Unifying Gravity and EM

[sweetser]

Submission for the 2007 Essays on Gravitation

Hello:

For the last 50+ years, there has been a contest for the most "provocative" paper dealing with gravity. I entered the 2007 contest, but alas, did not win, nor did it receive one of a few dozen "honorable mentions". I have been trading emails with the guy who won the top prize of $5k. I though I'd plug in my entry, even though it is far longer than is the norm here, coming in just under 1500 words, per the rules of the contest. I hope you enjoy.Submission for the 2007 Essays on Gravitation
Sponsored by the Gravity Research Foundation

Title: Geometry + 4-potential = Unified Field Theory
Author:
Douglas B. Sweetser
39 Drummer Road
Acton, MA 01720
sweetser@alum.mit.edu

Abstract:
Gravity is the study of geometry. Light is the study of potentials. A unified field theory would have to show how geometry and potentials could share the work of describing gravity and light. There is a long list of criteria that must be satisfied to have a reasonable hypothesis, from recreating the Maxwell equations, to passing the classical tests of gravity, to demonstrating consistency with the equivalence principle, and working well with quantum mechanics. This essay works through many of the common objections.

Paper:

Geometry without a potential is like a bed without a lover. The Riemann curvature tensor, with its divergence of two connections, is exclusively about geometry and all about the bed sheet. Newton's scalar potential theory was the first math to reach and direct the motion of the stars. It is only about the scalar potential. Unfortunately, it is too small, being inconsistent with special relativity. I will try to construct a unified field theory for gravity and electromagnetism as a compromise between Newton and Einstein, the potential and the metric, in a way that will get along with quantum mechanics. My guiding principle is provided by Goldielocks, who might find a scalar theory too small, a rank 2 theory too large, so perhaps a rank 1 proposal will be just right.

I honestly love Newton's potential theory. It is still in use today by rocket scientists who do not put an atomic clock onboard their ship. It gets half the answer right about light bending around the Sun. When a theory comes up short, we can either discard it or figure out the simplest way to lend a hand. A gravitational theory with a thousand potentials instead of one will be able to match every experimental test of gravity. Use Occam's razor: a 4-potential should be more than adequate to match how gravity makes the measurement of time get a little smaller, while the measurement of 3-space gets a little larger.

I am torn between two lovers, Newton and Einstein, feeling like a fool. Thugs from Ulm will insist that gravity must be a metric theory. They have the experimental tests of the equivalence principle to prove it. They punch home the fact that the way to take the derivative of a connection that transforms like a tensor is through the Riemann curvature tensor. Drop the Ricci scalar into an action, vary it with respect to the metric, and out from the heavens flies Einstein's field equations.

What's wrong with that? A metric theory isn't silly at all. One must be able to express gravity in terms of a metric. Based on my respect for Newton, I wonder if it is possible to find a compromise between a larger 4-potential and a metric theory?

When we were young, we would write a covariant tensor as $A_{\nu}$. The differential $\partial_{\mu}$ also transforms like a tensor. When we bring these two together, the 4-derivative of a 4-potential, $\partial_{\mu} A_{\nu}$, the result does not transform like a tensor. The reason is that as we move around a manifold, the manifold - not the potential - might change. A means of accounting for a changing surface must be made. Here is the definition of a covariant derivative all students of gravity learn:

$$\nabla_{\mu} A_{\nu} = \partial_{\mu} A_{\nu} - \Gamma_{\mu \nu}^{\sigma} A_{\sigma}$$

Can you spot the symmetry and identify the group implied by this definition? Imagine we make a measure of one of these terms, say $\nabla_0 A_0$, and it happens to be 1.007. If one worked in flat Euclidean spacetime, the connection would be zero everywhere, and everything would come from the change in the potential, $\partial_0 A_0$. One could also decide to use a constant potential, so the dynamic metric's connection would account for all the change seen, $-{\Gamma}_{00}^{\sigma} A_{\sigma}$. One has the ability to continuously change the metric and thus the connection so long as there is a corresponding change in the potential which leaves the resulting covariant derivative invariant. This sounds like the group Diff(M) of all diffeomorphisms of a 4D spacetime with the additional constraint that there are changes in the 4-potential such that the covariant derivative is unaltered.

Born background free, as free as general relativity, one must find a differential equation whose solution will dictate the terms of the dynamic metric. That is what the derivative of the connection in the Riemann curvature tensor does: there are second derivatives of the metric whose solutions under simple circumstances can be found. I have chosen to study the simplest vacuum 4D wave equation:

$$\square^2 A_{\mu} = 0$$

It is vital to note that I did not write the D'Alembertian operator, which would have been a box without the 2. Instead this is a covariant derivative acting on a contravariant derivative acting on the 4-potential. The first derivative will bring in a connection, and the second derivative will take the derivative of the connection, resulting in a second order differential equation of the metric, precisely what is needed to be background free. Can we find interesting combinations of metrics and potentials that solve this differential equation and is consistent with all tests of gravity to date?

Say we used a constant potential, where all the second derivatives were zero. Make the problem simple: a static, spherically symmetric, and non-rotating mass. For those skilled in the arts of differential geometry, it should be straightforward to show that the divergence of the connection of the exponential metric (below) is a non-trivial, entirely metric solution to the 4D wave equation. Compare the exponential metric in isotropic coordinates:

$$d \tau^2 = exp(-2 \frac{G M}{c^2 R}) d t^2 - {\frac {1}{c^2} exp(2 \frac{G M}{c^2 R}) (d x^2 + d y^2 + d z^2)$$

a nicely matched pair of exponentials, with the Schwarzschild solution in isotropic coordinates:

$$d \tau^2 = (\frac{1 - \frac{G M}{2 c^2 R}}{1 + \frac{G M}{2 c^2 R}})^2 d t^2 - \frac{1}{c^2} (1 + \frac{G M}{2 c^2 R})^4 (d x^2 + d y^2 + d z^2)$$

which is inelegant enough to rarely be seen in books on general relativity. Theorist prefer the Schwarzschild coordinates while experimentalists must work with isotropic ones. Beauty may be in the eye of the beholder, but an exponential is the calling card of a deep insight into physics.

Either metric satisfies all tests of the equivalence principle because the solution is written as a metric. Either metric satisfies all tests of the weak field because their Taylor series is the same to the terms tested. Either metric satisfies all strong field tests because it is entirely about a metric, so there is no other field to store energy or momentum. For an isolated system, the lowest mode of emission is the quadrapole moment. The metrics differ in second order effects by twenty percent in how much light is bent around the Sun, so it is a shame no one has been funded to get the data.

The 4D wave equation has been quantized, and written up in most books on quantum field theory, in the section on relativistic quantization of the Maxwell equations. Two of the modes of emission are the transverse spin 1 fields of light. That is no surprise. The scalar and longitudinal modes are banished to a virtual state using a ``supplementary condition'' because the scalar mode would allow negative probabilities, a no-no. That is the way it is for a spin 1 field theory where like charges repel. The field strength reducible asymmetric tensor $\nabla_{\mu} A_{\nu}$ for this proposal can be split in two: an irreducible antisymmetric rank 2 tensor to do the work of electromagnetism with a spin 1 field so like electric charges repel, and an irreducible symmetric rank 2 tensor to do the work of gravity with a spin 2 field so like mass charges attract. Gravity couples to the 4-momentum, not the rank 2 stress-energy tensor. All forms of energy go into both sources, except one: the energy of a gravitational field. To be consistent with electromagnetism, gravity fields do not gravitate. Should a gravity wave ever be detected and measured along six axes, the polarization of that wave will be transverse if general relativity is correct, but not if this unified field proposal is accurate. Such data will be hard to get, but the difference would be unambiguous.

The speed of gravity is the speed of light, and so its field strength tensor must be gauge invariant. The field strength tensor $\nabla_{\mu} A_{\nu}$ is only gauge invariant if its trace happens to be zero. That is where the massless graviton lives. When the trace is not zero, then the scalar field formed from the trace of $\nabla_{\mu} A_{\nu}$ will break the U(1) symmetry of electromagnetism. The Higgs particle is unnecessary. There is a quantum expression of the equivalence principle, a link between the spin 2 particle ($\nabla_{\mu} A_{\nu}$ when $tr(\nabla_{\mu} A_{\nu} = 0$) that mediates gravity and the scalar field needed to establish inertia ($tr(\nabla_{\mu} A_{\nu})\neq 0$).

There is an important benefit to splitting the load for describing gravity between the connection and the changes in the potential. By using Riemann normal coordinates, an arbitrary point in spacetime can have a connection equal to zero. For that point, the energy will be zero. That has remained a technical problem for people trying to quantize general relativity. For this proposal, the energy contributed by the connection could be zero, but that contributed by the potential would be non-zero. Localized energy is a good thing.

Einstein had a great respect for Newton's towering body of work. He might have appreciated this compromise between geometry and potentials which allows light to lay down with gravity in the same equation.

fini

 

[CarlB]

Doug, I'm glad you put this up here. That concept of "splitting the load between the connection and the potential" gives me a better intuitive understanding.

I will get back to the equations of motion soon. Right now there is a lot of furious paddling underneath the surface, so don't think that this duck isn't going anywhere.

 

[sweetser]

Meaning of "gauge"

Hello Carl:

That concept of "splitting the load between the connection and the potential" gives me a better intuitive understanding.

Thanks. The jargon word - gauge - I know is going to create issues. This is a German word for "measure" which we first get exposure to with model trains. In field theories, it means there is a field that can be added in that will not change a solution to some field equations, just the numbers that pour out. The ability to add in the field means one can impose a gauge constraint - I wish to make the first term always equal to zero, so I'll add in this gauge field.

