Skip to main content

Hyperbolic Motion, Rindler, Minkowski, Kruskal, and Karapetoff

I found something interesting yesterday, well interesting to me anyway.  What follows is a bit of a historical ramble and reference-fest.  Hyperbolic motion which is usually attributed to Wolfgang Rindler was actually first shown by Hermann Minkowski.  Rindler himself references Minkowski[2] in the first paragraph of his paper where hyperbolic motion under uniform acceleration in terms of special relativity is derived[1].  Not only did Minkowski show hyperbolic motion in his first lecture on spacetime, he also pointed out that you'd have to work kind of hard, actually setting all your acceleration components to zero to not exhibit hyperbolic motion, (picture 1).


Hyperbolic motion, I think, is attributed to Rindler because of his fleshing out and full development of the idea including the concept of event horizons.  Rindler pointed out that when an object undergoes uniform acceleration in spacetime, that a light signal sent out after the object will never be able to catch up.  And that, in a very small nutshell is how an event horizon works.


In picture 2, you might have noticed that what looks like Rindler hyperbolic motion in a Minkowski spacetime is referred to by Rinder as a Kruskal diagram.  Kruskal[4] worked on the hyperbolic nature of the Schwarzschild metric and it was his work that Rindler commented on leading up to what would become known as Rindler coordinates.  In addition to the Schwarzschild metric, Kruskal also discussed Einstein-Rosen wormholes, (picture 3).


There are two more interesting notes about Kruskal's paper.  First, Kruskal was working for project Matterhorn at the time of the publication of the article.  Project Matterhorn was a cold-war project to control thermonuclear reactions.  Second, at the recent Texas Symposium on Relativistic Astrophysics, Charles Misner told the story of how the Kruskal article was actually submitted under Kruskal's name by John Wheeler.  Kruskal had described his work to Wheeler earlier and Wheeler had told him he should write it up. Apparently Wheeler got tired of waiting for Kruskal and just went ahead and wrote it up for him!

Picture of the Day:
Sunrise over Texas A&M


References:
1.  Rindler on hyperbolic motion (not open access)
http://journals.aps.org/pr/abstract/10.1103/PhysRev.119.2082

2.  Minkowski's spacetime lecture, (not open access, not even online, but a cheap book):
http://books.google.com/books?id=S1dmLWLhdqAC&sitesec=reviews&rf=ns:5

3.  Rindler on  Kruskal Space (not open access)
http://scitation.aip.org/content/aapt/journal/ajp/34/12/10.1119/1.1972547?ver=pdfcov

4. Kruskal on the Schwarzild metric
http://journals.aps.org/pr/abstract/10.1103/PhysRev.119.1743

5. Project Matterhorn
https://en.wikipedia.org/wiki/Princeton_Plasma_Physics_Laboratory


Labels:

Comments

Post a Comment

Please leave your comments on this topic:

Popular posts from this blog

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...
1 comment
Read more

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 ...
11 comments
Read more

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:...
Post a Comment
Read more