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Suppose you are going to a distant star, and there is an observer on earth.

Usually the answer to the question how long would it take to travel to a distant star is explained this way :

  • You cannot go faster than the speed of light
  • Time slows down for you relative to the observer
  • It explains why you feel you go faster
  • You do not go faster than light because of length contraction

After seeing this answer to the question "do we know why there is a speed limit in our universe?", this is now the way I see it :

  • You can actually go at any 'speed' (or rather rapidity)
  • Time slows down for you relative to the observer
  • It explains why the observer feels you go slower
  • No need of length contraction to justify light speed limit

So for any observer, speed is indeed limited by the speed of light, but for the traveler there is no limitation. It makes more sense for me.

Is it wrong to see it this way?

If it is not, is there any reason for the first explanation to be commonly used?

EDIT : Thank you for your answers ! I understand now that the way I see things is not a possible interpretation of the theory of relativity. I still wonder if it is absurd though.

You point me out that rapidity is not speed. I thought that rapidity was at least proportional to the speed of a traveler relative to this same traveler. Is it wrong? If a traveler has a constant acceleration, even though an observer will see the traveler accelerate less and less, is the acceleration constant for the traveler?

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  • Did you mean to put "Time slows down for you relative to the observer" in both lists? – Pato Raimundo Jan 24 '17 at 11:38
  • Yes. The difference is in the interpretation of this relative time dilation. The usual explanation is from the point of view of the observer : you cannot go faster than light but you feel you go faster than light. My explanation is from the point of view of the traveler : I can go at any speed I want but you feel my speed is limited by the speed of light. –  Jan 24 '17 at 14:15

4 Answers4

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Both, time dilation and length contractions are consequences of the postulates of special relativity, so it is incorrect to say that time dilation is a consequence of length contraction. It is also wrong to say that there is a speed limit because of length contraction or of time dilation, it is the other way around, they are both consequences of the constancy of $c$. Length contraction and time dilation by themselves are not enough to show that c is a constant, you need additional assumptions to show that.

In your example length contraction is useful to understand why a fast traveller can reach another star in a time that, for him, is less than the time that light would take to travel from Earth to the star if he were at rest (that is, for an observer on Earth). For the traveller the distance between Earth and the star has contracted.

To put it in your own terms:

-Nobody can see another observer (or anything) moving faster than the speed of light.

-one consequence is that any observer sees that time slows down for an observer moving relative to him

-another consequence, length contraction, explains why, for the moving observer, it takes less time to travel than it would take light to travel if he were at rest.

  • Thank you ! So if I understand you, I can't see it the way I described it because the theory of special relativity does not explain things in that order (to keep it simple). So the question that remains : Supposing the traveler has a constant acceleration, is this acceleration constant from the point of view of the traveler? –  Jan 24 '17 at 13:52
  • The traveller can experience constant acceleration, which for him means what he reads in an accelerometer (it is called the proper acceleration). He will not see the Earth accelerating (if you define it as dv/dt) in the opposite direction at that rate though. He will see that Earth accelerates less and less to asymptotically reach c. –  Jan 24 '17 at 14:27
  • Alright I think I finally understood :D Thanks again. –  Jan 24 '17 at 14:47
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The rapidity is not a speed. It is a useful variable to use because rapidities add in a linear way while velocities have to be added using the relativistic equation for velocity addition. However the rapidity does not measure distance travelled per unit time. It is certainly true that the rapidity can become infinitly large, but the physical meaning of this is just that the speed is approaching the speed of light.

There is a discussion of the origin of time dilation in What is time dilation really?

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John Rennie
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Although Rapidity is not speed (distance/unit time) - It is true that regardless of your speed you can continue to accelerate. Consider any small acceleration relative to a object O in the travellers frame will initially seem classical. I think this describes your idea that for the traveller there is "no limitation" because acceleration is always possible for them. This answer is based on that interpretation of your question.

Never-the-less, your distant observer with relative velocity near c will see a different and much smaller difference in velocity between the initial traveller and the object O he just accelerated away from. (This difference will tends to zero as the travellers relative speed tends to c.)

This smaller $\delta v$ is consistent with the distant observer seeing everything in the travellers frame as being length contracted - which is to say that as the traveller's "spaceship" appears length contracted they don't have to go as fast to get from one end of it to the other.

As you point out the distance observer will never see the traveller exceed c.

I hope that helps.

JMLCarter
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No need of length contraction to justify light speed limit

Wrong way round.

Once you say that the speed of light is constant for all observers, you end up having to admit that time dilation and length contraction occur.

The speed of light was known to be constant for all observers in Einstein's day and that's one of the main reasons that prompted his development of the special theory of relativity.

So for any observer, speed is indeed limited by the speed of light, but for the traveler there is no limitation. It makes more sense for me.

Is it wrong to see it this way?

Yes, it's wrong.

As you said yourself (and other's have explained) rapidity is not speed. Your mistake is ignoring your own realization that this is so. One of the hardest things to do in modern physics is let go of your mental anchor of common sense. Relativity (and quantum theory) force you to dump a lot of your common sense ideas and that's hard to do. Everyone finds it hard.

If it is not, is there any reason for the first explanation to be commonly used?

Because It Works.

Lots and lots of physical theories are proposed every day. They range from the daft to the much rarer brilliant.

To be rated brilliant a theory must be proven to match reality by carefully vetted and repeated experiments and observations.

The theories of relativity have proven to be good matches to reality within the context of their intended uses. Most theories will fail in some way.

That's why we accept the mainstream view. Not because we all voted on it or something like that, but because it is proven to work within it's intended context better than the alternatives.

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StephenG - Help Ukraine
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