My proposal involves an issue of measurement, but of a different character. It comes straight out of the definition of a covariant derivative. You, the constructor of the numerical description, get to decide how much of the ball dropping to the Earth is due to a 4-potential, and how much is due to spacetime curvature. Go from one extreme to the other in a continuous way, you make the call.You are a courageous duck, and I am very patient. Those equations of motion look nasty to me, particularly since they are all derivatives with respect to the interval. I'm glad there are both the GR and GEM equations, because those results should differ only at the second order PPN level of accuracy, and even then only by 20%. My sense is that one would need to be hyper paranoid about error management before any difference between the two was "real" and not an artifact. I have no idea how one distinguishes between two big, complicated calculations that are different by a wee little bit, and rounding error.

doug

 

[sweetser]

Spin 2 current-current interactions

Hello:

I have been having an email discussion with a physics professor about my proposal. He did not agree that the second rank symmetric tensor would be the place where a spin 2 field would live. Instead he said it might be a place for a rank 0 field. That mystified me, since a quality of a scalar field is the complete lack of indexes, and a rank 2 tensor has 2. He also was talking about currents, something I had never done anywhere, not even in this very long thread! He scolded me, told me to read "Feynman Lectures on Gravitation", chapter 3.

Feynman rocks! I completely understood where the critical professor was coming from. His objection was reasonable. Then, once I really get a complaint, a small twist of an under-used math tool is usually all it takes to pull out the thorn. Here goes...

Feynman's analysis deals directly with 2 currents that interact, a viewpoint I have not used often. The charge coupling term, $j'^u A_u$ has one current. Where is the other? One can take the Fourier transform of the 4-potential $A_u$ and in the momentum space representation rewrite the potential like so:

$$A_u = - \frac{1}{K^2} j_u$$

This is how the 2 currents interact:

$$interaction = - \frac{1}{K^2}j'^u j_u$$

Make things simpler by having the current move along z:

$$K_u = (\omega, 0, 0, k)$$
$$K^2 = \omega^2 - k^2$$

Write out the interaction by its components:

$$- \frac{1}{K^2}j'^u j_u = - \frac{1}{\omega^2 - k^2}(\rho' \rho - j_x' j_x - j_y' j_y - j_z' j_z)$$

Charge is conserved, so:

$$K^u j_u = 0 = \omega \rho - k j_z$$

Use this to eliminate $j_x$:

$$- \frac{1}{K^2} j'^u j_u = \frac{1}{k^2}\rho' \rho + \frac{1}{\omega^2 - k^2}(j_y' j_y + j_z' j_z)$$

If we are in the rest frame of j' or j, then only the charge density matters. Move relative to that reference frame, and the other terms come into play.

Feynman now focuses on the jx and jy terms. This is where physics becomes math magic. These two currents always involve virtual photons. Further, Feynman works with the poles, where $\omega->k$. These virtual photons are the sum of two independent terms, $j_x' j_x$ and $j_y' j_y$. A different way to say this is that there are 2 independent polarities for photons.

That was fun, but I wanted to think more precisely about the product of two currents. I'm going to use quaternion algebra, but if you are more comfortable with the Dirac algebra - only a twist of i away - go ahead.

$$(0, j_x', j_y', 0) (0, j_x, j_y, 0)^* = (j_x' j_x + j_y' j_y, 0, 0, j_x' j_y - j_y' j_x)$$

The phase term is in the z slot. It will require a 2 pi rotation to get back to go. The current-current interaction is a spin 1 photon, so like charges repel. Good.

It occurred to me that there might be another distinct product of these two currents. Consider the conjugate operator, which flips all the signs except the first one. It is known by mathematicians that there is more than one anti-involutive automorphism. Let's break down that jargon. The automorphism means that the function maps back to the same space. Taking two operations brings the function back home. The final bit is (a b)* = b* a*. Big words, but here is a simple idea: fix a term other than the first one, and flip the signs of all others. This little algebra trick is missing from many professional physicists tool drawer. Let me define the second conjugate like so:

$$(t, x, y, z)^{*2} === ( (0, 0, 1, 0) (t, x, y, z) (0, 0, 1, 0) )^* = (-t, -x, y, -z)$$

Put this tool to work for two interacting currents:

$$(0, j_x', j_y', 0) (0, j_x, j_y, 0)^{*2} = (j_x' j_x - j_y' j_y, 0, 0, j_x' j_y + j_y' j_x)$$

Take a peek at page 39, and you'll realize this product has the character of spin 2! I think the idea is that the two parts of the phase term can add together, able to race back to their initial spot in pi radians. This product describes with two degrees of freedom a current interaction where like charges attract. Cool. A 4-current has 4 degrees of freedom, 2 for a spin 1 current where like charges repel, 2 for a spin 2 current where like charges attract.

This calculation made my memorial day weekend memorable.

 

[sweetser]

Subtle correction to GEM current-current interaction

Hello:

In the game of unified field theory, terms involving gravity must be similar to those for EM, but not exactly the same. In the above post, I made the gravity current-current interaction exactly in EM's image, looking only at the $j_x$ and $j_y$ currents. This indicates that gravity is a transverse wave.

Gravity is not a transverse wave. The transverse modes of emission for a 4D wave is light, the photon. That leaves the scalar and longitudinal modes for gravity. Something has to be different to justify the different modes of emission. I decided to see if I could somehow involve the current $j_z$, since that would be a real current along the direction of motion. Going back over the calculation, instead of eliminating $j_z$, I tossed out $\rho$:
$$- \frac{1}{K^2} j'^u j_u = \frac{1}{\omega^2}j_z' j_z + \frac{1}{\omega^2 - k^2}(j_x' j_x + j_y' j_y)$$
Now I can form products with $j_x$ and $j_z$:
$$(0, j_x', 0, j_z') (0, j_x, 0, j_z)^{*1} = (j_z' j_z - j_x' j_x, 0, - j_x' j_z - j_z' j_x, 0)$$
Likewise for $j_y$ and $j_z$:
$$(0, 0, j_y', j_z') (0, 0, j_y, j_z)^{*2} = (j_z' j_z - j_y' j_y, j_y' j_z + j_z' j_y, 0, 0)$$
This still has the spin 2 symmetry Feynman refers to on page 39 of his lectures on gravity. What is revealing is to add up these current products:
$$(0, j_x', j_y', 0) (0, j_x, j_y, 0)^{*} + (0, j_x', 0, j_z') (0, j_x, 0, j_z)^{*1} + (0, 0, j_y', j_z') (0, 0, j_y, j_z)^{*2}$$
$$=(2 j_z' j_z, j_y' j_z + j_z' j_y, - j_x' j_z - j_z' j_x, j_x' j_y - j_y' j_x)$$
The first term, the current density, is all the real current, while the phase has both spin 1 and spin 2 symmetry. This appears to be an improvement to me.

doug

 

[sweetser]

Report from the 10th Eastern Gravity Meeting

Hello:

This is another L O N G post. I went to the 10th Eastern Gravity Meeting expecting to give a 12 minute talk like I had since EGM 7. It didn't happen because there were too many people attending. I wrote up this note, and will be shipping it out on Monday to the organizers. This is life for an "independent researcher". I do try to do polite body slams.


Subject: Notes from the fringe of the physics community

Alan Lightman discussed progress in physics as a result of community. I thought I would share a few observations I made based on my experiences at EGM 10 from the position on the edge of the community. The hope is to improve the situation in the future, which is why I cc'ed the past organizers.

I define a fringe physicist as someone not employed as a physicist currently, or not on a path towards a degree in physics, who feels compelled to make a contribution. The definition can be applied without passing judgment on the value of the claimed contribution. By this definition, I am a fringe physicist, working for a software company with degrees from MIT in biology and chemical engineering, hoping to unify gravity with the other three fundamental forces of nature.

One defining characteristic of any community is how it treats the elements at the edge. The issues are never easy: is it better to let people live hard but free lives in the streets of the Northeast or force them to take shelter from an institution? The way to handle fringe physicists is not physically dire, but many have a great emotional devotion to their perspective projects. My own approach is to reflect Nature, which doesn't care if we get answers right or wrong, but she gives us an unmatched opportunity to be. Analytical indifference pulls against human hope, and at least for this email, hope has a lead.

At the bulletin boards provided by physicsforums.org, they grew tired of repetitive posts from fringe physicists and created a special place for "independent researchers". They have eight criteria for admission. My work met the criteria, and a discussion going on over there has been accessed twenty two thousand times, a high number for that site. That discussion has improved my proposal. I know where I've made mistakes in my Lagrangian. Dialog can make a difference.

The EGM has been small enough in the past to let everyone hear from the few east coast fringe physicists willing to make the journey. I had presented my work three times before, receiving a few questions at the end. My intent this year was to answer some of those questions, such as a demonstration that the exponential metric I work with satisfies the data for the precession of the perihelion of Mercury. For me, that was a difficult problem to solve. Although I had flipped through a good number of written solutions, none of them made sense in detail. For more than three years I had lived with the weight of feeling inadequate to answer the question. I am persistent to a fault. Sean Carroll's notes helped me step into the problem on the ground floor. I did complete the calculation. I was pleased enough with the twenty four step process that when I got married in October, for my groomsmen I made lunch boxes with the derivation on the side panels. A box made it to Ithaca, but stayed in my bags. The lunch box is both a prized possession and embarrassing.

EGM 10 had to take a different approach. There were too many talks to do in two days. We can hope that next year a smaller armada will fly out from the west coast. The wind was strong due to the stars appearing for Saul's symposium. People being creatures of habit, I would not be surprised if the attendance stays at this level for EGM 11.

The decision was made to provide a poster session for the fringe physicists. There is nothing inherently wrong with such an approach. My guess was such a decision was made very late in a chaotic and hectic game. There is almost no need to point out that people whose talks were shifted to the poster session should have been sent an email, but that detail of execution was missed. I checked the website on Wednesday at work and was able to print out my slides. At Ithaca during the lunch break on Thursday, a $34 trip to the bookstore plus some high paced snipping led to a colorful poster I was pleased with.

At the end of a long day of talks on Thursday, the session was closed without announcing the poster session. There were more fringe physicists with posters than members of the physics community proper. Although Alan Lightman feels guilty about his baby seal clubbing incident with Webber, being utterly ignored can be as bad. Between these two extremes, is there something better?

This is what I did: I practiced my skills as a skeptic. This turned out far better than I expected. It was trivial to spot errors and glaring omissions. It requires discipline to resist clubbing. The bigger challenge was to find the roots of the person's area of study. It is a great exercise to find the pea under the pile of mattresses that is bothering this person. Since the list is not long, I will tell you the positive things I learned from three of my fellow fringe physicists.

