The Problem
Meet you.
You’re an aspiring young roof tiler. Your boss is horrible,
your coworkers are stuck up veterans, and you make below the minimum wage,
despite working overtime. But you need to pay off that college debt from your
master’s degree in theoretical physics, so you put up with it all anyway.
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| This is You (representation) |
When you arrive home one night, you realize that the new
ladder you bought didn’t fit into your garage; it was simply too long, and the
folding mechanism was broken. Since you’re already behind on your electrical
bills, you can’t afford to have it stolen in the middle of the night. Getting
it in the garage is your top priority, and you’d do anything to do it.
But then you have an idea.
You tap into the grueling 8 years of your life you spent studying
theoretical physics and formulate a solution. You understand that according to
the principles of special relativity, when an object is moving relative to
another observer, that object will be horizontally compressed in the direction
that the object is moving. This is known as Lorentz contraction, and it can be
calculated exactly. The closer the object moves to the speed of light, the more
compressed the object will become.
Operating under this principle, you call two of your best
buddies, Bob and Mark, to come and manage the doors on either side of the
garage. The plan is that you will accelerate the ladder close to the speed of light
towards the garage. You know that relative to Bob and Mark, you and the ladder
will contract in length to the point where the ladder could fit inside the
garage. Once you and the ladder enter the garage, the two of them will slam the
two doors shut, trapping the ladder inside.
But no! Mark, who also has a master’s degree in theoretical
physics, says. If you do that, sure, the ladder’s length will contract relative
to us. But relative to you with the ladder, the garage will be the one that
contracts in length! How do you expect us to close the doors when the garage
will be even smaller than before!
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| What you think Mark will see |
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| What Mark thinks you will see |
But you know that there can only be one possible outcome:
either the ladder will fit, or the ladder won’t fit. You can’t have two
conflicting outcomes in the same timeline. So that being said, will your method
work in getting the ladder inside the garage?
The Solution
The solution to the apparent paradox lies in the relativity
of simultaneity. This means that what one observer (Bob or Mark) considers to
be two simultaneous events may not, in fact, be simultaneous to another observer
(you). When you want the ladder to “fit” inside the garage, you essentially want,
at one specific time, for the position of the back of the ladder and the
position of the front of the ladder to be both inside the garage simultaneously.
But since simultaneity is relative, both you and Mark can disagree whether or
not the ladder fits without contradiction.
To imagine this, let’s say that at one point, Mark and Bob
both close the doors and open them immediately again as the ladder passes
through the garage. In the frame of reference of Mark and Bob, the ladder would
fit, as they could close the doors. From your perspective however, it would be
rather different. You would first see the front end of the ladder approach the
exit door of the garage. The exit door closes, and opens again allowing for the
ladder to pass through. At the same time, the back of the ladder passes through
the entrance door, which closes and then opens. In this case, the two doors
closing were not necessarily simultaneous relative to you, and the ladder didn’t
fit inside the garage.
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| From your frame of reference, the doors will not necessarily close at the same time |
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| From Mark and Bob's frame of reference, the doors will close simultaneously around the ladder |
So in the end, will you be able to succeed in locking the
ladder into the garage? Well to do that, it means that the exit door will have
to remain closed and the front door will also have to remain closed as well. Let’s
take a look at this from the frame of the garage. If we assume that the exit
door is immovable, it means that once the ladder hits the exit door, it will
instantaneously stop. By this time, the entrance door has closed, and the
ladder is stuck inside the garage. Since its relative velocity is now zero, and
is not length contracted, it is now longer than the garage. The ladder must now
bend, snap, or explode. So the question here, this time, is how did we ever get
to close the doors to trap it inside in the first place?
The answer again lies with the relativity of simultaneous
events. In the garage’s frame, all parts of the ladder decelerated at the same
time. However, in your frame of reference, the ladder accelerates sequentially.
First the front decelerates, and then the middle, and then the back end of the
ladder. By the time the back end of the ladder has decelerated, the entire ladder
is already in the garage, after which you can close the back door. Only issue
is, after this, that ladder will have to bend or snap to accommodate its new
restrictive garage.
So is it a viable method?
Well assuming you can accelerate to about 95% the speed of
light, have an immovable garage, extremely fast and hyper coordinated people, and
too much time on your hands…
Probably.
