Unifying Gravity and EM

Lawrence B. Crowell · Feb 13, 2006

Follow on with octonions

This might be a bit outside of the GEM theory, but I figured I would try to clarify a couple of things.
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Let b_q~\rightarrow~\phi_q b_q for q~=~\pm k and \phi_q a scalar field that obeys
 [\phi_q,~\phi_{q^\prime}^\dagger]~=~f(q,~q^\prime), 
 [\phi_q,~\phi_{q^\prime}]~=~g(q,~q^\prime) 
Then the commutator of \phi b_q is
 [\phi_q b_q~,\phi_{q^\prime}b_{q^\prime}]~=~[\phi_q,~\phi_{q^\prime}]b_q b_{q^\prime} 
 =~1/2/{b_q,~b_{q^\prime}/}g(q,~q^\prime)~=~e_i, 
for the appropriate i on the table. In this way one can treat fermions as nonassociative
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The division algebra is one that want e_ie_j = e_k =/=0. If e_k is zero without either e_i or e_j being zero then there is said to be no algebra. At the level of octonions it is said that this is the final algebra.
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The whole process of construction from reals, complexes, quaterions and octonions involves a pairing of each other. The simplest of course is the complex plane where z~=~(x,~y) and the defined multiplication
 c*z~=~(a,~b)*(x,~y)~=~(ax~-~by)~+~i(bx~+~ay) 
 (a,~b)*(x,~y)~=~(ax~-~by,~bx~+~ay) 
The same goes for the quaternions, they are a pairing of complex numbers. Octonions are then in turn a pairing of quaternions. At each level one loses ordering, commutivity and finally associativity. This also reflects the so called Cayley numbers and the multiplication rule with pairing defines what is called the Cayley-Dickson algebra.
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The octonions are pairs of quaternions. Consider the octonion O~=~(A,~B) and O^\prime~=~(X,~Y). The multiplication of these two is then
 O\cdot O^\prime~=~(AB~+~e^{i\phi} Z^\dagger B,~BX^\dagger~+~AP). 
where the argument is usually taken as \phi~=~\pi/2. For a system of quaternions \sigma_i and \bf 1 this defines four additional elements e_i.
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Now for quaternionic valued operators as fields which satisfy the BRST quantization condition Q^2~=~0, where \psi~\in~ker(Q)/im(Q) gives the field as purely topological. In other words \psi~\ne~Q\chi. In this way it is possible to have the square of an element in the octonions being zero without it in a strict sense being unalgebraic. Of course in the octonions of operators e_je_j =/=0 for i =/=j.
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What is this good for? I might go on about this in greater detail in another post, but the Dirac operator \Gamma^a\partial_a has the same topological information as the field. Further, in general the Dirac matrices (the quaterions) may have a representation which depends upon the chart on the base manifold. Thus a more general Dirac operator is \partial_a\Gamma^a\_. In this way the quaternions are operators.
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In the case of sedenions one has e_ie_j~\ne~0 and algebra is lost. There are eight octonions within it that are "islands" of algebra, but "outside" of them appears to be algebraic "chaos." However, I think that by Bott periodicity there is structure there and I think it is involved with some sort of topology. The sedenions define S^7\times S^7\times G_2, which probably constrains this topology.
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Lawrence B. Crowell

sweetser · Feb 15, 2006

A symmetry, not a field

Hello:

I spend time wondering why I have trouble communicating this proposal. The action looks like the simplest one that can be made! I have fun finding tension between what we now know and the GEM proposal. In this post I will outline an example.

Why is energy conserved in EM? Take a look at the action:
S_{EM}=\int \sqrt{-g} d^{4} x (-\frac{1}{4 c^{2}}(\nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu})(\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}))
Vary time:
\delta S_{EM}=\int \sqrt{-g} d^{4} x (-\frac{1}{4 c^{2}}(\nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu})(\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu})) \delta t
There is no time in the action, so it is not going to be changed in the slightest: integrate over a nanosecond or a century, the integral will be the same under a variation in time. That makes the action symmetric under time transformations. Where there is a symmetry, there is a conserved quantity. In this case, that is energy, E_{cons.}=m \frac{\partial t}{\partial \tau}. The definition of energy conservation says that there is no change in the derivative of time with respect to the interval \tau.

Why is linear momentum conserved in EM? Look at the action above. No distance R appears in it, so like time, varying R will keep the variation in the integral at an extremum, and the conserved quantity is momentum, P = m \frac{\partial R}{\partial \tau}. The same story.

We know what mass is from special relativity, the difference between the square of energy and the square of momentum. Since energy and momentum both arise from a symmetry in the action, mass which is calculated from this two must also arise exclusively as a symmetry in the action.

That is not how it works for the Hilbert action of general relativity. Instead of hunting for a symmetry, the metric is treated as a field and varied.

In the GEM proposal, one looks at the definition of a covariant derivative to spot a symmetry. One can vary changes in the potential or changes in the metric (the connection). So long as the changes offset each other, there would be no difference in the integral of the action with those changes. There must be a conserved quantity. Since the metric is changing, it is reasonable to propose mass is conserved. Logical consistency appears to favor the GEM proposal.

Convincing someone who has made the investment in understanding general relativity at the nuts and bolds level may be beyond my reach, but I still like the view from my chair: it's drop dead gorgeous.

doug

Lawrence B. Crowell · Feb 24, 2006

There is something rather mysterious here. You say there is no time in the action for the Lagrangian [/itex]{\cal L}~=~1/4 F_{ab}F^{ab}. Agreed there is no explicit function of time, but one does have elementsF_{0j}~=~-\partial A^j/\partial t~-~\nabla_jA^0which are the electric field components. A variation with respect to time is going to pick out\partial_0F^{0j}terms of the form\partial E/\partial t, and\partial_0F^{ij}of the form\partial B/\partial t. These are just the time derivative parts of the Maxwell equations. In fact your\delta S_{EM}is wrong, for the variation with time is not going to involve just a multiplication by\delta t, but partial derivatives of this with time. // Lawrence B. Crowell

Lawrence B. Crowell · Feb 24, 2006

retry: symmetries & fields

There is something rather mysterious here. You say there is no time in the the Lagrangian {\cal L}~=~1/4 F_{ab}F^{ab}. Agreed there is no explicit function of time, but one does have elements F_{0j}~=~-\partial A^j/\partial t~-~\nabla_jA^0 which are the electric field components. A variation with respect to time is going to pick out \partial_0F^{0j} terms of the form \partial E/\partial t, and \partial_0F^{ij} of the form \partial B/\partial t. These are just the time derivative parts of the Maxwell equations. In fact your \delta S_{EM} is wrong, for the variation with time is not going to involve just a multiplication by \delta t, but partial derivatives of this with time.
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Lawrence B. Crowell

sweetser · Feb 24, 2006

Hello Lawrence:

There are partial derivatives with respect to time, but nothing that is just time. A delta t is not a t. One can imagine a Lagrangian where the effects dissapate after a certain amount of time. That is not what happens with the classical EM Lagrangian. If one comes back in 10 years, the delta t's will still be the same. That is the source of energy conservation as I see it.

I believe my S_{EM} is standard, although most people write it in flat spacetime.

Looks like your first itex bracket should not have a /.
doug

Lawrence B. Crowell · Feb 24, 2006

Conservation of energy

The variation of the action results in the Euler-Lagrange equations, which are the equations of motion. Conservation of energy for fields comes from the momentum-energy tensor
 T^{ab}~=~\partial{\cal L}/\partial g_{ab}~-~g^{ab}{\cal L} 
with the continuity condition \partial_cT^{ab}~=~0. The variation of the action gives dynamical equations, or F~=~ma stuff, and the momentum-energy tensor gives the conservation laws a'la Noether's theorem.
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Lawrence B. Crowell

sweetser · Feb 24, 2006

Equations of motion and forces

Hello Lawrence:

My equation about \delta S_{EM} above is wrong. Partial derivatives with respect to t are required.

I can see that my description of the issue was informal, but not unconventional. Here is a quote from http://en.wikipedia.org/wiki/Noether's_theorem:

wikipedia said:

The word "symmetry" in the previous paragraph really means the covariance of the form that a physical law takes with respect to a one-dimensional Lie group of transformations which satisfies certain technical criteria. The conservation law of a physical quantity is usually expressed as a continuity equation.

The most important examples of the theorem are the following:

* the energy is conserved iff the physical laws are invariant under time translations (if their form does not depend on time)
* the momentum is conserved iff the physical laws are invariant under spatial translations (if the laws do not depend on the position)
* the angular momentum is conserved iff the physical laws are invariant under rotations (if the laws do not care about the orientation); if only some rotations are allowed, only the corresponding components of the angular momentum vector are conserved

A Noether charge is a physical quantity conserved as an effect of a continuous symmetry of the underlying system.

The form of the GEM and EM Lagrangians do not depend on time, so energy is conserved.

Landau and Lif****z gave me a really great lesson on this topic. I knew that once the Lagrangian is set, then everything can flow from that, be it equations of motion or dynamical equations or stress tensors. But how are these related? Let's start with the GEM Lagrangian:
 \mathcal{L}_{GEM}=\frac{-\rho_{m}}{\gamma}-\frac{1}{c}(J_{q}^{\mu}-J_{m}^{\mu})A_{\mu} -\frac{1}{4c^{2}}(\nabla^{\mu} A^{\nu}-\nabla^{\nu} A^{\mu})(\nabla_{\mu} A_{\nu}-\nabla_{\nu} A_{\mu}) - \frac{1}{4c^{2}}(\nabla^{\mu} A^{\nu}+\nabla^{\nu} A^{\mu})(\nabla_{\mu} A_{\nu}+\nabla_{\nu} A_{\mu}) 
Vary A and its derivative, keeping the velocity field fixed, and that generates the equations of motion, the stuff of the Maxwell equations. The second, third, and fourth terms come into play.

