The Garden Fence Paradox
Come on, let’s take a ride on the Einstein express!
I got the inspiration for this Gedankenexperiment from a poem by nonsense-lyricist Christian Morgenstern, titled “Der Lattenzaun” (The Picket Fence), that goes like this:
One time there was a picket fence
with space to gaze from hence to thence.
An architect who saw this sight
approached it suddenly one night,
removed the spaces from the fence,
and built of them a residence.
The picket fence stood there dumbfounded
with pickets wholly unsurrounded,
a view so loathsome and obscene,
the Senate had to intervene.
The architect, however, flew
to Afri- or Americoo.
© Knight, Max, http://www.alb-neckar-schwarzwald.de
We will not build a house, we’ll use the gaps in another way. Our fence is rather special: It’s infinitely long. If that’s too long for you, then make it “arbitrarily long”. In practice this makes no difference. Its slats are 10 cm wide, the gaps slightly larger, 15 cm. You may exchange centimeters with “inches” or “yards” or “cubits”, whatever you like.
Now we get a ball with a diameter of 10 cm. It fits well through the gaps. And if we stand very near to the fence, we can easily throw the ball through any gap to the other side of the fence.
Along the fence, fairly close to it, are tracks that are as long as the fence. The Einstein Express runs on them, a very special train. It was designed by Albert Einstein to illustrate his ideas. What’s special about it: It may accelerate until its speed is close to that of light.
We get on the train and drive off slowly. We can still throw our ball through the gaps in the fence without any problem; we just have to take into account the deflection angle caused by airspeed. But, as the saying goes, it will be “negligibly small” if we let the train pass very close to the fence. After all, it’s only a thought-experiment, not necessarily possible in reality.
Now we accelerate to a speed close to the speed of light. And lo and behold: We can no longer throw the ball through the gaps in the fence, because, according to length contraction, the fence — and with it the space in between — is getting smaller: The ball no longer fits through.
This in itself is not surprising. We cannot expect the same state (we are used to) to prevail under extreme conditions.
But now let’s do something very simple: we change our point of view. Instead of riding along, we now crouch behind the fence and hand the ball to the conductor of the Einstein train with the instruction to throw it through the fence at high speed. And that’s easy to do: Because according to Einstein’s principle of relativity, we experience the same thing behind the fence as we did on the train: the train and everything that travels on it shrinks lengthwise. The ball gets thinner and now fits easily through the gaps in the fence:
The paradox starts: Changing view = changing reality
What happened there? Has changing the point of view made something possible that was previously impossible? No, we forgot something only recently detected: An object (here: the ball) rotates before passing through the space between the garden fence. According to Sexl/Schmidt (“Raum — Zeit — Relativität”. Vieweg, Braunschweig 1991) it looks like this:
Sorry for the satire …
Explanations are numerous. But don’t forget:
- There is no necessity for synchronization. You need synchronization for two observers that communicate with each other. But here, there is only one observer.
- You cannot invoke general relativity. Although we talked about “acceleration”, length contraction is solely dependent on speed.
- You cannot shrug off these contradictions by declaring everything as “not real”, “apparent”, “illusory”. If that’s the case, any observation gets absurd. Besides, for illusions you don’t need an elaborate physical theory.
- As you see: Rotating the ball doesn’t help!
More than 100 years since inception of this phenomenon, there are still articles published in renowned journals explaining the paradox, which is no paradox, but a logical inconsistency. If such a simple situation needs more than 100 years not to be resolved — is it really that simple? And why does it need mostly very complicated explanations?
This was an excerpt from my book “Einsteins einmalige Einsichten: Die Relativitätstheorien und wie es dazu kam”, Books on Demand 2022.