1. Ed "negative mass" Miksch was the man who drove up from Pittsburgh with his wife. The organizers certainly know, but other folks may not be aware, how assertive he and his wife were about the need for Ed to be heard by this gathering of physicists. He was the guy who gave the three minute speech at the end of Friday's session. He claims to have shown that we should all be working with negative mass because he's done the calculation, one no physics professor at Reed College in the 50s could find an error in. The reason is that from where his calculation starts, there is no trivial math error.

This issue was understood by none other than James Clark Maxwell, but has not reached the wider physics community. The story was written up well at this URL: http://www.mathpages.com/home/kmath613/kmath613.htm. In GR books, they note the link between Newton's potential theory and the Schwarzschild metric, g_00 = 1 - 2 G M/c^2 R = 1 - 2 phi, and so ignoring the constants, phi = - M/R. What Maxwell understood was that such a potential plugged into a field theory like that for electromagnetism implies that like mass charges repel. The correct answer for a Newtonian potential where like charges attract is phi = 1 - G M/c^2 R!. Ed started from the wrong place because he was instructed by everyone that phi = -M/R is OK. Maxwell did see the correct way out - add a HUGE positive constant - he just couldn't justify it. Einstein's metric theory gets the potential theory right. Ed should be proud that he saw a problem realized by Maxwell. It is unfortunate that this potential theory is taught incorrectly to this day, because there will be people in the future that will walk down this wrong bend in the road.

To appreciate the consequences of how Ed's issue is misunderstood at the highest levels of the physics community, I asked Clifford Will why a 4-potential theory was not listed even as a possibility in his Living Review article on GR. He claimed one could make a potential theory get h!alf of the light bending around the Sun, but not all the bending measured by experimental tests. Will is correct for a scalar potential theory, the g_00 term. A 4-potential theory could easily match the data, with A = (1 - 2 GM/c^2 R, -1 - 2 GM/c^2 R, -1, -1), so the g_00 and g_11 terms will contract time measurements and expand space measurements as seen in tests.

2. Fred "MM" Pierce had a poster on the Michelson-Morley experiment. I was not hopeful as he went into his story. He talked about relativity, not once distinguishing between special and general relativity, two very different theories. He pointed out that if the interferometer was placed vertically instead of horizontally, the vertical machine would have interference fringes, both here and at the antipode, while the horizontal machine would not. I told him that was consistent with our current understanding. He then claimed this was connected to a misunderstanding about an ether. I told him I saw no need for an ether.

He also focused on a spinning satellite. He pointed out how the satellite would provide an accelerated reference frame. Go a different distance out from the spinning hub, and your weight would change although your mass would stay constant. Again it sounded consistent with our knowledge. He claimed it shows a link between motion and the cause of gravity. Gravity must be motion since it is the same as being on a spinning satellite. I told him I'd think about it, and left it at that.

Both Newton and Einstein spent quality time thinking about the spinning bucket. I don't think that debate has been settled. There is the standard elevator thought experiment. Gravity has tides, but the elevator does not. Some say that tides are the only real effect of gravity. The spinning satellite has tides, therefore it is a more faithful representative of gravity than the rocket. Once going, the satellite can maintain its "fake" gravity via angular momentum inertia, unlike the rocket which requires ever increasing amounts of energy.

Consider this thought experiment. Someone has replaced the Earth with a thin shell made of dense material such that the mass of the Earth is the same. You don't notice the change until you find a hole in the shell. Curious, you climb through, and almost by accident because you didn't hang on, you end up (or down?) flying across the hollow inside, there being no gravity field. After quite some time (the Earth is large), you get to the other side, and spring on back. This time you grab on to the edge. You got to collect physical data on Gauss' law that there is no gravity field inside a hollow sphere, how cool. You hear a noise, some creaking, and then stand up. The shell of an Earth is moving. It is picking up speed, and finally, it feels like you weigh your usual weight due to the spinning of the Earth. You can crawl back out the hole, and feel your weight due to gravity, or go into the hole, and feel your weight due to the spinning. Do an experiment with tides, and you realize one is a continuation of the other, no matter if you are inside or outside the shell.

I have come to the conclusion that any proposal for gravity must make clear the connection between gravity and rotational dynamics.

3. John "Two Timing" Kulick works with the two dimensions of time. I have written command line programs to add, subtract, multiply, divide, take sines, cosines, and apply group theory to events in spacetime. None of these will work if there are two times. John's work made no sense to me. Although I challenged him on traveling faster than the speed of light, that discussion went nowhere. His mantra was geometry, and I didn't get it.

What I did understand was his complaints about cosmology. From the rotation profiles of disk galaxies, to the big bang, basically anything big or old, physics fails. I am too skilled a skeptic to believe in two dimensions of time, dark matter, or dark energy. It is our mathematical description of nature that will have to change.

4. Doug "Rank 1" Sweetser has a unified field theory. He has also worked on leprosy (cloning genes from the mycobacteria that cause the disease) which somehow seams appropriate: no one wants to hang out with someone who has anything to do with leprosy or unified field theory. It turns out that leprosy is near impossible to transmit, but is the most visually frightening disease and thus the most feared because we are predominantly visual. Since Einstein worked more than thirty years on a unified field theory and failed, this is a topic to avoid, a kiss of death for an academic career.

Prof. Steve Carlip said through an email exchange that he thought my action could only involve a spin 1 and a spin 0 field. I told him if that was true, my proposal would be wrong. I had trouble following his logic. My field equations are rank 1, but my field strength tensor is rank 2. The part that does the work of gravity is a rank 2 symmetric tensor, so I couldn't understand how it could describe a spin 0 field that arises from an index-free tensor. Steve held his ground however, giving up on the discussion, telling me to go read chapter 3 of the Feynman lectures on gravity. That's what I did during EGM 10.

There are two separate reasons why one can spot a spin 1 field in the EM Lagrangian. The first is the rank 2 antisymmetric field strength tensor, A_u;v - A_v;u. The second arises from the charge coupling term, J^u A_u, that can be rewritten using a Fourier transform in momentum space as a current-current interaction. Take one current and the conjugate of another current, form the product, and the phase of the resulting term will return after 2 pi rotations, thus is spin 1. See section 3.2 for details.

My relationship with Steve broke because I was utterly unaware of this standard approach to field theory. I spent many fun hours going through the details of chapter 3, picking up nuances, working on my speed of creating the logic flow. I had his critique based on the coupling term down pat. I have been fortunate that once I understand an issue, I can see two others: how to get around the problem and why it has stumped people before.

If you are given a 4D vector space, list the anti-involutive automorphisms. Sounds too mathematical, sorry. A conjugate is an example. My guess is that nearly all well-trained physicists believe there is nothing other than the conjugate. That is true in a 2D space like complex numbers. A conjugate has three properties: you stay in the same place, two operators in a row is like doing no operator, and (a b)* = b* a*. One could imagine an operator that flips the sign for all but the x. I call this the first conjugate because it is the first part of the 3-vector. The second conjugate is defined similarly. Although one could define a third conjugate, it does what the other three could do in combination, so I don't include it. My bet is that all who read this note have never used a first or second conjugate, but it should be a simple idea to absorb.

I redid the Feynman current-current interaction calculation, and when the time was right, used a first conjugate instead of the Plain Jane conjugate. The resulting product is written on page 39 for a spin 2 particle. Getting the details solid on that calculation made the trip.

My unobserved poster had two themes. One board was nothing but the Lagrangian, a partial derivative party. I included the recently worked out details of the current-current spin 2 interaction. The second part tried to answer a difficult question asked of me: what does your theory do that's really different? A 4D Lagrangian is not different.

I decided at this meeting to come out of a mathematical closet, and admit publicly that I use quaternions. My observation is that half of technically trained people are familiar with the word, and few have done any serious calculations with them. The exceptions are rocket scientists and game designers who do 3D calculations without the problem of gimbal lock. There are a few people who play near quaternions: Connes with non-commutative geometry, Penrose with twistor theory, Alder with quaternionic quantum mechanics, and Baez with octonions. Out here on the edge, I learned how to take a quaternion expression and make an animation. I figured out how to make a ten second animation of SU(2), the group sitting in the middle of the standard model. I can say with complete confidence you don't know what it looks like, but my ipod does. Small steps from there have led to animations of U(1), U(1)xSU(2), SU(3), and Diff(M)xSU(3). I think a visual justification of the standard model with gravity qualifies as "really different".


FUTURE LINE OF ACTION

The organizers of EGM 11 will face the same issues you have. Please feel free to pass on any of these observations and suggestions.

We should admit that there is a physics fringe. That fringe needs to interact with physics skeptics. I could see a later presentation session, or a poster session for "alternative approaches". We would need to get a good dozen or more grad students and professors, including the organizer of the meeting. The reward for the professionals would be a chance to work on their skills as skeptics. It is a balance of asking, connecting, sifting through the history of physics, and criticizing. The session should not be promoted as such, rather we find a different way to accommodate a group that will hopefully remain on the same scale of a half dozen. I could serve as a liaison or chair of the session.


Was the conference worth my time and eleven hundred dollars? The funds came from an estate left by my mother back in August. She would have wanted me to go but to be careful someone did not steal my ideas. This unified field theory is the elephant in my life. I am proud of its latest trick with a spin 2 current-current interaction. I want to know if this elephant is real or a technical mirage. My own limitations are painfully glaring to me. This note, long as it is, has brought clarity on my feelings concerning the fringe of physics. Sorry for the fire hose of an email and attachments, but I am from MIT.

Thank you for all the effort you put into this meeting.

doug

 

[CarlB]

Thanks for the report. About "The exceptions are rocket scientists and game designers who do 3D calculations without the problem of gimbal lock." The situation in geometric algebra is similar. Half the users are computer programmers.

 

[sweetser]

Why like charges repel in EM, like charges attract for gravity

Hello:

I posted this in the newsgroup sci.physic.research. It was a clear explanation about like charges in EM and gravity, so am adding it to this thread.

