five = d/t = c. In other words, the velocity is c, the speed of light. This blazon of interval is therefore called lightlike and it is represented in the blitheness past the two yellow worldlines representing light passing through consequence A.

Excercise 7: Given that upshot A as usual is in the origin, determine which of the intervals are spacelike, timelike and lightlike for the post-obit events B. Use both the numerical and the graphical method to determine. Likewise use the "Play worldline"-function and various velocities to confirm your choice.
a - d = 3, t = 5. b - d = -three, t = -six. c - d = 5, t = 1.five. d - d = -three.5, t = 3. eastward - d = ii.0, t = 2.1. f - d = 4.v, t = 4.5.

8) Observations

The two observers will encounter event A simultaneously, at t = d = 0. When will they run into (observe) result B? To find out, ane must summate the travel time for light from event B to the worldline of the two observers. The push: Show/hide low-cal path to observers activates the adding and shows the light paths to the observers. Input v = 0.iv c, d = 4 and t = 4.5. Calculate and toggle the Show/hide push. Notice that the nonmoving observer measures event B to happen at t = 4.five but he sees the event at t = viii.5. The moving observer measures event B to happen at t' = three.1642 and she sees the event at t' = 5.5646. Find the angle of the lite path. Discover that the same calorie-free path passes through both observers.

Excercise 8a: What is the angle of the calorie-free path relative to the time axis t?

9 The twin paradox

The twin paradox is the well-known fact that someone travelling at a relativistic speed ages more slowly than someone at residue. This can be demonstrated by an experiment with a pair of twins where 1 twin travels an interstellar altitude and dorsum at high speed.

Allow result A₁ exist the separation of the twins and event A₂ the reunion. These events are both on the t worldline of the twin that stays at dwelling (ie, at rest). Outcome B is the turn around signal for the travelling twin. This effect must therefore exist on the t' worldline. To achieve this nosotros must prepare the values such that d / t = relative velocity. Let united states of america try v = 0.75, d = 3 and t = 4. Notation that t' = 2.6458, this is the elapsed time for the traveller. Elapsed time for the stay-at-abode twin is of grade 4. At present the travelling twin turns effectually. The return trip therefore has five = -0.75 and d = 3 and t = -4. t must be negative to achieve the negative relative velocity. An alternate reasoning is that the issue A_ii which is the common event when the ii twins are dorsum together again has the values t = d = 0, and to achieve this t and t' for upshot B on the return trip must be negative. Note that t' = -2.6458 and t = -4. Time elapsed must be the negative of this result since nosotros started the home lap with a negative time value. Full elapsed time for the travelling twin is thus two.6458 * 2 = 5.2916. And total elapsed time for the stay-at-home twin is 4 * ii = eight.

Conclusion: the travelling twin ages slower! This is of course not actually a paradox: The stay-at-dwelling twin is staying in the aforementioned inertial reference frame simply the travelling twin is switching reference frame at the plow around point. This tin can also be seen from the fact that the worldline of the stay-at-dwelling-twin is continuous (vertical both times) but the travelling twin has ii unlike worldlines with - in this example - an bending of almost 74^o.

Excercise 9a: Utilise the animation to calculate how much younger the travelling twin is after travelling 24 light years and back at 0.96 c. (Hint: Start past finding the travel fourth dimension at 0.96 c!)

Excercise 9b: Employ the blitheness to calculate how far you lot can travel and render while 500 years laissez passer at home and you travel at 0.9999 c. What is your elapsed time as a traveller?

10) Length wrinkle

To use the Minkowski diagram to summate and prove the Lorentz-Fitzgerald length contraction requires some special consideration. Suppose we have an object of a sure length, say vii units. What would it entail for the two observers to measure the length of this object? The nonmoving observer has no problem, the object is not moving relative to him, so he can employ his distance axis d. We know that all points on this centrality are simultaneous in the nonmoving reference frame. The moving observer also needs to measure the length of the object by measurements simultaneous at both ends of the object. That is only possible forth the d' axis or a line parallell to this axis. We know that all points on the d' centrality are simultaneous for the moving observer. In the case of the d' axis we as well know that the time t' at all points is 0.

