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Showing posts from March, 2014

Fulling-Davies-Unruh-Hawking Radiation

G+'er +Jonah Miller pointed out yesterday that Unruh had shown that the radiation predicted by Fulling produced a black body thermal spectrum in the same manner as Hawking radiation[1].  For some reason, I had it in my head that Unruh had actually pointed out prior to Hawking, so I wound up on another search through the research.  The necessary papers are usually fairly close at hand since the main thrust of my current theoretical research is looking at the spectrum of particles created, (or not), when an observer is rotating rather than accelerating in a linear fashion, but I digress.  Here's the chronology of papers to the extent I was able to work them out today.  By they way, Jonah was right!

Fulling[2] was the first to point out that an event horizon coudl muck with your particle creation and annihilation operators producing what might look like particles to the uniformly accelerating observer.

Davies[3] certainly discussed the temperature of the radiation in the same yea…

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 …

Gudermannian Special Relativity, with a tip of the hat to Zigzag numbers

The day's running long, so I'm logging a few thoughts and notes on Gudermannian relativity here.  When I started trying to derive formulas in hyperbolic relativity, as I mentioned[1], I was inspired by Brian Greene's explanation that all objects move at the constant speed of light and can transfer their light speed from the time dimension, (where it points when an object is at rest), to the space dimension if they'd like, (leading to time dilation, slower time), in the rest frame.  I built diagrams like the following, (picture 1), which showed what I imagined the relationship to be in terms of circular trigonometry.



However, this never got me to the proper relationships which  I knew involved the hyperbolic tangent. What I was missing was the inverse Gudermannian which gives a relationship between circular and hyperbolic angles, (picture2).  The super-cool bit is that sigma shown in picture 2 corresponds to the actual speed that the travelling object would measure for…

Takeno Scooped by Thirty Years and Keeping the Line Element Simple Special Rel Style

One major mistake was pointed out in my work on the Takeno line element from yesterday[6].  When working on a theory that describes the behavior of physical quantities with respect to velocity, (like special relativity), don't decompose the velocity into its components like distance and time, or in my case into angular displacement, and time.  Just leave omega, (angular velocity), as omega!  Since omega isn't one of the variables that the derivatives are being taken with respect to, the many terms due to chain ruling out the innards of the hyperbolic trig functions disappeared and the phi and time components of the line element as calculated by Takeno fell out pretty simply.


I also found out last night while reading through one of the Takeno referencing articles[5] I mentioned a few days ago that Takeno was scooped on his transform by about thirty years.  As it turns out, Phillip Franklin[2], (picture 2), a then recent PhD from Princeton, beat Takeno to the punch in 1922[3, op…

Re-Calculating the Takeno Line Element

I'm still working on re-deriving Takeno's results for the line element given in his paper.  I'm slogging through the work having to remember where to divide by factors of the radius squared to get back to differential angle elements and where to multiply by factors of the speed of light squared to transform from time elements to distance elements.  The following is a rather abstruse set of instructions on how I'm doing all of this, left mostly for me should I ever forget, or need to do it again.  I'm not sure if it's even all correct yet, so a warning...  if you're not into spending your Sunday morning reading distracted notes on differential calculus stop now :)

Despite all the differential calculus shown in the board work in the picture, the line element basically describes how the Pythagorean theorem for finding the length of a line works in a given coordinate system.  The familiar example from middle/high school is that that length squared equals the s…

TAMU Physics Festival and Rotating Discs in Special Relativity

The Texas A&M Physics Department Physics Festival is today[5]!!!  If you're in town, you should wonder over to the Mitchell Physics building, (see the picture), on University Drive.  Nobel laureates will speak, there will be three showings of a physics circus, and there will be tens and possible over a hundred hands on demos to take a look at.  Lately, the chemistry, mathematics, and engineering departments have gotten in on the fun, so things could get crazy!  Oh, and there's almost certainly going to be exploding bottles of liquid nitrogen used to erupt water out of barrels, a perennial crowd pleaser.