>give some examples of your calc that do occur in, or do reflect our real world applications. (mech/dynamics/electr)

>I ask this because the EM force carrier is supposed to be of a spin 1 nature, whereas the gravitational force mediator is supposed to be a spin 2 entity.

You are correct, the photon is spin 1, and the graviton which we won't be detecting any time soon is spin 2. A fundamental property of EM is that like charges repel. That is consistent with the force mediating particle being spin 1. Likewise, in gravity like charges attract, so the force mediating particle must be spin 2. Brian Hatfield gave a good explanation of this in the introduction to "The Feynman Lectures on Gravity". Force mediating particles must have integral spin. To always act one way, the gravity mediating particle must be even spin.

Since light is bent by a gravity field, a spin 0 particle will not work (anyone want to provide the reason, I've seen that written a few times, but am not clear on the logic). Ergo the simplest particle would be spin 2.

In EM where like charges repel, the photon must be odd spin. I don't know that I have ever heard a spin 3 particle discussed, but a photon is spin 1.

A Lagrange density describes all the ways a system can trade energy per unit volume. Integrate a Lagrangian over space and an arbitrary amount of time. If you can find something that can be varied without changing the integral, that is a conserved quantity for the action. From the Lagrangian, one can crank out the force equations by varying the 4-velocity keeping the 4-potential fixed, or the field equations by varying the 4-potential and keeping the 4-velocity fixed.

In the real world of EM, how can we look at the Lagrangian and tell that like charges repel and the force mediating particle is spin 1? Here is the Lagrangian:

$$\begin{equation}\mathcal{L}_{EM} = - \rho_m/\gamma\end{equation}$$
$$\begin{equation*}\tag{2}- J_q^{\mu} A_{\mu}/c\end{equation*}$$
$$\begin{equation*}\tag{3}- \frac{1}{4 c^2} (\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu})(\partial^{\mu} A^{\nu} - \partial^{\nu} A^{\mu})\end{equation*}$$

To generate the Lorentz force equation, only (1) and (2) matter since they are the ones that have a 4-velocity inside. Because they have the same sign, the resulting force law will indicate that like charges repel.

To generate the Maxwell field equations, only (2) and (3) matter since they have the 4-potentials. Here again, because they have the same sign, like charges repel.

There are 2 ways you can spot the spin 1 force mediating particle. The first way focuses on (3). If one switches the order of the indexes on the antisymmetric field strength tensor, the sign of the field strength tensor will flip. That is a signal that the spin is odd.

What I did not understand was how to look at term (2) and pick out a spin 1 field. Section 3.2 does a great job of it. The idea is transform the charge couping term into a charge-charge interaction. Look at the product of a charge-charge interaction, and focus on the phase. That takes $2 \pi$ to get back to where it started as expect for a spin 1 particle.

So to review, there are 4 reasons while like charges repel in the $\mathcal{L}_{EM}$ Lagrangian: the force equation (1&2), the field equation (2&3), the spin 1 particle in the antisymmetric field strength rank 2 tensor (3), and the spin 1 particle in the current-current interaction (2). Nothing like logical consistency!

If you wanted to make a Lagrangian that could do the work of gravity, all four of these chips must stack up. Here is a Lagrangian I play with:

$$\begin{equation*}\tag{4}\mathcal{L}_G = - \rho_m/\gamma\end{equation*}$$
$$\begin{equation*}\tag{5}+ J_m^{\mu} A_{\mu}/c\end{equation*}$$
$$\begin{equation*}\tag{6}- \frac{1}{4 c^2} (\partial_{\mu} A_{\nu} + \partial_{\nu} A_{\mu})(\partial^{\mu} A^{\nu} + \partial^{\nu} A^{\mu})\end{equation*}$$

The force equation will have like charges attract because the signs of (4) and (5) are different.

The field equations will have like charges attract because the signs of (5) and (6) are different. These two are easy and well known.

Look at (6), and you seen an indication of an even spin field because if the indexes are changed, the sign does not change. Because there are indexes, the second rank field strength tensor cannot be spin 0. The trace of this tensor would make a spin 0 field. Effectively there will always be spin 0 field associated with this higher spin field. Cool. If the trace is zero, then the particle characterized by (6) will travel at the speed of light. If not, the particle will have a non-zero mass.

In Misner, Thorne, & Wheeler, problem 7.2, they consider an antisymmetric tensor for (6) which cannot work because it would indicate an odd spin mediating particle, and thus not self-consistent.

The memorial day weekend calculation was about term (5). One has to be able to spot a spin 2 particle in the current-current interaction. That will have a phase that looks like 2 jx jy, so that in $$\pi$$ radians it will get back to where it started. I used a different kind of conjugate that many physicists are not aware of, basically one that tosses in a pair of basis vectors.

So to review, there are 4 reasons while like charges attract in the $\mathcal{L}_G$ Lagrangian: the force equation (4&5), the field equation (5&6), the spin 2 particle in the symmetric rank 2 field strength tensor (6), and the spin 2 particle in the current-current interaction (5).

Steve Carlip was correct to complain about my lack of understanding about the current-current/spin 2 issue. Hopefully I have made progress on it.

doug

 

[Mentz114]

Interesting

[Edit] I'm up to page 13 and it's looking great.

The attached paper is not peer reviewed, and I am not suggesting its content should be compared with your own work. Only the proposed tests of the metric might be of interest.

 

[sweetser]

Hello Hai-Long:

I've printed out your paper, and will hopefully figure out a comment which I will send via email, not this forum. I hope that reading through this LONG thread you will see that some progress has been made.

doug

 

[sweetser]

Betting Against the Higgs

Hello:

I have decided that until I identify a show stopping problem, I should try and promote this work with established physicists once a week. This is a sales job. Sales is difficult, because the answer is always no, no, no, no, no, no, and no. If I try weekly, that is at least 52 knocks on the door. As an example, no one responded to the the EGM 10 note posted earlier here. Presume no one responds to these notes unless I tell you otherwise. Nothing in this is personal, it is how people respond to pitches, whether it is a door-to-door vacuum cleaner salesman or an internet unified field theory promoter.

There was a show stopping problem recently that Prof. Steve Carlip pointed out, the current-current interaction having to be spin 2 when Feynman had shown it was spin 1. I think that one has been resolved by using a different kind of conjugate.

So the first pitch was to a Harvard professor. If you want to read a great, long article on the search for the Higgs by the Large Hadron Collider, click here:

http://www.newyorker.com/reporting/2007/05/14/070514fa_fact_kolbert

Many good people are in on the effort to detect the Higgs. My objections is only mathematical. I believe that gravitational mass breaks the symmetry of the standard model, so no Higgs mechanism with its false vacuum is needed. I work with a unification group theory that is smaller than the standard model - U(1)xSU(2)xSU(3) - which has 12 elements in its Lie algebra. Instead I work with SU(3)=(U(1)xSU(2))* U(1)xSU(2). One of the cool things about this smaller model is that it has the chance to explain why we don't see quarks: we already see particles for the photons and the weak particles. I hope people have clicked through the youtube visualization here:

http://youtube.com/watch?v=ExNPiMcVXww

So these are the reasons I don't accept the math - and it is only about the math, nothing personal.

Here was the email I sent to the good professor:


Hello Prof. Arkani-Hamed:

Based on a recent New Yorker article, I see you are an enthusiastic
believer in the Higgs particle, that we should be able to detect it at
the LHC. If that happens any time in the next ten years, I will fill
out the scanned check for the number of GeV/c^2 for the Higgs. You
have an incentive to hope for a heavy Higgs :-)

I used to play against someone who went on to win the World Series of
Poker (at that time, we were about even in skill for dealer's choice).
It might appear this is a high risk without reward. There is a small
chance that despite how busy you are, you might click through
some of the attached work I've done which makes me believe that the
Higgs is unnecessary.

We know what U(1) looks like, a circle in the complex plane. Do you
know what the group SU(2) looks like? I bought an Ipod because I
figured out how to do this (a unique reason for the purchase, I still
forget to bring the earphones with the device, visualizing the
standard model being more important than a few mp3's). I also have
electroweak symmetry, SU(3), and Diff(M) in the machine. I happen to
be an independent guy who has been playing games with quaternions, the
kind of number sitting at the center of the standard model. Analytic
animations using quaternions led directly to these result.

I'm a lab technician by training. I report what I have. I have a
rank 1 field theory that unifies gravity and EM. That statement can
be supported by the Mathematica notebook that goes from the Lagrangian
out to a metric that is consistent with first order PPN tests of weak
field gravity, and predicts 0.8 microarcseconds more bending of light
around the Sun than the Schwarzschild metric of GR. The exponential
metric is manifestly more elegant than the Schwarzschild metric
because exponentials appear in fundamental laws of physics.

It is clear that if this proposal is correct, there will be a new
stable constant velocity solution to classical gravity. I have yet
to cow rope the equation to problems like the rotation profile of
spiral galaxies or issues with the standard big bang (I've had trouble
both understanding the numbers being used, and how to do the numerical
integration). I point this area out because it may provide a new way
to solve big problems in cosmology and to show I don't overstate my
case.

Stay busy.
doug

ps. The check was written yesterday, before the details of the work at
Fermilab on Cascade B, not the Higgs, came out. This is all about the
math, nothing about the good people who do both the experimental and
theoretical work.

 

[rewebster]

Just as an aside--In what area of work are you a 'lab technician' ? Is it sort of like a patent clerk?

 

[Mentz114]

Doug,

in post #313 you give the EOM for the exponential metric. You're in 3 dimensions and theta is missing so I asume you've made a simplifying assumption like theta = pi/2, or dtheta = 0. Can you please confirm exactly which metric you used so I can compare with my calculations ?

Thanks,
M

[edit]I sorted it out. I get the same equations now.

 

[sweetser]

The aside

Hello Rewebster:

I am a oversensitive to your question. This is the Independent Research Forum, which makes it fringe physics turf with a veneer of respectability, a veneer that is important not to scratch. John Baez created the "Crackpot Index", http://math.ucr.edu/home/baez/crackpot.html. For example, I get points for this one:

30. 30 points for suggesting that Einstein, in his later years, was groping his way towards the ideas you now advocate.

