DOING PHYSICS WITH MATLAB

RELATIVITY

SPECIAL RELATIVITY

MINKOWSKI SPACETIME DIAGRAMS

 

Ian Cooper

 

matlabvisualphysics@gmail.com

 

 

 

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GR_001.m        Minkowski spacetime diagrams

 

 

This article will examine how the Minkowski spacetime diagram can be used to enhance the study of the special relativity.

 

A Minkowski spacetime diagram is a geometric representation of motions in spacetime. The

vertical axis is usually plotted as the time axis. Any point in spacetime is called a world-point,

and a series of world-points representing the motion of some object is called a worldline.

Any individual event is uniquely represented by some point B. The description of this event

is described in the O frame by the coordinates (x, ct) and in the O’ frame by the coordinates (x',

ct'). The origins of O and O' are chosen so as to coincide at ct = ct' = 0, and the relation between

(x, ct) and (x', ct') is contained in the Lorentz transformations. The worldline of a light signal

starting out at x = 0, ct = 0, is a bisector of the angle between the axes. This holds good in both

the O and O' frames.

 

The starting point is the Lorentz transformations between the inertial reference frames (IRF) O (observer Steve) and O’ (observer Mary) where the IRF O’ is moving in the + x direction with uniform velocity v. The velocity v will be expressed by the factor

(1)          

 

and the Lorentz factor  is

     (2)         

 

Spacetime coordinates:

               O (Steve)           O’ (Mary)  

 

Lorentz transformations:

                      

     (3)            

                             

 

 Coordinate axes:

          O (Steve)      x-axis  ct = 0 and ct-axis x = 0        Cartesian axes   

          O’ (Mary)    x’-axis   and ct’-axis 

          Angle between x-axis and x’-axis    

          Angle between ct-axis and ct’-axis  

 

The coordinate axes are referred to as worldlines.

 

Light cone:

       

 

Light on a Minkowski diagram will always travel at 45o to the time axis along the surfaces of a 45o cone, which is called the light-cone (figure 11).

 

Spacetime interval, an invariant quantity called the Lorentz invariant:

        

 

The units for all axes are metres [m]. The ct and ct’ axes give the distance that light would travel in times t and t’. respectively. For example, a 1.0 m interval corresponds to a time interval of (1/c) s.

 

Steve observers Mary travelling a constant velocity v in the direction of the + x-axis. Therefore, the displacement of Mary w.r.t. Steve is

        

 

Hence, Steve observes Mary travelling along the ct’ worldline.

 

A point defines an event that occurs at a particular place at a particular instant in time.

 

The Minkowski spacetime diagrams shown represent a model of two spacetime events, event A (green dot) and event B (red dot). Two observers in two inertial reference frames pass each other at the Origin, this being the event A (green dot). In the input section of the Script GM_001.m, the value of and either the coordinates in O  or the coordinates in O’   are specified for event B (red dot). The Minkowski spacetime diagrams can be used to show time dilation, the relativity of simultaneity and other effects of special relativity. Figure 1 shows an example of a Minkowski spacetime diagram.

 

Fig. 1.   The yellow lines at 45o shows the light-cone worldlines for . The black lines are the coordinate axes in the IRF O  and the blue lines the coordinate axes for the IRF O’ . The spacing between black or blue dots is equal to one unit of distance [m]. In the IRF O, the components of event B are given by the Cartesian coordinates which can be read off the grey grid. In the IRF O’, the components of event B are read off the blue wordlines where the magenta lines intersect the worldlines (the magenta lines are parallel to the worldlines for x’ and ct’). The x’ worldline makes an angle  with the x-axis while the ct’ worldline makes an angle with the +ct axis.

 

 

Figure 2 shows the change in worldlines as  is incremented . The spacing between the worldlines x’ and ct’ become narrower since the angle increases as increases .

Fig. 2.  Change in worldlines as  is incremented . The spacing between the worldlines x’ and ct’ become narrower as  increases.

 

 

The Minkowski diagrams show the events as measured by the two observers as designed by Einstein - each observer has a set of synchronized clocks and previously established distance points. The time measurements for event B are what the synchronized clocks would display and so the observed times do not take into account the travel time of light from the events to the observers. One observer (Steve) follows the worldline of ct, the other (Mary) follows a worldline ct'.

 

Time intervals between event (temporal order) and simultaneity

The two observers agree that event A occurred when both their clocks recorded 0 . However, for event B depicted in figure 1, their clock readings are not the same . They measure different time intervals between events A and B.

 

The two events A and B are simultaneous in both reference frames when their relative velocity is zero  as shown in figure 3.

 

Fig. 3. When v = 0 the two events A and B occur simultaneous in both IRFs.

 

When there is relative motion between the two IRFs then the two events A and B are simultaneous in O but they are not simultaneous in O’ as shown in figure 4 where event B occurred before event A.

 

Fig. 4.  Events A and B are simultaneous in O but not in O’.

 

 

With input values ct’ = 0 (ctBD = 0) and x’ = 2 (xBD = 2) as shown in figure 5, event B is now simultaneous to event A in the reference frame of the moving observer O’, but not in the non-moving reference frame O, where event B happens 1.50 time units after event A. Notice that event B is placed on the axis x' (ct’ = 0). All points on this axis are simultaneous to the origin in the moving reference frame. Likewise: all points on any line that is parallel to x' are equidistant in time from the Origin in this reference frame.

 

Fig. 5.  Events A and B are simultaneous in O’ but not in O.