On the theory side of things this week, I'm working on re-calculating the line element of the Takeno transform.  The transform was derived by Takeno in 1952 in an attempt to explain what happens to a rotating disc when it rotates at special relativistic speeds.  There's been a conundrum here almost since the inception of special relativity in 1905.  It was pointed …

The Calibrated Leak and AVS Awards

The leak detector is up and running!  After draining,flushing, and filling the roughing vacuum pump and cleaning the liquid nitrogen trap, we put the system back online.

Once all that was done, I found out something else that's very cool!  You can make your system leak in a calibrated manner to test the leak detector!  The gadget for doing this is called a calibrated leak!  I'd read about these in the documentation, but never expected to actually see one in the field!  Here's a picture



The calibrated leak is the silver cylinder with the white label in the center of the picture.  A few weeks ago when the I first started playing with the leak detector[1], I mentioned that it detects leaks by looking for helium using a built-in atomic mass spectrometer.  The calibrated leak has a small container of helium that is released at a calibrated rate into the vacuum system once the black valve at the bottom is opened.  After that, the leak detector gauge reads the rate at which heliu…

MAA Focus, A Writing Contest, Doubling the Cube and Danica McKellar

The latest open access and completely free issue[1] of the +Mathematical Association of America's Focus magazine is out!!!  There's a lot of information about the recent joint meeting where it looks like a great time was had by all and among other things, the JPBM Award for Communications was presented to Danica McKellar[3], who I thought looked a lot like the actress from Wonder Years, and who in fact is.  She's written a number of math books for middle and high schoolers and was a co-author of a mathematical physics theorem that bears her name, the Chayes–McKellar–Winn theorem.

Fernando Q. Gouvêa wrote an excellent history of mathematics article tracing the problem of doubling the cube back to ancient Greece.

Finally, there's a reminder about the March 19th deadline fort the SIGMAA – History of Mathematics 2013 Student Paper Writing Contest[2].  If you've already gotten started on your entry, don't forget to submit it.  If not, and you can throw together 5,0…

Cosmology from Newtonian Mechanics and Teaching Special Relativity in the First Week of Freshman Physics

The first page of an arXiv[3] article I came across this morning has a very nice explanation of what a Newton-Hooke spacetime is.  As it turns out you can model expanding and contracting cosmological theories using Newtonian mechanics without going all the way to general relativity.  This was most recently demonstrated by Elisha Huggins in Physics Teacher[1], but seems to have been around for quite a long time.  I've found it as far back as the '60s in Wolfgang Rindler's excellent book "Essential Relativity"[2].  Finally, there's what looks like a pretty excellent video of Huggins in a colloquium where he advocates teaching special relativity as well as Fourier analysis and quantum mechanics in the first course of freshman physics very early on[4].

*References*
1.  http://scitation.aip.org/content/aapt/journal/tpt/51/6/10.1119/1.4818374

2.  http://books.google.com/books?id=0J_dwCmQThgC&lpg=PP1&dq=essential%20relativity%20rindler&pg=PP1#v=onepage&a…

The Care and Feeding of anti-Gudermannians: Spotting Them in the Wild

I mentioned yesterday that the anti-Gudermannian had come up in several articles I’d read, but that the authors hadn’t recognized, or pointed out the anti-Gudermannians lurking in their formulas.  This is a brief set of instructions on how to recognize anti-Gudermannians in their natural state which can be pretty  messy looking.  The first thing you’ll need to know how to spot the multiplicative inverse of something known as the quotient function[5].  It looks like this, 

but may be frequently disguised as

The quotient function is very interesting in its own right and turns up all over in things like electrical transmission line formulas, quantum mechanical transmission and reflection coefficients, and in optically active material formulas.  For complete coverage of the quotient function, see Lindell in AJP[5], (sorry I couldn’t find an open access version, so you’ll have to head to your closest university library).  As cool as it is though, it’s only the start of the Gudermannian.
N…

Karapetoff, Oblique Angle Special Relativity, and the Gudermannian

I've really missed writing for several days.  I was off studying about the Gudermannian yet again in hopes that I could tie up a research project over spring break.  It turns out, (as pointed out by Shahin), that the launch angle that gives the maximum range for the catenary trajectory of a relativistic projectile in a  gravitational field is the same launch angle that will maximize the arc length along the parabolic trajectory of a classical (non-relativistic) projectile.  The derivatives that have to be solved in both cases contain Gudermannian functions, hence all the recent study of hyperbolic math at the expense of my writing.  What follows is a free-writing dump capturing everything I’ve been up to for the last several days.  This relates to our poster from December[7] that I have yet to fully describe and more about Karapetoff the subject of my history of physics presentation that will come up in a few weeks.

