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I have been reading the section on Relativity in the ninth edition of University Physics by Young and Freedman. They include the following proof that no observer can move at the speed of light.

University Physics states:

Einstein's second postulate immediately implies the following result: It is impossible for an inertial observer to travel at c, the speed of light in vacuum. We can prove this by showing that travel at c implies a logical contradiction. Suppose that [a] spacecraft S' [...] is moving at the speed of light relative to an observer [E] on Earth, so that [the velocity of S' with respect to E equals c]. If the spacecraft now turns on a headlight, the second postulate now asserts that the Earth observer E measures the headlight beam to be also moving at c. Thus this observer measures that the headlight beam and the spacecraft move together and are always at the same point in space. But Einstein's second postulate also asserts that the headlight beam move at a speed c relative to the spacecraft, so they cannot be at the same point in space. This contradictory result can be avoided only if it is impossible for an inertial observer, such as a passenger on the spacecraft, to move at c.

The main point of the proof is that the postulate implies an inconsistency in the relative location of objects in space. The Earth observer sees the light beam in the same location as the spacecraft, but the observer in the spacecraft sees the location of the light beam to be ahead of the spacecraft.

But couldn't a similar line of reasoning be used to show that no observer can move at the speed 1 m/s if the postulate is to hold? In this case, the Earth observer sees the light beam ahead of the spacecraft, but the observer in the spacecraft would see the light beam in a location further ahead of the spacecraft. There is an inconsistency in the relative locations of the light beam and spacecraft.

I found that the 1981 book Discovering Relativity for Yourself by Sam Lilly gives a proof with similar reasoning to show an observer cannot move at the speed of light.

What am I failing to understand in this line of reasoning? Why does it apply to observers moving at the speed of light but not other speeds?

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  • $\begingroup$ I do not understand it at all. What is the soeed of headlight beam if no the speed of the headlight moving with the ship? In other words what should be ahead the observer in the ship according to himself? $\endgroup$ – Alchimista Mar 10 '18 at 19:33
  • $\begingroup$ Have you tried drawing a spacetime diagram of both cases? $\endgroup$ – Alfred Centauri Mar 10 '18 at 19:39
  • $\begingroup$ @Alfred Centauri I have just started learning about the details of relativity. I do not know how to draw a proper spacetime diagram. $\endgroup$ – Dschumanji Mar 10 '18 at 21:39
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In the case of $1 m/s$, the earth observer would still see the spacecraft and beam of light having a certain growing distance between them, which is coherent with the argument that the observer in the spacecraft measures the light beam having any positive velocity ($c$) relative to him. But in the case of the spacecraft having a velocity of $c$, the earth observer isn't observing any growing distance between them, but the observer in the spacecraft must see the distance growing, because the light beam moves with velocity $c$ relative to him.

Basically the difference is, that if the spacecraft is moving even at $0.99999*c$ the earth observer and the observer in the spacecraft, both see the distance growing.

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    $\begingroup$ I guess I would just add to this: the point is that in relativity that two observers can measure different non-zero distances between two objects, but if two objects are literally touching at the same point that must be the same for all observers to avoid a logical contradiction. $\endgroup$ – Rococo Mar 10 '18 at 19:53
  • $\begingroup$ The line of reasoning you introduced by tyler_house and @Rococo makes sense, but it seems to me that more needs to be assumed about measuring the speed of light in an inertial frame of reference than the original postulate states. Do either of you find the Young and Freedman proof somewhat lacking? $\endgroup$ – Dschumanji Mar 10 '18 at 21:50
  • $\begingroup$ I believe Young and Freedman’s proof is false. The observer “moving” with the speed of light will see no time pass regardless of how much time passes for the “stationary” observer. Since no time passes for him, the light beam will not advance in front of him. This is exactly what the “stationary” observer sees, hence no contradiction. $\endgroup$ – Polhode Mar 11 '18 at 3:00
  • $\begingroup$ Dschumanji, the proof you are citing is not rigorous of course and is just trying to support the most widely recognized opinion on that matter. There still are debates whether it is possible to move with the speed of light and as @Polhode mentioned, whether time can stand still, if you are moving with the speed of light. The reason this debate even exists, is that when you put the speed of light as the relative velocity to the stationary observer you get singularities in the Lorentz transformations and other formulas of Special Relativity. $\endgroup$ – tyler_house Mar 11 '18 at 7:57
  • $\begingroup$ @tyler_house, I feel the same way about the proof. I suppose should just look at it as a motivation to abandon the Galilean transformation as reference frames approach the speed of light. It would be better to understand how the postulates lead to the Lorentz transformation and then look at the consequences of this new transformation. $\endgroup$ – Dschumanji Mar 11 '18 at 16:03
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Young and Freedman (YF) here have a true conclusion but arrived at by an argument that does not work. The conclusion here is that it does not make any sort of physical sense to apply the phrase "observer" or "frame of reference" to something moving at the speed of light. This is correct. However, the argument fails as follows. What YF name "Einstein's second postulate", i.e. the one about the speed of light, is a postulate that applies to observations made with respect to any physically allowed frame of reference. It has nothing to say one way or the other about what would be observed in "a frame of reference moving at the speed of light" (i.e. moving at that speed relative to some allowed inertial frame) because there is no such thing as "a frame of reference moving at the speed of light". You can say the words, but as soon as you try to give meaning to them, by stating how distance and time are to be measured or indicated in such a "frame", you are faced with infinities and impossibilities, such as rigid bodies contracted to zero length, and clocks that do not advance at all. The "impossibility" here is then in giving meaningful definitions of space and time measurements using instruments of no length and no evolution.

It is a good and useful lesson to know that the notion of a frame traveling right at the maximum speed is an unphysical notion, but the way to prove this is not the way suggested by Young and Freedman. Their argument does indeed suffer from the kinds of defects that commenters here have pointed out.

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