Hi all. Up until now i thought i had a pretty good grasp on basic relativity. That was until i stumbled over this paradox.
Imagine a huge frozen lake. Now imagine a circular hole in this lake. Next to the hole lies a piece of ice exactly the same shape as the hole.
Two hightech eskimoes comes by. One of them fits his newly designed rocket engine to the loose piece of ice and accelrates to something near the speed of light. The other eskimoe remains back at the hole and observes.
The man racing along on the free piece of ice is now on a collision course with the hole (in the plane of the lake). Now the question is. Will he fall in when he travels over it (read: will the piece fit the hole)?
For the man standing still (with respect to the hole) there is no doubt. The rocket driven ice looks contracted and so is smaller than the hole. The racing eskimoe will fall in, and he would have no matter the speed.
For the eskimoe racing along the ice the hole appears to be contracted along his vector of movement. He will expect to race over the hole.
Who is right ?
Im pretty sure the answer is something like the “barn and pole” paradox. But i just cant figure it out.
This has been bugging me all day, so a huge thanks to anyone who can explain.
You’re right. It is barn and the pole, only this time with ice it seems that you have one object (the lakehole) whereas with the barn it was clear that you had a front door and a back door and they either were simulateously open or simultaneously closed in one reference frame whereas they weren’t in the other. Seem confusing? Well, at the heart of this is the concept that one cannot be in two places at the same time so there is some ambuiguity as to the “when” of the events.
Work it out with the front end and the back end of the hole and look for the contradiction in simultaneity. It’s subtle, to be sure, but what you will find is that either the ice is too large for both observers to see it in the front part of the hole and the back part of the hole over the same proper time interval or it is small enough to be seen as such. Also for the simultaneity to work as being subjective, you have to take into account the fact that the ice is not allowed to “instantaneously” drop all at once into the hole. Nor is it allowed to suddenly change reference frames either (less you plunge ahead into more paradoxes). These things conspire to maintain the laws of physics.
If you keep all these facts in mind, I think you’ll find there really is no paradox.
The version of the barn paradox I heard was this: There is a barn exactly the size of a rocket, which is passing through the barn at relativistic speeds. The observer in the barn sees the rocket length-contracted, and thus can close the barn doors simultaneously while the rocket is completely inside the barn. The observer in the rocket sees the barn length-contracted, so at no time can the rocket be completely inside the barn. So the question is, what does the observer in the rocket observe when both barn doors are closed
The solution of this paradox is that the observer in the rocket sees the barn doors close at different instants in time because they are in different locations. So the back door of the barn closes and opens before the front door of the barn closes.
So similarly, the moment when the earth-bound eskimo believes that the front and back of the ice are within the bounds of the hole is not a single moment for the rocket-bound eskimo. What seems co-temporal to the earth-bound eskimo does not seem co-temporal to the rocket-bound eskimo, so again, to the rocket-bound eskimo, there are two different instants, one where the front of the ice is over the hole, and one where the back of the ice is over the hole, and for the earth-bound eskimo, these are the same moment. This happens because of the diameter of the hole.
So clearly, at relativistic speeds the ice is not going to “fall into” the hole, but it still seems puzzling. What would happen if the ice lake was very close to an event horizon, and the gravity was strong enough to pull the ice into the hole before it passed over the hole from the perspective of the planet-bound eskimo? It seems like before the ice was completely over the hole it would begin to pivot on a balance point. It seems the observers would agree on the pivot point, since it is derived from a ratio of the length. But once the ice actually underwent acceleration due to gravity I have no idea what would happen. Any physicists?
Again, the event horizon problem is invovling changing of reference frames. You are necessarily changing the speed at which your object is travelling. This causes our laws of special relativity to fly out the window. If the ice was going over a blackhole, it would appear to an observer who was in constant acceleration out of the blackhole, that the ice was accelerating into the blackhole… thus causing it to length contract and fit nicely. Likewise, the ice would shoot happily into the black hole in its reference frame. If the ice doesn’t make it into the black hole than its velocity is great enough to only take a scattering geodesic (wrt the observing accelerating their way away from the blackhole) and so it wouldn’t go into the black hole for either the observer or the ice block.
Well, this is a bit harsh. What is better to say is that you have to take into account that the reference frame isn’t moving at a costant velocity anymore, so your barn-paradox considerations cannot be made (as was rightly pointed out).
