Oct 25, Gravity is a Mystery (in words, no equations)
Snarky puzzle
Is there a handedness to getting older? Is there a handedness to communication? If there is a handedness to either, how would that effect the arrow of spacetime?

The Back Story:
Clocks tick on. The universal march of time has no handedness. Put me in an isolation chamber and I will get older at the same rate as having Thanksgiving with the family. [clarification: as noted in the comments, there is no such thing as a complete isolation chamber, it is one of those unreachable ideals.]

Space is filled with arrows: this over here, that over there. There is a handedness to communication. Reach out to shake hands that are handed.

Answer:
There is a handedness to spacetime, all of the handedness coming from space.

Discussion:
There are a few scholars such as Julian Barbour  who make their daily bread pondering the "Arrow of time" problem. If they were forced to work on the arrow of spacetime, the problem could be solved already, and people could move on.

Nov. 1, RETRACTION: Deriving And Fixing The Force Equations (6/5+1)
Snarky puzzle.
Show how to derive the Coulomb force equation if you start from Gauss's law. If you get stuck, read chapter 28-5 of David Halliday and Resnick, "Physics", part 2.

The Back Story:
The most important thing about this blog was the RETRACTION. Not many people do that for their own work. The few examples I know about usually involve fraud. It is too bad this is not more commonplace. How many papers on super symmetry will be retracted should no super particles be found at the LHC? Will any papers be withdrawn if the Higgs stays in hiding?  The number of retractions will probably equal the number of these particles found, zero.

There were two reasons that motivated my retraction. The first is that I got utterly confused about force. The second was I could not see a way to a proposal with local gauge symmetry. Why is that such a big deal for me? If you have a local gauge symmetry, then there is a conserved charge. Mass is conserved, so that conservation requires a local gauge symmetry in the action. I wanted my proposal to join the club of gauge theories like EM, the weak force, the strong force, and general relativity.

The snarky puzzle comes from why I don't think I have utterly misled the Science20 community. A valid proposal must be consistent every way one looks at it. I didn't pull that off. The thin slice that is only about fields looks like it got the basics right, that like charges attract for the Newtonian field equations, while like charges repel for Gauss's law.

Answer:
Assume a point charge at the center of a sphere. This is a situation where the integral form of Gauss's law makes the calculation easier:

$q = \epsilon_0 \oint \vec{E} \cdot d \vec{S}$

where S is the surface vector of the sphere. Because the source is a point at the center of the sphere, the two vectors E and S are always in line, and the dot product can be dropped, the angle between the two is always the same (zero). The electric field will always be constant over the surface, so it can be pulled outside the integral. The integral ends up being the surface of a sphere, pure geometry:

\begin{align*} Q& = \epsilon_0 \oint \vec{E} \cdot d \vec{S} \\ &= \epsilon_0 E \oint d S \\ &= \epsilon_0 E 4 \pi R^2 + C\\ or\\ E &= \frac{1}{4 \pi \epsilon_0} \frac{Q}{R^2} \end{align*}

I forget what assumptions are needed to set the constant C equal to zero.

A force law is used to define the electric field E like so:

$\vec{F} = q \vec{E}$

Plug the electric field expression from Gauss's law into this definition of an electric field:

$\vec{F} = \frac{1}{4 \pi \epsilon_0} \frac{q Q}{R^2}$

This is Coulomb's law. Like charges will have a positive force, meaning they will push away from each other.

Discussion:
There are always details to worry about. One can define the integral form of Gauss's law so that the factor of 4 pi epsilon bows out of the way. Nature has a habit of using integration constants we might toss away. Should we be concerned about how the vectors are handled in the two approaches? And there are those assumptions one doesn't know about, but might hear about in the comments :-)