Now start with the same Lagrange density, but keep A and its derivative fixed, and vary the velocity V, which appears in the terms \frac{-\rho_{m}}{\gamma} and in the current densities J_{q}^{\mu}-J_{m}^{\mu}. That will generate the Lorentz force. One can also get to the Lorentz force through the stress tensor. Only the first and second terms come into play.

One important thing to notice is the role played by the second term, the so-called charge coupling term. In EM, the equations of motion indicated like charges repel, which is the same message as arises from the Lorentz force equation. By having the opposite sign for the gravitational charge in the second term, like charges attract for the field and force equations.

doug

Chronos · Feb 25, 2006

Pardon my dumb question sweetser, but is this a quantum treatment?

sweetser · Feb 25, 2006

Possible to do quantum calculations

Hello Chronos:

I hope to give a sophisticated answer to your dumb question. Bur first, the short answer: their are concrete technical reasons to hope the GEM field equations can be quantized, but due to my own limitations, I have not done any quantum calculations, such as a scattering cross section of an electron fired at a proton. A well-behaved, finite calculation of a scattering cross section would demonstrate the proposal is consistent with quantum mechanics.

How does one know if a theory can or cannot be quantized? There are fancy answers, but I prefer simple ones (but not too simple). Zeroes for observables of a classical theory are bad. In a classical field theory, an observable such as energy or momentum is a number. To make the same theory a quantum theory, an operator must be found for that observable that acts on the wave function. The operator gets plugged into a commutator that should be some multiple of Planck's constant. If a classical observable is zero, the quantum operator is zero, and the commutator will be zero, not a multiple of Planck's constant. Failure.

That is exactly what happens to the classical EM Lagrange density:
\mathcal{L}_{EM}=-\frac{1}{4 c^{2}}(\nabla^{\mu}A^{\nu}-\nabla^{\nu}A^{\mu})(\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}))
What is the energy/momentum 4-vector? One thing not appreciated widely enough is the answer to this question is automatic: all it takes is a calculation, no thought involved, a no-brainer. One takes the derivative of this with respect to the time derivative of the potential A, and out comes the canonical energy momentum. I don't know if you can buy a book that does this in detail. It is not a hard calculation, but it is painful to do the LaTeX. I like to see all the details of an easy calculation, so here it is. First, this is the EM Lagrangian written out in all its component parts:
\mathcal{L}_{EM}= - \frac{1}{2} ( ( - ( \frac{\partial \phi}{\partial x} )^2 - ( \frac{\partial \phi}{\partial y} )^2 - ( \frac{\partial \phi}{\partial z} )^2 \\ - ( \frac{\partial A_x}{c \partial t} )^2 + ( \frac{\partial A_x}{\partial y} )^2 + ( \frac{\partial A_x}{\partial z} )^2 \\ - ( \frac{\partial A_y}{c \partial t} )^2 + ( \frac{\partial A_y}{\partial x} )^2 + ( \frac{\partial A_y}{\partial z} )^2 \\ - ( \frac{\partial A_z}{c \partial t} )^2 + ( \frac{\partial A_z}{\partial x} )^2 + ( \frac{\partial A_z}{\partial y} )^2 \\ - 2 \frac{\partial A_x}{c \partial t} \frac{\partial \phi}{\partial x} - 2 \frac{\partial A_y}{c \partial t} \frac{\partial \phi}{\partial y} - 2 \frac{\partial A_z}{c \partial t} \frac{\partial \phi}{\partial z} \\ - 2 \frac{\partial A_y}{\partial z} \frac{\partial A_z}{\partial y} - 2 \frac{\partial A_z}{\partial x} \frac{\partial A_x}{\partial z} - 2 \frac{\partial A_x}{\partial y} \frac{\partial A_y}{\partial x} )$ 
Now take the derivative of the Lagrangian with respect to 4 things: \frac{\partial \phi}{\partial t}, \frac{\partial A_x}{\partial t}, \frac{\partial A_y}{\partial t}, \frac{\partial A_z}{\partial t}. Looks painful. Start with the first one, \frac{\partial \phi}{\partial t}, and you will notice there is no such term in the classical EM Lagrangian. Even if one's calculus is rusty, if a variable is not there, the derivative with respect to there's no there there is zero. This classical EM Lagrangian cannot be quantized for that reason alone.

So what do people do? They tack in a term to cover up this problem, and surround the patch job with discussions of gauge theory. The sport is called fixing the gauge. There are all kinds of super sophisticated things to say about this topic. I prefer the simple question: why is there this problem? The answer is also simple: there was a subtraction step in the Lagrangian, so of course you are missing something. All the GEM proposal does is keep all the parts of the 4-derivative of a 4-potential together so nothing is missing. Let's look at the GEM Lagrangian written out explicitly:
 \mathcal{L}_{GEM}=- \frac{1}{2} ( ( \frac{\partial \phi}{c \partial t} )^2 - ( \frac{\partial \phi}{\partial x} )^2 - ( \frac{\partial \phi}{\partial y} )^2 - ( \frac{\partial \phi}{\partial z} )^2 \\ - ( \frac{\partial A_x}{c \partial t} )^2 + ( \frac{\partial A_x}{\partial x} )^2 + ( \frac{\partial A_x}{\partial y} )^2 + ( \frac{\partial A_x}{\partial z} )^2 \\ - ( \frac{\partial A_y}{c \partial t} )^2 + ( \frac{\partial A_y}{\partial x} )^2 + ( \frac{\partial A_y}{\partial y} )^2 + ( \frac{\partial A_y}{\partial z} )^2 \\ - ( \frac{\partial A_z}{c \partial t} )^2 + ( \frac{\partial A_z}{\partial x} )^2 + ( \frac{\partial A_z}{\partial y} )^2 + ( \frac{\partial A_z}{\partial z} )^2 ) 
That certainly looks complete. Calculate the canonical momentum:
 \pi^{\mu} = h \sqrt{G} \frac{\partial \mathfrak{L}}{\partial ( \frac{\partial A^{\mu}}{c \partial t} )} = h \sqrt{G} ( - \frac{\partial \phi}{c \partial t}, \frac{\partial A_x}{c \partial t}, \frac{\partial A_y}{c \partial t}, \frac{\partial A_z}{c \partial t} ) 
Nothing is zero, so quantizing the theory is possible. [For fun, notice the units required to get mass*length^2/time^2.]

Being skeptical of my own skills, the only time I get confident is when I see that others have already done a nearly identical thing. The field equations written in the first post have been quantized, the energy and momentum turned into operators. The problem is that this was done for a massless spin 1 field of EM only. There are 4 modes of emission, but only 2 of them could be described by a massless particle (something about constraints on polarization because the particle is travel ling a the speed of light). The scalar mode is non-sense, allowing for negative probability. So Gupta makes the ad hoc supplementary condition whose entire purpose is to eliminate the scalar mode of emission as well as the longitudinal one.

In the GEM approach, the same operators for energy and momentum used by Gupta and Bleuler are used. Instead of just a spin 1 field doing only EM, GEM has two fields, a spin 2 for gravity, a spin 1 for EM.

The fact that I can open a graduate level quantum field theory book, go to the section on the Gupta/Bleuler quantization method, and point to the operators for energy and momentum, says the theory can be quantized. The next step is to do a calculation, for example the scattering of an electron off of a proton. It is known how to do this for EM. This is where Feynman diagrams come in. There would be two changes to go from EM scattering to gravity scattering: the coupling constants and the propagator. The couple constants are easy: e^2 -> G m_{e} m_{p}. If you plug in numbers, the coupling goes from weenie to extra-super-wimpy: 2.56 \times 10^{-38} C^2 -> 1.01 \times 10^{-67} C^2. The propagator, the squiggly line between two nodes of a Feynman diagram, must change from a spin 1 to a spin 2 propagator. Weinberg wrote papers on this topic in the early 60's. I Xeroxed them, and knew I would never understand the contents. I gave those papers to a friend with a recent Ph. D. in cold matter physics from MIT over his Christmas break, and he could not decipher the information. For someone actively doing scattering calculations in quantum field theory, this calculation is probably simpler than what they are getting paid to investigate: it is a variation on a calculation they are trained to do, and variations are easier to perform than making up something new. I have no idea how to reach a person who could do that work, and at this time have no authority to compel them to crank through it.

doug

Chronos · Mar 6, 2006

Thanks for the thoughtful reply, doug. I didn't phrase the question very eloquently, but you went right to the heart of the matter. Quantization is a difficult prospect in the best of circumstances. It's not pretty, much less elegant, and that rubs me wrong. I've always been intrigued, and am curious what you think of this paper:

That strange procedure called quantisation
http://www.arxiv.org/abs/quant-ph/0304202

sweetser · Mar 23, 2006

Preparing for a talk

Hello Folks:

I have been preparing for a 15 minute talk at the 9th Eastern Gravity Meeting happening this Friday and Saturday at MIT. I am slated for noon, March 25. The title is "Unifying Gravity and EM, a Riddle You Can Solve". The slides are available, and the pdf has additional text approximating the main points I will say. Please feel free to check out the slides and give feedback.

I am also moving into a new home which takes mountains of time and effort, hence my delay in replying to Chronos.

doug

The slides:
http://theworld.com/~sweetser/quaternions/talks/riddle/riddle.html
[note: The page uses css & javascript, use the click, arrows, pageup and pagedown to advance]

The slides + comments:
http://theworld.com/~sweetser/quaternions/ps/riddle.pdf

sweetser · Apr 11, 2006

Response to talk

Hello:

The 9th Eastern Gravity Meeting at MIT was full of folks who work with Einstein's field equations for their daily bread. Some are working on trying to make gravity wave detectors, others work with computers trying to develop models of spiraling black holes that crash into each other. There was one string theory guy who still does not have any way to test the proposal.

My observation of the meeting in general was there was very little communication between different workers. It was only clusters of folks who happened to be working on related problems that could bring up decent questions. That may just be the nature physics: it's too technical today to be a generalist.

I can say with some confidence that no one trained in physics today works with a vector proposal for gravity anywhere in their education. It is either Newton's scalar theory or Einstein's rank 2 theory. I don't consider the two paragraphs a piece in MTW, or papers by Gupta and Thirring to count (I am embarrassed by the logical weaknesses involved in those short comments).

There was one work on general relativity - I cannot remember which one - that after introducing the covariant derivative, pointed out that one could take the divergence of the connection:
\partial_{\mu}\Gamma_{\sigma}{}^{\mu\nu}A^{\sigma}
The author then quickly pointed out that there is no need to ever do this sort of calculation because the term does not transform like a tensor. If we want second order derivatives of the metric that transform like a tensor, then we MUST use the Riemann curvature tensor. The logic is that simple and straightforward.

What's wrong with this logic? It is an omission. If you put the divergence of a connection in the right place, it transforms like a tensor:
\partial_{\mu}\partial^{\mu}A^{\nu}- \partial_{\mu}\Gamma_{\sigma}{}^{\mu\nu}A^{\sigma}
If the potential happens to be constant, the result is the divergence of the connection, the thing that was dismissed as silly to bother to calculate. This has second derivatives of the metric in a term far simpler that the Riemann curvature tensor.

My sense of the project is this: I will not be able to communicate this proposal until by some chance combination I find someone else who actually does that calculation for the Rosen metric. If you don't go through the exercise, you don't see it. That was the [non]reaction I got. Oh sure, I can be a bit more entertaining, the graphics are easier to read than the average presentation, but there was no effective communication about the core idea of the talk. I don't think there is anything too unusual about that.

doug

Lawrence B. Crowell · Apr 18, 2006

talk & tensors

To be honest the objection appears valid. Consider this term
 \partial_a\Gamma_b^{ac}A^b 
then the gravity connection transforms by U(1) by
 A^b~=~U^{-1}A_b'U~+~U^{-1}\partial_b U 
the partial on the gravity connection will transform homogeneously by local Lorentz transformations. However, this whole term does not transform homogeneously under the gauge action
 \partial_a\Gamma_b^{ac}A^b~=~\partial_a\Gamma_b^{ac}(U^{-1}A_b'U~+~U^{-1}\partial_b U) 
The second order term \partial\partial A (indices suppressed) does not cure this, since this will transform homogenously under the gauge action and thus contains no inhomogenous terms which cancel out the inhomogeneous term U^{-1}\partial_b U. So this is not a tensor. Thus the objection is valid.

Lawrence B. Crowell

sweetser · Apr 20, 2006

Hello Lawrence:

We both agree that \partial_a\Gamma_b^{ac}A^b does not transform like a tensor. That is a non-issue.

I will always start instead with this:
\partial^a A^c-\Gamma_b^{ac}A^b
This is a tensor. This is the definition of a covariant derivative I am starting with. I choose to work with a potential A^b that is constant, so the derivatives of A^b are zero. Then I act on it with a partial derivative, \nabla_a (Note: I used the triangle there because it too has a partial derivative and a connection). The second derivative of the potential does not get stapled in at the end. Instead I start only with valid tensors, and act on those tensors with tensor operators.

U(1) is an approximate symmetry of this proposal, not an exact symmetry. For the special case of massless particles, the U(1) symmetry is exact. Mass breaks U(1) symmetry extremely weakly (1 part in 10^16 for an electron). You are correct that the connection breaks the U(1) symmetry, but that is not a constraint on the proposal. If one only works with tensors throughout, the result must be a valid tensor expression.

Breaking symmetry with in this way is a good thing. It means that the Higgs particle and Higgs mechanism are unnecessary. In the standard model, the mass of an electron has to be included via the Higgs and the spontaneous symmetry breaking of a false vacuum. I use something we already associate with mass - the metric, and the changes in the metric or connection - to break very slightly U(1) symmetry with something physically relevant.

doug

sweetser · May 14, 2006

Hello:

I think I am going to be a bit more assertive in the coming months about what I may have accomplished. Unifying gravity and EM in a way that can be quantized should get people's attention. In practice, delivering that line is not enough. This confuses me, because whenever someone makes that claim to me, being a good skeptic, I investigate, and a little digging usually determines there is no math there (and thus for me, no content).

Prof. Clifford Will is going to MIT to give a talk. He is a fellow who is an expert on experimental tests of general relativity. He wrote a great review article on the topic in living reviews (long paper, can be found via Google). Someone suggested I read it because it was an exhaustive survey of approaches to gravity. I read it carefully, and like everyone else, he did not mention the possibility of a vector field theory for gravity. He did cite vector-scalar and vector-tensor theories, but no plain old vector. See, Newton's field theory for gravity is the simplest scalar theory that can be written. It is still a darn good theory, used in all rockets except those that carry atomic clocks. The simple scalar theory has a technical flaw: it does not respect the speed of light limit.

A better theory is the simplest rank 2 field theory, that is general relativity. It too has a technical flaw: it cannot be quantized.

The only thing between the simplest rank 0 and the simplest rank 2 field theory is the simplest rank 1 field theory. It should be a big, honking red warning light that someone like Will has this sort of sin of omission. So I will be going to his talk and seeing if I can point this out to him. The odds are very bad - it is like pointing out a shade of green to someone who is color blind. He has zero experience with a vector field theory for gravity, Jq^{\mu}-Jm^{\mu}=\nabla^{2}A^{\mu}.

doug

nrqed · May 17, 2006

sweetser said:

Hello:

I think I am going to be a bit more assertive in the coming months about what I may have accomplished. Unifying gravity and EM in a way that can be quantized should get people's attention. In practice, delivering that line is not enough. This confuses me, because whenever someone makes that claim to me, being a good skeptic, I investigate, and a little digging usually determines there is no math there (and thus for me, no content).

Prof. Clifford Will is going to MIT to give a talk. He is a fellow who is an expert on experimental tests of general relativity. He wrote a great review article on the topic in living reviews (long paper, can be found via Google). Someone suggested I read it because it was an exhaustive survey of approaches to gravity. I read it carefully, and like everyone else, he did not mention the possibility of a vector field theory for gravity. He did cite vector-scalar and vector-tensor theories, but no plain old vector. See, Newton's field theory for gravity is the simplest scalar theory that can be written. It is still a darn good theory, used in all rockets except those that carry atomic clocks. The simple scalar theory has a technical flaw: it does not respect the speed of light limit.

A better theory is the simplest rank 2 field theory, that is general relativity. It too has a technical flaw: it cannot be quantized.

The only thing between the simplest rank 0 and the simplest rank 2 field theory is the simplest rank 1 field theory. It should be a big, honking red warning light that someone like Will has this sort of sin of omission. So I will be going to his talk and seeing if I can point this out to him. The odds are very bad - it is like pointing out a shade of green to someone who is color blind. He has zero experience with a vector field theory for gravity, Jq^{\mu}-Jm^{\mu}=\nabla^{2}A^{\mu}.

doug

Good luck Doug.

I am just starting to read this long and fascinating thread so I will read more before starting to ask questions that I am sure have already been asked by others. Of course, the first comment is that the force between like charges interacting through a vector field repel so it seems as if a gravitational force based on such a field would lead to gravitational repulsion. But I am sure you have debated that with others so I will read on.

Regards

sweetser · May 24, 2006

Hello:

There are two papers in the literature that also claim that a vector field equation must have like charges attract just like EM. That result comes from copy EM too closely! My replies to the question are in posts 23 and 33.

The key is one perfectly placed minus sign in the Lagrange density, which then appears in the subsequent field equations and in the Lorentz force laws. The charge coupling term for EM has the same sign as the rank 2 field strength tensor contraction. That is what generates like electrical charges that repel in Gauss's law. The same sign for the coupling term and the inertial mass term lead to the Lorentz force law where like electric charges repel.

By changing the sign of the charge coupling term, the couping and field strength tensor terms have opposite signs, and like mass charges attract. The different sign for the coupling term and the mass term lead to a Lorentz force law where like mass charges attract.

The signs are actually required due to the asymmetric field strength tensor. See, that gets split into two, an irreducible symmetric field strength tensor, and an irreducible antisymmetric field strength tensor. The symmetric rank 2 field strength tensor will be represented by spin 2 particles, and those must attract (read that in Hatfield's introduction to the Feynman lectures on gravity, go there if you want a good explanation). The antisymmetric field strength tensor will be represented by spin 1 particles, and like charges will repel.

