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In summary, clocks on a spacecraft will appear to run very slowly, but the motion of the whole spacecraft will still appear to move close to the speed of light.f

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Wrong! (Well, at the very least: imprecisely stated).As a spaceship approaches the speed of light time itself slows down for everyone and everything on the spaceship

Precisely what you said. The spaceship will whoosh past at almost the speed of light and all the clocks on the spaceship will appear to run very, very slowly.What would this look like from the earth?

The people on the spaceship themselves wouldn't notice anything out of the ordinary, except that when the Earth whooshes by at almost the speed of light all the clocks on it would appear to run very, very slowly.

So let me stress again: clocks which are stationary with respect to their observer (such as clocks on the spaceship viewed by a passenger, as well as clocks on Earth viewed by an inhabitant) do not run slower or faster.

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That wasn't what I said... my question was- would the motion of the whole spacecraft actually appear to slow down? (sorry if this was unclear)Precisely what you said. The spaceship will whoosh past at almost the speed of light and all the clocks on the spaceship will appear to run very, very slowly.

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except that when the Earth whooshes by at almost the speed of light all the clocks on it would appear to run very, very slowly.

Actually, time dilation is a coordinate effect. Visual appearances are given by the Doppler effect, so moving clocks can to run fast or slow, depending on the relative direction of travel. I worked a couple of examples in

https://www.physicsforums.com/showthread.php?p=2186296#post2186296.

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... my question was- would the motion of the whole spacecraft actually appear to slow down? (sorry if this was unclear)

The answer is no, the spaceship would still appear to move close to the speed of light.

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my question was- would the motion of the whole spacecraft actually appear to slow down? (sorry if this was unclear)

Going a little deeper into your question...

You talk about a spacecraft "approaching the speed of light" (as measured in the Earth frame, I guess). So it is accelerating, isn't it? If so, the answer is contained in your question: as you have been told, in the Earth frame it's always going faster and faster.

But maybe what you are implicitly asking also is what peculiar thing SR says with regard to this situation as opposed to Galilean relativity. If so, the peculiar thing is that in the Earth frame the spacecraft never reaches the speed of light, no matter how much it accelerates.

You can think of your example with a variation that clarifies that postulate: initially there's a mother craft traveling at u = 0.5 c with regard to the Earth from which a second craft takes off at v = 0.5 c with regard to the mother craft; then a third craft accelerates away at w = 0.5 c with regard to the second craft; and so on.

In Galilean relativity you would say that the second craft travels wrt the Earth at u + v = 0.5 c + 0.5 c = c (the speed of light) and the third craft travels wrt the Earth at (u + v) + w = 1.5 c and so on.

Instead, in SR the formula for addition of velocities is (u + v)/(1+uv/c^2).

If you make the calculations with this formula, you will notice that the second craft travels wrt the Earth not at 1 c but at 0.8 c; the third at 0.92 c, the fourth at 0.97 c, the fifth at 0.99 c and succesive ones at always higher fractions of c but never c.

Thus "somehow", in a very loose way, you could say that the motion of the accelerating spacecraft is "slowed down": it is slowed down vis-à-vis what you'd expect to hear as solution to the same problem in a classical framework...

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I have been thinking of my own answer to the problem and found an objection. This is the case of a spacecraft chasing a light beam. The spacecraft is constantly accelerating but, in accordance with SR postulates, it never catches up with the light beam. Another way to present the same problem is the variation I proposed above: a spacecfraft is traveling wrt the Earth at v = 0.5 c, a 2nd spacecraft takes off from the first at v = 0.5 c wrt to the former and so on and so on, like in a game of Russian dolls. Following the relativistic law for the addition of velocities, the new crafts travel closer and closer to the speed of light (wrt the Earth) but never reach c.

So far, so good. But then I've thought of Zeno's paradox. You know, Achilles is chasing the tortoise. The tortoise starts with a certain advantage (say X metres) but Achilles travels at double the speed of the tortoise wrt the ground. Achilles wonders: while I have traversed a distance of X metres, the tortoise has traversed X/2. While I traverse the distance X/2, the tortoise covers a distance X/4 and so on. So I can never reach the elusive tortoise.

The standard solution is: Poor Zeno had trouble with the idea of an infinite series. Today we know that the addition of infinite terms leads, in the limit, to a finite number. He didn't have that state of the art mathematical solution. Had he been enlightened with this knowledge, he would not have created so much trouble.