Nov. 8, Can't Buy a Gauge Symmetry
Snarky puzzle
The oh-so-familiar conjugate operator is a tool of the quaternion Wall Street elite hiding in a velvet vector dress. The rectangular box world of Z2xZ2 prefers to divide the world in two. Doodle with these:
Let the Z2 conjugate *i, *j, and *k do flip two signs:
$\\ (\phi, Ax, Ay, Az)^{*i} \equiv (\phi, Ax, -Ay, -Az) \\ (\phi, Ax, Ay, Az)^{*j} \equiv (\phi, -Ax, Ay, -Az) \\ (\phi, Ax, Ay, Az)^{*k} \equiv (\phi, -Ax, -Ay, Az)$
Form the following products:
$\\ (\rho/\sqrt{3}, Jx, Jy, Jz) \boxtimes (\phi/\sqrt{3}, Ax, Ay, Az)^{*i} \\ (\rho/\sqrt{3}, Jx, Jy, Jz) \boxtimes (\phi/\sqrt{3}, Ax, Ay, Az)^{*j} \\ (\rho/\sqrt{3}, Jx, Jy, Jz) \boxtimes (\phi/\sqrt{3}, Ax, Ay, Az)^{*k}$
Just for fun, take the sum.

The Back Story:
I have limited experience doing calculations that use the Klein 4-group to form a product. Most well-educated people have no such experience. This is a straight forward calculation.

Answer:

\begin{align*} (\rho/\sqrt{3}, Jx, Jy, Jz) &\boxtimes (\phi/\sqrt{3}, Ax, Ay, Az)^{*i} \\ &=(\rho/\sqrt{3}, Jx, Jy, Jz) \boxtimes (\phi/\sqrt{3}, Ax, -Ay, -Az) \\ &=(\frac{1}{3}\rho \phi + Jx Ax - Jy Ay - Jz Az, \\ &\quad \frac{1}{\sqrt{3}}\rho Ax + Jx Ax - Jy Az - Jz Ay, \\ &\quad -\frac{1}{\sqrt{3}}\rho Ay + Jy \phi + Jz Ax - Jx Az, \\ &\quad -\frac{1}{\sqrt{3}}\rho Az + Jz \phi - Jx Ay + Jy Ax) \\ \ \end{align*}

\begin{align*} (\rho/\sqrt{3}, Jx, Jy, Jz) &\boxtimes (\phi/\sqrt{3}, Ax, Ay, Az)^{*j} \\ &=(\rho/\sqrt{3}, Jx, Jy, Jz) \boxtimes (\phi/\sqrt{3}, -Ax, Ay, -Az) \\ &=(\frac{1}{3}\rho \phi - Jx Ax + Jy Ay - Jz Az, \\ &\quad -\frac{1}{\sqrt{3}}\rho Ax + Jx Ax - Jy Az + Jz Ay, \\ &\quad \frac{1}{\sqrt{3}}\rho Ay + Jy \phi - Jz Ax - Jx Az, \\ &\quad -\frac{1}{\sqrt{3}}\rho Az + Jz \phi + Jx Ay - Jy Ax) \\ \end{align*}

\begin{align*} (\rho/\sqrt{3}, Jx, Jy, Jz) &\boxtimes (\phi/\sqrt{3}, Ax, Ay, Az)^{*k} \\ &=(\rho/\sqrt{3}, Jx, Jy, Jz) \boxtimes (\phi/\sqrt{3}, -Ax, -Ay, Az) \\ &=(\frac{1}{3}\rho \phi - Jx Ax + Jy Ay - Jz Az, \\ &\quad -\frac{1}{\sqrt{3}}\rho Ax + Jx Ax + Jy Az - Jz Ay, \\ &\quad -\frac{1}{\sqrt{3}}\rho Ay + Jy \phi - Jz Ax + Jx Az, \\ &\quad \frac{1}{\sqrt{3}}\rho Az + Jz \phi - Jx Ay - Jy Ax) \\ \end{align*}

\begin{align*} \sum_{*n=*i, *j, *k} J \boxtimes A^{*n}&= (\rho \phi - Jx Ax - Jy Ay - Jz Az, \\ &\quad -\frac{1}{\sqrt{3}}\rho Ax + \sqrt{3} Jx Ax - Jy Az - Jz Ay, \\ &\quad -\frac{1}{\sqrt{3}}\rho Ay + \sqrt{3} Jy \phi - Jz Ax - Jx Az, \\ &\quad -\frac{1}{\sqrt{3}}\rho Az + \sqrt{3} Jz \phi - Jx Ay - Jy Ax) \end{align*}

Discussion:
Don't read too much into this calculation.