Sorry for my delay in replying, the site forgot to send me email. This is an issue my proposal must get right, or it is not worth bothering anyone.

doug

sweetser · May 26, 2006

Interaction with Prof. Clifford Will

Hello:

In this note, I will describe my interaction with Prof. Clifford Will of the University of Washington, St. Louis, an authority on experimental tests of gravity, that happened during his visit to MIT Thursday, May 18, 2006.

I am conflicted about my own body of work. On one hand, I consider the equations to be as gorgeous as the come. I've taught my mailman the field equations which he remembers to this day with the mnemonic, "Always give 2 Brownies to Jim", the reverse of J=Box^2 A. The exponential metric is prettier that the algebraic fragment of the Schwarzschild metric in the Schwarzschild coordinates, and completely trumps the Schwarzschild metric in isotropic coordinates with its 4th power terms (most folks studying GR probably aren't even familiar with it, but it is the form used in experimental tests). I can see the symmetries in the GEM Lagrangian that lead to energy, momentum, and mass conservation. As long as I keep focused on the equations, I can feel joy in their elegance.

On the other hand, I am familiar with every misstep made along the way, even in this forum. I am aware of my own inadequacies as a messenger, without any degree in physics, just the ad hoc collection of graduate-level courses audited from MIT, Harvard, and BU (Boston may be the best town to audit courses). It feels like a cruel cosmic joke that someone with such limited skills in math and physics should have found this cache of rocket fuel. As I age and see those limited skills shrink, so does my confidence.

For about a week, I thought about the questions I would like to ask Will. That was fun. Not fun was the self-doubt which can derail the task. I decided to wear the Turquoise Einstein t-shirt featured at the top of quaternions.com for two reasons: I like the shirt enough that it boosts my self confidence, and it has the Always Give 2 Brownies to Jim equation which might be useful in a technical discussion.

Will was to be at an MIT physics lounge at 3:45, with the talk in 10-250, a large hall, at 4:15. I showed up right on time. Will looks much like Ted Turner, with white hair and mustache. He was seated on a couch, the room sparsely occupied by small clusters of graduate students talking about grading undergrads. I went straight for the cookies, wondering when I should say hello. Rather than have a long debate, I decided to get it over with sooner rather than later.

Eye contact made, I told Will that he was the center of a small debate on the Internet (sci.physics.research several years ago). I was investigating a simple approach to gravity in 4D. A fellow suggested I read Will's Living Reviews article on tests of gravity because it covered EVERY respectable approach to gravity, bar none. I read it. It was a very good article (that brought a smile). The paper cited vector/scalar theories, and vector/tensor theories, but no simple vector-only theories. Newton's law of gravity is the simplest rank 0 field theory one can construct. It is a darn good theory, still used in the guidance systems of most rockets. If those rockets carry atomic clocks, we realize two technical limitations of Newton's law: that the speed of gravity is wrong (no speeds are infinite) and gravity bends spacetime more than Newton's law predicts. A better theory is the simplest rank 2 field theory that can be constructed to explain how gravity works - Einstein's field equations for general relativity. It really is a darn simple approach, with the Ricci scalar sitting alone in the Hilbert action. The speed of light is respected. All weak field tests are passed. All strong field tests are passed. The rank 2 field theory also has a technical problem: it cannot be quantized. The brightest folks in physics have all tried to no avail.

Between the simplest rank 0 field theory and the simplest rank 2 field theory is the simplest rank 1 field theory. This formal possibility was not discussed in this otherwise exemplary review article. I asked him what he thought about that omission.

He replied that there was no need to disprove a vector theory because we already had proof that a metric theory was required. All the tests of the equivalence principle were effectively tests that gravity must be a metric theory.

I nodded. I was aware that for my own efforts to succeed, I would have to precisely deflate this position. I may do that in a subsequent post here, but that was not my purpose in the discussion with Will. Rather, I had a rare chance to talk technically with an expert, and wanted to find out his perspective on several issues. I had it, so it was time to move on.

I pointed to my t-shirt, saying this was the 4-vector field equation I happened to study. Most physicists if asked would think it was the Maxwell equation in the Lorenz gauge, where like charges repel, and thus not applicable to gravity. If the metric had a +2 signature instead of -2, then like charges would attract. The key to understanding the proposal is that the box^2 is not a D'Alembertian operator (a scalar operator consisting only of the second time derivative minus a Laplacian operator). Instead the box^2 represents two covariant derivatives. A covariant derivative has a normal derivative minus the connection, so the equation reads: normal derivative minus a connection, applied to a normal derivative minus a connection, applied to a 4-potential. One could have a differential equation that was the divergence of the connection of the potential. I claimed that the Rosen metric (the one at the start of this forum) solves that very differential equation.

He shrugged his shoulders. That was too dense to follow, and he didn't. I offered to send him a copy of my paper. He told me he was extremely busy, and there was no way he would have the time to look at it. I took him at his word, and promised him I would not email him. I am not going to beg for people to look at elegant equations.

He asked if the theory was a full one. I told him I had a Lagrange density and the field equations from the action, along with solutions to the field equations that were physically relevant.

An objection I was sure people would raise is that the proposal has linear vacuum field equations. Like EM, once particles start interacting, things become nonlinear. My impression was that physicists believe that the vacuum field equations had to be nonlinear, that non-linearity was a way to decide if a proposal was mature or just a toy.

Will said there was no experimental data on the topic. After a bit of going back and forth, he said essentially that the collection of good candidate theories for gravity happened to all be nonlinear, so it was reasonable given that observation for there to be a sense that nonlinear field theory was the correct approach. The Schwarzschild metric was linear in one coordinate system, and nonlinear in another, so linearity is coordinate dependent, and of limited value.

I asked if there were any plans to do gravity tests to second order PPN accuracy. My proposal is identical at first order PPN accuracy, but about 12% different at seccond order PPN accuracy. He said those experiments are very demanding. There were nothing coming soon. He wondered if I had shown my proposal worked for the precession of the perihelion of Mercury. I claimed that it did, a four page calculation (available at quaternions.com). I also asserted that the lowest mode of gravity wave emission should be a quadrupole because the theory conserves momentum, and there is no other field to store momentum.

He nodded along, but was not engaged, so I thanked him for his time, shook his hand, and went back for another cookie. He eventually got off the couch and chatted with a few professors. I realized that had I been delayed, there would have been no way to have had such a long and detailed private discussion. He left at 4:05 to go to the lecture hall.

The talk itself was good. He covered tests of the equivalence principle, solar tests (weak field), binary pulsars (strong field), and gravity waves. He was one of the folks who developed the PPN system. For general relativity, \gamma=1, \beta=1, all 8 others are equal to zero. At the end of the lecture, I asked if there were any other theories where \gamma=1, \beta=1, all 8 others are equal to zero. [I was interested because this is a property of my own proposal.] He said Rosen, of Einstein, Poldolski, Rosen fame, came up with such a theory. It works for some tests, but for binary pulsars, the theory predicts the frequency should increase instead of the decrease observed. It is vital to check a proposal in all regions.

Someone asked a question on MOND, the Modification of Newtonian Dynamics that is consistent with some data on the velocity profile of disk galaxies (as well as other astronomical data sets). He said that someone had figured out a Lagrange density for MOND, he had seen it, and it was UGLY. I found that funny. The strength of the GEM proposal is that the Lagrangian is fit and trim. Can it match all the data collected in eighty years? Not in a brief private discussion.

I did feel it was a good day for the GEM proposal. I certainly did not convert Will: that would require a formal demonstration that the proposal works for all weak field tests, all strong field tests, all tests of the equivalence principle, and gravity waves as a starting point. On an informal level, it was not dismissed out of hand. It felt like there was stuff worth talking about. I was glad I had attended and asked as many questions as I did.

doug

sweetser · Jun 9, 2006

Metric OR Potential Theory

Hello:

One of the things that gives me confidence is when my points of contention with standard theory become subtle instead of confrontational. Too many folks brighter than myself have thought about these issues before. It is unreasonable to expect them to be wrong. It is natural for a blind spot to remain unseen.

Clifford Will has argued that gravity must be explained using a metric theory. He is not the first one to state that position. I've seen it espoused in Misner, Thorne, and Wheeler (the beginning of chapter 40 to be more precise). I agree, gravity must be explained with a metric theory.

Here is where the subtle issues start. The question not asked is how many ways are there to implement a metric theory? The standard approach is to work with the Riemann curvature tensor. If you are not familiar with this rank 4 monster, it is essentially the difference between two curved paths. It measures how much the curvature of spacetime is changing. I call it a monster because one time I tried to write it out in terms of metrics and it was a futile exercise, there were just too many terms. [More detail than necessary: the Riemann curvature tensor is the difference between two derivatives of a Christoffel symbol, and the difference of two products of two Christoffel symbols, and a Christoffel symbol itself has three derivatives of a metric that get contracted with another metric, so the details of the Riemann curvature tensor can be overwhelming.]

There is an implied message behind the pitch for a metric theory: that the approach cannot use a potential. I forget who did the calculation (someone between Newton and Einstein), but he showed that gravity will bend light. Einstein also did the calculation in 1911. The answer was exactly half right. It did make exactly the same prediction for bending the time term of the metric as does general relativity, the g_00 term of the Schwarzschild metric, which gets a little smaller than one. The error is that Newton's theory leaves the space part unchanged. Experimental tests show the g_11 term is a little greater than one.

