Frames of Reference and Galilean Transformations

We now need to be more precise about some of the terms that we will be using. Observers are simply people or instruments capable of making and recording measurements. Books on Special Relativity often refer to frames of reference. A frame of reference can be thought of as a set of three orthogonal meter sticks, with a clock attached. The meter sticks provide coordinates with respect to which we can measure the position of objects or events, and hence their displacements as they move. The clock is needed to measure time, so that we can measure velocities and accelerations. Of course, such a frame of reference is useless unless it also holds an observer. Since both the observer and the frame of reference are crucial and hence inseparable for the following, we will often use the terms interchangeably. Both terms will refer to a system of measuring sticks and a clock, with the mechanism to record observations. For simplicity, we will restrict consideration to motion in one direction, so that each frame of reference only contains one meter stick and all the meter sticks in every frame of reference point along the same direction. Fig.11.1 illustrates an event (an explosion) as viewed in a pair of one dimensional reference frames in relative motion with velocity v.

  

Figure 11.1: Two Frames of Reference in Relative Motion

For concreteness, you can think of the frames of reference as trains, each with meter stick and clock attached, traveling at constant speeds along parallel tracks (Strictly speaking for the analysis to be one dimensional they should be on the same track, but this would give rise to practical problems as they pass each other.) In Fig.11.1, the coordinates x' and t' gives the location and time of the explosion as measured in the ``moving'' frame O' whereas x and t denote the coordinate and time of the same event relative to the ``stationary'' frame. Of course, ``moving'' and ``stationary'' have no absolute meaning. In this context they just mean that the diagram is drawn in the same frame as O, instead of O'.

Prior to Einstein it was assumed that identical clocks of any two observers could be synchronized so that they would always agree. In terms of equations it was expected that:

 

t' = t (11.1)

In other words, time was expected to have an absolute meaning, independent of the motion of the observer.

With regard to spatial coordinates, it is clear that if two frames of reference are moving relative to each other, they will measure different coordinates on their respective meter sticks for the same event, and also different velocities for the same object. For example, if the frames in Fig.11.1 were exactly lined up at t = t' = 0, then the coordinate of the explosion in frame O would be equal to the coordinate x' as measured by O' plus the distance that the frames had moved relative to each other in that time interval:

 

x = x' + vt (11.2)

Eqs[11.1] and [11.2] constitute the so-called Galilean transformations relating coordinates as measured in two different frames. Common sense tells us that they must be correct. It was therefore a complete shock to the scientific community when Einstein realized that they contradicted the speed of light postulate, and suggested that they were, in fact, incorrect. To see where they fail, we need to look at what they imply for the addition of velocities. Suppose that just as the frames O and O' coincide, their clocks are synchronized to read t = t' = 0. At precisely this instant the observer in O' throws the ball to the right with a velocity of u' = 3 cm/s relative to his frame. (See Fig. 11.2 below).

  

Figure 11.2: Relative Velocity - 1

Fig. 11.3 shows the situation at a later time t.

  

Figure 11.3: Relative Velocity-2

The ball is now at the position labelled by x' = ut as measured by O'. However, the position along the meter stick of O is, according to the above arguments, given by the expression in 11.2 (with plus sign) giving a velocity u relative to O of

 

u = x/t = u + v (11.3)

Eq.11.3 is consistent with our intuition: the velocity relative to O is the sum of the velocity of the ball relative to O' and the velocity of O' relative to O.

We now have to extend our discussion to a situation which is easy to imagine, but rather difficult to realize in nature. Such situations are called gedanken or thought experiments. Suppose that O' is moving relative to O, at three quarters of the speed of light, instead of 2 cm/s. Moreover, instead of throwing a ball forward O' points a flashlight in the forward direction and turns it off and on quickly. This sends out a pulse of light moving at 300 million meters/second as measured relative to O'. How fast would O' see the pulse of light moving relative to his frame? In order to save writing, we will henceforth use the symbol c to denote the speed of light. Everytime you see this letter, you should think ``300 million meters per second''. According to the above discussion the speed of the pulse relative to O' would be c + 3c/4 = 7c/4, or about 500 million meters/second. This contradicts the speed of light postulate, which says that the speed of light should be the same in every frame of reference.

So what has gone wrong? The easiest conclusion to draw is that the speed of light postulate is incorrect. However, Einstein realized that one cannot give up this postulate without giving up either Maxwell's wave theory, or the postulate of uniform motion, and he was willing to give up neither. He therefore made a brilliant intuitive leap and concluded that our common sense is wrong. Einstein's faith in the speed of light postulate turned out to be well founded. Its strange consequences have since been verified experimentally. We will now discuss them in turn.