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Say I have a ball at 0.999999% the speed of light going past the Sun toward Earth. Now from the ball's reference frame, the distance between Earth and Sun is the same length as the ball's diameter. Why is the ball occupying the entire space between the Earth and Sun? What happened if a comet was between the Earth and Sun in the ball's path? Would the comet be inside the ball?!

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  • $\begingroup$ The ball does not magically "occupy" in one go. If there is a comet there it gets hit just like a automobile would hit something in the road as it moves along. There is no mystery there. $\endgroup$ Dec 22, 2013 at 19:05
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    $\begingroup$ 0.999999% of the speed of light isn't that much. Length of contraction isn't going to noticeable. $\endgroup$
    – Stan Liou
    Dec 22, 2013 at 20:53
  • $\begingroup$ @StanLiou I bet he meant 99.999999% instead of 0.999999%! Otherwise it really doesn't make any sense. $\endgroup$
    – Gonenc
    May 4, 2015 at 15:21
  • $\begingroup$ @gonenc yuh; I was just obliquely pointing that out. ;) $\endgroup$
    – Stan Liou
    May 4, 2015 at 21:33

3 Answers 3

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You should check out the barn paradox! It's about the same thing.

The problem is that there's an extra effect in relativity you haven't accounted for: observers don't agree on the order of events. For example, in the earth frame, we may have the ordering

  1. Back of ball at sun
  2. Something passes between sun and Earth
  3. Front of ball at Earth

In the ball's frame of reference, 1 and 3 happen at the same time. So it appears to mean that 2, which is between 1 and 3, must also happen at that time. But there's actually no such guarantee! The ordering may be

2'. Something passes between earth and sun

1'/3'. Ball occupies space between earth and sun

which resolves the paradox.

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If you change the question slightly and instead of the ball consider a spherical region of space, then your spherical region of space would indeed include the Earth, the Sun and and everything in between such as Venus, comets, asteroids etc.

Well, not quite. Remember that the contraction is only in the direction of motion so the Earth, Sun etc are flattened into disks from your point of view. Your spherical region of space only includes the centres of the disks and it wouldn't include any other celestial bodies unless they lay within one ball radius of the line the ball is moving along. The probability of this is astronomically (pun intended) low.

If you swap the ball shaped region of space for a real ball you get a huge explosion as soon as your ball hits the first thing in its path.

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  • $\begingroup$ What about the space that the ball occupies? The fairly large one... I mean from the ball's perspective it has a length of 1 meter just like the distance between Earth and Sun... $\endgroup$ Dec 22, 2013 at 19:12
  • $\begingroup$ Yes, the ball occupies a large volume of the other frame, but it's a solid ball. You can't have things inside it. Assuming the ball starts out in space as it nears the Solar System it doesn't engulf the Earth. It would just hit it and explode. Interestingly, the Earth dwellers would see the length of the ball decreased i.e. the ball volume would decrease. You're going to have to reach for your Lorentz transformations to see why ... $\endgroup$ Dec 22, 2013 at 19:17
  • $\begingroup$ @JohnRennie "so the Earth, Sun etc are flattened into disks from your point of view." That depends on the direction in which they move: an approaching sphere appears flattened, a receding sphere appears elongated, and a passing sphere appears rotated. See Terrell rotation. $\endgroup$
    – Pulsar
    Dec 22, 2013 at 19:19
  • $\begingroup$ How can the same ball be near the Sun and not near the Sun at the same time? $\endgroup$ Dec 22, 2013 at 19:19
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    $\begingroup$ @BinaryBurst: you are making the common misstep of applying intuition to relativity. To understand what is going on you need to do the sums. Search the site for lorentz contraction for many questions exploring this area. $\endgroup$ Dec 22, 2013 at 19:30
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This sort of effort at constructing a paradox usually gets answered by correctly defining the frames of reference and also considering the relativity of simultaneity. It's not just length contraction or expansion that is needed for proper resolution but also consideration of what it means to say "at the same time". I don't think you have defined the various viewpoints fully. From the Earth frame, the "ball" would really appear to be a disc and there would be plenty of space to avoid the ball, From the ball's frame, the comet would either be passing through the region of space before or after the ball was "there".

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