Copasetic Flow

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 th...

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 b...

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 meas...

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 t...

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...

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. ...

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 g...