Paradox of the car and the driver in SR

[Hak]

I would like to propose a relativistic problem that I cannot understand.

A car of rest length L travels at close to the speed of light towards a garage, of length L, which also has an opening at the bottom. For a person next to the garage door, the car has a length less than L, due to the contraction of lengths, so it is possible to close both garage doors at the same time with the car inside. According to the driver, however, it is the garage that has a length less than L, due to the contraction of lengths, and therefore it is impossible for the two doors to be closed at the same instant with the car inside the garage. Are the two conclusions contradictory? Can you provide an explanation?

The only explanation I can think of is that if you close the two doors at the same time in the system in which the car is moving, it means that you don't close them at the same time in the system of the car. Even if the car is longer than the garage, in the driver's system the back door of the garage closes late, and the car (or at least the half further back of the car, if the front has crashed) meanwhile keeps moving.
If the explanation of what happens in the system in which the car moves is correct, one can place a car at rest long L in a garage smaller than the car. It stands to reason that for this to be possible, the car must somehow get off the ground, at least when you try to stop it. But why should the two doors close non-simultaneously for the driver? And why should the front one close first and not the rear one first?
Thank you for any clarification.

 

[Orodruin]

The only explanation I can think of is that if you close the two doors at the same time in the system in which the car is moving, it means that you don't close them at the same time in the system of the car.

Spot on. This is relativity of simultaneity in a nutshell.

 

[Hak]

Spot on. This is relativity of simultaneity in a nutshell.

Thank you very much. So this would be the explanation for this paradox? Isn't it a very specific paradox?

 

[Hill]

Looks the same as this: https://en.wikipedia.org/wiki/Ladder_paradox

 

[Orodruin]

If the explanation of what happens in the system in which the car moves is correct

Both frames are correct. All frames will agree on physical things such as if the doors closed before/after the car passed. Where they do differ is in which events are simultaneous.

one can place a car at rest long L in a garage smaller than the car.

Only in that frame will the car be fully inside the garage. Unless it stops moving (which comes with its own problems) it will quickly crash (or exit the garage if you open the back door again.

And why should the front one close first and not the rear one first?

That is simply how the relativity of simultaneity works between those two frames.

Thank you very much. So this would be the explanation for this paradox? Isn't it a very specific paradox?

It is a pretty standard ”paradox”. I put it in quotations because it really only seems a paradox if you don’t understand the relativity of simultaneity.

”Fits in the garage” is a statement that refers so the entire car fitting at some given time, ie, simultaneous events of the car’s front and back are within the garage doors. Since what simultaneous means differs between frames, it is just natural that the fit or not conclusion may change between frames.

 

[Orodruin]

Looks the same as this: https://en.wikipedia.org/wiki/Ladder_paradox

It is. Different people just like presenting it using different objects. Different names for the same thing.

 

[Nugatory]

Thank you very much. So this would be the explanation for this paradox? Isn't it a very specific paradox?

It’s a very common teaching example, often described using a barn or a tunnel instead of a garage, and a pole, ladder or train instead of a car - but the key idea is the two doors closing at the same time in one frame but not the other. Google for “pole-barn paradox” and “ladder paradox relativity” and you’ll get a ton of examples.

I find that the best way of understanding it is to draw a Minkowski spacetime diagram plotting the four relevant paths through spacetime: path of the front end of the car, path of the rear end of the car, path of the near door, path of the far door. Two of these will be vertical lines, two will be parallel and slanted. I’m not going to bother drawing/posting one because that Google search will find several bazillion already online. Some will even be here at PhysicsForums.

 

[hutchphd]

For the OP. This puzzler is so venerated that it has myriad forms: ladder carried through barn, train in tunnel, pole vaulter, grate covering hole, magician wand and ring, etc and etc..

 

[Hak]

For the OP. This puzzler is so venerated that it has myriad forms: ladder carried through barn, train in tunnel, pole vaulter, grate covering hole, magician wand and ring, etc and etc..

Thank you very much. Are there any articles I can refer to for these "paradoxes"? I'm not talking about the Wikipedia page, I just saw that you yourselves advised against it because of the "ladder paradox," exposing all its flaws.

 

[Hak]

It’s a very common teaching example, often described using a barn or a tunnel instead of a garage, and a pole, ladder or train instead of a car - but the key idea is the two doors closing at the same time in one frame but not the other. Google for “pole-barn paradox” and “ladder paradox relativity” and you’ll get a ton of examples.

