Skip to main content

Proper Velocity!!! and Getting Index Notation Worked Out: EM II Notes 2014_09_09

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:  Does this obviate the need for the $\eta$ metric?

A:  But wait!  There's so much more!  This is a way to write things without the $\eta$ cruising through everywhere, but it also explains the oddball spacing of the indices.  Maybe I just don't remember this from MacConnel?  First, the Lorentz transform is not a tensor.  Now, we hve that down.  The next bit is how to arrive at the above expression.

$\eta_{\mu\nu} \Lambda^\mu_{\;\rho} \Lambda^\nu_{\;\sigma} = \eta_{\rho\sigma}$

The next step is to plow the $\eta$ on the l.h.s. in and lower the $\nu$ on the second transform.

$\Lambda^\mu_{\;\rho} \Lambda_{\mu\sigma} = \eta_{\rho\sigma}$

We then raise the sigma

$\eta^{\sigma\lambda} \Lambda^\mu_{\;\rho} \Lambda_{\mu\sigma} = \eta^{\sigma\lambda}\eta_{\rho\sigma}$

$\eta^{\sigma\lambda} \Lambda^\mu_{\;\rho} \Lambda_{\mu\sigma} = \eta_\rho^\lambda = \delta_\rho^\lambda$

$\Lambda^\mu_{\;\rho} \Lambda^{\;\lambda}_\mu = \eta_\rho^\lambda = \delta_\rho^\lambda$

Now, since we can see that the first two indices are $\mu$s, we can call the above statement a transpose.  The index locations matter to get the transpose to be a transpose, making sure the rows and columns are handled properly.

NOTE:  This doe arise in MacConnel.  His notation is slightly different.  Instaed of $\Lambda^\mu_{\;\rho}$, he writes $\Lambda^\mu_{.\rho}$

We're going to need the delLambertian soon and it's important to note that it is

$\Box = -\partial_0\partial_0 + \partial_i\partial_i = \partial^\mu \partial_\mu$

A few notes follow on why the D'Alambertian is written with one index up and one down.  It has to do with the negative sign in the first entry of the Minkowski metric.  As it turns out, the index up version of the Kronecker delta is the same as the index down version, but not so for $\eta$ because of the $-1$ $00$ entry.  This is all much simpler if you never start writing your indices in the 'wrong' location in the first place.


OK, now for the interesting stuff.  First, with the choice of signature for the Minkowski metric here, we wind up having to write down $d\tau$ as \

$d\tau = -ds^2 = dt^2 - dx^2 - dy^2 - dz^2$

Given $d\tau$ we define the four velocity to be

$U^\mu = \dfrac{dx^\mu}{d\tau}$

Which is technically mixing frames, but then, that's what proper velocity does.  Intersting that we're starting with four velocity, proper velocity here.  It's really nice to get this out of the way early on and benefit from it.

Picture of the Day
Presenting the Takeno metric line element.  Hopefully one of the payoffs of all this will be understanding this more fully, and possibly using it to explain the circular Unruh effect.  However, a far more productive use is in explaining the Thomas Precession.


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…

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 concept very simpl…

The Javascript Google URL Shortener Client API

I was working with the Google API Javascript Client this week to shorten the URLs of Google static maps generated by my ham radio QSL mapper. The client interface provided by Google is very useful. It took me a while to work through some of the less clear documentation, so I thought I'd add a few notes that would have helped me here. First, you only need to authenticate your application to the url shortener application if you want to track statistics on your shortened urls. If you just want the shortened URL, you don't need to worry about this. The worst part for me was that the smaple code only showed how to get a long url from an already shortened rul. If you follow the doucmentaiotn on the insert method, (the method for getting a shortened url from a long one), there is a reference to a rather nebulous Url resource required argument. It's not at all clear how to create one of these in Javascript. The following example code shows how:
var request = gapi.clie…