Karapetoff [1]began writing about special relativity in the 1920s.…

Rapidity, History and Vladimir Karapetoff

This April, I’m giving a presentation at the APS April meeting about Vladimir Karapetoff, an electrical engineering professor who worked at Cornell University in the early 1900s.  He wrote rather prolifically and one of the many subjects he covered in depth was special relativity, specifically the use of hyperbolic geometry in special relativity.

I only have ten minutes for the talk.  There are two ways I can put the talk together.  One would be to use science to call attention to the history.  Although he doesn't mention Karapetoff, this approach has been done really nicely and fairly completely by Scott Walter[pdf][1].  The other approach is to cache science in a little bit of history.  This is the approach I’ll take.  The first little bit of the talk will focus on Karapetoff the person.  He’s actually quite interesting.  He was a member of the socialist party when socialism was cool.  He ran for New York state offices on several occasions although he never won an election.  As …

Cookie Monster Numbers, Journal Reading, and Mini-Blogging

I just spent the last several minute taking it easy at the end of my day reading a little bit about set theory.  It wasn't the dry and/or indecipherable stuff some of you might remember from first grade thanks to the 'new math'.  Nope, this article on set theory was on something called the Cookie Monster Number, the least number of moves that Cookie Monster can take to empty a set of jars of cookies if he can only take the same amount of cookies from every jar on each move.  Since I'm headed off to bed for the night, rather than rehash the subject here, I'll spend competitora little bit of time telling you about where you can read more on the subject.

For a brief intro to the Cookie Monster Number, check out +Richard Green's mini-blog on G+.  Specifically, you'll want the post on Cookie Monster[1].  For those reading this on my blog as opposed to G+, the Facebook competitor from Google has actually become quite the mini-blogging site for scientists of all …

Rapidity as a Decomposition and a Writing Exercise

Back in December I posted a picture of the poster that I was taking to the Dallas Astrophysics Symposium.  Lots of people asked what it all meant and I promised I’d answer eventually.  Well, I've finally found the time to at start expanding on the poster in small chunks.  The following is a first attempt and will make far more sense to physicists than it will to anyone else, (at least I hope it’ll make sense to someone)..  I’m using these posts as a practice ground for explaining these concepts simply and the explanations should get better as time progresses.  In the meantime, thanks for being my practice audience if you decide to read this!  If you have questions, please ask them, and if you can see an obviously better way of expressing the ideas, and you’d like to share it that would be awesome and help a lot!

Physicists in the early 1900s defined a quantity called rapidity that serves as a way of measuring how fast a particle is traveling in special relativistic terms.  Rapidity…

Leak Detector at Last

I've wanted to work on an AMS 10 leak detector since I was in high school,and now, in grad school, finally, I get to!  My relationship with the AMS 10 starts when I was 17 touring a lab at New Mexico State University.  I was working on building a cyclotron back at our highschool lab, this guy specifically, (picture one):



and the AMS 10 looked like it could be insanely helpful in tracking down the leaks in the accelerator vacuum chamber.  The vacuum chamber is the small brass box at the end of the copper pipe sandwiched between the poles of the magnet in the picture.  Although I had youthful dreams of pulling the perfect vacuum with the help of the ASM 10, in reality, the vacuum we were pulling wasn't really good enough for us to benefit from the services of such a precise instrument.  Its main use is for finding the last leak that has eluded you.  We had a variety of leaks, so it was debatable if we would have benefited at all.  Still, it was such a sexy piece of equipment.  T…