It is clear that Einstein, in his later years, was looking for a way to unify gravity and EM. There is great documentation for this. I would not say he used my approach. He did try all kinds of different ways. For a good discussion of this, read chapter 26 of A. Pais' "Subtle is the Lord...". The difference technically is clear: Einstein used the Riemann curvature tensor which is kind of like a divergence of connections, a connection being a measure of how metrics change. The Riemann curvature tensor is thus a measure of the second order changes in metrics that transforms like a tensor.

That works for gravity. The problem is that there is no place for a 4-potential. Potential theory is essential for EM, the weak, and the strong force. I am trying to find the compromise between changing potentials and changing metrics.

Now to your aside. I was a molecular biologist. I cloned and sequenced DNA from the mycobacteria that causes leprosy. It was a remarkable job at many levels. The most relevant part for this thread was the way I would go in every day, feeling utterly inadequate for the scale of what was ahead, yet trying to do a little bit. I have avoided going for top positions in my area of work so there would be time to work on things I cared about that are utterly irrelevant to how I make a dollar. The same holds true today, although now I work for a software company. I am not trying to climb any ladders here, no 60 hour work weeks for me.

doug

 

[rewebster]

Please forgive me if there was any 'itch' caused by my comment. There is/was a comparable nature of someone pursuing and doing the all fine work that you ARE doing while still laboring at a job that isn't related to your main passion. AND, if ANYONE was to receive the 30 points, they should be allotted/ascribed/deposited into MY account. I KNOW that Einstein was a physicist FIRST. The day to day way that people make a living should, as even in Einstein's case, should be taken as an 'aside', rather than a defining attribute (patent clerk), to a person's greater goal. Too often he is described as a patent clerk who wrote relativity and, also won a Nobel prize for his photoelectric effect, when he was always a TRAINED PHYSICIST first, working for a while to make a living for his family at a job, I think, just to minimize what he had done as if 'anyone' could do it. If your work does succeed, someone else (besides me) will make a similar comparison. (I was trying to make a small compliment of the intensive work you were doing.)

 

[sweetser]

Fair enough. I am not a trained physicist. I am a trained scientist, and try to remain skeptical about my own efforts in physics because I admit the odds of success are very low.

This was an interesting phrase: "intensive work" (and thanks for the compliment, I am not so gracious at accepting them, my bad). Due to the structure of my life, I cannot devote big blocks of time to the work, forcing me to be efficient. The amount of partial differential equations makes it LOOK pretty darn scary. Yet the nuts and bolts of it are actually a bit easier - or at least on par - with the Maxwell equations. My former mailman Jim committed the field equations to memory (backwards): Always give 2 Brownies to Jim, or J = Box^2 A.

The entire reason why is also simpler: it is about being a 4D slinky, super tiny oscillations around doing nothing. Although I understand some of the math behind GR, I don't have any sense why mass should tell spacetime how to curve.

doug

 

[rewebster]

Fair enough. I am not a trained physicist. I am a trained scientist, and try to remain skeptical about my own efforts in physics because I admit the odds of success are very low.

This was an interesting phrase: "intensive work" (and thanks for the compliment, I am not so gracious at accepting them, my bad). Due to the structure of my life, I cannot devote big blocks of time to the work, forcing me to be efficient. The amount of partial differential equations makes it LOOK pretty darn scary. Yet the nuts and bolts of it are actually a bit easier - or at least on par - with the Maxwell equations. My former mailman Jim committed the field equations to memory (backwards): Always give 2 Brownies to Jim, or J = Box^2 A.

The entire reason why is also simpler: it is about being a 4D slinky, super tiny oscillations around doing nothing. Although I understand some of the math behind GR, I don't have any sense why mass should tell spacetime how to curve.
doug

1) you're welcome (again, nothing in the way of a 'bad' remark was intented)

2) agreed, as to 'something' doesn't seem to 'fit' to me either


and it's /(your's) not string theory! ('not that there's anything wrong with that!'--to quote Seinfeld)

 

[sweetser]

Zhao's paper

Hello:

Here are two specific objections to Hai-Long Zhao's paper.

In equation 4, he claims
$$E = m_0 c^2 (1 - exp(GM/c^2 R))$$
In support, he notes that it gives the gravitational potential energy everyone uses, $\phi = -GM/R$ which solves the Laplace equation, $\nabla^2 \phi = 0$. While both true mathematically, we recently discussed here the problem Maxwell had with the idea, namely it leads to like charges repelling if you use a field theory like the one for EM. Bad, very bad, but oh so common.

The way around this is to add in a big honking constant because $\phi = K - GM/R$ also solves Laplace's equation but gets the energy situation right. This is what happens in GR. When Zhao does the precession of the perihelion equation, he actually is defining the energy like this:
$$E = (1 - 2 GM/c^2 R) \frac{d t}{d \tau}$$
When I do the same calculation, I use:
$$E = exp(-2 GM/c^2 R \frac{d t}{d \tau}$$
The Taylor series expansion is the same as the line above to first order in M/R. To me, it looks like Zhao uses two definitions of energy that are not the same. The work does not look logically consistent.

Zhao recreates another common error. Gravity has to involve symmetric rank 2 tensors. Why? Because the metric is a symmetric rank 2 tensor. Do something with an antisymmetric tensor, and it makes not a bit of difference to a metric. Now think of a B field - a totally antisymmetric animal. Change the order of indexes, and the sign flips. Oops.

If you want a fun exercise involve LOTS of partial differential equations, write out all the components for this tensor:
$$\partial^{\mu} A^{\nu} + \partial^{\nu} A^{\mu}$$
Full credit is ONLY given if you include all 16 Christoffel symbols. I had to use my eraser a lot to get it, terms everywhere, signs flipping, not flipping. If you like physics, that will be a fun, time consuming puzzle, particularly if you want the result to look pretty.

Even if you are insecure about getting each sign right - I know I was - one thing is very clear: all the terms that go into what I call the little b field have the same sign, like $b_x = -\frac{\partial A_y}{\partial z} - \frac{\partial A_z}{\partial y}$. The gravitomagnetism B field in Zhao's paper can only be represented by an odd spin field where like charges repel. The small b field I work with can be represented by a spin 2 field, where like charges attract. Bingo, bingo.

doug

 

[Mentz114]

Doug, thanks for the review. I didn't think his derivations were elegant or convincing.


Are these right ? I got 18 Christoffel symbols to be non-zero.
Metric signature is -+++ , co-ords are t,r,theta, phi
$$g_{00} = -\exp(-kr^{-1})$$
$$g_{11} = \exp(kr^{-1})$$
$$g_{22} = r^2$$
$$g_{33} = r^2\sin(\theta)^2$$

Christoffel symbols.
$$3-23 = \cot(\theta)$$
$$3-13 = r^{-1}$$
$$2-33 = -\cos(\theta)\sin(\theta)$$
$$2-12 = r^{-1}$$
$$1-33 = -r\sin(\theta)^2\exp(-kr^{-1})$$
$$1-22 = -r\exp(-kr^{-1})$$
$$1-11 = -0.5kr^{-2}$$
$$1-00 = 0.5kr^{-2}\exp(-2kr^{-1})$$
$$0-01 = 0.5kr^{-2}$$

 

[sweetser]

The Christoffel Symbol

Hello Lut:

I have done this calculation in the past, but thought I would repeat it, without looking at your answer, so it is independent. I am using the book "Gravitation and Spacetime" by Ohanian and Rufini, page 329. They use a metric signature of +---. This does matter to me because that signature also shows up if one wants to work with quaternions (there are three imaginary basis vectors, each one squared being -1).

They write the metric tensor like so:

$$g_{\mu \nu }=\left( \begin{array}{cccc} \text{Exp}[N] & 0 & 0 & 0 \\ 0 & -\text{Exp}[L] & 0 & 0 \\ 0 & 0 & -R^2 & 0 \\ 0 & 0 & 0 & -R^2\text{Sin}^2[\theta ] \end{array} \right)$$

Outside a spherically symmetric, non-rotating, electrically neutral mass where one has chosen to work with a constant potential, I claim the exponential metric is the solution to the GEM field equations.

$$N=-\frac{2G M}{c^2R}$$

$$L=\frac{2 G M}{c^2R}$$

There are 13 non-zero terms:

$$\Gamma _{\text{ }01}^0=\Gamma _{\text{ }10}^0=\frac{1}{2}\frac{\partial N}{\partial R} = \frac{G M}{c^2 R^2}$$

$$\Gamma _{\text{ }00}^1=\frac{1}{2}\frac{\partial N}{\partial R} \text{Exp}[N-L]} = \frac{G M}{c^2 R^2} \text{Exp}[-\frac{4 G M}{c^2 R}]$$

$$\Gamma _{\text{ }11}^1=\frac{1}{2}\frac{\partial L}{\partial R} = -\frac{G M}{c^2 R^2}$$

$$\Gamma _{\text{ }22}^1=-R \text{Exp}[-L] = -R \text{Exp}[-\frac{2 G M}{c^2 R}]$$

$$\Gamma _{\text{ }33}^1=-R \text{Sin}[\theta ]^2\text{Exp}[-L] = -R \text{Sin}^2[\theta ] \text{Exp}[{-\frac{2 G M}{c^2 R}}]$$

$$\Gamma _{\text{ }12}^2=\Gamma _{\text{ }21}^2=\frac{1}{R}}$$

$$\Gamma _{\text{ }33}^2=-\text{Sin}[\theta ]\text{Cos}[\theta ]}$$

$$\Gamma _{\text{ }13}^3=\Gamma _{\text{ }31}^3=\frac{1}{R}$$

$$\Gamma _{\text{ }23}^3=\Gamma _{\text{ }32}^3=\text{Cot}[\theta ]}$$

Now I'll compare with your results...They are identical, even down to the signs! We are looking at how the metric changes, so the signature does not matter, cool. I suppose I should have "expected" that, but I like these kinds of surprises.

doug

Note to the casual reader of this thread: this is a "look it up, plug it in'' sort of calculation. At this time, I don't have a feel for what individual terms here mean. I basically accept that this is the way a math person measures how changes happen from one place in a manifold to another if the metric is dynamic - meaning it depends on R and theta, where your happen to be.

