Question

Discuss the concept of simultaneity, referring to the appropriate equation.

Answer 1

As you can guess from the equations for the lorentz transformation the concept of time becomes a relative concept since we no longer have the simple relation t=t'. What we actually mean here is that the concept of simultaneity will depend on which frame you are in. This is a simple consequence of the fact that the speed of light  is the same no matter which inertial frame you are in.

Consider a moving train with a light bulb in the middle. If you turn the light bulb on, light will travel both toward the front of the train and also toward the back of the train with speed c=3×10⁸m/sec.

From the point of view of the observer riding on the train, the distances from the light bulb to the front and back ends of the train are the same so light will reach at both ends at the same time .

However, from the point of view of the person on the ground, the front of the train is moving away from the light coming toward it while the back of the train is moving closer to the light coming toward it. This means that the distance covered by light going forward will be longer than the light going backwards. And since the speed of light is c in both directions for the observer on the ground also, the light will reach the back of the train before it reaches the front of the train.

What I am saying is that 2 events that seem to have happened at the same time for the observer on train do not happen at the same time for the observer on the ground. In fact, from the point of view of another observer moving faster than the train, like the observer in the sports car above, the light reaches the back of the train after it reaches infront of the train since in that observer's frame, it is the back of the train that is moving away from the light and the front of the train that is moving toward it.

So the concept of before and after actually depends on the observer. In order to clarify what we are trying to say here, Light from the lightbulb reaches the back end of the train at A, while it reaches the front end of the train at B. All observers are observing the same two events A and B. The spacetime points at which they occur are frame independent. However, the chronological order of the events are frame dependent:

• In the frame fixed to the ground (x,t), A happens before B.
• In the frame fixed to the train (x',t'), A and B happen at the same time.
• In the frame fixed to the sports car (x'',t''), A happens after B.

  Consider how we measure elapsed time. If we use a stopwatch, for example, how do we know when to start and stop the watch? One method is to use the arrival of light from the event, such as observing a light turning green to start a drag race. The timing will be more accurate if some sort of electronic detection is used, avoiding human reaction times and other complications.

  Now suppose we use this method to measure the time interval between two flashes of light produced by flash lamps.

  Two flash lamps with observer A midway between them are on a rail car that moves to the right relative to observer B. Observer B arranges for the light flashes to be emitted just as A passes B, so that both A and B are equidistant from the lamps when the light is emitted. Observer B measures the time interval between the arrival of the light flashes. According to postulate 2, the speed of light is not affected by the motion of the lamps relative to B. Therefore, light travels equal distances to him at equal speeds. Thus observer B measures the flashes to be simultaneous.

  Observer B measures the elapsed time between the arrival of light flashes as described in the text. Observer A moves with the lamps on a rail car. Observer B perceives that the light flashes occurred simultaneously. Observer A perceives that the light on the right flashes before the light on the left.

  Now consider what observer B sees happen to observer A. Observer B perceives light from the right reaching observer A before light from the left, because she has moved towards that flash lamp, lessening the distance the light must travel and reducing the time it takes to get to her. Light travels at speed relative to both observers, but observer B remains equidistant between the points where the flashes were emitted, while A gets closer to the emission point on the right. From observer B’s point of view, then, there is a time interval between the arrival of the flashes to observer A. From observer B’s point of view, then, there is a time interval between the arrival of the flashes to observer A. In observer A’s frame of reference, the flashes occur at different times. Observer B measures the flashes to arrive simultaneously relative to him but not relative to A.

  Now consider what observer A sees happening. She sees the light from the right arriving before light from the left. Since both lamps are the same distance from her in her reference frame, from her perspective, the right flash occurred before the left flash. Here a relative velocity between observers affects whether two events are observed to be simultaneous. Simultaneity is not absolute

  This illustrates the power of clear thinking. We might have guessed incorrectly that if light is emitted simultaneously, then two observers halfway between the sources would see the flashes simultaneously. But careful analysis shows this not to be the case. Einstein was brilliant at this type of thought experiment (in German, “Gedankenexperiment”). He very carefully considered how an observation is made and disregarded what might seem obvious. The validity of thought experiments, of course, is determined by actual observation. The genius of Einstein is evidenced by the fact that experiments have repeatedly confirmed his theory of relativity.

  In summary: Two events are defined to be simultaneous if an observer measures them as occurring at the same time (such as by receiving light from the events). Two events are not necessarily simultaneous to all observers.