Nov. 15, Analyzing Actions: Newtonian G (1/3?)
Snarky puzzle
Take Newton's gravitational theory and make it consistent with special relativity, nothing more. Oops, that is a totally unfair question. You have to be as good as  Gupta, Thirring, Feynman, Weinberg, or Deser to take that one technical requirement and end up at general relativity.

The Back Story:
This issue was discussed in Chapter 7 of "Gravitation" of Misner, Thorne, and Wheeler. Newton's instantaneous gravity theory had to be wrong. Einstein knew that too. I had thought of general relativity as novel, the key new idea being the general equivalence principle, which is actually two principles, the weak and strong equivalence principles. Yet the way it was presented in MTW, it sounded like one could view GR as a repair of broken Newtonian theory. That strikes me as a disturbing idea.

<more detail>

Here are the papers cited in Chapter 7:

Gupta, S. N., 1954, "Gravitation and electromagnetism," Phys. Rev 96, 1683-1685.

Gupta, S. N., 1957, "Einstein's and other theories of gravitation," Rev. Mod. Phys. 29, 337-350.

Gupta, S. N., 1962, "Quantum theory of gravitation," in Recent Developments in General Relativity, Pergamon, New York, pp. 251-258.

Kraichnan, R. H., 1955, "Special-relativistic derivation of generally covariant gravitation theory," Phys. Rv. 55, 1118-1122.

Thirring, W. E., 1961, "An alternative approach to the theory of gravitation," Ann.Phys. (USA) 16, 96-117.

Feynman, R. P., 1963, Lectures on Gravitation.

Weinberg, S., 1965, "Photons and gravitons in perturbation theory: Derivation of Maxwell's and Einstein's equations," Phys. Rev. B 138, 988-1002.

Deser, S., 1970, "Self-interaction and gauge invariance," Gen. Rel.&Grav. 1, 9-18.

I see where I got the repair notion. On page 178, in one of those side comments, the three amigos write:

Best modification (tensor theory in flat space-time) is internally inconsistent: when repaired it becomes general relativity.

<end detail>

Answer:
$R_{\mu \nu} - \frac{1}{2} g_{\mu \nu}R = \frac{8 \pi G}{c^4} T_{\mu \nu}$

Discussion:
<begin ???>
Any proposal for gravity must have Newton's law as a classical limiting case. The issue is not Right Law (GR) to Useful Limiting Case (Newton), it is this all-star team of physicists appears to do it the other way (I think I looked at two of the papers, but had trouble following the logic). We have plenty of experiments to tell us that gravity changes both time and space. Newton's scalar theory has only one position to store any information about gravity, and it is how gravity changes time. It sounds to me like the first move from Newtonian theory should be to add potentials such that space can be changed by gravity because we know it is.

This is one of the core reasons I battle on (and it is a battle, I am frustrated at my limited skills, time, and resources). The first move is critical. The standard opening for GR feels wrong in my gut.
<end ???>

Nov. 22, Analyzing Actions: EM (2/3?)
Snarky puzzle
Construct a spin 2 projection operator. Two factors of 2 will be required.

The Back Story:
Why bother with spin? This was a bullet to the head issue.  A spin 1 field indicates like charges repel for a static field theory, while spin 2 will allow like charges to attract. So says Brian Hatfield in his introduction to Feynman's book on gravity. I was forced to embrace that idea when a well-regarded researcher in gravity let me know I should already know that. I don't think anyone is born knowing such details, but now I must respect this bit of physics.

Answer:
On page 39 of "Feynman Lectures of Gravitation", he is discusing a rank 2 tensor, which he then says is behaving like this complex product:

\begin{align*} (x + i y)(x + iy) + (x - i y)(x - i y) &= (xx - yy + 2 x y i) +(xx - yy - 2 x y i) \\ &= 2(xx - yy) \end{align*}

The projections are +/-2 and a way to represent spin 2.

Discussion:
The math is not too bad. I am glad that professor pointed me to this gem from Feynman, or I might never have been able to approach this subject.

Doug

Google+ hangout: 11:00-11:45pm Eastern time, Tuesday-Wednesday. http://gplus.to/sweetser

This could be an efficient way to exchange a few ideas. If you have a question or two, hangout.

Next Monday/Tuesday: A Local Gauge Symmetry Found