 

[Mentz114]

Hi Doug,
I got in close with GEM and calculated the field tensors and force equations for the Rosen metric in polar and cartesian coordinates. The cartesian is easy to work with as you've observed and all the b field terms disappear. On inspection I can see that choosing a potential A = ( 1,0,0,0) eliminates the EM fields and leaves force equations (signature is -+++ so x^0 is t )

$$f^0 = \frac{-4GM}{c^2R^3}q_m(x\beta_x+y\beta_y+z\beta_z)$$

$$f^\mu = \frac{-4GM}{c^2R^3}q_mx^\mu\gamma$$

which in the limit gamma ->1 and beta-> 0 give Newtons gravity. I haven't yet examined what paths the equations will give in relativistic cases. Also on the plus side, if you look at the first equation in polar coords,

$$f^0 = -\frac{4GMq_m}{c^2R^2}$$

this could be interpreted as gravitational redshift - not predicted by Newton obviously.

Doing this has raised some conceptual problems for me. It seems to me you are using the geodesic equations of motion implicitly or explicitly in GEM. But they are derived by extremizing an action based on Ricci curvature, wrt the metric, something you abjure. Planetary precessions and deflection of light should be based on the GEM equations of motion.

Which brings me to the question - how does one describe the paths of light (null geodesics?) from the force equations ? Because there is a factor of gamma in the spatial forces, naively putting v=c will not do, because no deflection can ever take place if gamma=0. There's also the question of the mass charge of light.

I'm going to have a play with the force laws later.

Regards,
Lut

 

[sweetser]

Precession of the Perihelion of Mercury

Hello Lut:

I don't think this logic holds true, or at least let me give you my slant:

> but they are derived by extremizing an action based on Ricci curvature, wrt the metric, something you abjure.

My father would have called "abjure" a Word Wealth word. It sounded right, but I checked: "to reject solemnly". Bingo! The Riemann curvature tensor, and its contractions, the Ricci tensor and scalar, have done much good. I am hoping to find something better.

> Planetary precessions and deflection of light should be based on the GEM equations of motion.

While this could be done, it is not what I did do. The precession calculation is not easy. After getting over much fear, I finally did figure out all the steps, available here:

http://theworld.com/~sweetser/quaternions/gravity/precession/precession.html

If you go there, you'll see I based on off of the Rosen/exponential metric. Once I do the Taylor series approximation, the derivation is exactly the same. I understand bending of light based on the exponential metric.

The way I first got to the exponential metric used the force equation. First I needed to find a relevant potential, one that was "classical", and only involved light. As I told you, I did find it, but it is not trivial. The potential is a linear time perturbation of a 1/R solution. If the spring constants are chosen with care, one get a symmetric field strength tensor like so:

$$\nabla^{\mu}A^{\nu} = \frac{\sqrt{G} M}{R^2}I(4)$$

where I(4) is the 4x4 identity matrix. I was able to solve the force equation and get a 4-velocity solution. I rearranged that solution, and there was the exponential metric. I know that approach is unorthodox, but it was the first way I did it: force eq->velocity solution->metric. I was presuming the equivalence principle was valid, and dropped the inertial and gravitational masses. The photon does have a zero mass and electric charge. A more careful student of the mathematical arts would be concerned about this, getting it in a limit process. I am more practical. Initially, I only discussed the road using the force equation to the exponential metric since it was the only path. Now that I figured out the divergence of the Christoffel is a solution to the field equations, I talk about that path. This is the ignored path because people presume I am using a D'Alembertian operator, not a covariant followed by a contravariant derivative, which has a divergence of a Christoffel in it. What is really encouraging is that two very different paths - one using the force equation in an unusual way, the other from the field equation - lead to exactly the same metric which is consistent with weak field experimental tests, strong field tests, and tests of the equivalence principle.

Hope that helps the conceptual issues, which never go away entirely.

doug

 

[sweetser]

Potential Theory and Diffeomorphisms

The email below is part of my "sales" activities, efforts to contact people of some stature. This process will involve some repetition, but I hope to adjust the location of the argument so there are some surprises for readers of this thread.

Clifford Will is a leading expert on experimental tests of gravity who happens to look like Ted Turner. Here is the email I sent off ten minutes ago...


Hello Prof. Will:

I have criticized your Living Review article "The Confrontation between General Relativity and Experiment" for being incomplete, both in private emails to the Eastern Gravity Meeting organizers, and in a public forum at physics.forums. It is my responsibility to inform you of this entirely technical issue, which you may or may not act upon.

<quote>
[I was discussing the value of work by people working on the fringe of physics, who clearly misunderstood a number of issues, yet might be raising a topic that is worthy of addressing.]

1. Ed "negative mass" Miksch was the man who drove up from Pittsburgh with his wife. The organizers certainly know, but other folks may not be aware, how assertive he and his wife were about the need for Ed to be heard by this gathering of physicists. He was the guy who gave the three minute speech at the end of Friday's session. He claims to have shown that we should all be working with negative mass because he's done the calculation, one no physics professor at Reed College in the 50s could find an error in. The reason is that from where his calculation starts, there is no trivial math error.

This issue was understood by none other than James Clark Maxwell, but has not reached the wider physics community. The story was written up well at this URL: http://www.mathpages.com/home/kmath613/kmath613.htm. In GR books, they note the link between Newton's potential theory and the Schwarzschild metric, g_00 = 1 - 2 G M/c^2 R = 1 - 2 phi, and so ignoring the constants, phi = - M/R. What Maxwell understood was that such a potential plugged into a field theory like that for electromagnetism implies that like mass charges repel. The correct answer for a Newtonian potential where like charges attract is phi = 1 - G M/c^2 R. Ed started from the wrong place because he was instructed by everyone that phi = -M/R is OK. Maxwell did see the correct way out - add a HUGE positive constant - he just couldn't justify it. Einstein's metric theory gets the potential theory right. Ed should be proud that he saw a problem realized by Maxwell. It is unfortunate that this potential theory is taught incorrectly to this day, because there will be people in the future that will walk down this wrong bend in the road.

To appreciate the consequences of how Ed's issue is misunderstood at the highest levels of the physics community, I asked Clifford Will why a 4-potential theory was not listed even as a possibility in his Living Review article on GR. He claimed one could make a potential theory get half of the light bending around the Sun, but not all the bending measured by experimental tests. Will is correct for a scalar potential theory, the g_00 term. A 4-potential theory could easily match the data, with A = (1 - 2 GM/c^2 R, -1 - 2 GM/c^2 R, -1, -1), so the g_00 and g_11 terms will contract time measurements and expand space measurements as seen in tests.
</quote>

It is almost as if on one understands what diffeomorphism means in the context of a potential theory. Write a field equation, Del^2 A = J, and people think they can write down an answer for a simple current. People presume Del^2 is the d'Alembertian, a scalar operator, when it is not (unless the space is flat and Euclidean). It is a covariant derivative applied to a potential, followed by a contravariant derivative. Einstein's summation convention should NOT be applied to two covariant derivatives. It is necessarily the case that one could find 2 solutions to the field equations: a flat background metric and the potential contains all the information, or a dynamic metric describes every bend and kink in spacetime, while the potential does nothing. The divergence of the connection is in the field equations, which means there is a second order differential equation involving the metric. Solve that equation, and the metric is determined by the physics of the problem at hand. The choice of how to measure - a changing potential or changes in the connection - is up to the observer. One can choose to make a 4-potential field equation into a metric theory, and thus consistent with all tests of the equivalence principle (the Rosen metric is the solution, but without the fixed background that creates problems for the strong field tests).

I hope you get a chance to think about your deliberate omission. There is a lot of fun math going on there.

doug

 

[sweetser]

MIT Professional Institute Class cancelled

Hello:

I had a choice this summer: go to the big meeting on GR in Australia,http://www.grg18.com/ or sign up for a 4 day class on GR at MIT's Professional Institute. I voted for the latter, which would have had 6 hours of lectures each of 4 days, with discussions. Nerd out!

Unfortunately, Prof. Joss has had a medical emergency, and the class was canceled. This is quite a bummer for me because I had high hopes of making a personal connection to a professional in the field of GR. I want to know why I can leave this project alone, and the class might have provided a path.

If anyone reading this thread is going to GRG 18, I would appreciate a report back on the section: A4 Alternative Theories of Gravity – Gilles Esposito-Farese. My bet it will be on things like scalar-tensor and vector-tensor stuff, certainly nothing like vector-all-alone like we've been discussing here.

doug

 

[Mentz114]

But they are derived by extremizing an action based on Ricci curvature, wrt the metric...

Wrong ! I withdraw this obviously incorrect statement.

But I'm still not happy mixing geodesics and forces.

Bad luck with the GR course.

 

[sweetser]

The action, fields, and forces

Hello Lut:

Let's try and form a few correct statements, always a good exercise. I remember my surprise at seeing the simplicity of the Hilbert action of GR:

$$S_{Hilbert} = \int \sqrt{-G} d^4 x R$$

Not many symbols, but let me explain the few that are here. The action S is an integral over spacetime of all the energy interactions of a system per unit volume. Integrate over a fixed amount of space, but arbitrary amounts of time, and you will likely get arbitrary integrals. The game is to use the calculus of variations to find what things can be varied such that the integral stays the same, no matter what time interval is used.

The integral is in 4D spacetime because we live in a 4D Universe. In my opinion, this is all that is needed to reject work done with strings because they integrate over ten or eleven dimensions which is not the way the world is. Notice where I put the $\sqrt{-G}$. It turns out the the spacetime volume element, $d^4 x$ does not transform like a tensor. In curved spacetime, it will have a different value. The square root of the determinant of the metric compensates for this, so that $\sqrt{-G} d^4 x$ does transform like a tensor.

