Lab Notes for a Scientific Revolution (Physics)

August 20, 2007

Lab Note 1, Part 2: Einstein’s 1915 General Relativity Paper, §20: Maxwell’s Electromagnetic Field Equations for Free Space

Filed under: Physics,Science — Jay R. Yablon @ 6:57 am

If you would like the following in a PDF file, you may obtain this at lab-note-1-part-2.pdf.
In the first part of this lab note, I talked about how, in my view, the identity
eq-i-2-1.jpg
is one of the most important, yet least-utilized identities in (classical) physics. Here, to lay the foundation for a classical unification of electromagnetism in the absence of potentials, with gravitation, I will show how this identity is at the heart of §20 of Einstein’s 1915 paper on General Relativity. To do this, let us use the version of (1) above which is obtained by setting A–>F and B–>F to obtain:
eq-i-2-2.jpg
Now, showing the full calculation, it is possible to directly recast identity (2) into:
eq-i-2-3.jpg
by simply renaming and reordering indexes and by applying the product rule, and, in the final line, employing the Maxwell energy tensor T^\mu\,_{\nu (Maxwell)}=-\frac{1}{4\pi}(F^{\sigma\mu}F_{\sigma\nu}-\frac{1}{4}\delta^\mu\,_\nu F^{\sigma\tau}F_{\sigma\tau}), the expression \kappa_\mu\equiv F_{\mu\sigma}J^\sigma=\frac{1}{4\pi}F_{\mu\sigma}F^{\tau\sigma}\,_{;\tau} which represents the energy-momentum transfer between the electric masses and the electromagnetic field per unit of time, and a magnetic monopole charge density specified by P^\sigma\equiv \frac{1}{4\pi}^*F^{\tau\sigma}\,_{;\tau}.
Now, at the moment, of course, the magnetic charge need not be zero, because we have not yet introduced a potential. But, as soon as we introduce the Abelian potential:
eq-i-2-4.jpg
equation (2) immediately decouples into P_{\tau\sigma\mu}=\frac{1}{4\pi}(F_{\tau\sigma;\mu}+F_{\sigma\mu;\tau}+F_{\mu\tau;\sigma}) and P^\sigma\equiv \frac{1}{4\pi}^*F^{\tau\sigma}\,_{;\tau}=0 which means that the magnetic monopoles are zeroed out, and equation (3) simplifies to:
eq-i-2-5.jpg
This expresses the exchange of energy-momentum between the electric masses and the electromagnetic field. When \kappa_\mu=0, then so too, T^{\mu\nu}_{(Maxwell)}\,_{;\mu}=0, that is, the Maxwell energy tensor is separately conserved, i.e., does not exchange energy with the electric charges. In this manner, §20 of Einstein’s 1915 paper on General Relativity is easily re-cast on the basis of particular version of identity (1) given by identity (2). In particular, when one employs a potential (4), then one immediately and directly obtains (5) from identity (2).
First, it is important to note from equation (5) that identity (2) contains a term T^{\mu\nu}_{(Maxwell)}\,_{;\mu}, which expresses energy conservation / transfer. More to the point, identity (2) specifically, and identity (1) more generally, is of the same differential order as an equation for conservation of energy, and the single free vector index in (1) and (2) corresponds with the single free vector index of an energy conservation equation.
Second, equation (3) is identically equal to zero whether or not there exists a potential at all, whether the magnetic charges are zero or non-zero, whether or not the potential should it exist is Abelian or non-Abelian, and whether or not energy is exchanged between the electric charges and the electromagnetic field.
Therefore, it becomes quite useful to think of equation (1) as providing the mathematical basis for energy conservation in the electromagnetism, in very much the same manner that the Bianchi identity (R^\mu\,_\nu - \frac{1}{2}\delta^\mu\,_\nu)_{;\nu}=0 provides the mathematical and geometric basis for energy conservation in gravitational theory. It is out of this observation that classical gravitation, and classical electromagnetic theory in the absence of a potential, may be fully unified into the single, simple electro-gravitational equation
eq-i-2-6.jpg
This will be demonstrated in Part 3 of Lab Note 1.

Advertisements

6 Comments »

  1. Maybe if I put a comment here it will influence you to keep posting.

    Comment by carlbrannen — September 24, 2007 @ 3:53 am | Reply

  2. Yet another comment meant to influence you to keep up this interesting and useful blog.

    Comment by carlbrannen — November 26, 2007 @ 4:42 am | Reply

  3. Hi Carl:

    Thanks for your comment. I have been extremely overloaded lately and have not had time for physics. I am hoping to be able to return in early 2008 and get this blog moving again, and I am very appreciative of your interest.

    Jay

    Comment by Jay R. Yablon — December 13, 2007 @ 11:30 pm | Reply

  4. Well? It’s now “early 2008”.

    Comment by carlbrannen — January 1, 2008 @ 11:38 pm | Reply

  5. Very early. Happy New Year!

    Comment by carlbrannen — January 1, 2008 @ 11:38 pm | Reply

  6. Hello to all:
    I hope to start up posting again soon. I just started playing around with a Word to Tex program and ran some successful tests, which will greatly streamline the process of getting equations posted here in a good format.
    I need to go back and prioritize my past research, and I have to fit this in with everything else, but I am ready to (slowly) turn back to physics and over then next few months will try to begin really pulling together all of my “lab notes.”
    Happy new year!
    Jay.

    Comment by Jay R. Yablon — January 19, 2008 @ 7:34 am | Reply


RSS feed for comments on this post. TrackBack URI

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

Blog at WordPress.com.

%d bloggers like this: