How Can Gravity and Electromagnetism Be Unified Through a Rank 1 Field Theory?

[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}<br />

 

[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:

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

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 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&#039; law and the boundary of a boundary is zero. The other two together make the field equations:<br /> <br /> R_{\mu \nu} - 1/2 g_{\mu \nu} R = 0<br /> <br /> 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.<br /> <br /> Let&#039;s compare that with GEM. If we only want to get the field equations in a vacuum, this action should be sufficient:<br /> <br /> S_{GEM} = \int \sqrt{-G} d^4 x \frac{1}{c^2} \partial_{\mu} A_{\nu} \partial^{\mu} A^{\nu} R<br /> <br /> 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.<br /> <br /> 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&#039;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&#039;s law - the signs don&#039;t flip in ways that would &quot;make it so&quot;.<br /> <br /> For GEM, the road to an expression involving force is direct. The action involved is different than the one used for fields:<br /> <br /> 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})<br /> <br /> 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.<br /> <br /> 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 &quot;force&quot; is zero.<br /> <br /> 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.<br /> <br /> 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

 

[sweetser]

Light Bending Around the Sun

Hello:

The Bad Astronomy discussion is going OK. If I write up a note over there that I like, I'll port it over here. The following deserves "double posting" because it is a simple calculation with consequences. I was asked to derive the light bending around the Sun effect which is different between GR and GEM. Here goes...

The calculation comes directly out of a paper: “Post-post-Newtonian deflections of light by the Sun” by Reuben Epstein and Irwin Shapiro, Physical Review D, 22:2947, 1980. They write a generalized Schwarzschild metric like so:

dτ² = A(R) dt² – B(R) dR.dR

where

A(R) = 1 – 2 G M/c² R + 2 β (G M/c² R)² + O(3)

and

B(R) = 1 + 2 γ G M/c² R + 3/2 ε (G M/c² R)² + O(3)

There are three Greek letters in play here: β, γ, and ε. This is a research paper, so naturally they write "Calculating the deflection angle in the usual manner, we find for the ppN contribution

Δθ_ppN = π (2 + 2 γ - β + 3/4 ε) (G M/c² R)²"

For the Schwarzschild metric, all the Greeks are 1, so these constants add up to 3.75. For GEM, the constants add up to 4. Now collect the numbers needed. I fetch mine online, http://physics.nist.gov/cuu/Constants/index.html, http://en.wikipedia.org/wiki/Sun

G = 6.674 28 x 10^-11 m³ kg^-1 s^-2
M = 1.989 x 10³⁰ kg
R = 6.955 x 10⁸ m
c = 299 792 458 m s^-1
π = 3.14159
206265 arcseconds/degree

Go to Google, and type it in:
206265 * pi * 3.75 * 6.67428^2 * 10^-22 * 1.989^2 * 10^60 / 299792458^4 / 6.955^2 /10^16
and the answer is 10.96 microarcseconds.
For GEM, the answer is 11.69.
The difference is 0.73 microarcseconds.

That's how its done. GEM predicts more bending that GR, as earlier GR predicted more bending than Newtonian theory. Tradition!

doug

 

[CarlB]

I noticed that the GPB is indicating frame dragging effects that are centered about 25 or 35% higher than GR would indicate, but with large enough error bars that GR is consistent. Does GEM get higher frame drag than GR?

 

[sweetser]

Hello Carl:

GEM always predicts a bit more bending since the coefficients on the Taylor series are a bit larger. I have NOT tried to do a frame dragging calculation. Do you have a reference that spells it all out in detail?

doug

 

[sweetser]

Note: by "a bit more", this applies only to second order PPN effects, not first order PPN where the two are exactly the same. I don't have a metric for a rotating reference frame, so the short answer is I don't know at this time.

 

[Don J]

Just by pure curiosity I want to know if GEM can describe "spin orbit coupling"

Apparently from Wikipedia,
As the master theory of classical physics general relativity has one known flaw: it cannot describe "spin orbit coupling"
So they have developed
http://en.wikipedia.org/wiki/Einstein–Cartan_theory

 

[sweetser]

Spin Orbit Coupling

Hello Don:

Great question! If I was a stellar student of GR, I would know about this issue. Now I know something based on the wikipedia article.

All the players in the Einstein field equations are symmetric: the Ricci tensor and metric tensor are both symmetric. That creates a problem. Things like spin and angular momentum need an antisymmetric tensor. This is a fundamental problem for GR since there is no place to put such energy that exists in the real world.