What the data unambiguously proves is that a scalar potential theory cannot explain light bending around the Sun. The coefficient in front of the dt part of the metric is less than one, whereas the coefficient in front of the dR part of the metric is greater than one. A one parameter potential can not be both greater and lesser than one. Proof complete.

Let's shine light on the blind spot. A potential that has more than one parameter could be consistent with experimental data. If one used 100 parameters, it would be trivial to match any data set. The question is what is the smallest number of parameters needed? It has been established that one is too small. The next thing to try after a scalar tensor potential is a vector tensor potential. In spacetime, that would have 4 parameters. There is sufficient freedom with 4 parameters to match the experimental results.

Another subtle issue is that people if pressed would view this discussion as a metric theory versus a 4-potential theory. It cannot be that, since I have already agreed the theory must be a metric theory - up to a diffeomorphism. That is a fancy way to say I am proposing a metric theory and/or a 4-potential theory. Some might say that is a slimy political move, but I believe that is a beautiful symmetry slight of hand. It's magic! Remember, general relativity is magic too, saying gravity is exclusively about spacetime geometry, no force needed. Incorporating diffeomorphisms correctly to a vector theory allows me to say gravity is all about spacetime geometry or potential, or any combination of geometry and potential you choose. The GEM proposal is not only a unification of gravity and EM, but the union of Newton's exclusively potential theory with Einstein exclusively metric theory. That rocks!

This work keeps getting sexier than string theory :-)
doug

sweetser · Jun 12, 2006

Visualizing GEM

Hello:

More of our brain is devoted to visual information processing than any other, so it is vital to have a visualization of the GEM proposal. It can be found in this link:

http://TheWorld.com/~sweetser/quaternions/gravity/em2gem/figure_1.jpg

The image is a collage of two figures most people are familiar with. The Newtonian potential looks a little like a ladle, and particles hang out at the bottom of the dip. The lines on the graph paper in the background for the x and y coordinates are always as straight as can be drawn.

For Einstein's explanation of gravity, artists use a rubber sheet, with the mass stretching the sheet. Now a "straight line" - one that stays within the warped lines - looks curved to us from afar.

What the GEM theory proposes is that either or a combination of both images is correct. Newton's potential theory for gravity does not work because it can only bend one way, and spacetime needs at least four parameters to describe its bending as is possible with a 4-potential versus a scalar potential theory.

Newton's theory gets the light bending around the Sun half right. Specifically, it gets the bending of the time portion correct. If one used a metric that was flat for the time part (g_{00} = 1), but was bent the appropriate amount in the space part (g_{11} = 1 + 2 GM/c^{2}R), then the combination of a Newtonian potential with a curved-only-for-R metric would be consistent with experimental tests of gravity to first order PPN accuracy. The curved-only-for-R metric would not be a solution to Einstein's field equations which explain gravity as exclusively arising from spacetime curvature. It is unfortunate, but the GEM theory says that exclusive metric theories like general relativity are too restrictive. The potential well/rubber sheet symmetry must be embraced to unify gravity and light.

doug

sweetser · Jun 19, 2006

Pitching the program

Hello:

People have suggested I submit a paper to a physics journal. I have set a specific goal to achieve before I go through that process. I would like to find someone who understands general relativity and the Maxwell equations at the level of the action, have that person read my draft paper, and tell me if s/he thinks it is valid. Based on the ensuing give and take, I would either submit the paper, or post to my website (and here) why the proposal was flawed. In this post I will show my latest effort to find someone with the skills required to review my work on a technical level.

A few posts ago on this thread, I told how I had a productive discussion with Clifford Will, but he was too busy to read my work. The Internet was developed by physicists at CERN for physicists working together over the globe. It is my sense that in measurable ways, the pace of life for theoretical physicists is more frantic than most other professions. I expect doors to be closed not due to aloofness - Will was both approachable and polite - but because there is too much stuff crammed in the room to open the door.

Well-known physicists at MIT, my alma mater, are frantic. I've seen it up close when I worked at the bench for Prof. Richard Young. It is part of my ethics not to bother a busy scientist unless I meet one of two criteria: either I have a well-formed question or I have a result. Being a skeptic, it is important to me that at the very least Mathematica has checked the algebra contained in the draft (Mathematica did not alway approve of my efforts, but that is a different story). Professors that are less well-known are probably even more frantic, trying to establish a name.

What follows is my last email to a less well-know physicist, which references an email to a well-known physicist. The short story is the less-well known prof. has not written back, and the well-known prof. documented why he cannot read my draft. I am not mad or frustrated because I understand why they behave as they do. I remain persistent.

doug

--------
From: Doug <dougsweetser@gmail.com>
To: slloyd
Date: Jun 8, 2006 7:22 AM
Subject: Fwd: The work of the stand-up physicist

Hello Seth:

Dave Pritchard gave me your name (email exchange at the end). I am an
amateur physicist who has a Lagrangian for a unified field theory.
There have been a few times in the history of science when an amateur
- after a few pitches by mail - has mad a contribution. What is not
part of the lore is the number of appeals of limited value. I hope
the details in my note to Dave and the attached pdf file show it is
possible my work has some of the claimed significance.

I hope you get a chance to review the work. Thinking about the
action, the field equations, or the solutions to the field equations
gives me a quiet sense of joy (unusual for such abstractions).

Have a good day despite the rain,
doug

---------- Forwarded message ----------
From: Dave Pritchard
Date: Jun 7, 2006 12:30 PM
Subject: Re: The work of the stand-up physicist
To: sweetser@alum.mit.edu
Cc: Sarah Smith

Doug,

Right now I have about 6 papers of which I'm a coauthor (often senior
authro)to read through and correct - I can't take on another (I spent
the weekend dusting off 3 overdue referee reports ahd nave a big
proposal to review, and another one to do also). So adding to this list
is impossible.

Your ideas sound like something theorists might have tried.

[I'll reply here since it would have been impolite to reply to Dave.
The action would certainly have been tried back in the 19th century.
This was before the notion of a diffeomorphism was developed, a key to
riddle. Clifford Will wrote a Living Review article on the modern
view, and despite being over a hundred pages and quite thorough, it
never brought up the simplest vector theory, as a resonable proposal
between the simplest scalar theory, Newton, and the simplest rank 2
theory, GR. Will recently visited MIT, and I asked him why his paper
had this omission. He said that there is more than enough data to
indicate that gravity must be a metric theory. A potential theory
will only generate half the bending of light that has been measured.
He, like Pritchard, said he was too busy to view my work. Gravity
must be explained via a metric theory, I agree on that point. An open
but overlooked question is how to implement a metric theory. All
efforts to date have relied on the Riemann curvature tensor in some
way or another. I work from a simpler starting point, the connection
as it appears naturally in a covariant derivative. It is true that a
scalar potential theory of gravity can only get the smaller than one
g_00 term correct (the g_11 term of the Schwarzschild metric is
greater than one, and a single parameter cannot do both). It would be
imprecise to presume that a 4-potential theory would be insufficient
to achieve the results seen in experiment.]

[...back to Dave]

I think you might try to talk to someone in the field who's not so famous (Seth Lloyd at MIT has been working on GR and QM at MIT).

I wish you luck, and am sorry I can't help.

Dave

David E. Pritchard 617/253-6812
Associate Director, Research Laboratory of Electronics
Cecil and Ida Green Professor of Physics
Room 26-241
MIT
Cambridge, MA 02139
617/253-4876 fax
http://www.rle.mit.edu/pritchard
Education Research: http://relate.mit.edu

