Copasetic Flow

Our cosmology course is well under way and it's  a lot of fun so far!  The class direction overall is towards describing the inflationary universe by means of quantum field theory, but this week we're focused on relativity.  We're allowed to work on our homework together, however, I'm spending most of my time in the lab this semester, so I'll be posting my homework notes here.  If you'd like to grab bits and pieces, make suggestions, or contribute, the whole shooting match will also be archived on github . Our first homework contains a problem that involves accelerating reference frames.  The question is, given the transformation between the lab and the accelerating reference frame, figure out if the line element $ds^2$ is preserved.  There are a few interesting aspects to this problem.  First, while the transform looks similar to Rindler coordinates, it's not, (as ar as I can tell.)  Second, looking into Rindler coordinates a bit, they seem to ...

Here's today's special relativistic EM question.  Can the Thomas precession be shown to be a special case of the perihelion advance of relativistic elliptical orbits?  Any ideas?  Here's what's going on: We've been deriving the special relativistic  orbit of a charged particles around another fixed charged particle.  At the end of the day, you wind up with a perihelion advance which is a fancy way to say that major axis of the elliptical orbit won't stay put.  It swivels around, (orbits), the charged particle as well.  The advance angle of the major axis winds up being\\ $\delta\phi = 2\pi\left[\left(1 - \dfrac{\kappa^2}{l^2}\right)^{-1/2} - 1\right]$ Which is very, very, similar to the Thomas angle for the spin precession, or gyroscopic precession along a circular orbit at special relativistic speeds:\\ $\delta\phi = 2\pi\left[cosh\left(w\right) - 1\right]$ $= 2\pi\left[\left(1 - \dfrac{v^2}{c^2}\right)^{-1/2} - 1\right]$ In the expression...

Summary:  It looks like I'll finally get a good understanding of the gamma notation for moving proper velocities to lab velocities and back.  It'll be nice to know it inside and out, but a little irksome given all that can be done with the hyperbolic notation we're not using.  I want to maintain my fluency in both. There may be a subtle second notation for inverted Lorentz transforms.  As it turns out, the subtle notation difference of moving around indices in the top and the bottom with spaces is meant to keep track of which index comes first when you go back to side by side notation. First, we cover Lorentz transforms, (which are not in fact tensors), and contractions and arrive at the interesting result in equation 1.99: $\Lambda^\mu_\rho \Lambda^\sigma_\mu T^\rho_\sigma = \delta^\sigma_\rho T^\rho_\sigma$ Which indicates the transpose of the Lorentz transform times itself follows a sort of orthogonality rule making use of contravariant indices. Q: ...

Summary:  Continuing notes on the tensor version of the Lorentz tranform.  It's time to start on the second set of examples. The interval in four space is invariant under Lorentz transforms and is called the Lorentz scalar. The Lorentz transform also applies to differential distances as, $dx^{\prime\mu} = \Lambda^\mu_\nu x^\mu$ We were asked in class to work out $x^2+y^2+z^2-t^2 = x^{\prime 2}+y^{\prime 2}+z^{\prime 2}-t^{\prime 2}$ The transforms we'll use are: $x = \gamma\left(x^\prime + vt^\prime\right)$ $t = \gamma\left(t^\prime + vx^\prime\right)$ Substituting these into the l.h.s. gives $\gamma^2\left(x^\prime + vt^\prime\right)^2 - \gamma^2\left(t^\prime + vx^\prime\right)^2 = x^{\prime 2} - t^{\prime 2}$ $ = \gamma^2\left(x^{\prime 2} +2vtx + v^2t^{\prime 2}\right) - \gamma^2\left(t^{\prime 2} + 2vxt+v^2x^{\prime 2}\right)= x^{\prime 2} - t^{\prime 2}$ $ = \gamma^2\left(x^{\prime 2} + v^2t^{\prime 2}\right) - \gamma^2\left(t^{\prime 2} + v^2x^{...

