Particular omission
Hello Lawrence:
Lawrence B. Crowell said:
I indicated this by way of comparison.
Fair enough. I felt it necessary to point out that this particular comparison was not accurate on a technical level. You still have your serious reservations, but this is not one of them.
Lawrence B. Crowell said:
The "procedures" you go through are comparatively elementary, which would suggest that if these were appropriate that some physicists would have discovered and applied them by now. After all people such as Feynman were quite on the bright side of the intelligence scale.
The logical conclusion based on accepting this analysis is that one should not bother to try and do physics, unless one could demonstrate in a measurable way they were smarter than Feynman. One of Feynman's chief characteristics is to challenge anybody, no matter what their station. It is ironic that you place Feynman on an unreachable pedestal.
I do not put Feynman there. He was amazing, and human. Let's challenge Feynman specifically on the completeness of his analysis. Do you have "Feynman Lectures on Gravitation" in your possession? If not, one can look inside at amazon.com to review the pages in question (pages 31-34). He does an analysis of the current coupling term in EM, J^{\mu} A_{\mu}. He takes the Fourier transform of the potential to get a current-current interaction. He restricts the analysis to a current moving along the z axis. The goal is to figure out the phase of the transverse current, the Jx and Jy terms (he used J1 and J2 in his lectures). The equation on page 34 is the basis of the statement that for EM, the transverse wave will take a 2\pi radian change in phase to get back to where it started, a property of spin 1 particles. Spin 1 particles mediate a force where like charges repel, so photons can do the work of gravity.
His reasoning is both flawless and incomplete. He does not considered the rho-Jz current coupling (or J3-J4 in the lecture). For that term, the J and J' work together, and because they work together, it will require a change of \pi radians to get back to the start. That is a property of particles with spin 2. Feynman's analysis was not complete, which is different from wrong.
During my first three years at MIT, I played poker against a guy named Rob that went on to win the World Series of Poker. We were roughly the same level, although we both knew Dean was better. Rob kept up and intensified his study of the game, although I did not. I respect the conservative bet that it is unlikely that anyone posting to Independent Physics at Physics Forums has not found something new. Aware of these odds, what I actively look for are areas that have not been explored. I know neither Feynman or Einstein worked with quaternions. I know P.A.M. Dirac was asked if he was interested in a formulation of relativistic quantum field theory with quaternions (in other words, the Dirac equation), and after pausing a really long time as was his way, said he would only be interested if they were the real-valued quaternions (the person asking the question was crushed, since he had worked with complex-valued quaternions). This implies that Dirac did not work with 19th century quaternions.
Rob won that year on the flip of the last card, where is opponent got a flush, but he pulled a full house. In this particular instance, I have documented how Feynman's published analysis was not complete. It is nice it ties in so closely with the content of this thread.
I always get skeptical when I see "quotes" around "words". It usually indicates a breakdown in communication. A differential equation written with a division algebra should always be invertible. This would be a great time to cite such a proof for that assertion, or just do the proof myself, but I know my limitations.
The games with gauges are precise. One needs EM theory as well as gravity to be invariant under a gauge transformation for two reasons. First, both the particles that mediate the forces - the photon and the graviton - travel at the speed of light. Second, according to Lut, it is much easier to demonstrate a gauge theory conserves energy.
So what exactly do I mean by gauge theory in the context of the GEM proposal? Consider this scalar field:
g=\frac{\partial \phi}{\partial t} - \frac{\partial Ax}{\partial x} -\frac{\partial Ay}{\partial y} -\frac{\partial Az}{\partial z}
Not a one of these terms ends up in the field equations in GEM field equations derived in most 438, 442, or 457. You are free to let \nabla . A = 0, known as the Coulomb gauge. You could work in the static gauge, setting \frac{\partial \phi}{\partial t} = 0, or the Lorenz gauge, \frac{\partial \phi}{\partial t} + \frac{\partial Ax}{\partial x} + \frac{\partial Ay}{\partial y} + \frac{\partial Az}{\partial z} = 0.
Peter Jack was the first person to write operators with real-valued operators to generate the Maxwell field equations. A year later, I did that trick independently. We has the mark of independent researchers: we did not do it the right way, just a way that worked. Post 438 is significant because the derivation is the first to use a quaternion to generate the Lagrange density. Once one gets E
2 - B
2, the rest is completely triple grade A standard field theory. And I got the Poynting vector as a freebee. An accident? I wouldn't bet against that one. Elegance is an essential guide in the search for truth.
One of the steps used there is familiar to anyone who decides to play with quaternions, and that is to eliminate the scalar using a conjugate, q - q*. That is what was done on the road to E
2 - B
2. We know the Maxwell equations is gauge invariant, and when the Lagrangian is formulated with quaternions, the
reason is clear: it got subtracted away. Nice. Do the same exercise with the Even representation of quaternion - an idea from February 2008 - and one gets the field equations which are the natural relativistic form of Newton's law of gravity.
There was no way I could have planned it, but to find the unified field theory, do the exact same as EM and G separately, just skip the q - q* business. Then both Maxwell and G toss in the squared gauge, but with opposite signs, so they drop. An accident? See above comment again.
I also need to formulate GEM in a way that is not invariant under a gauge transformation. This will apply to the multitude of particles that do not travel at the speed of light. I haven't done that yet here, I got distracted defending the virtue of this work, and preparing talks, and living life (broken arm managements, yadda, yadda).
It is clear that the word symmetric is of concern to someone steeped in the technical nature of approaches to GR. As often repeated, I am not doing a variation on GR, I am doing a variation on the Maxwell equations. As you may know, one needs to supply the background metric as part of the mathematical structure of the Maxwell equations to put it to use (people usually use a flat metric, but it is a choice). There is no differential equation to solve that constrains what the metric can be.
What I am doing is a variation on the Maxwell equations, just barely enough to provide a differential equation whose solution is a dynamic metric based on the physical conditions.
It is a fine thing to question how a symmetric component could integrate into the math structure of GR to provide a non-zero result. As a variation on GEM, the concern is silly. Here are the undoubtedly non-zero terms found in the fields of Maxwell:
E = -\frac{\partial A}{\partial t} - \nabla \phi
B = \nabla \times A
And from the same soil, here are the two fields I refer to as the symmetric analogues needed for gravity:
e = \frac{\partial A}{\partial t} - \nabla \phi - 2 \Gamma_\nu^{\mu 0}A^{\nu}
b = -\nabla \Join A - 2 \Gamma_\nu^{i j}A^{\nu}
where \Join is defined as the symmetric curl, composed of the same terms as the curl, but all the signs are positive.
The fields of E and B are manifestly free to be non-zero. The fields e and b, no matter what labels we attach to them, are also free to be non-zero.
So how are these four different? If one decides to work with a metric compatible, torsion-free connection, then the way a dynamic metric changes will not change a calculation of the fields E and B, but will change e and b.
There are many claims on the Internet about Maxwell and quaternions. The idea of the curl was due to quaternions, as was the gradient, the dot product, the divergence, scalars and vectors. In the first edition, he used pure quaternions, where the scalar is equal to zero. That is a different way to write a 3-vector. The pure quaternions were removed by the third edition. In the introduction, he predicts that someone will someday figure out how to do all the work with quaternions, a point of pride for me, completing a task Maxwell defined.
Maxwell would not have been concerned about the issue of spin 2 symmetry, it was before his time. He certainly wouldn't be concerned with the divergence of the Christoffel of the Rosen metric. Smart guy, but imperfect at seeing into the future, a common problem.
Doug