Doug wrote:
> Hello Dave:
>
> I was having lunch at Mary Chung's with Sarah Smith. I was lamenting
> my plight in theoretical physics, and she suggested I drop you a note.
>
> It would be great if gravity and EM could play nicely with each other.
> That is not the case today, with general relativity standing outside,
> refusing all efforts to be quantized.
>
> Mathematica and I think we have a solution: a Lagrange density, the
> field equations generated by varying the action, physically relevant
> solutions to those field equations, the force law, a dynamic metric,
> an appreciation of the group theory perspective for the proposal, and
> two experimental tests to distinguish the approach from general
> relativity. That's a long list. My problem is that I am an amateur
> physicist, no funding from the government or industry. I know it
> takes an hour and a half to go from the classical EM Lagrangian to the
> Maxwell equations. In my approach, I toss in another current and a
> symmetric field strength tensor to do the work of gravity right next
> to EM. The vacuum field equations for gravity are every bit as linear
> as EM, and so quantization is just like EM, but with a spin 2 field so
> like charges can attract.
>
> Of course I should just publish, but a man has to know his
> limitations. I am an amateur. I do not have a Ph.D., a masters, or
> even an undergraduate degree in physics. In the 90's, I sat in on one
> graduate physics class a semester. I worked at a lab bench, and know
> how to shut up and get more data. Writing a technical paper in
> physics is not a craft I have mastered. I have attached to this email
> a pdf draft of my best effort to date. It is highly probably that I
> have a garbled line or two in the text that I am unable to spot. That
> is all that is needed for a paper to be rightly rejected.
>
> I know you are not an expert in the study of gravity. If an amateur
> and a symbolic math program can figure it out, so can you. It is a
> variation on the Maxwell equations, so it requires just as much work
> as the Maxwell equations to understand. You probably know from
> teaching that most folks can get Newton's scalar law of gravity.
> Electromagnetism is far more difficult to teach. No one bothers to
> teach general relativity to undergraduates (almost no one at least,
> Edwin Taylor is giving it a try). My work requires an amount of
> effort between Maxwell and GR, which is to say it takes considerable
> work.
>
> In some ways, the thesis is all about doing no work. That is what
> life in spacetime is like, flat as a pancake. Only there is a little
> bit of other stuff around, so everyone does the very least they can,
> which would be simple harmonic oscillation. Do a SHO in 4D, and there
> are two transverse modes for light, longitudinal and scalar modes for
> gravity. The rest is details.
>
> There are a lot of details. At APS meetings I get all of 12 minutes
> to explain five hours of differential equations. You want to write a
> novel and put it in a fortune cookie format? I have a web site, but
> who could possibly click through that much math? I am the kind of
> fellow who prefers a creative solution no matter what the cost.
> People will listen to someone telling stories, to someone teaching
> with a passion. So I decided to create a community access TV show,
> "The Stand-Up Physicist", and it has a companion web site,
> www.thestandupphysicist.com. The show serves as an outlet for me.
> These equations for unifying gravity and EM are drop-dead gorgeous.
> It all works in 4D, no ten or eleven dimension BS needed. It makes
> sense physically and mathematically. If experimentalists could
> measure the bending of light three orders of magnitude better than
> done today, they might see the 0.8 microarcseconds more bending my
> theory predicts than general relativity. It is funny and a little
> tragic that someone with my limited skill set has uncovered these gems
> (my one strong suit is creativity, and that can be measured by
> painting, piano, French pastries, swing dancing, and recumbent bike I
> designed and ride).
>
> I hope you can scan the paper and see if it looks logically coherent,
> something that cannot be faked. Like "The Old Man and the Sea", I am
> trying to land a fish that I know is too big for me to handle by
> myself. I am willing to talk about it at anytime at your convenience
> (I work for a software company in Waltham, live in Acton, and like to
> eat at Mary Chung's).
>
> Have a good day, elegance governs the heavens.
> doug

I attached a file which can be viewed either as a pdf or in HTML
http://TheWorld.com/~sweetser/quaternions/ps/em2gem.pdf
http://TheWorld.com/~sweetser/quaternions/gravity/em2gem/em2gem.html

Thanks for reading,
dougd

CarlB · Jun 19, 2006

Doug,

Could there be any chance that this new result has anything to do with an interaction between gravity and E&M that is consistent with what you are doing?

New Experimental Results on the Lower Limits of Local Lorentz Invariance
Fabio Cardone1, 2, 3, Roberto Mignani4, 5, 6 and Renato Scrimaglio1

(1) Istituto per lo Studio dei Materiali Nanostrutturati (ISNM-CNR), via dei Taurini 19, 00185 Roma, Italy
(2) Istituto di Radiologia, Facoltà di Medicina, Università di Roma “La Sapienza”, Roma, Italy
(3) I.N.D.A.M.—G.N.F.M., Sesto Fiorentino, Italy
(4) Dipartimento di Fisica “E. Amaldi”, Università di Roma “Roma Tre”, Via della Vasca Navale, 84, 00146 Roma, Italy
(5) Sezione di Roma III, INFN, Roma, Italy
(6) Dipartimento di Fisica, Università dell’Aquila, Via Vetoio, 67010 Coppito, L’Aquila, Italy

Received: 7 October 2004 Published online: 9 May 2006

An experiment aimed at detecting a DC voltage across a conductor induced by the steady magnetic field of a coil, carried out in 1998, provided a positive (although preliminary) evidence for such an effect, which might be interpreted as a breakdown of local Lorentz invariance. We repeated in 1999 the same experiment with a different experimental apparatus and a sensitivity improved by two orders of magnitude. The results obtained are discussed here in detail. They confirm the findings of the previous experiment, and show, among the others, that the effect is independent of the direction of the current. A possible interpretation of the results is given in terms of a geometric description of the gravitational and the electromagnetic interactions by means of phenomenological, energy-dependent metrics.

Foundations of Physics, Volume 36, Issue 2, Feb 2006, Pages 263 - 290, DOI 10.1007/s10701-005-9014-z, URL:
http://dx.doi.org/10.1007/s10701-005-9014-z

I still haven't forgotten about putting together a program to compare your gravitation with Einstein and Newton. But I've been too busy to get it written. When it's done, it will be written in Java. As a test to see if you can compile my variety of Java, try downloading the java source code for my Sudoku solver, which also has a link to where you can get the free Java development program from Borland:
http://www.brannenworks.com/SU/

Carl

sweetser · Jun 25, 2006

Invariant and Covariants in GEM theory

Hello Carl:

Thanks for the reference to the paper. Based only on the abstract, I was not able to justify the $30 cost for downloading the pdf, so I cannot comment on the specific content.

Here's a skeptical (in the postive sense of the word) view. In the 1800s, people thought that electric and magnetic phenomena should somehow be linked. When Oelmsted found that link, experimentalist got excited and did a barrage of work. Experimentalists are like that today. On the positive side, there was a frenzy to find ever higher temperature superconductors. On the negative side, there was a squall of work to detect a 5th force and to see the signs of cold fusion.

I have not picked up a buzz concerning a local violation of Lorentz symmetry. The experiment was first done in 1998, then apparently repeated by the same folks in 1999, and published some 5 years later. I believe the journal "Foundations of Physics" has a reputation for publishing work of questionable long term value.

What does the phrase "violate local Lorentz symmetry" mean? I'm not sure, but will discuss this symmetry as best as I understand it, and how it relates to my work.

Empty spacetime is governed by the Minkowski metric. Square the spacetime difference between any two events, and all inertial observers agree on the value of the interval (Lorentz invariance), and the all disagree about time and space measurements in a way all observers understand is related to their relativistic velocities (Lorentz covariance).

A complete answer to an observation always included both the invariant and the covariant quantities. This pairing is often forgotten. For example, people will point out that the speed of light is an invariant, and forget to mention that the frequency and wavelength of light are covariant, changing by a relativistic doppler equation. Many popular sciences sources will talk about rulers and clocks disagreeing (Lorentz covariance), without mentioning the invariant interval.

So I have an ecletic rule: I must always talk about the invariant and the covariant measurements in order to be complete. Once I have a rule, I try not to break it ever, so let's see how this goes...

In spacetime with mass in GEM theory, either one continues to use a flat spacetime metric with a dynamic potential, or one uses a dynmaic metric with a constant potential, or some combination of the two (in GR, it must exclusively be the metric that changes). There are only an infinite number of choices to be made! That's diffeomorphic symmetry for you. Let's choose to work with the boring potential, so everything is due a dynamic metric. Another way to say it is that the metric is different depending on where you are. We know how it changes. So the interval is now a covariant quantity, not an invariant one, because the interval changes in a way we understand.

That raises the question: what is the invariant quantity? I know exactly what it is, but I don't know its name! If you recall the metric that appeared in the first post of this thread, the dt term had a exponential with a negative exponent, while the dx, dy, dz all had exponentials with a positive exponent. It turns out that the products dt dx, dt dy, and dt dz are invariant under the introduction of mass into empty spacetime!

I don't know what to call dt dx, dt dy, or dt dz, so for now I'll make up a name: a 3-rope. Inertial observsers conserve the interval, and folks in a Universe with mass conserve the 3-rope. Is there any connection between an interval and a 3-rope?

This is where the story gets downright unbelievable. They have a simple, direct connection. Write an interval as a quaternion (if you are unfamiliar with quaternions,visit my website devoted to the topic, quaternions.com, to learn about the next division algebra after the real and complex numbers, which as 4 parts that can be added, subtracted, mutliplied or divided). Here is an interval written as a quaternion:
\xi=(dt, dx, dy, dz).[/itex] Now square the interval quaternion: \xi^2=(dt^2 - dx^2 - dy^2 -dz^2, 2 dt dx, 2 dt dy, 2 dt dz). In special relativity, it is the first term of this square that is invariant, while the 3-rope is covariant because we know exactly how it changes. Now toss in a mass and for the GEM theory, the first term is covariant while the 3-rope is invariant. If one wanted to be more precise, I would have to add a caveat that all the theory could claim is that there exists a choice of coordinates such that the 3-rope is invariant (I think there is also a caveat like this for special relativity, that pathological coordinates are pathological and mess nice statements up). If anyone reading this knows the offical name for the 3-rope symmetry, it would be a big help to me. If I knew the name of the beast, then I could read up on it.To get back to Carl's question, the electric and magnetic fields live inside the same irreducible field strength tensor, the antisymmetric tensor \nabla^{\mu}A^{\nu} -\nabla^{\nu}A^{\mu}. The reason there are all those interactions between E and M is because they live inside the same irreducible tensor. The fields for gravity live in a different irreducible tensor, the symmetric \nabla^{\mu}A^{\nu} +\nabla^{\nu}A^{\mu}. I identified three gravitational fields, and those should be able to mix with each other as happens for E with B. The gravity and EM fields cannot mingle so directly according to GEM theory. To be completely honest, I still am unclear about their relationship. Both are caused by the same 4-potential, something that cannot be measured directly. What can be measured is the change in the 4-potential, and that change falls into these two irreducible tensors. The bottom line at this point looks like gravity and EM should not mix together in an obvious way according to GEM theory, and the 3-rope symmetry in the context of quaternions is a fun way to wonder what is going on. doug

CarlB · Jun 25, 2006

sweetser said:

Here is an interval written as a quaternion:
\xi=(dt, dx, dy, dz).[/itex] Now square the interval quaternion: \xi^2=(dt^2 - dx^2 - dy^2 -dz^2, 2 dt dx, 2 dt dy, 2 dt dz). In special relativity, it is the first term of this square that is invariant, while the 3-rope is covariant because we know exactly how it changes.