Summary :  This one took awhile.  I got busy in the lab  These notes start with rotation matrix properties and the transpose products of matrices.  special relativity via tensors also begins.  Specifically, the Lorentz transformation tensor components are reviewed and the number of independent parameters are counted. Did a few concrete checks that a matrix times its transpose is symmetric.  Sure enough, it is. $\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}\begin{pmatrix} a & c \\ b & d \\ \end{pmatrix} = \begin{pmatrix} a^2 + b^2 & ca + bd \\ ca+bd & c^2+d^2 \\ \end{pmatrix}$ $\begin{pmatrix}a & b & c\\ d & e & f \\ g & h & i \\ \end{pmatrix}\begin{pmatrix} a & d & g\\ b & e & h \\ c & f & i \\ \end{pmatrix} = \begin{pmatrix} a^2 + b^2 + c^2 & ad + be + cf & ag + bh  + ci\\ ad + be + cf & d^2 + e^2 + f^2 & gd + eh + fi \\ ga + hb + ic ...

Summary of what's gone on before.  In the last two days, unbeknownst to me at the time, we first showed that the Lorentz/Fitzgerald contraction only happens in the direction of the velocity of a frame.  Coupling this with rotational symmetry, yesterday, it was shown how the vector version of the Lorentz transform could be derived.  Yesterday's treatment needs to be repeated from end to start to truly  count as a derivation though, I suppose. Picking up where I left off yesterday, the second half of the vector Lorentz transform, the expression for time in the primed frame can be written as $t^{\prime} = \gamma\left(t - \vec{v}\cdot\vec{r}\right)$ Once again, this shows that only the component of the position vector parallel to the velocity gets to play.  However, here you should notice that the velocity vector is not normalized, not divided by it's magnitude.  In other words, while only the component of the position vector parallel to the velocity gets...

Summary: Starting with the vector version of the Lorentz transform for a frame's position, we work backwards and arrive at the expressions and assumptions needed to derive it.  The derivation is not in the notes.  The transform is simply given along with a statement that it can be easily checked. The notes derive the generic vector form of the Lorentz transforms in three dimensions.  The position transform is simply stated as: $\vec{r}^{\prime} = \vec{r} + \dfrac{\gamma - 1}{v^2}\left(\vec{v}\cdot\vec{r}\right)\vec{v}-\gamma\vec{v}t$ A little massaging will make it more clear what's going on.  First, the mystery of the $v^2$ $\vec{r}^{\prime} = \vec{r} + \left(\gamma - 1\right)\left(\dfrac{\vec{v}}{v}\cdot\vec{r}\right)\dfrac{\vec{v}}{v}-\gamma\vec{v}t$ OK, so, we're taking the component of the $\vec{r}$ displacement vector that lies along $\vec{v}$ and then laying that in the $\vec{v}$ direction.  The $v^2$ was just to get us back to unit vectors in ...

I'm working on a new writing project as I prepare for Electricity and Magnetism II this semester.  I'll be reviewing the class lecture notes each day leading up to the start of the semester.  After each half hour review, I intend to write up a summary of the notes.  (Yes, I took a reflective writing workshop this summer, and I'm trying it out.) Pages covered today:  7 & 8 We consider light as a spherical wavefornt $x^2 + y^2 + z^2 - c^2t^2 = 0$ $x^{\prime 2} + y^{\prime 2} + z^{\prime 2} - c^2t^{\prime 2} = 0$ We also look at the usual situation with two frames, $S$ and $S^{\prime}$ that are moving parallel to each other where $S^{\prime}$ has positive velocity $v$ in the x direction with respect to $S$. We have the usual postulate that c, the speed of light is constan in all frames. We need a way to transform the distance and time coordinates in the two frames.  Because we want to preserve invariance with respect to translation, (conserva...

Meetings, Magnets, and Special Relativity Spent the morning attending a meeting and getting to do just a tiny little bit of special relativity research, regarding proper acceleration in the instantaneously moving rest frame.  The rest of the day was spent debugging, first the liquid nitrogen Dewar, and then the leak detector.  The leak detector is still not up to snuff.  Prep work was done for lifting the magnet that will supply the quenching field for the superconducting samples. New to these posts? The hole theory of superconductivity, a BCS compatible modle, predicts that when a superocnductor is brought back to its normal state quickly, it will emit x-rays.  We plan to experimentally verify this or set a new lower limit for its detection.  Scroll to the bottom for more complete information of what this physics experiment is about and what we hope to achieve. Relativistic range work: Reviewed and followed Brehme’s derivation of proper accelerati...

