Special Relativity: Answers and Questions

The speed of light in a vacuum is 186,282 miles per second. It is a central tenet of the special theory of relativity (SR) that the speed of light is the same for every observer, regardless of the motion of an observer relative to the source of the light being observed. The meaning of the latter statement is not obvious to a non-physicist (like me). In an effort to understand it, I concocted the following thought experiment (TE), which I will call TE 1:

1. There is a long train car running smoothly on a level track, at a constant speed of 75 miles per hour (mph) relative to an observer who is standing close to the track. One side of the car is one-way mirror, arranged so that the outside observer (Ozzie) can see what is happening inside the car but an observer inside the car cannot see what is happening outside. For all the observer inside the car knows, the train car is stationary with respect to the surface of the Earth. (This is not a special condition; persons standing on the ground do not sense that they are revolving with the Earth at a speed of about 1,000 mph.)

2. The train car is commodious enough for a pitcher (Pete) and catcher (Charlie, the inside observer) to play a game of catch over a distance of 110 feet, from the pitcher’s release point to the catcher’s glove. Pete throws a baseball to Charlie at a speed of 75 mph (110 feet per second, or fps), relative to Charlie, so that the ball reaches his glove 1 second after Pete has released it. This is true regardless of the direction of the car or the positions of Pete and Charlie with respect to the direction of the car.

3. How fast the ball is thrown, relative to Ozzie, does depend on the movement of the car and positions of Pete and Charlie, relative to Ozzie. For example, when the car is moving toward Ozzie, and Pete is throwing in Ozzie’s direction, Ozzie sees the ball moving toward him at 150 mph. To understand why this is so, assume that Pete releases the ball when his release point is 220 feet from Ozzie and, accordingly, Charlie’s glove is 110 feet from Ozzie. The ball traverses the 110 feet between Pete and Charlie in 1 second, during which time the train moves 110 feet toward Ozzie. Therefore, when Charlie catches the ball, his glove is adjacent to Ozzie, and the ball has traveled 220 feet, from Ozzie’s point of view. Thus Ozzie reckons that the ball has traveled 220 feet in 1 second, or at a speed of 150 mph. This result is consistent with the formula of classical physics: To a stationary observer, the apparent speed of an emitted object is the speed of that object (the baseball) plus the speed of whatever emits it (Pete on a moving train car).

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So far, so good, from the standpoint of classical physics. Classical physics “works” at low speeds (relative to the speed of light) because relativistic effects are imperceptible at low speeds. (See this post, for example.)

But consider what happens if Pete “throws” light instead of a baseball, according to SR. This is TE 2:

1. The perceived speed of light is not affected by the speed at which an emitting object (e.g., a flashlight) is traveling relative to an observer. Accordingly, if the speed of light were 75 mph and Pete were to “throw” light instead of a baseball, it would take 1 second for the light to reach Charlie’s glove. Charlie would therefore measure the speed of light as 75 mph.

2. As before, Charlie would have moved 110 feet toward Ozzie in that 1 second, so that Charlie’s glove would be abreast of Ozzie at the instant of the arrival of light. It would seem that Ozzie should calculate the speed of light as 150 mph.

3. But this cannot be so if the speed of light is the same for all observers. That is, both Charlie and Ozzie should measure the speed of light as 75 mph.

4. How can Ozzie’s measurement be brought into line with Charlie’s? Generalizing from the relationship between distance (d), time (t), and speed (v):

  • d = tv (i.e., t x v, in case you are unfamiliar with algebraic expressions);
  • therefore, v = d/t;
  • which is satisfied by any feasible combination of d and t that yields v = 110 fps (75 mph).

(Key point: The relevant measurements of t and d are those made by Ozzie, from his perspective as an observer standing by the track while the train car moves toward him. In other words, Ozzie will obtain measures of t and/or d that differ from those made by Charlie.)

5. Thus there are two limiting possibilities that satisfy the condition v = 110 fps (75 mph), which is the fixed speed of light in this example:

A. If t = 2 seconds and d = 220 feet, then v = 110 fps.

B. If t = 1 second and d = 110 ft, then v = 110 fps.

6. Regarding possibility A: t stretches to 2 seconds while d remains 220 feet. The stretching of t is a relativistic phenomenon known as time dilation. From Ozzie’s perspective, the train car slows down. More exactly, a clock mounted in the train car would seem (to Ozzie) to run at half-speed from the moment Pete releases the ball of light.