Now we get to the heart of the action, the Ricci scalar R which is a contraction of the Ricci tensor, itself a contraction of the Riemann curvature tensor. If the action is varied with respect to the metric tensor [itex]g_{\mu \nu}[/tex], that generates 3 terms (here my knowledge becomes less precise: I know what happens, but have not looked at it closely enough to understand all the details). One of these three is zero, something about Gauss' law and the boundary of a boundary is zero. The other two together make the field equations:

$$R_{\mu \nu} - 1/2 g_{\mu \nu} R = 0$$

It should be clear why general relativity is a theory only about gravity, having nothing to do with EM: there was only the Ricci scalar R in the action.

Let's compare that with GEM. If we only want to get the field equations in a vacuum, this action should be sufficient:

$$S_{GEM} = \int \sqrt{-G} d^4 x \frac{1}{c^2} \partial_{\mu} A_{\nu} \partial^{\mu} A^{\nu} R$$

A basic question is how can we even hope that this will do both EM - which requires a spin 1 force mediating particle, and gravity - which requires a spin 2 field. I hope I am recalling this correctly and not making it up, but there was a group theory jock who said with great disdain that this idea was silly because the tensor $A_{\mu \nu}$ is reducible, and thus cannot be use to represent a fundamental force of nature. The reducible tensor can be written as the sum of two irreducible tensors, $\partial_{\mu} A_{\nu} = 1/2(\partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu}) + 1/2(\partial_{\mu} A_{\nu} + \partial_{\nu} A_{\mu})$. The first of these is straight-out-of-the-book EM field strength tensor: swap the order of the indexes, the sign changes, as one would expect for a spin 1 field strength tensor. The second one is a rank 2 symmetric tensor, the kind that if its trace is equal to zero, could be a home for a graviton.

Vary the GEM action with respect to the 4-potential, and one gets a 4D wave equation. That part is direct. Rewriting the 4D wave equation in terms of fields, that detail gets more complicated. The answer is NOT the Maxwell equations with gravity fields that are just clones of EM (as happens in gravitomagnetism). The easiest way to see this is to consider the gravity field which has terms like $b_x: \frac{\partial A_y}{\partial z}, \frac{\partial A_z}{\partial y}$. What happens in gravitomagnetism is they cheat, copying EM too directly, claiming there should be a minus sign involved between those two terms. Swap the order, and $b_x$ would flip signs, not good for a representative of a spin 2 field which doesn't have that property. In GEM, the symmetric analog is $b_x = - \frac{\partial A_y}{\partial z} - \frac{\partial A_z}{\partial y}$. This is not a curl operation. There will be no vector identity laws such as no gravity magnetic monopoles, or Faraday's law - the signs don't flip in ways that would "make it so".

For GEM, the road to an expression involving force is direct. The action involved is different than the one used for fields:

$$S_{GEM} = \int \sqrt{-G} d^4 x (- \frac{\rho}{\gamma} - \frac{1}{c} Jq_{\mu} A^{\mu} + \frac{1}{c} Jm_{\mu} A^{\mu})$$

Vary this action with respect to the 4-velocity, and one gets the Lorentz force law. There is a velocity in the gamma and the current terms, but not in the field strength tensor, $A_{\mu \nu}$, which is why it can be ignored when thinking about the force law. For every point in the spacetime manifold, there is a force, there is an energy.

In GR, the story is different. There is no force law. If there was a simple force law, we could go to a simple expression about the energy of the field. The Riemann curvature tensor is the source of the problem, since it involves the difference of two paths in spacetime. Instead one solves the field equations and gets a metric solution. Looking at the metric, one can pick out conserved quantities, the Killing vectors. I think of a geodesic as the easiest path through spacetime. The "force" is zero.

I do need to spend more time thinking about the GEM force law, how it can be a Lorentz force law with the potential doing all the work, and one where the force is zero, but the curved metric is where the action happens.

doug

 

[Mentz114]

Doug:
thanks for the exposition ( nobody expects the Spanish exposition !).
To save you typing in the future, you may assume I know GR well enough, including the EH action and variations thereof.

Your actions are new to me. The first one is obviously the GR action with a vector potential and seems a direct route to including the potential. But now you have you extremize wrt variations in the potential and the metric, simultaneously ( rather than assuming a metric). This will mean introducing constraints between the potential and the metric, which could be interesting.

The second action also seems to require a background metric.

...how it can be a Lorentz force law with the potential doing all the work, and one where the force is zero, but the curved metric is where the action happens.

Bingo ! You have articulated my 'conceptual' problem. Also the practical problem of expressing forces as 'curvature'.

The river model cited by Carl earlier is the nearest I've seen to anyone doing this.

Regards,
Lut

 

[sweetser]

No variation with respect to the metric

Hello Lut:

I can tell you are well schooled in the arts of GR. My long posts are both to test my own self-taught knowledge, and to bring along other readers of this forum with less experience in this area.

> But now you have you extremize wrt variations in the potential and the metric, simultaneously ( rather than assuming a metric).

I don't think this proposal makes sense. The 4-potential is a rank 1 tensor, but the metric is rank 2. I have yet to say one should vary the action with respect to the metric because that would yield rank 2 field and force equations, but the field and force equations in GEM are rank 1. The metric must be fixed.

A fixed metric does not mean a flat metric. All it means is that in the variation of the action, it was not varied. The fixed metric can be dynamic. Let's see how this works in EM. In EM, if you want to generated the Lorentz force equation, you vary the 4-velocity keeping the 4-potential fixed. Now you have the Lorentz force equation, you can use a dynamic potential to solve problems. If you want to generate the Maxwell field equations, the potential is varied, keeping the 4-velocity fixed. Solve a problem using the Maxwell equations, and the velocity of charges is likely to be dynamic.

I see you slipped in a killer word for me, the "background" metric. A fixed metric also does not mean a background metric. In EM, there are no differential equations that can be used to determine what the metric should be. As such, one must add a metric as part of the background mathematical structure in order to solve problems.

In the GEM proposal, there are differential equations that can be solved to figure out what the metric is according to physical properties of the system. This is the take the divergence of the Christoffel of the exponential stuff I cite from time to time. Because there is a second order partial differential equation involving the metric that depends on conditions, the metric is not part of the background math structure, but instead something that can be solved for.

EM is a gauge theory. GR is a gauge theory. In both cases, there is a field that can be added in without changing a solution. "Gauge" means measure, so arbitrary fields can be included in the measure of the fields.

GEM is a gauge theory of a different class. Here a measurement can involve either the changes in the 4-potential, or changes in the connection. One cannot add in a gauge field as happens in EM or GR because that would change the system. Once can instead decide to make the measurement all about the 4-potential or all about the connection, or any combination of the two.

doug

 

[Mentz114]

Hi Doug:

...well schooled in the arts of GR.

Not as well schooled as I think, obviously. But my point is central to what bothers me - the relationship between the metric and the potential. How do you decide how much of what goes where ?

If I begin with flat space-time and assume a matter current, then try to solve for a potential - I get a lot of EM fields from the solution that were not ordered by anyone. If I start with a charge current and do the same thing, the potential I get will contain gravitational fields I didn't want.

I'm not sure what you mean by "decide to make the measurement all about the 4-potential or all about the connection, or any combination of the two".

I appreciate you taking the time to answer my questions - even though they are starting to sound like whinges and quibbles.

Lut

 

[sweetser]

The covariant derivative bridge

Hello Lut:

Let me adjust your question just a little:

> But my point is central to what bothers me - the relationship between the metric and the potential. How do you decide how much of what goes where ?

It is the relationship between the connection - derivatives of the metric - and the derivatives of the potential. The metric and the potential don't have a relationship to each other until a covariant derivative comes into play. The field equations then involve the second order derivatives of the metric, and the second order derivatives of the potential.

The relationship between the derivative of the metric and the derivative of the potential is in the standard definition of a covariant derivative:

$$\nabla^{\mu} A^{\nu} = \partial^{\mu} A^{\nu} - \Gamma^{\mu \nu} _{\space\sigma} A^{\sigma}$$

This equation shows what I am calling "gauge choice", the ability to choose how much of the covariant derivative is due to $\partial^{\mu} A^{\nu}$, and how much is due to $\Gamma^{\mu \nu} _{\sigma} A^{\sigma}$. There is not a new equation, just a new way at looking at, and really using, an old definition.Again, this is a great question:

> If I begin with flat space-time and assume a matter current, then try to solve for a potential - I get a lot of EM fields from the solution that were not ordered by anyone. If I start with a charge current and do the same thing, the potential I get will contain gravitational fields I didn't want.

One thing I have specifically avoided is "really new physics", because it is far too easy to get lost and confused. The simplest kind of solution to the 4D wave equation that uses a potential with factors of $1/\sigma^2 = 1/(x^2 + y^2 + z^2 - c^2 t^2)$. That leads to a $1/distance^3$ force law. That constitutes "really new", so new no one will listen. Since I am both a skeptic and a fringe physicist, I have volumes of self-doubt. One way I manage this is to always try and get my units right. The smart kids in the class and in tenured positions use natural units so they can skip these details. If you work out the units for that most general relativistic potential solution, it looks like so:

$$A_0 = \frac{\sqrt{G} h}{c^2} \frac{1}{x^2 + y^2 + z^2 - c^2 t^2}$$

That has the units of relativistic (c), quantum (h), gravity (G) as an inverse cube force law. It is a clear outcome of the math, but I don't know what to do with it.

Hope you appreciate my fears.
doug

 

[sweetser]

Note added in proof: The Maxwell equations written in the Lorentz gauge also has this problem. It has inverse distance squared solutions, and thuse inverse cube force laws. Does anyone know what the standard approach is to such solutions? My bet is they claim it is "unphysical", which strikes me as empty.

In a second form of denial, anyone who works with the antisymmetric field strength tensor could also calculate the symmetric field tensor from the same terms, and then have to claim that Nature has a means of never ever using any such information in any way. That does not strike me as reasonable.

doug

 

[sweetser]

Another pitch

Peter Woit wrote a book recently called "Not Even Wrong" which was a critique of string theory. Most of the book was a darn good description of quantum field theory at a deeper level than one usually gets without plowing through the technical literature.