I find that the best way of understanding it is to draw a Minkowski spacetime diagram plotting the four relevant paths through spacetime: path of the front end of the car, path of the rear end of the car, path of the near door, path of the far door. Two of these will be vertical lines, two will be parallel and slanted. I’m not going to bother drawing/posting one because that Google search will find several bazillion already online. Some will even be here at PhysicsForums.

Thanks.

 

[Orodruin]

Two of these will be vertical lines, two will be parallel and slanted. I’m not going to bother drawing/posting one because that Google search will find several bazillion already online. Some will even be here at PhysicsForums.

For those too lazy to google (from my lecture notes):

The sheets are the car (red) and garage (green) in their rest frames $S'$ and $S$ respectively. The dashed lines show the simultaneities for both frames.

 

[Orodruin]

Are there any articles I can refer to for these "paradoxes"?

Most "paradoxes" boil down to the statement in my signature.
(Turn device to landscape if you are using mobile to see it.)

 

[Hak]

For those too lazy to google (from my lecture notes):
View attachment 334530
The sheets are the car (red) and garage (green) in their rest frames $S'$ and $S$ respectively. The dashed lines show the simultaneities for both frames.

Thank you very much. Are your lecture notes available somewhere? Could I find them and read them too, or is that not possible? Let me know, thank you very much.

 

[Hak]

Most "paradoxes" boil down to the statement in my signature.
(Turn device to landscape if you are using mobile to see it.)

Nice one!

 

[Sagittarius A-Star]

But why should the two doors close non-simultaneously for the driver?

In a typical Minkowski diagram, for all events on the t' axis is valid:
$x' = 0$ and $x=vt$ (relativity of "same location", also known from Galilean relativity/intuition).

In a typical Minkowski diagram, for all events on the x' axis is valid:
$t' = 0$ and $t=\frac{v}{c^2}x$ (relativity of "simultaneity", not known from Galilean relativity).

In Galilean relativity, the x' axis would be horizontally: $t'=t =0$ (absolute time).

 

[Nugatory]

Are there any articles I can refer to for these "paradoxes"?

You really need a intro textbook to work through this stuff. Just about all the "paradoxes" will appear as exercises because they were invented as teaching tools: "Here's an apparent paradox, figure out how to resolve it". Working through a proper textbook will also show you how the math (just high school algebra as long as you avoid problems involving accelerations) makes sense of all this.

But since I don't expect you to pay any attention to this advice, I'll suggest a few paradoxes to get started on:
1) The one you brought up in this thread, basically the ladder paradox.
2) The time dilation paradox: A and B are moving relative to one another. The time dilation says that both clicks are slower than the other. How can this be?
3) Bug-rivet paradox, or why you can't beat relativity of simultaneity using rigid objects

Until you can clearly explain all three of these, preferably by drawing Minkowski diagrams, you have no business taking on more advanced topics.

 

[Hak]

You really need a intro textbook to work through this stuff. Just about all the "paradoxes" will appear as exercises because they were invented as teaching tools: "Here's an apparent paradox, figure out how to resolve it". Working through a proper textbook will also show you how the math (just high school algebra as long as you avoid problems involving accelerations) makes sense of all this.

Right. Can you recommend any?

 

[Hill]

If you prefer an online free course, this is a good one for starters, with exercises etc.:
https://www.coursera.org/learn/einstein-relativity

 

[Hak]

If you prefer an online free course, this is a good one for starters, with exercises etc.:
https://www.coursera.org/learn/einstein-relativity

Thank you very much, but I always prefer paper texts.

 

[Hill]

Thank you very much, but I always prefer paper texts.

Well, me too.

 

[Hak]

Well, me too.

If you know of any that are complete and detailed... your advice is always welcome.

 

[Sagittarius A-Star]

If you know of any that are complete and detailed... your advice is always welcome.

I like the following SR books:

A. Einstein 1916 "Relativity: The Special and General Theory" (high school level, not complete)
Online version: https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory

L. Susskind 2017: "Special Relativity and Classical Field Theory"
Online videos: https://theoreticalminimum.com/courses/special-relativity-and-electrodynamics/2012/spring

W. Rindler 1991: "Introduction to Special Relativity" 2nd edition
Disadvantage: Use of out-dated terminology "relativistic mass"
Online short version: http://www.scholarpedia.org/article/Special_relativity

 

[Vanadium 50]

I always prefer paper texts

There is always the PRINT command.

Lots of books out there: Taylor and Wheeler is popular, French maybe a little less so. But don't wait for the perfect book to start.

 

[Hak]

Thanks to all.