7. Regarding possibility B: d contracts to 110 feet while t remains 1 second. The contraction of d is a relativistic phenomenon known as length contraction. From Ozzie’s perspective, it appears that the distance from Pete’s release point to Charlie’s catch (which occurs when Charlie is adjacent to Ozzie) shrinks when Pete releases the ball of light, so that Ozzie sees it as 110 feet.

8. There is no reason to favor one phenomenon over the other; therefore, what Ozzie sees is a combination of the two, such that the observed speed of the ball of light is 75 mph.

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Here is TE 3, which is a variation on TE 2:

1. The train car is now traveling leftward at 110 fps, as seen by Ozzie. The car  is a caboose, and Pete is standing on the rear platform, whence he throws the baseball rightward (relative to Ozzie) at 110 fps (relative to Pete).

2. Ozzie is directly opposite Pete when Pete releases the ball at t = 0. According to classical physics, Ozzie would perceive the ball as stationary; that is, the sum of the speed of the train car relative to Ozzie (- 110 fps) and the speed of the baseball relative to Pete (110 fps) is zero. In other words, Ozzie should see the ball hanging in mid-air for at least 1 second.

3. Do you really expect the ball to stand still (relative to Ozzie) in mid-air for 1 second? No, you don’t. You really expect, quite reasonably, that the ball will move to Ozzie’s right, just as a light beam would move to Ozzie’s right if switched on at t = 0. (This is analogous to the behavior of a light beam emitted from a flashlight that is switched on at t = 0.)

4. Now suppose that Charlie is stationary relative to Pete, as before. This time, however, Charlie is standing at the front of a train car that is following Pete’s train car at a constant distance of 110 feet. According to the setup of TE 1, Charlie will be directly opposite Pete at t = 1, and Charlie will catch the ball at that instant. How can that be if the ball actually moves to Ozzie’s right, as stipulated in the preceding paragraph?

5. If Pete had thrown a ball of light at t = 0 — a very slow ball that goes only 110 fps — it would hit Charlie’s glove at t = 1, as seen by Charlie. If Ozzie is to see Charlie catch the ball of light, even though it moves to Ozzie’s right, Charlie cannot be directly opposite Ozzie at t = 1, but must be somewhere to Ozzie’s right.

6. As in TE 2, this situation requires Pete and Charlie’s train cars to slow down (as seen by Ozzie), the distance between Pete and Charlie to stretch (as seen by Ozzie), or a combination of the two. Whatever the combination, Ozzie will measure the speed of the ball of light as 110 fps (75 mph). At one extreme, the distance between Pete and Charlie would seem to stretch from 110 feet to 220 feet when Pete releases the ball, so that Ozzie sees Charlie catch the ball 2 seconds after Pete releases it, and 110 feet to Ozzie’s right. At the other extreme (or near it), the distance between Pete and Charlie would seem to stretch from 110 feet to, say, 111 feet when Pete releases the ball, so that Ozzie sees Charlie catch the ball just over 1 second after Pete releases it, and 1 foot to Ozzie’s right. The outcome is slightly different than that of TE 2 because Pete and Charlie are moving to the left instead of the right, while the ball is moving to the right, as before.

7. In the case of a real ball moving at 75 mph, the clocks would slow imperceptibly and/or the distance would shrink imperceptibly, maintaining the illusion that the formula of classical physics is valid — but it is not. It only seems to be because the changes are too small to be detected by ordinary means.

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TE 2 and TE 3 are rough expositions of how perceptions of space and time are affected by the relative motion of disparate objects, according to SR. I set the speed of light at the absurdly low figure of 75 mph to simplify the examples, but there is no essential difference between my expositions and what is supposed to happen to Ozzie’s perceptions of time and distance, according to SR.

If Pete and Charlie actually could move at the speed of light, some rather strange things would happen, according to SR, but I won’t go into them here. It is enough to note that SR implies that light has “weird” properties, which lead to “weird” perceptions about the relative speeds and sizes of objects that are moving relative to an observer. (I am borrowing “weird” from pages 23 and 24 of physicist Lewis Carroll Epstein’s Relativity Visualized, an excellent primer on SR, replete with insightful illustrations.)

The purpose of my explanation is not to demonstrate my grasp of SR (which is rudimentary but skeptical), or to venture an explanation of the “weird” nature of light. My purpose is to set the stage for some probing questions about SR. The questions are occasioned by the “fact” that occasioned SR: the postulate that the speed of light is the same in free space for all observers, regardless of their motion relative to the light source. If that postulate is true, then the preceding discussion is valid in its essentials; if it is false, much that physicists now claim to know is wrong.

