Wolfgang Rindler and the Rod vs. Hole Lorentz Contraction Paradox

If a Lorentz contracted rod travels over a hole smaller than it's un-contracted length, does it fall in? Wolfgang Rindler, (picture 1), pointed out the answer in 1961. Dr. Rindler is my new favorite author for the week. His papers are paragons of clarity. Rather than immediately bogging down in formulas or complicated jargon, he spends a significant amount of space in every paper explaining the problem and its solution in everyday terms. Then, almost as an afterthought, he lays out the math that describes the situation. Sometimes though, you have to read into his math and then read in a little more. If anyone has a more clear explanation of the rod and hole contradiction, it would be great to hear.

Dr. Rindler has done interesting work on special relativity, general relativity, and cosmology. He moved to the Southwest Center for Advanced Studies in the mid '60s, and he's still there. The institute now bears a different name however: the University of Texas at Dallas.

Now, back to our Lorentz contracted rod. The answer to the question above is that the rod will in fact fall into the hole. There's a downward force acting on the rod due to gravity. The amount the rod moves downward depends on the amount of time it is subjected to the force. All of this seems straightforward, there's only one catch. In the rod's frame of reference, time is dilated, it moves more slowly. Hence, the normal distance moved downwards by an object under a gravitational force in the rest frame is, (picture 2),

which, in the frame of the moving rod becomes, (picture 3),

OK, so what does that mean? Basically, in the frame of reference moving with the rod, the rod appears to distort along a parabola that enters the hole. All of this is detailed quite nicely in a diagram from Rindler's article[1], (picture 4).

Note:
Science is fluid. There's was a second article written about Rindler's paradox 44 years later[2]. The authors felt that Rindler was in error regarding the 'bending' of the rod. My take on the Rindler article was that he did't infer a true bending of the rod as much as a perceived bending, but as I said, I don't know for sure.

I'm interested to hear what other physicists think of this, I'm a grad. student, and don't quite no what to think about it all yet. Just for reference I found a letter in AJP that treats a similar paradox without acceleration and describes the hole as appearing rotated in the rods frame of reference, (picture 5).

This hits a resonance with me per my own understanding of special relativity [4]. It seems simple to believe that if a hole moving at constant speed appears rotated up, as shown in the picture, then a hole moving at the equivalent of an accelerating speed would appear curved, hence it could be that the hole appears curved in the rods frame of reference, rather than the rod.

Apologies for the non open-access nature of the references here, (except the reference to my stuff which is always OA). At university libraries in the States, (at least those with physics departments), AJP should in all likelihood be sitting on the shelves.

References:
1. Rindler AJP paradox article
http://dx.doi.org/10.1119%2F1.1937789
Rindler W. (1961). Length Contraction Paradox, American Journal of Physics, 29 (6) 365. DOI: 10.1119/1.1937789

2. EJP article 44 years later
http://dx.doi.org/10.1088%2F0143-0807%2F26%2F1%2F003
Lintel H.V. & Gruber C. (2005). The rod and hole paradox re-examined, European Journal of Physics, 26 (1) 19-23. DOI: 10.1088/0143-0807/26/1/003

3. Letter with constant speed hole
http://dx.doi.org/10.1119%2F1.1941907
Shaw R. (1962). Length Contraction Paradox, American Journal of Physics, 30 (1) 72. DOI: 10.1119/1.1941907

4. My take on special relativity
http://copaseticflow.blogspot.com/2013/02/spheres-special-relativity-and-rotations.html

More Cowbell! Record Production using Google Forms and Charts

First, the what : This article shows how to embed a new Google Form into any web page. To demonstrate ths, a chart and form that allow blog readers to control the recording levels of each instrument in Blue Oyster Cult's "(Don't Fear) The Reaper" is used. HTML code from the Google version of the form included on this page is shown and the parts that need to be modified are highlighted. Next, the why : Google recently released an e-mail form feature that allows users of Google Documents to create an e-mail a form that automatically places each user's input into an associated spreadsheet. As it turns out, with a little bit of work, the forms that are created by Google Docs can be embedded into any web page. Now, The Goods: Click on the instrument you want turned up, click the submit button and then refresh the page. Through the magic of Google Forms as soon as you click on submit and refresh this web page, the data chart will update immediately. Turn up the:

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

Now available as a Kindle ebook for 99 cents ! Get a spiffy ebook, and fund more physics The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems , there seem to be fewer explanations of the procedure for deriving the conversion, so here goes! What do we actually want? To convert the Cartesian nabla to the nabla for another coordinate system, say… cylindrical coordinates. What we’ll need: 1. The Cartesian Nabla: 2. A set of equations relating the Cartesian coordinates to cylindrical coordinates: 3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system: How to do it: Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables. The chain

The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1. While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero. This is typically explained in the following way. While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare. That doesn't stop us from looking for them though! Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2]. Cabrera was utilizing a loop type of magnetic monopole detector. Its operation is in concept very sim

AnonymousDecember 1, 2014 at 2:51 PM
Keep in mind that no potential effect of a moving gravitational field is considered; the considerations are purely SR. Consequently, the same calculation is made as for a fast moving roller belt with a hole in it.

Thus, what do you think, will the relative speed of a hole to you, make you weak like a pudding? What kind of physics is that?

You can verify the calculations yourself; they are not complicated! ;-)

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