The problem is averted in GR by including torsion in the connection. This is tough stuff to follow for me. One no longer uses the Christoffel symbol of the second kind which again is a symmetric tool. Bring in torsion to the connection, and there is a place to handle angular momentum conservation in an antisymmetric tensor.

With GEM, spin orbit coupling is handled easily. It is in the EM part of GEM. The Lagrangian has the rank 2 antisymmetric field strength tensor. To quote from http://en.wikipedia.org/wiki/Spin_tensor:

Examples of materials with a nonzero spin density are molecular fluids, the electromagnetic field and turbulent fluids.

Spin orbit coupling will not be a problem for GEM. Unification does has some advantages!

doug

 

[sweetser]

Pitch to Peter Coles

Hello:

Here is another one of my pitches, but this time I included a t-shirt, one that has an oil painting I did called "Turquoise Einstein" along with the GEM field equations. I've invested $16.99, so I'll have to see if the investment pays off.

doug


Hello Peter:

I enjoyed your book, "Cosmology: A Very Short Introduction", which was assigned reading for a class I planned to take at MIT's Professional Institute on "Relativity, Gravity, and Cosmology" taught by Paul Joss. Unfortunately he had a significant medical issue which led to a cancellation of the class. Near the end of your book, you speculated that one might be able to put a central equation for understanding the Universe on a t-shirt. You should have by now, or within a few days, received from cafepress.com my effort toward that elusive goal. It should be popular due to the image of "Turquoise Einstein", even if people ignore the math.

I'll spend a few paragraphs describing my idea for how to unify gravity with the rest of physics. In my own experience, I have found it is usually quite easy to spot if someone is taking a logically coherent approach to a problem. If I can be coherent in my reasoning, perhaps like a laser I can project an idea over the Atlantic and share some of the profound excitement I feel.

Newton's law is still the most useful approach to gravity. It is what engineers use for most of their calculations. The exception are those rocket scientists that bring on board atomic clocks. Then the flaw in Newton's approach - that gravity works instantaneously - becomes apparent.

Over time, I have come to appreciate the sophistication of general relativity. I particularly like the Hilbert action, which only needs the Ricci scalar to do its magic. Yet more than eighty years of trying to quantize the approach have come up short.

Our very best theory today for gravity is a rank 2 field equation. Our most useful theory is rank 0. Goldielocks would wonder, what about a rank 1 field theory for gravity? This turns out to be a subtle theory in 4D.

Surely we have a nice, solid reason why a rank 1 field theory is dead on arrival. Someone told me to look up Clifford Will's Living Review article on tests of GR, that I would find the reason there. In the third section, a pure 4-vector theory was not mentioned. Go to his green book, p. 19, a vector theory by Kustaanheimo, which was not published (p. 360).

I found two papers in the 1950's, one by Gupta, the other by Thirring, that said it was not possible to make a rank 1 field theory where like charges attract. This was an assertion, it was not proved. If you were to copy the approach to the Maxwell equations keeping all the signs the same, then yes, like charges repel. If one flips the sign of the charge coupling term, both the force equation which results from varying the action relative to the velocity, and the field equations which result from varying the action with respect to the potential, have like charges attract. A well-placed sign change can work wonders.

Earlier in the Spring, I found another fun problem concerning the spin of the mediating particle for gravity. The graviton has to be spin 2 to bend light. In my action, I have a symmetric, rank 2 field strength tensor, a natural place for a spin 2 particle to call home. If you read chapter 3 of Feynman's "Lectures on Gravitation", he shows how to spot a spin 1 particle in the current coupling term. At a critical point, he multiplies to different currents together, J' J* (p. 34). Looking at the resulting phase, it will take 2 pi to come back around, as we expect of a spin 1 particle. What I realized was if a slightly different current product was used, J' (iJi)*, then the resulting phase will return home in pi, the signal that a spin 2 particle can live in the current coupling term (p. 39).

The immediate reaction to the equation on the t-shirt is to dismiss it as the Maxwell equations written in the Lorenz gauge (at least I care enough about details to know the difference between the Danish Ludwig Lorenz and the more famous Dutch Hendrik A. Lorentz). If I had meant the d'Alembertian operator, I would only need the box, not a box squared. The box squared is suppose to symbolize two covariant 4-derivatives, one taken after the other. A covariant derivative has a normal derivative and a connection, which is the derivative of a metric. Take two in a row, and there will be a derivative of a connection, which is a second order differential of a metric. The shirt has a second order differential equation that can determine the metric, so the metric does not have to be provided as part of the background mathematical structure.