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My first question is about the effect of a change in Charlie’s perception of movement:

a. Recall that in TE 1 and TE 2 Charlie (the observer in the train car) is unaware that the car is moving relative to the surface of the Earth. Let us remedy that ignorance by replacing the one-way mirror on the side of the car with clear glass. Charlie then sees that the car is moving, at a speed that he calculates with the aid of a stopwatch and distance markers along the track. Does Charlie’s new perception affect his estimate of the speed of a baseball thrown by Pete?

b. The answer is “yes” and “no.” The “yes” comes from the fact that Charlie now appreciates that the forward speed of the baseball, relative to the ground or a stationary observer next to the track, is not 75 mph but 150 mph. The “no” comes from the fact that the baseball’s speed, relative to Charlie, remains 75 mph. Although this new knowledge gives Charlie information about how others may perceive the speed of a baseball thrown by Pete, it does not change Charlie’s original perception.

c. Charlie may nevertheless ask if there is any way of assigning an absolute value to the speed of the thrown baseball. He understands that such a speed may have no practical relevance (e.g., to a batter who is stationary with respect to Pete and Charlie). But if there is no such thing as absolute speed, because all motion is relative, then how can light be assigned an absolute speed of 186,282 miles per second in a vacuum? I say “absolute” because that is what the speed of light seems to be, for practical purposes.

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That leads to my second question:

a. Do the methods of determining the speed of light betray an error in thinking about the how that speed is affected by the speed of objects that emit light?

b. Suppose, for example, that observers (or their electronic equivalent) are positioned along the track at intervals of 44 feet, and that their clocks are synchronized to record the number of seconds after Pete releases a baseball. The first observer, who is abreast of Pete’s release point, records the time of release as 0. The ball leaves Pete’s hand at a speed of 110 fps, relative to Pete, who is moving at a speed of 110 fps, for a combined speed of 220 fps. Accordingly, the baseball will be abreast of the second observer at 0.25 seconds, the third observer at 0.5 seconds, the fourth observer, at 0.75 seconds, and the fifth observer at 1 second. The fifth observer, of course, is Ozzie, who is adjacent to Charlie’s glove when Charlie catches the ball.

c. Change “baseball” to “light” and the result changes as described in TE 2 and TE 3, following the tenets of SR. It changes because the speed of light is supposed to be a limiting speed. But is it? For example, a Wikipedia article about faster-than-light phenomena includes a long section that gives several reasons (advanced by physicists) for doubting that the speed of light is a limiting speed.

d. It is therefore possible that the conduct and interpretation of experiments corroborating the constant nature of the speed of light have been influenced (subconsciously) by the crucial place of SR in physics. For example, an observer may see two objects approach (close with) each other at a combined speed greater than the speed of light. Accordingly, it would be possible for the Ozzie of my thought experiment to measure the velocity of a ball of light thrown by Pete as the sum of the speed of light and the speed of the train car. But that is not the standard way of explaining things in the literature with which I am familiar. Instead, the reader is told (by Epstein and other physcisists) that Ozzie cannot simply add the two speeds because the speed of light is a limiting speed.

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My rudimentary understanding of SR leaves me in doubt about its tenets, its implications, and the validity of experiments that seem to confirm those tenets and implications. I need to know a lot more about the nature of light and the nature of space-time (as a non-mathematical entity) before before accepting the “scientific consensus” that SR has been verified beyond a reasonable doubt. The more recent rush to “scientific consensus” about “global warming” should be taken as a cautionary tale for retrospective application.


  1. Space and time are hopelessly entangled not just locally (special relativity) but globally (general relativity). The paradox of accelerated observers who do not agree on time gets worse and worse. For instance, the Unruh effect implies that they don’t even agree on the vacuum. Whereas a freely falling observer sees a vacuum, the accelerated observer sees a bath of particles. In GR, I.e., curved space-time, gravity is curvature and even crazier things happen.

    We have enormous evidence to prefer relativity to Newtonian mechanics. And so far, we don’t have any credible alternatives.


  2. I am passingly familiar with the strangeness of GR, though some aspects (e.g., gravitation equivalent to acceleration, bending of light near a massive body) are easy enough to visualize. It seems to me that the difficult part of SR and GR, for a layperson, is that space and time (and therefore motion) are really one thing, and not the two things (or four dimensions) that our “common sense” perceives.


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