Since my background is in elementary particles, I'd prefer that you go with the flat space and explain it again from that point of view and would appreciate that. But your equation with quaternions is interesting from a particle point of view. Let's see how you can map your way of doing bizness into the Pauli algebra. If \xi = idt + \sigma_xdx +\sigma_ydy +\sigma_zdz then \xi^2 = (-dt^2 + dx^2 + dy^2 + dz^2) + 2i\sigma_x(dt\;dx) +2i\sigma_y(dt\;dy) +2i\sigma_z(dt\;dz) and all the other cross terms cancel because of anticommutation of the Pauli sigma matrices, giving a result very similar to your own. I suppose I should multiply through by i. Now as it turns out, the above use of the Pauli algebra is what I was pushing before I decided to abandon "Euclidean relativity" in favor of submitting to Einstein's forms. This was not because it was wrong, but because I got tired of being the nail that stuck out and kept getting hammered down. I was using an algebra where the natural differential operator \nabla is defined as: \nabla = \hat{t}\partial_t + \hat{x}\partial_x + \hat{y}\partial_y + \hat{z}\partial_z + \hat{s}\partial_s where \hat{t}^2 = -1 and all the other hats square to +1, and the t hat commutes with everything while the x,y,z and s hats anticommute with each other. The "s" coordinate is for proper time in a short cyclic coordinate. If you go to massive particles, the s coordinate goes away, so ignore it. This was written up in my original paper classifying the fermions. See the comment on page 5, "For those manifolds that do not explicitly include time, an extra commuting operator (...) accounting for momentum versus position must be included and this increases the value of k by one." <a href="http://brannenworks.com/a_fer.pdf" target="_blank" class="link link--external" rel="nofollow ugc noopener">http://brannenworks.com/a_fer.pdf</a> see What this boils down to is that if you want to model spacetime and you insist that time NOT be a geometric part of the manifold (as time is considered geometric by Minkowski), then you have to have an extra degree of freedom to distinguish between traveling backwards and forwards in time. That is, vectors in space-time can give directions forwards and backwards in time. Vectors in space can not, and thus need an extra degree of freedom. This you can accomplish either by having double sets of canonical variables, i.e. keeping track of Ps and Qs in the traditional classical mechanics manner, or you can accomplish this extra degree of freedom by adding an extra "notation" coordinate to your geometry. As a notation coordinate, it must commute with everything else. There are also some simple ways of embedding the quaternions into the Dirac algebra that might be less unsettling. The key is that you have to arrange for the time component to commute with the spatial components, while the spatial components anticommute. But every example of this will be analogous to the above Pauli example. That is, the algebra is completely defined by these commutation rules, along with the rules telling what the squares need to be. Carl

Don J · Jun 27, 2006

CarlB said:

Since my background is in elementary particles, I'd prefer that you go with the flat space and explain it again from that point of view and would appreciate that.

Hello Carl
I think this paper can complete your research ...because you work in that field access to this rare paper should be granted to you.
Your opinion will be appreciated
http://prola.aps.org/abstract/PR/v105/i2/p735_1

sweetser · Jun 28, 2006

Dirac algebra and quaternions

Hello Carl:

> Since my background is in elementary particles, I'd prefer that you go with the flat space and explain it again from that point of view and would appreciate that.

No problem, so long as it is clear the ability to choose between a metric theory like GR, and a potential theory like Newton's approach is at the core of what is "new" about GEM. The quotes are required, because all I am doing is exploiting a symmetry property of a covariant derivative, which has been around for a while.Quaternions have been discovered and rediscovered many times. The first to find them was Gauss, because Gauss discovered everything. Only much later did they realize this bit of one of his notebooks was an uncredited invention. Hamilton discovered them while trying to come up with a rule for multiplying triplets (can be done, but division does not work). Rodriguez discovered them while trying to do 3D rotations, their one wide use today.

The Pauli algebra is almost the quaternions. For doing calculations in physics, they make things a little easier, because there is an extra factor of i. It is simple to set up the Lorentz group using Pauli matrices. It is not simple to do with real quaternions. De Leo first did it in 1996 or so. Peter Jack was the first person to write the Maxwell equations using only real quaternions, and I repeated the trick independently a year later.

There is a cost to convenience, and it is very subtle. With the Pauli algebra, one can multiply two non-zero numbers and get zero. How many pairs of these are there? About an infinite amount, plus or minus 42. In some ways, these infinite zeroes don't do anything but complicate statistics. It is my belief, without evidence, that these bogus zeroes could be the reason one has to do regularization and renormalization in quantum field theory. This is 100% speculation. I could prove the point by doing a calculation using quaternions exclusively (no gamma matrices), and demonstrating that regularization and renormalization were not needed to make the calculation behave. I am quite certain I could not do such a calculation on my own, only able to act like a adviser for people skilled in the arts of quantum field theory. I have no expectation that this speculation with be confirmed or rejected in my lifetime. It is a favorite speculation of mine though :-)

I am a big fan of time! The reason is that time lies straight down the diagonal of a quaternion. With all of our incredibly tiny relativistic velocities, it says that we change hardly at all in space, that nearly all of the change we experience is through time.

It is one of those funny "size" issues: most people will say that the Dirac algebra is bigger since quaternions fit inside the Dirac algebra, along with all those bogus zeros. I claim that being a division algebra will allow you to do more, and it is the size of what can be done that matters most.

What is the biggest, most important algebra? The one the user thinks they can do the most in :-) In my case that would be the quaternions, in yours, a variation on the Dirac algebra. I actually have a limitation with quaternions, it is the pea under a mountain of mattresses for me. I have no idea how to handle the connection. It is also clear that far too few folks work with quaternions, so to communicate, I had to translate all the work I did initially in quaternions into tensors.For anyone interested in seeing good old technical conflicts, I posted something in slashdot.com about my work, and got in a classic nasty discussion (well, he was the one to toss insults, and I did learn something, that a simple model of mine was too simple. Apres the name calling part, he is back to the banal defense of the status quo).

HTTP://science.slashdot.org/comment...mmentsort=3&mode=thread&pid=15604834#15605147

Don J · Jun 28, 2006

sweetser said:

For anyone interested in seeing good old technical conflicts, I posted something in slashdot.com about my work, and got in a classic nasty discussion (well, he was the one to toss insults, and I did learn something, that a simple model of mine was too simple. Apres the name calling part, he is back to the banal defense of the status quo).

HTTP://science.slashdot.org/comment...mmentsort=3&mode=thread&pid=15604834#15605147

A good discussion netheless without too much math formulas...only pur discussion about the subject at hand...glad that you finally try to challenge your theory outside the relatively confortable independant research section of this forum...its hard to play in the major league Isn't it?
Maybe its time to try
http://www.bautforum.com/forumdisplay.php?f=18

Warning: -avoid presenting your theory in the against the mainstream section where the discussion can turn harsh about your ideas -

sweetser · Jun 29, 2006

I consider slashdot and this forum to be the random league. Someone random person comes in with a comment, which I usually have to get them to repeat a few times, and eventually I will see their point. I've learned three things from the first 12 pages of this thread:
1. the field strength tensor indices should both be up, \partial^{\mu}A^{\nu},
2. to get the Lorentz force from the action I need to include the inertial term in the Lagrangian, -\rho/\gamma,
3. the GEM action breaks U(1) symmetry for massive particles.

In the slashdot discussion, I learned that my simple argument for a transverse mode of emission is too simple. I have to rely on how the field is quantized to justify the gravity waves are longitudinal or scalar modes of emission.

Harsh? I don't care about that. I make a note of it, and focus on the technical content.

Let me list the initials of the major leaguers I have dealt with, and by that I mean professors who are good enough to have their research funded: AG, SD, and CW. There were so busy, I spent 45 minutes with #1, and ten minutes with the other two, which is not time to present the idea. Again, this is an observation. The pitch to Seth Lloyd is like triple A league. He might get to reading the paper.

I am going to outsource the critique. I know someone who was a physicist in Russia, so I'll try and get feedback from the other side of the globe. I also have a friend in India who is works in at an institute, and will try to work that too.

This forum has helped me in measurable ways, so thanks to all who have participated so far.

doug

CarlB · Jun 29, 2006

Doug,

The factorizations of zero in the Pauli algebra are of the form (1\pm \sigma_u)/2. To get equivalent factorization in the quaternions requires that we add an imaginary unit to it, as in:
(1 + k\sqrt{-1}) \; (1 - k\sqrt{-1}) = 0

So is the Pauli algebra equal to a complexified copy of the quaternions?

Carl

sweetser · Jun 29, 2006

bingo, bingo.