Input v = 0.3 c and d = seven. The meaning of this is that the rest length of the object is vii units, and this is the length the nonmoving observer measures. How can we discover t? This equation from the inverse Lorentz transformation will aid:

We know 5 and d, and we also know that t' = 0 since nosotros are only interested in points along the d'-centrality. This fact simplifies the equation to t - vd = 0 and t = vd. With v = 0.three and d = 7 we easily calculate that t = 2.1. Intput this result.

Observe that this gives d' = half-dozen.6776. This demonstrates that the object is shorter by appr 0.3224 when the velocity is 0.3 c.

At present input some new values: five = 0.4 c, t = 0.4 * vii = 2.8. The result is now d' = six.4156.

Excercise 10a: Find the length contraction at v = 0.5, 0.6, 0.7, 0.8, 0.nine, 0.95, 0.99, 0.999 and 0.9999 c.

Notice that the length wrinkle increases more speedily with increasing velocity. Notice also that the length wrinkle becomes farthermost when 5 approaches c.

Excercise 10b: Suppose the object is moving along with observer t'. Find out how to use the animation to calculate the length contraction when 5 = 0.3 c and d' = seven. (Hint: Apply the original Lorentz transformation, not the inverse!)

Answers to excercises

1b: The two angles are equal in size and contrary in management.

2a: v = -0.25 c, d = 5.164, t = ii.582.

2b: five = -0.8, d = -five, t = 5.five.

3a: Consequence B happens at d = two and t = 2.five in the rest frame. The rate of time dilation is and so ane.5 / 2.five = 0.6.

3b: For the nonmoving observer effect B happens exist at d = ane.7321 t = three.4641. The rate of time dilation is 3 / iii.4641 = 0.8660.

3c: The time dilation increases with increased velocity.

4a: One event is 4 units of fourth dimension before the other and they are separated by five units of infinite.

4b: The nonmoving observer measures d = two.5 and t = two.

5: At five = 0.ii c the events are simultaneous in the moving reference frame. For any velocity < 0.2 c event A happens before consequence B.

half dozen: iⁱⁱ = -24, i² = 32.

7: a & b - timelike. c & d - spacelike. e - timelike. f - lightlike.

8a: The bending is 45^o in this example, information technology may otherwise be -45^o. In any and all cases it is parallel to one of the two worldlines of light passing through event A.

9a: The stay-at-home twin ages 50 y, the travelling twin ages 14 y, the difference is 36 y.

9b: 249.975 ly and back, and the total travel time is appr vii.07 y, just a little over 1/100 of the stay-at-habitation elapsed fourth dimension!

10a: The post-obit tabular array shows the length contractions for several values of c when the length for the nonmoving observer is 7 units. (Rounded to 4 decimal places every bit in the animation.)

0.1 c d' = 6.9649
0.2 c d' = vi.8586
0.3 c d' = 6.6776
0.four c d' = 6.4156
0.5 c d' = half dozen.0622
0.6 c d' = five.6
0.7 c d' = iv.999
0.8 c d' = 4.2
0.9 c d' = 3.0512
0.95 c d' = 2.1857
0.99 c d' = 0.9875
0.999 c d' = 0.313(0)
0.9999 c d' = 0.099(0)

10b: Use this equation from the original Lorentz transformation: . Now y'all want to measure along the d axis and therefore t = 0. v and d' are given and nosotros only need to detect t'. Since t = 0, the equation reduces to t' = - vd' and nosotros easily calculate t' = -2.one. Input v, d' and t'. The animation confirms that d = 6.6776, equally expected.

Sources

• Mook & Vargish: Within Relativity. Princeton Univ. Press 1987
• Taylor & Wheeler: Spacetime Physics. 2nd ed. Freeman & co 1992
• Bais: Very Special Relativity. Harvard Univ. Press 2007

Animation and text: Kristian Evensen trell.org
Copyright © 2009-2020

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Source: http://www.trell.org/div/minkowski.html

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