That did convince me with regard to Zeno's problem. But if we applied the same solution to the accelerating craft (or the series of Russian crafts taking off from their mother crafts), shouldn't we say that modern maths rules that some craft reaches the light beam, because the addition of an infinite series gives out a finite result...?

So far, so good. But then I've thought of Zeno's paradox. You know, Achilles is chasing the tortoise. The tortoise starts with a certain advantage (say X metres) but Achilles travels at double the speed of the tortoise wrt the ground. Achilles wonders: while I have traversed a distance of X metres, the tortoise has traversed X/2. While I traverse the distance X/2, the tortoise covers a distance X/4 and so on. So I can never reach the elusive tortoise.

The standard solution is: Poor Zeno had trouble with the idea of an infinite series. Today we know that the addition of infinite terms leads, in the limit, to a finite number. He didn't have that state of the art mathematical solution. Had he been enlightened with this knowledge, he would not have created so much trouble.

That did convince me with regard to Zeno's problem. But if we applied the same solution to the accelerating craft (or the series of Russian crafts taking off from their mother crafts), shouldn't we say that modern maths rules that some craft reaches the light beam, because the addition of an infinite series gives out a finite result...?

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Sure if we had an infinite number of spacecraft with an infinite source of energy. But that's the reason it's impossible, because mathematically, the amount of energy required is infinite.

So far, so good. But then I've thought of Zeno's paradox. You know, Achilles is chasing the tortoise. The tortoise starts with a certain advantage (say X metres) but Achilles travels at double the speed of the tortoise wrt the ground. Achilles wonders: while I have traversed a distance of X metres, the tortoise has traversed X/2. While I traverse the distance X/2, the tortoise covers a distance X/4 and so on. So I can never reach the elusive tortoise.

The standard solution is: Poor Zeno had trouble with the idea of an infinite series. Today we know that the addition of infinite terms leads, in the limit, to a finite number. He didn't have that state of the art mathematical solution. Had he been enlightened with this knowledge, he would not have created so much trouble.

That did convince me with regard to Zeno's problem. But if we applied the same solution to the accelerating craft (or the series of Russian crafts taking off from their mother crafts), shouldn't we say that modern maths rules that some craft reaches the light beam, because the addition of an infinite series gives out a finite result...?

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Hey thanks for the replies

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Only some infinite series have finite sums. Others have infinite sums, and some don't even have a sum!because the addition of an infinite series gives out a finite result...?

Do not confuse "doesn't have a sum" with "sums to zero"

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Sure if we had an infinite number of spacecraft with an infinite source of energy. But that's the reason it's impossible, because mathematically, the amount of energy required is infinite.

Ok.

Only some infinite series have finite sums. Others have infinite sums, and some don't even have a sum!

Do not confuse "doesn't have a sum" with "sums to zero"

Ok.

I didn´t have a clear idea of what the comment meant, now it is clearer. Really, I did not want to discuss the postulate that the light beam cannot be overrun. It was only that I had always felt uncomfortable with the idea that Zeno’s paradox is solved with pure “mathematical technology”. The teaching I draw is:

- In Zeno´s example Achilles and the tortoise are traveling at constant velocity (as if inertially) or, if Achilles is accelerating, it is accelerating within the limits of what is physically possible. That is why what happens (Achilles reaches the tortoise) is expressed with a mathematical series that has a finite sum.

- Instead, in the SR example, the chaser, in order to reach the light beam, for some physical reason, would need an infinite amount of energy. That is why the series does not have a finite result…

You may ask, “what do you mean by a physical reason”? Well, I do not know and I admit it may not be necessary to know it. But I suppose you need at least to “postulate” a reason in order to choose to employ one mathematical tool or the other… More or less right…?

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As regards your scenario where each ship is launched with a speed of 0.5c with respect to the previous one. I have not done the mathematics, I am much too lazy ( I love mathematical concepts but hate computation ) but you would just have to work out the successive terms of the sequence using the velocity addition formula. You would of course not sum the terms of the sequence. You would have a series of increasingly large terms and I suspect that in the limit the individual terms of the sequece will approach c as a limit from below, with respect to the first ship or first observer.

Its fairly obvious that whatever figures you use in such compouded velocities the physics and mathematics of the situation dictate that c, with respect to the original observer, will not be exceeded and the limit of the sequence may for some values of the increment be less than c. Sequences and sums of series can give many surprising and elegant results, although this is not a surprising one. I am sure a mathematician can correct and expand on this as necessary.

Matheinste

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