Lubos Motl is a string theorist. He used to post in the newsgroup sci.physics.research. He is the strongest supporter of string theory I have ever encountered, so strong, I was embarrassed by the guy. He now has a faculty position at Harvard.

These two REALLY don't like each other. That's why I wrote them both.


Hello Peter and Lubos:

I am aware how tense the professional relationship between you two
happens to be. I thought it would be an interesting challenge to find
points of agreement, and perhaps spot some new physics.

I am a fringe physicist, which I define precisely to be one without a
job or on the way to a degree who still has specific hopes of making a
contribution to physics. I have read Lubos' strong support for work
with strings in sci.physics.research. Being a skeptic, I have
promised to deliver a check to Lubos if in any time in a decade
(April, 2014) the physics of gravity in more than four dimensions
becomes broadly accepted as the correct way to deal with gravity,
instead of merely promising. Responding to a plea for a testable
hypothesis, I forwarded some of my initial work to Peter. At this
time, neither of you have responded, busy as you are. Those are my
ethereal connections to you both.

I have taken all of two lines of text from "Not Even Wrong...". I
hope to show that we all agree with the first line, yet all - even the
author - agree that the second sentence is wrong. Given the
passionate disagreement of the value of the book, the two chosen lines
have nothing to do with work on strings!

The first line came in the back of the book, where Peter is searching
to find common themes in successful approaches to doing new math.
Peter wrote (p. 258):

"Traditionally, the two biggest sources of problems that motivate
new mathematics have been the study of numbers and the study of
theoretical physics."

Knowing some history of mathematics makes this statement sound
reasonable. What I happen to do is study quaternions, a type of 4D
number, in order to do standard physics. Quaternions are just
4-vectors that can also be multiplied and divided. Most of my work
has been prosaic: I can generate Newton's law for a centrally directed
force in a plane as a one line quaternion expression (it usually
requires a few pages of text). Yet there are times when I have been
pushed to do new physics. I came up with a definition of a quaternion
derivative that uses a two limit process reminiscent of L'Hospital's
rule that may justify why causality for classical physics (a
directional derivative along the real axis) is different from quantum
mechanics (a normed derivative is all that can be properly defined).
From my perspective, the quaternion derivative is a math issue with
implications for physics.

Now it is time to turn to the incorrect statement. It is not a
trivial error. Here is the line (p. 117):

"The ability to visualize the graph of the function [which depends
on two complex numbers] is now lost, since it would take four
dimensions to draw it."

Don't tell animators that they cannot draw in 4D, three for space, one
for time. I have written the software to animate quaternions, which
are three complex numbers that share the same real number. If three
complex numbers can be visualized, then doing two is easy.

Rene Descarte developed analytic geometry which is still in wide use
today. Our brains devote more hardware to visual analysis than
anything else, which may explain the lasting power of analytic
geometry. I am developing the tools for analytic animations.
Addition of quaternions is the simplest operation out there. The
result from a physics perspective is an inertial observer, at the
heart of special relativity. That is the first animation included in
this email.

Peter's book deals with the standard model. I have therefore included
animations of U(1), SU(2), and SU(3). My goal has become to work with
a smaller standard model, one that has the symmetries of U(1), SU(2),
and SU(3), but does not view the three as tensor products which would
depend on a Lie algebra with 12 independent players. It may be
possible to view SU(3) as composed of two electroweak symmetries. The
advantage of the smaller model is that it could provide a
justification for the confinement that happens only for the strong
force.

The group SU(3) looks like an expanding, then contracting, bumpy
tennis ball in the attached animations. Go to a different place in a
spacetime manifold, and the size of the tennis ball might change. To
continuously change the measure of distance would involve the group
Diff(M), a symmetry at the heart of our understanding of gravity.

A new approach to understanding the symmetries of nature should look
different. I had to write the software to take quaternion expressions
off the command line and generate animations. My work will not fit in
a PDF. I chose the simple GIF animation just to be sure anyone can
see it.

I have set this email up as a "Prisoner's Dilemma" problem. You both
can choose to ignore it, which is the easiest thing to do. There is a
potential cost, since I will be publishing this email publicly, so if
and only if quaternion animations are an important step forward for
physics, this email would document how difficult it is to bring a new
way to look at the world to life. The second possibility would be
that one of you would start playing with what is visually new math. I
bought an Ipod only because I wanted to see SU(2), and be able to
carry it around with me. The person who "discovered" this new branch
of work would forever have bragging rights over the other, quite a
payoff.

The third possibility is that both of you were interested by the
animation. Given the diversity of your world views, that would speak
to the power of the animations.

Good luck in your areas of research. I hope to hear from either or both of you.
doug

65 MB 10 second GIF animations:
1. Addition/inertial observer:
http://quaternions.sourceforge.net/inertial_obs.povray.animation.scan.100.1002.gif

2. 3 ellipses in the complex planes, the group U(1) of EM:
http://quaternions.sourceforge.net/u1.povray.animation.scan.100.1000.gif

3. A unitary quaternion, the group SU(2) of the weak force:
http://quaternions.sourceforge.net/su2.povray.animation.scan.100.1000.gif

4. An evenly expanding/contracting 4-sphere, the group SU(3) of the
strong force:
http://quaternions.sourceforge.net/u1xsu2xsu3.povray.animation.scan.100.1000.gif

5. Two different 4-spheres, the group Diff(M) of gravity:
http://quaternions.sourceforge.net/u1xsu2xsu3_delta_scale.povray.animation.scan.100.1000.gif

 

[Mentz114]

Doug:

The relationship between the derivative of the metric and the derivative of the potential is in the standard definition of a covariant derivative:

$$\nabla^{\mu} A^{\nu} = \partial^{\mu} A^{\nu} - \Gamma^{\mu \nu} _{\space\sigma} A^{\sigma}$$

This equation shows what I am calling "gauge choice", the ability to choose how much of the covariant derivative is due to , and how much is due to . There is not a new equation, just a new way at looking at, and really using, an old definition.

Understood. I'll have think about this. Feels dodgy at the time of writing.

I'm not sure I follow the next bit of your post #345 (!). Again, I'll have to reflect on it. The only 1/R^3 law I can remember is the electric field of a dipole.

I don't think fear should come into this. If your theory proves to be inconsistent mathemaically or unphysical - what's to fear ?

In a second form of denial, anyone who works with the antisymmetric field strength tensor could also calculate the symmetric field tensor from the same terms, and then have to claim that Nature has a means of never ever using any such information in any way. That does not strike me as reasonable.

Tensors don't exist, numbers don't exist. They are abstractions of the human mind. Trying to second guess nature is not on. Every physical theory has defects. Little dark corners of unphysicallity. What is the meaning of negative energy, what are advanced solutions, why do energy conservation laws not transform relativistically ?

This adds up to one message for me - it can't be done. There is no complete physical theory. So I'm not betting - hence no fear.

For an amusing read, see "Why the laws of physics lie" by Nancy Cartwright ( no, not that one, this one was prof. of logic(?) at London U some time ago).

Regards,
Lut

PS I admire your chutzpah, in your challenge to Motl and Woit.

 

[sweetser]

GEM dipoles

Hello Lut:

I have had a pleasant day thinking about this comment:

> The only 1/R^3 law I can remember is the electric field of a dipole.

Let me first set the stage, which goes back to Heisenburg. When he first tried to formulate quantum mechanics, he made sure it was completely relativistic. Makes sense, but it did not match the result of Bohr's atom. Working with an equation that was not relativistic, a connection to the data of the day was possible. It took another five years for Dirac to repeat the effort, and figure out new physics, like all the antiparticles.

The GEM equation is fully relativistic. It took me more than a year to figure out how to correctly break relativity's grip, and make a connection to physics we know is true, specifically the exponential metric that is consistent with weak field tests of gravity.

Your questions are focused on the "Dirac"-like aspect of the GEM field equations. I like your read of it: this is a dipole. Here is my speculation of what may be happening.

When I wrote the charge coupling term on a black board for a friend,

$$-(Jq^{\mu} - Jm^{\mu})A_{\mu}$$

he wondered why I didn't just redefine a new current density, $Ju^{\mu}=Jq^{\mu} - Jm^{\mu}$. I said the problem with that approach is that the sign difference is required so the force laws have like electric charges repel and like mass charges attract. The sign difference is also required for the field equations, where like electric charges repel and like mass charges attract.

The two irreducible field strength tensors, the antisymmetric one and the symmetric one, each require their own charge because they are separate fundamental forces. If you want to have some sense of what the field strength tensors mean, the symmetric tensor is the average amount of change in the 4-potential with respect to t, x, y, and z, while the anti symmetric tensor is the deviation from the average amount of change. I call it "Average Joe and the Deviants". It should be clear that the average is a different sort of thing from the deviation of the average.

Although we have two types of charges, one can be up to sixteen orders of magnitude larger than the other, as is the case for an electron. For every charged particle, there are necessarily two types of charge: the electric charge where like repel, and a mass charge where like attract. In other words, every charged particle is a dipole. Further, the particle will behave like a permanent dipole because the charges are so different in size.

A fundamental property of particles like electrons and protons is the charge to mass ratio. It is a late night speculation, but it seams to me that ratio may be linked to the permanent dipole of electric and mass charges. I am going to study up on dipoles...

doug

ps. Fear really wasn't the right emotion. It is more strategic: make connections to known physics, and the odds are higher it will be right and get listened to.

 

[sweetser]

Bad Astronomy Forum

Hello:

My weekly effort to pitch the proposal involved writing to a forum on Bad Astronomy. This site is a minor Internet phenomenon run by Phil Plait where he shoots down odd claims, as well as discusses the good work going on in astronomy today.

I had to clip my post there to get it under 15,000 words. Here's the URL if you want to read it:

http://www.bautforum.com/against-mainstream/61876-gem-rank-1-unified-field-proposal.html

Here's my summary. Much of the stuff on that board is BAD, hard to follow the limited logic presented. I spent about half the post discussing the EM action, pointing out the 4 reasons why like electric charges repel. I then go through and show the sign changes necessary so like charges attract as happens with gravity. I finish with a discussion of the field equations. I invite the reader to "bust my pinata".

We will see if we get a spirited debate over there.

doug