I have this speculation that no one of your station in life will find this body of work of interest until they do a simple calculation: calculate the Christoffel of the Rosen metric. I prefer to call that metric the exponential metric for obvious reasons: static exponentials live on the diagonal. Only one of the three metric derivatives matter, and the contravariant exponential walks into its covariant cousin leaving a charge/distance that would look familiar to Poisson himself. Math magic like that means something.

I'll stop here, although there is much more. I've done my responsibility. I have no doubts you are overwhelmed with the standard pace of life. This is a low-odds, high rewards game. At least you have a t-shirt out of it. Feel free to ask any questions.

doug

 

[sweetser]

The Big Idea

Hello

As it was presented, GEM looks like the fish none of the other fishermen saw. Since this cast of casters is made up of all the brightest kids from all the brightest classrooms on the planet, that doesn't sound likely.

Work in 11 dimensions - oh my! - that will impress the unknowning crowd. Say it is so tiny no one will ever see the stuff claim is amazing! Might as well use three bangs since it will forever be beyond reach.

Let me tell you how I dreamed up this old dream. I was preparing to go to the Second Meeting on Quaternionic Structures in Mathematics and Physics, in Rome, September 1999. As usual, I had to pay my way, but me and my traveling partner, Prof. Guido Sandri from BU added a week on to spend time looking at the art that has been gathering at the birthplace of western civilization. I was making up my transparencies for my talk which was how to write the Maxwell equations using only real-valued quaternions. It turns out to be easy to write the Maxwell equations with complex-valued quaternions, which are not a division algebra, and therefore of no more interest than any other arbitrary Clifford algebra. Making things work with real quaternions, that was a trick. James Clerk Maxwell himself speculated that someday someone would be able to do the trick. Knowing I accomplished something Maxwell himself wanted will always be one of my more ridiculous achievements. I should be noted that a year earlier, Peter Jack, another non-professional figured out all the hoops that have to be passed through.

The homogeneous equations:
<br /> (\frac{\partial}{\partial t}, \nabla)((\frac{\partial}{\partial t}, \nabla)(, A) - (\phi, A)(\frac{\partial}{\partial t}, \nabla)) + ((\frac{\partial}{\partial t}, \nabla)(\phi, A) - (\phi, A)(\frac{\partial}{\partial t}, \nabla))(\frac{\partial}{\partial t}, \nabla)
+(\frac{\partial}{\partial t}, \nabla)((\frac{\partial}{\partial t}, \nabla)^*(\phi, A)^* + (\phi, A)^*(\frac{\partial}{\partial t}, \nabla)^*) - ((\frac{\partial}{\partial t}, \nabla)^*(\phi, A)^* + (\phi, A)^*(\frac{\partial}{\partial t}, \nabla)^*)(\frac{\partial}{\partial t}, \nabla) = (0, 0)<br />

The source equations:

<br /> (\frac{\partial}{\partial t}, \nabla)((\frac{\partial}{\partial t}, \nabla)(\phi, A) - (\phi, A)(\frac{\partial}{\partial t}, \nabla)) - ((\frac{\partial}{\partial t}, \nabla)(\phi, A) - (\phi, A)(\frac{\partial}{\partial t}, \nabla))(\frac{\partial}{\partial t}, \nabla)
-<br /> (\frac{\partial}{\partial t}, \nabla)((\frac{\partial}{\partial t}, \nabla)^*(\phi, A)^* + (\phi, A)^*(\frac{\partial}{\partial t}, \nabla)^*) + ((\frac{\partial}{\partial t}, \nabla)^*(\phi, A)^* + (\phi, A)^*(\frac{\partial}{\partial t}, \nabla)^*)(\frac{\partial}{\partial t}, \nabla) = 4 \pi (\rho, J)<br />

Impressively ugly. I can justify why it took me six months to find this particular combination of terms, and why Maxwell did not find them.

Now imagine me with a transparency and a sharpie, trying to come up with something I could be proud to travel several thousand miles to present to a half dozen people. This was my one result, and it was so bulky particularly in comparison to the Maxwell equations themselves. The technical struggle was to toss away just the right stuff.