Lab Book 2014_05_16     Hamilton Carter Summary There's a minor setback.  The fiberglass Dewar has a leak.  On the theory side of things, the relativistic range equation is shown to be proper x velocity times the rapidity of the y component of velocity divided by the acceleration due to gravity which is about what you'd expect it to look like if you first looked at the classical result and then squinted.  Work is being done to determine what, if anything, to make of the result.  Notes and a brief Mathematica file are included. Leak testing the fiberglass Dewar today.  I’m also looking through the second Tehran paper . There is a leak at the Teflon joint that is away from the Dewar.  It’s a rather small leak that can’t be detected using the roughing pump gauge. I've very slightly opened the Dewar valve and begun pumping on the large volume.  After about five minutes, the vacuum was back down to 3 * 10^-1. ...

Lab Book 2014_05_15     Hamilton Carter Summary Almost the entire day was spent finally actually using the leak detector!  The new stopcock was attached to the glass helium Dewar early this morning.  After that, I attended a theory meeting.  After cleaning and vacuum greasing a few fittings we found out that the glass liquid helium Dewar is leak tight!!!  The next step, glass Dewar-wise, will be to modify the table that it sits in so that it can be placed between the poles of the electromagnet.   I did some more thinking about the relativistic trajectory problem and found some possilbe symmetries in Shahin's expression for the y vs. x.  Both the expressions for hang time and the maximum range equations are interesting. Hang time is interesting because it's actually the same for the relativistic and classical cases.  Range is interesting because the expression involves the vertical component of the projectiles rapidity. ...

Lab Book 2014_05_13     Hamilton Carter                Summary After fixing the crusty battery cable in the car this morning, I got to do a little bit of work around the lab before the end of the day.  We met and discussed the special relativity work today as well.  Fermi-Walker transport makes sense as just being the acceleration normal to the tangential velocity that changes the direction of the tangential velocity. The hose between the auxiliary roughing pump and the leak detector was attached.  The auxiliary pump is used to rough out the volum to be leak detected before using a valve on the leak detector to attach the built in diffusion pump to the volume to attain a much higher vacuum, (in the range of 10E-8 Torrs). The added hose used the fittings I built a few weeks ago to attach the system.  There’s an intermediate piece that contains an O-ring that fits between the KF fitting on the hose...

Lab Book 2014_05_11     Hamilton Carter Summary Most of the day was spent documenting work.  A few new articles inspired more work on the hyperbolic Thomas precession derivation that's been in the hopper for the last few months. Most of my work today was summarizing the Letaw and Pfautsch vs. the number operator arguments I built yesterday and getting the results out to all the interested parties. I also found a rather complete reference about the many ways to derive the Thomas precession.  The method that I hope to publish which involves using the Walter methodology to arrive at the Takeno metric was not included!  I should start writing this up, once again, soon. There are a few other articles regarding Fulling-Unruh radiation in rotating frames that have been added to the reading queue.  The first of these is from Nokic , and the second is from Nelson .  Interestingly, Nokic also published a paper where he opined on the va...

Lab Book 2014_05_10     Hamilton Carter Summary A seemingly simple question about why magnetic forces act at right angles to their associated field lines led me to derive that the transverse forces on a charged particle moving in a circular path have no gamma terms associated with special relativity.  This seems to tie nicely into why the quantum mechanical number operator predicts no spectrum of Fulling-Unruh radiation from a particle moving in a circular path, but a Fourier decomposition of the wave solution does as shown by Letaw and Pfautsch.  L and P left the lack of spectrum predicted by the number operator as an open question. Someone asked an interesting question on stackexchange regarding why the Lorentz force from a magnetic field acts at right angles to the direction of the magnetic field.  The simple offhanded answer is that in the fame of the moving particle, the magnetic field transforms into an electric field that is parall...