Browsers that typeset this document correctly:
One of the axioms in the theory of relativity is that all events occur in all frames of reference identically. The apparent amount of time between events may be different between frames of reference, but the sequence remains the same for all observers.
Another phenomena predicted by the theory of relativity is known as Lorentz contraction
– that an object will be compressed along its axis of motion when moving near the speed of light.
Say, for example, the starship Enterprise travels by at ¾ warp:
This stretch factor
is represented by
The flickering bulb was a thought experiment proposed to disprove
The aparatus for demonstrating the paradox is:
A lamp and battery in series connected to two long rails:
A frictionless slider with conducting arms and an insulating body:
The lamp circuit is open at one end, so the lamp will only light if the slider is touching both the rails.
One of the rails has a square vertical offset the same length as the slider. This does not cause the bulb to flicker as the slider moves, however, because as one side breaks contact, the other picks it up.
This concept of contraction of length introduces an apparent paradox. Say for instance that the slider is moving at
At
So, the length of the slider is reduced by a factor of two, making it 1m.
If the slider is contracted then there will be a length of time during which the circuit is completely broken and the light goes off.
The theory of relativity rests heavily on the concept that there is no absolute frame of observation, so it is equally valid to see the experiment as the slider holding still and the rails moving in the opposite direction at
The gap is now only 1m, so the slider maintains contact the entire time and the bulb will never go off. Relativity does not allow for different events to occur for different observers. The order of the events can be different, but the bulb either goes off or it doesn't and that fact is the same for all frames of reference. So, there must be some way to resolve this paradox, or the contraction of length does not exist.
The resolution comes with the recognition that the electromotive force, ε, like everything else, cannot move faster than the speed of light. This means that when a rail makes (or breaks) contact there is a time lapse of at least 4 light-meters before that change in force will reach the bulb. (It is 4m to the side of the gap closest to the bulb, 6m to the far side.) To better examine this, it is convenient to define some events. These events will take place in all frames (though not necessarily at the same time or even in the same order.) They are shown here in the frame with the rails at rest, though again this is arbitrary and could have been shown just as easily in the frame with the slider at rest.
Event #1: Bar A enters the gap (breaking contact with the rails):
Event #2: Bar B exits the gap (reestablishing contact with the rails):
You will notice that the bulb is not out in either event even though during the interval between then there was no circuit. This seems incorrect at first, but again the changes in ε are not simultaneous; they require time for the change is force to propagate through the wire. These changes in ε are represented with *s. To better analyze this we will define two more events.
Event #3: The drop in ε from event #1 reaches the bulb:
Event #4: The restoration of ε from event #2 reaches the bulb:
If event #4 occurs at or before the time event three occurs then ε is never diminished and the bulb will not flicker.
The bulb will be off for the period of time between events 3 and 4 (). Using the following notations:
Obviously there are relationships between events 1 and 3, 2 and 4 since 1 causes 3 and 2 causes 4. The time between 1 and 3 is just:
Since those events are separated by the time that it takes the pulse to cover the distance from
The time between 1 and 2 is a little more sophisticated. The length if the gap is
Since is the first event, it is convenient to assign:
So:
Substituting in the values that were given:
In spacetime it is not uncommon to express lengths of time in units commonly associated with distance. The two converge with the speed of light which is constant in all frames of reference at .
This scenario is very similar to the one where the rails are at rest: the time that the bulb will be off is still the time between events 3 and 4 (). Using the following notations:
One important difference in this scenario is that relative velocity between the change in ε and the bulb (
Based on this change, the previous equations are altered slightly:
The other major difference in this frame of reference is that event #2 happens before event #1. This means that the time difference between 1 and 2 is the time it takes bar A to traverse the distance
Since event #2 is now the first, it is convenient to combine the equations with
Substituting in the values that were given:
The proper time is the time separation in the frame where the space separation is 0m. Event #3 and #4 occur in the same place in the frame where the rails are at rest. The rails frame is therefore the frame with the proper time.
According to my figures:
The relationship between two frames of reference is represented by
This is agrees with the computed value of
The transmission of electrical impulses through different mediums is generally less than the speed of light. For the next questions, assume that the pulses travel at a speed of .75c in the frame where the rail is at rest.
This is found using the law of combination of velocities. The velocity of C relative to B:
When there is no flicker, the resumption of ε reaches the bulb before the drop, i.e. .