Special Relativity Explained by Diagrams

Richard Beauchamp

Department of Physics, College de Bois-de-Boulogne, Montreal, Qc.

The following is a presentation of the main elements of special relativity using diagrams whose main property is to preserve the interval (ct)²-(x)²=(ct')²-(x')². They use only two axes for the coordinates (x and ct), and two perpendicular projections on these axes for the coordinates (x' and ct'). First, the reader will understand the notion of an event using classical mechanics diagrams and, then, will understand that it needs just a simple modification in special relativity.

I. INTRODUCTION

The presentation of the main elements of the theory of special relativity becomes possible with this approach in only a few hours. We use diagrams : two axes and two perpendicular projections will be required. In the first section, we present the Galilean relativity and the associated space-time diagrams to illustrate various events. The constant value of the speed of light will require a slight modification of these diagrams; this is the subject of our second section. The diagrams will clearly show the relativity of simultaneity, time dilation and length contraction. The diagrams will also allow to establish the equations of the Lorentz transformation. We hope that the diagrams will represent an educational tool which will help in quickly understanding the notions of special relativity.

II. GALILEAN RELATIVITY

A. Event and Coordinates of an Event

In relativity, an event is "something" that happens at a given place in space and at a given moment in time. We indicate this event by means of spatio-temporal coordinates (x,y,z,t) relative to an inertial reference frame K (called K-frame below), the origin (O) of which is at position (0;0;0); temporal coordinate (t) can be obtained by using a clock located where the event takes place. Consider the two frames K and K' which coincide at t=0; the K'-frame moves with speed (v) along the x axis relative to the K-frame. In the K-frame, the coordinates of any event are (x,y,z,t) and are (x',y',z',t') in the K'-frame. (Fig. 1)

Fig. 1. Coordinates of event and Galilean transformation

For any event E, we will always obtain x'=x-vt and, if the progress of time is the same in both frames, we can write t'=t. As there is no motion along the y axis and the z axis, we will write y'=y and z'=z. These four equations constitute the Galilean transformation. It is valid only when the speed (v) is much less than the speed of light c=3x10⁸ m/s.

B. Space-Time Diagram in Classical Mechanics

To illustrate the motion of a particle, we use a space-time diagram. (Fig. 2) To represent several events for the two frames on the same diagram, we tilt the time axis at angle a. The time axis, associated with an arbitrary constant C (having units of speed), carries the same units as the x axis. It is tilted at angle a toward the right hand side so that sin(a) = v/C. We can explain this angle a by considering that any event taking place at position x'=0 in the K'-frame should be at x=vt in the K-frame.

Fig. 2. Coordinates of event (E) for two frames related by Galilean transformation

The x coordinate of any event E is obtained by a perpendicular projection on the x axis. Coordinates x' and Ct are obtained by drawing a parallel to the x axis. The value of x' corresponds to the horizontal length between the event and the tilted time axis. Time coordinate Ct corresponds to the length defined by the projection on the time-axis. This diagram clearly shows the relation x'=x-vt (Galilean transformation).

C. Speed of Light Contradiction

Consider that a flash of light is produced at x=0 at the same moment that both frames coincide. The position of the light wave front along the x axis in the K-frame is given by x=ct, where c is the speed of light in this frame. According to Galilean transformation, that would give x'=(c-v)t. Since t'=t, we also get x'=(c-v)t'. The speed of the wave front in the K'-frame would be (c-v), a value different from the one observed in the K-frame. This is contradictory to the second postulate of special relativity. (See next section.) Measurements confirm the constant value of the speed of light in space, whatever the motion of the observer. Therefore, it is necessary to review our equation x'=x-vt and the diagram to obtain x'=ct' for the K'-frame.

III. SPECIAL RELATIVITY

A. Postulates of Special Relativity and Its Consequences

i) Physical laws are identical for all inertial frames.

ii) The speed of light has a constant value in all inertial frames.

Consider the two frames K and K'. K' is moving with great speed v along the x axis relative to K. Two clocks measure the times: t in K-frame and t' in the K'-frame. Consider that the two clocks indicate t=t'=0 when O' and O coincide. Finally, at this moment, a flash of light is emitted in all directions.

Shortly after, the light wave front is at a distance r=ct from origin O in all directions, r being the radius of the sphere of light centered at O. Because of the constant value of the speed of light in all directions, it is also necessary that the light wave front be at a distance r'=ct' from origin O'. So r=ct and r²=x²+y²+z² => (ct)²=x²+y²+z².

We also have r'=ct' and (r')²=(x')²+(y')²+(z')² => (ct')²=(x')²+(y')²+(z')². Since y'=y and z'=z, we obtain the very important equation :

(ct)²-(x)²=(ct')²-(x')²

which is the only one required to build the transformation of coordinates in special relativity .

B. Special Relativity Space-Time Diagram.

Let any event E take place at coordinates (x,y,z,t) in the K-frame and at coordinates (x',y',z',t') in the K'-frame and, then, consider the following diagram where the time axis is tilted toward the right hand side at angle (a). (Fig.3)

Fig. 3. Space-time diagram for the Special Relativity

In this diagram, the x coordinate is obtained by a perpendicular projection onto the x axis. The ct coordinate is also obtained by a perpendicular projection on the time axis. The x' coordinate corresponds to the shorter length between the time axis and event E. Also, the ct' coordinate corresponds to the shorter length between the x axis and event E. All these coordinates can be positive or negative. The time axis is tilted at angle a so that sin(a)=vt/ct; that is, sin(a)=v/c where v is the speed of K'-frame relative to K-frame, and c is the speed of light in vacuum. We can explain angle a by considering that any event taking place at position x'=0 in the K'-frame should be at x=vt for the K-frame. With the two right angle triangles sharing the same hypotenuse OE in this diagram, we obtain (ct)²+(x')²=(x)²+(ct')² which is completely equivalent to (ct)²-(x)²=(ct')²-(x')² for any event E.

