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


Comments

Popular posts from this blog

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

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 rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

Lost Phone

We were incredibly lucky to have both been in university settings when our kids were born.  When No. 1 arrived, we were both still grad students.  Not long after No. 2 arrived, (about 10 days to be exact), mom-person defended her dissertation and gained the appellation prependage Dr. 

While there are lots of perks attendant to grad school, not the least of them phenomenal health insurance, that’s not the one that’s come to mind for me just now.  The one I’m most grateful for at the moment with respect to our kids was the opportunities for sheer independence.  Most days, we’d meet for lunch on the quad of whatever university we were hanging out at at the time, (physics research requires a bit of travel), to eat lunch.  During those lunches, the kids could crawl, toddle, or jog off into the distance.  There were no roads, and therefore no cars.  And, I realize now with a certain wistful bliss I had no knowledge of at the time, there were also very few people at hand that new what a baby…

Lab Book 2014_07_10 More NaI Characterization

Summary: Much more plunking around with the NaI detector and sources today.  A Pb shield was built to eliminate cosmic ray muons as well as potassium 40 radiation from the concreted building.  The spectra are much cleaner, but still don't have the count rates or distinctive peaks that are expected.
New to the experiment?  Scroll to the bottom to see background and get caught up.
Lab Book Threshold for the QVT is currently set at -1.49 volts.  Remember to divide this by 100 to get the actual threshold voltage. A new spectrum recording the lines of all three sources, Cs 137, Co 60, and Sr 90, was started at approximately 10:55. Took data for about an hour.
Started the Cs 137 only spectrum at about 11:55 AM

Here’s the no-source background from yesterday
In comparison, here’s the 3 source spectrum from this morning.

The three source spectrum shows peak structure not exhibited by the background alone. I forgot to take scope pictures of the Cs137 run. I do however, have the printout, and…