But does Nature toss away anything? I felt the answer had to be a flat "no". At this point, I was focusing on EM. Why bother doing all this work to throw things away? It appeared to me that the EM equations liked things that were antisymmetric, and things that were symmetric were getting disposed of (which now makes sense, understanding the antisymmetric field strength tensor are the heart of EM). I recalled a quote at the start of one of the chapters of Misner, Thorne, and Wheeler dealt with symmetry. Those quotes are often the only part I understand, so I have read a decent fraction of them. I pulled the black phone book off the shelf, and started to hunt for the quote. Here it is, Chapter 17:

The physical world is represented as a four-dimensional continuum. If in this I adopt a Riemannian metric, and look for the simplest laws which such a metric can satisfy, I arrive at the relativistic gravitational theory of empty space. If I adopt in this space a vector field, or the antisymmetrical tensor field derived from it, and if I look for the simplest laws which such a field can satisfy, I arrive at the Maxwell equations for free space...at any given moment, out of all conceivable constructions, a single one has always proved itself absolutely superior to all the rest...

After reading that, I thought it was possible that if I did not toss away information, the field equations might be able to do the work of both gravity and EM. I traveled to Rome and had a grand time, even if no one in the group of six appeared that interested in the talk.

I had my unified field equations:


<br /> Jq^u - Jm^u = \square^2 A^u<br />

This is when I had the meeting with Prof. Alan Guth (24, #355). He told me I needed to figure out the action, derive the field equations from the action, find solutions consistent with current tests and different for more refined ones. That several year march took me away from quaternions. There also was a strategic decision. I knew my intended audience is trained and comfortable with tensors. I rewrote it all to use tensors, and did not mention quaternions. I found it amusing that quaternions could play such a subtle role.

My experience is that theoretical physicists are frenetically busy. They are only comfortable getting involved in a discussion close to their area of expertise. Since no one works on rank 1 field theories for gravity as documented by the lack of coverage in the literature, there is no one to target.

The discussion here has convinced me that I have to return this proposal to its quaternion roots. The technical reasons have to do with discoveries made only in this calendar year. The first was to address Steve Carlip's complaint about a spin 2 particle in the current coupling term, which was dealt with in post p 22, #319 and #320. One cannot multiply one 4-tensor by another 4-tensor and get a third 4-tensor unless all the machinery for automorphic multiplication is there, which is set if one works with quaternions.

The second reason has to do with the weak and the strong forces. To this day, I still have not seen anyone write something like: "...and this force equation does a similar thing to Coulomb's law but for the weak force". There are excellent reasons for this, but still, it makes the weak and the strong forces feel unapproachable. My one handle on them is their symmetries. In particular, the weak force has the symmetry of SU(2), known as the unit quaternions (I am not making that up). If I formulate the GEM proposal in terms of quaternions, then where I can place a unit quaternion will be the symmetric house where the weak force can do its work. That is so direct and simple, odds are good that it is true.

The sexy math idea is that the properties of quaternions - when done right - dictate every fundamental aspect of Nature, bar none. There are four forces of Nature because when you consider the symmetries of two quaternions interacting:


<br /> (\frac{J}{|J|} exp(J - J^*))^* (\frac{J&#039;}{|J&#039;|} exp(J&#039; - J&#039;^*)) = 1 + \delta<br />

this one expression has the symmetries of U(1), SU(2), SU(3) and Diff(M), EM, weak force, strong force and gravity respectively. The video is on YouTube because the new math cannot be shown in a PDF.

doug

 

[Mentz114]

Hi Doug, I hope you are going well.

I've been catching up in the arXiv and I wonder if you might be interested in these ( take a break from GEM ! ).

[40] arXiv:0708.1507 [ps, pdf, other]
Title: Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter
Authors: C Meusburger, B J Schroers
Comments: 34 pages
Subjects: General Relativity and Quantum Cosmology (gr-qc)

arXiv:0708.1154 (cross-list from hep-th) [ps, pdf, other]
Title: Antisymmetric-Tensor and Electromagnetic effects in an alpha'-non-perturbative Four-Dimensional String Cosmology
Authors: Jean Alexandre, Nick E. Mavromatos, Dylan Tanner
Subjects: High Energy Physics - Theory (hep-th)

 

[sweetser]

Hello Mentz:

The body is not doing so well. Looks like I have forgotten how to eat. I have diabetes, and one of the possible complications is to mess up neurons that signal things, such as "hey, let's empty the belly". I am working on living with gastroparesis. Life is always a challenge.

I did download both papers, and honestly understood about nothing. I was pleased to see a Lagrange density early one in one paper, but was unable to parse it.

GEM is in a good state. Personally, I am not aware of a "killer" problem with the proposal as it is today. There have been times when there were killer problems. Initially, the current for gravity had the same sign as the one for EM, and I got slapped for writing that one. Just this Spring I had the spin 2 current coupling scare that was avoided by the use of the first conjugate (iqi)* at just the right time.

doug