 

 

Time dilation effect and the slowing of moving clocks

For an observer in an inertial frame of reference, a clock that is moving relative to them will be measured to tick slower than a clock that is at rest in their frame of reference. This phenomenon is called the special relativistic time dilation effect. The faster the relative velocity, the greater the time dilation between one another, with time slowing to a stop as one approaches the speed of light. The equation for time dilation is

     (4)        

 

which expresses the fact that the moving observer's period of the clock is longer than the period in the frame of the clock itself (figure 6).

 

Fig. 6.  The moving clock of Mary as observed by Steve runs slower. Mary’s clock has advanced 1.00 units of time while Steve’s clock has advanced 1.25 units of time . Thus, a clock in the moving frame will be seen to be running slow, or dilated. The time will always be shortest as measured in its rest frame. The time measured in the frame in which the clock is at rest is called the proper time.

 

 

Input values for figure 7:  . This indicates that both events A and B are on the worldline of the moving observer O’. Suppose both events represent a tick of a clock that ticks at regular intervals. From the moving observer's point of view O’, the clock shows that 2 time-units has passed. Notice that the non-moving observer O measures that 2.50 time-units has passed on their clock. The results are in agreement with the predictions of equation 4. The non-moving observer O will conclude that the moving clock is slow.

 

Fig. 7.   The event B occurs on the worldline ct’ for O’ and thus ctDB is the proper time and ctB is the dilated time in O.

 

 

Input values for figure 8:  . This indicates that both events A and B are on the worldline of the observer O. For observer in O, clock shows that 2 time-units between events A and B, whereas the observer in O’, 2.50 time-units has passed. The observer O’ will conclude that the moving clock in O is slow.

 

Fig. 8.   The event B occurs on the worldline ct for O and thus ctB is the proper time and ctDB is the dilated time in O’.

 

Both observers will conclude that the other clock is slow by a factor of .

 

 

Lorentz-Fitzgerald Length Contraction

The length of any object in a moving frame will appear foreshortened (contracted) in the direction of motion. The length L0 is maximum in the frame in which the object is at rest (proper length) and the length L observed is in the moving frame is the contracted length. The equation for the contraction in length is

     (5)      

 

Example:  

 

Length contraction increases more rapidly with increasing velocity and also the length contraction becomes extreme when v approaches c.

 

The use of a Minkowski diagram to calculate and show the Lorentz-Fitzgerald length contraction requires some special consideration. Suppose we have an object of length 2 units. How can this length be measured? In the non-moving IRF there is no problem, since the object is not moving it is easy to make the length measurement along the x-axis, since we know that all points on this x-axis are simultaneous in the non-moving IRF. In the moving IRF the length measurement needs to be made by making measurements simultaneous at both ends of the object. That is only possible along the x' axis or a line parallel to this x’-axis. We know that all points on the x’-axis are simultaneous for the moving observer.  Figures 9 and 10 show simulations for length contraction.

 

Fig. 9.  Inputs: 

           Outputs:  proper length xB = 2.50 in O and contracted length xDB = 2.00 in O’.

 

 

Fig. 10.   Inputs: 

              Outputs:  proper length xDB = 2.50 in O’ and contracted length xB = 2.00 in O.

 

 

Both observers in the two IRF would conclude that length is contracted in the direction of motion.

 

 

Twin Paradox

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

 

Let event A1 be the separation of the twins and event A2 the reunion at the Origin. These events are both on the ct worldline of the twin that stays at home at rest. Event B is the turn around point for the travelling twin, and this event must therefore be on the ct' worldline.

Outward journey

Inputs:

Outputs:

 

For the return journey, the travelling twin turns around, therefore the velocity and displacement must be negative 

Inputs:

Outputs:

 

Fig. 11. Total elapsed time for the travelling twin is thus 6.0 a.u. while for the stay-at-home twin, the total elapsed time is 7.5 a.u. (3.75 + 3.75).

 

Hence, we can conclusion that the travelling twin ages slower! This is of course not really a paradox, since the stay-at-home twin is staying in the same inertial reference frame but the travelling twin is switching reference frame at the turn around point. This can also be seen from the fact that the worldline ct of the stay-at-home-twin is continuous (vertical) but the travelling twin has two different ct’ worldlines.

 

 

Timelike and spacelike intervals

The Minkowski spacetime diagram is divided into four equal sectors by the two lightlines. Two sectors are timelike and the other two sectors are spacelike (figure 12). The two lightlines occur when the spacetime interval is zeros, .

Fig. 11. Timelike and spacelight sectors.

 

If event B occurs in a timelike sector than the spacetime interval is negative,  . In this case the sequence in time between events A and B cannot be reversed. With timelike intervals the separation in time dominates, and there is no fixed directional relationship between the two events, left and right can be exchanged depending on the motion of the observer.

 

If event B occurs in a spacelike sector than the spacetime interval is positive,  . With spacelike intervals the spatial separation dominates, , and there can be no causal relationship between the two events A and B, i.e., the spacetime interval is such that there is not enough time for light to pass from event A to event B.

 

Fig. 12.  Timelike (left) and Spacelike (right).

 

 

Spacetime interval contours

We can plot lines of constant spacetime intervals on a Minkowski spacetime diagram. The contours shown in figure 13 are:

                            yellow lightlines

                           timelike region   

                           spacelike region   

Fig. 13.   Spacetime interval contours: invariant hyperbolae

 

It is not possible to define ‘space’ uniquely since different observers identify different sets of events to be simultaneous. But, there is still a distinction between space and time, since temporal increments enter with the opposite sign from spatial ones.

 

 

References

Many of my examples were based upon the article:

 

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