The diagram show that if x = ct, then x' = ct' according to the second postulat of special relativity for the speed light.

This diagram is similar to the Brehme¹ diagrams but simplified. (See annex.)

C. Relativity of Simultaneity

Consider the two events E₁ and E₂ illustrated on the space-time diagram in Fig. 4.

Fig. 4. Events simultaneous K' and not simultaneous for K

Given that both events take place at the same time in the K'-frame, ct'₁=ct'₂, the events are said to be simultaneous in the K'-frame. The diagram clearly shows that these events are not simultaneous for the K-frame because ct₁<ct₂. Consequently, the simultaneity is proper to a given frame. The interval of time Dt can be obtained by means of the equation cDt=Dx sin(a). For example, imagine a train speeding at v =108 km/h and two events separated by a distance Dx'=30 m in this train. If the events are simultaneous in the train, they will be separated by an interval of time Dt = 10^-14 s for an observer at rest!

The next diagram shows that events E₂ and E₃ are simultaneous for the K-frame because ct₂=ct₃ but are not simultaneous for the K'-frame because ct'₂>ct'₃.

Fig. 5. Events simultaneous K and not simultaneous for K'.

D. Lorentz Transformation

Consider the two frames K and K'. K' moves at constant speed (v) along the x axis relative to the K-frame, and t = t' = 0 when O' coincides with O and event E is happening somewhere in space at a given moment in time.

Fig. 6. Lorentz transformation for positions

Lets :

The diagram in Fig. 6 directly shows that : .

The diagram in Fig. 7 directly shows that :

Fig. 7. Lorentz transformation for time

Lets :

These equations are called the Lorentz transformation. They can be reduced to the Galilean transformation when v<<c.

E. Time Dilation

Consider the two events E₁ and E₂ illustrated on the space-time diagram in Fig. 8. Both events take place at the same point x'₁=x'₂. The time Dt' between the events is called proper time in the K'-frame. The diagram clearly shows that the time Dt in the K-frame is longer than Dt' observed in K'. This is time dilation. In the triangle of Fig. 8, we obtain :

Fig. 8 Proper time in (K') and dilated time for (K)

On the other hand, in the Fig. 9 diagram because x₂=x₃ ,the proper time is in the K-frame and we obtain

Fig.9 Proper time in (K) and dialted time for (K')

F. Length Contraction

The distance between the two oblique lines in Fig. 10, represents a constant length Dx' at rest in the K'-frame; since it is at rest, there is no need for a particular pair of events on these lines to measure their distance. But for that length to be significant for an observer placed in the K-frame, it is required that he has access to the two extremities of the distance segment at the same moment t₁=t₂. It is, therefore, necessary to imagine the two events E₁ and E₂ to be simultaneous in the K-frame at the extremities of the length which he wants to determine.

Fig. 10. Length at rest in K', contracted in K

The diagram clearly shows that a length Dx' in the K'-frame becomes Dx in the K-frame and that Dx is shorter than Dx'. This is called length contraction. In the triangle of Fig. 10, we obtain :

On the other hand, in the Fig. 11 diagram it is in the K-frame that the length Dx is at rest. It is up to the observer in K' to find two simultaneous events at the extremities of the distance to measure. The diagram shows that it is Dx' which is shorter than Dx, and we obtain :

Fig. 11. Length at rest in K, contracted in K'

G. Example

A particle moving towards the Earth crosses 900 m in 5 ms.

Find the duration in the particle frame? Find the length in the particle frame?

The speed of the particle being and c = 300 m / ms we have sina=0,6 and cosa=0,8. In Fig. 12, the time-axis was tilted such as sina = 0,6. The distance travelled by the particle is illustrated in the straight line AB.

Fig. 12. A particle moving towards the Earth

The diagram shows that.

For the distance to be is significant in the frame-particle, it is necessary to imagine two simultaneous events in this frame, A and C for example. The diagram shows

IV. Conclusion

The main element of this paper is certainly the diagram in Fig. 3. We show two projections on both axes necessary to establish the coordinates (x, ct) and (x', ct'). It seems rather obvious that two events separated by a certain distance Dx, and are simultaneous for a frame (Fig 4 and 5), cannot be simultaneous for another frame. With the triangles of Fig. 6 and 7, we use only a single line to establish the Lorentz transformation. And the same goes for time dilation and length contraction. The main advantage of this approach is certainly to understand the main elements of special relativity in a shorter period of time. Let us add that these diagrams can be used to demonstrate the other equations of special relativity, the relativistic Doppler effect, etc.

V. AKNOWLEDGEMENTS

I want to thank my family for encouraging me to write this paper. I also want to thank my colleagues for their much needed advice and help in reviewing it. Last, but not least, I would like to thank all my of students who constantly challenge my ability to find new methods to explain physical models.

VI. References

Robert W. Brehme, Am. J Phys. 30, 489 (1962)

Annex

Fig. 13. The Brehme diagram for special relativity

The x' coordinate is obtain by a projection onto the x'-axis and corresponds to the shorter length between the time axis and even E. Also, the ct' coordinate is obtain by a projection onto the ct' axis and corresponds to the shorter length between the x axis and even E.

Richard Beauchamp, 10555 Avenue Bois-de-Boulogne, Montreal, Qc. Canada. H4N 1L4

Email: richard.beauchamp@bdeb.qc.ca