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Propagation of Light in Empty Space

Einstein asserts that "light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body." We will investigate the meaning of this statement will the aid of an example taken from Einstein's 1905 paper, with added commentary.

Suppose you are standing at the top of a tall tower. A friend of yours comes along in his rocket ship at 60% of the speed of light, passing just a few feet above you. At the instant the ship passes overhead, a lamp on the tower sends out a pulse of light.

The pulse of light starts out as a point and emanates in all directions, forming a hollow sphere. The radius of the sphere grows at the speed of light. Therefore, after one second the shell of the sphere is 186,000 miles away from the center of the sphere.

Question: Where is the center of the sphere after one second? The obvious answer is that the center of the sphere is at the point from which the pulse was emitted—the point from which you are now observing the expansion of the sphere. This answer is correct. But it is not the only correct answer.

There are two observers in this example: you and your friend. The postulate asserts that the pulse of light propagates at the same fixed speed for all observers, moving or stationary. This means that the pulse of light is moving away from your friend at 186,000 miles per second. So, after one second, your friend, who at this time is 111,600 miles away from you, is also at the center of the sphere.

Thus, according to Einstein's postulate, the stationary observer is at the center of the light sphere, and so is the moving observer.

Of course, it’s impossible for the center of a sphere to be in two places at one time. Here it is helpful to remember the earlier discussion of Lorentz's electron theory, in which we learned that problems involving moving bodies were simplified by use of the Lorentz transformation. The result of the transformation was a contracted length and a local time. Einstein's theory also makes use of the Lorentz transformation, with a different interpretation. In Einstein's theory, there is no fixed reference in space, so there is no universal time. Time advances at its own rate in each moving body, according to the relative velocity of the body as judged from a "stationary" body.

We saw the equations of the transformation earlier. Interested readers will find a derivation of them, based on this scenario, in Menzel, Mathematical Physics, page 375. Here are the equations, with notes for this example:

Xb = (Xa - v • ta) / Sqrt(1 - v2 / c2)

tb = (ta - v / c2 • Xa) / Sqrt(1 - v2 / c2)

where:

Xa = position in coordinate system A (the tower)

Xb = position in coordinate system B (the rocket ship)

ta = time on a clock in the tower

tb = time on a clock in the rocket ship

v = velocity of the rocket, relative to the tower.

c = speed of light, 186,000 miles per second

Sqrt represents the square root function

The use of the equations will be explained with the help of the following table:

EventTime
(you)
Position
(you)
Time
(ship)
Position
(ship)
10000
25930,0002.5465,000
3101,860,0005.0930,000
4152,790,0007.51,395,000

There are four ‘events’ in the table. Each event consists of two pairs of values, one pair for you on the tower, and one pair for your friend in the ship. As you can see, each pair consists of a time value and a position value. The time value is the number of seconds since the pulse of light was emitted by the lamp, and the position value is the number of miles that the pulse has traveled in that time.

On your side of the table, the entries are spaced five seconds apart. The position associated with each time is simply the time value multiplied by the speed of light.

On your friend’s side of the table, the time and position values were calculated using the transformation equations.

The data begins at the instant the pulse of light is emitted and your friend passes the tower. No time has elapsed, so the light has not traveled any distance. That is why all times and distances are zero for Event 1.

The next entry, Event 2, is five seconds later by your watch, and the pulse of light has traveled 930,000 miles.

However, Event 2 is 2.5 seconds later by your friend’s watch; therefore by his reckoning the light has traveled 465,000 miles.

This disagreement continues at every event: time and position as reckoned by your friend are half the time and position reckoned by you. (The ship’s speed was specifically chosen at 60% of the speed of light to give this round-numbered result).

When we began the analysis of Einstein’s postulate, it seemed obvious that it is impossible for the center of the sphere to be in two places at one time. But we see from the table that we are not talking about “one time”. We are talking about "one event"—an event that is defined equally well by two distinct combinations of place and time. The theory is self-consistent.

While the theory does not contradict itself, the notion that two observers at different locations can be at the center of one and the same sphere of light certainly does contradict our intuitive perception of reality. The theory introduces a new understanding of simultaneity, and demands that we adjust our view of reality accordingly.



Last edited: 2015-04-04 04:43:30

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