Sunday, June 9, 2013

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:

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:

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:

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

1 comment:

Anonymous said...

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! ;-)