Here is my ontological problem with that approach. The quaternion (a, b, 0, 0) behaves with other quaternions of this form EXACTLY like a complex number. So it would be fair to call this a+bi. The quaternion (a, 0, c, 0) also behaves EXACTLY like a complex number. Then there is (a, 0, 0, d), which also could be written a+di, and nothing about how it plays with (e, 0, 0, f) would be different from complex numbers. Now when we call a quaternion a+bi+cj+dk, and have 3 distinct imaginary numbers, it looks like cheating to add in a forth i that behaves like the first one used to. One could claim that as long as we make the rules, it is a fine and very productive thing to do for a mathematician. Thing is, I don't care about mathematicians, I care about Nature, and she is more demanding of silent perfection in how thing work (if they don't work, they die, so that is as demanding as it gets). I understand that there are gaggles of humans that are skilled at manipulating complexified quaternions. It is my belief that Nature does not use that algebra. An odd belief, but quaternions.com demonstrates my practice.

doug

sweetser · Jul 5, 2006

Black holes (not)

Hello:

In a different forum, someone asked me about black holes and this proposal. For politeness sake, if you have any questions related to GEM theory, please do so right here.

The short answer is that I know the math behind very dense gravitational sources is going to be fundamentally different from what appears in general relativity, but I do not understand the details (and the details matter). The differences will be so significant that current efforts to understand black holes will have to be dismissed should GEM theory succeed both on experimental and theoretical grounds.

I have two ways so far of finding physically relevant applications of the GEM
theory. The first is the exponential metric that solves the field equations for a single spherical source with the choice of a constant potential. The second is a potential that in an approximation has a derivative with a 1/R^2 dependence. The potential gets plugged into a Lorentz force law to get to te same exponential metric expression (odd but true). In these two approaches, there is a point in the derivation where I say that the change in time is tiny compared to the changes in distance, so assume the small time contribution approximation. That is how one gets to the kinds of solutions we see often in Nature: changes in space dominate the near vacuum of a Universe we live in 13.6 billion years apres the big bang, with enough of a nod to changes in time that special relativity is respected.

Now consider the case where the changes in time are as significant as those that happen in space, a condition which may appear for very dense gravitational sources. The potential whose derivative is 1/R^2 via an application of perturbation theory will no longer be applicable. The potential will have a derivative that is 1/R^3, and then have a force that had the same inverse cubic dependence on distance. How odd! I have heard it said that an inverse cubic law is unphysical. That is true for a classical law of gravity. The math may give a different story for dense sources.

I recall from classes on differential equations taken decades ago that to say an equation was singular had a precise technical meaning. Nearly all researchers consider the point singularity that appears in general relativity as something worth working on, not a thin ice warning that could open up and drown a large body of work. There is a singularity for my field equations, but it is not a point singularity. Instead the equation blows up lightlike intervals, when tau^2=0. That may turn out to be a better deal, because we know there are particles like the

photon and graviton that live on the lightlike surface.

Should the GEM hypothesis get a following, the behavior of singular solutions will be a fun area of study.

doug

A_Wanderer · Jul 5, 2006

Doug Sweetser type-person:

This is not in the nature of a reply, yet now and then a question is worth a thousand words - did Godel and his progeny ever have anything discrete to say about completeness:consistency as related to GUT and so your little GEM?

sweetser · Jul 5, 2006

Logical consistency

Hello Wanderer:

My first introduction to Godel was through the book "Godel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter. That inspired me to buy "Godel's Proof" by Nagel and Newman, a book that got to the point in 102 pages instead of 777. I have reread that book more times than any other on my bookshelf, so I have a feel for the technical detail. There is no technical connection whatsoever between GEM (or as far as fields, G, E, B) and Godel, Escher, Bach, other than a happy accident.

I tried a logical consistency argument on Hofstadter himself. Darn, I wish I could find my notes from his talk he gave at Harvard, but I recall it was an overview of physics history. His point may have been - my memory is fuzzy - about how physics has gone for consistency over being radical. Work that goes against consistency loses.

I got all excited by the thesis, and wrote out a question. We have a logically consistent theory for EM, that would be the Maxwell equations. We have a logically consistent theory for gravity, that would be general relativity. These two theories are in conflict with each other. So we have to choose between the two, either the Maxwell equations or general relativity, and if we choose one, the other will become a historical footnote, and is wrong. Given the connection between Maxwell, quantization, and the standard model, I would say the weight of history is for the consistency of Maxwell equations over general relativity. Granted we don't know which one is correct, but given his sense of the history of physics, which theory does he think may be shown to be most consistent in the future.

A little bit about the setting: it was a packed large lecture hall at Harvard because Hofstadter has some media draw what with the Pulitzer Prize for the book most physicists have read and enjoyed. The big names in physics in Boston were in the house. There were plenty of young people, and a few really old ones who you know probably figured out some very important things in physics or math in their prime. I was excited by my question, because it would press Hofstadter on his own thesis to say something edgy: either Maxwell's field equations were flawed, or Einstein's field equations were flawed. Logic is that tough.

After his talk was done, I got to asked my question first. I recall feeling the question was audacious: I was uncomfortable with saying in public there was a choice, and with that choice, one would say Maxwell remains right, GR is wrong, or GR remains right, and the Maxwell equations are wrong. I was that clear in the confrontation.

Hofstadter played defense. He noted he as a historian of science, and was not a researcher. He was unwilling to speculate on any future direction for physics research. As far as I can tell, no one in the audience found my question of interest. I know someone else in the audience who heard the question did not think the choice between Maxwell and GR made sense (he was a string theorist).

Einstein and Godel both worked at the Institute for Advanced Studies at Princeton. They went on walks together, discussing physics. This was late in Einstein's career, so all he was working on was unified field theory and the logical foundations of quantum mechanics. I know they collaborated on a paper together, one on closed loop solutions to the Einstein field equations. I don't know more of the history than that.

GEM is not grand, it is extra ordinary.

doug

CarlB · Jul 5, 2006

sweetser said:

The short answer is that I know the math behind very dense gravitational sources is going to be fundamentally different from what appears in general relativity, but I do not understand the details (and the details matter). The differences will be so significant that current efforts to understand black holes will have to be dismissed should GEM theory succeed both on experimental and theoretical grounds.

Doug, I've got my Java applet for gravitation simulation almost done. It now produces correct Newtonian orbits and the vast majority of the user interface works perfectly. I'm working on the equations of motion for the general relativity version, which will assume a flat space coordinate system as done by Lasenby, Gull and Doran as in:
http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/grav_gauge.html

And what does this have to do with you? Because the Lasenby Doran and Gull gauge version of gravity works on flat space, it is very natural to compare with the Newtonian version. And I'm guessing that since your theory also works on flat space it will also be easy to simulate. And therefore I've also got a spot in the simulation for the Sweetser version of flat space gravity.

What I need from you, if and when you've got it, is a flat space version of force (i.e. change in velocity per unit time) as a function of position and velocity (i.e. phase space). To get you started, here is the Java code for the Newton theory. (All units are natural.)

private DeltaV Newton(PhaseSpace PS) {
DeltaV DV = new DeltaV();
double R = Math.sqrt(PS.X*PS.X+PS.Y*PS.Y);
R = R*R*R;
DV.DVX = -PS.X/R;
DV.DVY = -PS.Y/R;
return DV;
}

The above routine computes "DV" or change in velocity, from "PS" or position in phase space. The calculation is made in two dimensions (i.e. the z direction is ignored and all orbits stay in the same plane). The change in velocity therefore has two components, (DVX,DVY), and the phase space has four components, (X,Y,Vx,Vy).

For the Newtonian calculation, the force does not depend on the velocity, so there is no use of things like "PS.Vx" or "PS.Vy".

Anyway, the orbit is found by integrating the acceleration numerically. The applet allows you to change the initial condition and see how the particle orbits for the different physics assumptions differ.

I've written it so that the physics is encapsulated in a simple subroutine as shown above so that the Java will be easily modified by someone who doesn't want to learn the ins and outs of object oriented programming of a user interface.

Now when this is done, I'll put the applet, along with its source code, up at my website //http:www.GaugeGravity.com[/URL] , which is intended to promote the Lasenby, Gull and Doran version of GR. I've put my current (test only) version up on the net here:
[PLAIN]http://www.gaugegravity.com/testapplet/SweetGravity.html

The buttons Einstein, Newton and Sweetser don't work, and I need to readjust it so that it runs faster, which you can accomplish (with some loss in numerical precision) by hitting the "faster" button a couple times. The numerical precision isn't adjusted according to how close you are to the gravitating point, so it can get a bit dodgy on near misses. And when you alter the initial parameters (with a carriage return), it stops the simulation and to start it again you have to press "go". I'll make the stop/go button be red / green so it is more obvious what is going on.

By the way, to translate from LGD's version of the Schwartzschild metric back to the usual GR theory, the reason that they work on a flat space (i.e. coordinates are x,y, z and t), is that their ds^2 is not diagonal, but instead includes a drdt term. Thus their solution is not symmetric with respect to time, which I think is a great thing.

Carl

A_Wanderer · Jul 6, 2006

sweetser said:

Hello Wanderer:

My first introduction to Godel was through the book "Godel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter. That inspired me to buy "Godel's Proof" by Nagel and Newman, a book that got to the point in 102 pages instead of 777.

sweetser said:

Hi.

Thud. Seven hundred and 77 pages of a POPular derivation of a technically and imaginitively profound mathematics. You have more patience and tolerance than I could muster in a hinayana trance! Thank heavens you found the Nagel/Newman. Better yet would be to first extract Godel from that framework and wrestle with it unencumbered by focal bias.

What I was suggesting was: Does not Godel's grand reductio establish that any GUT which expression is more complete and and so (inversely?) LESS consistant (in comparison to any theory on an opposing gradient) shall be closer to the truest description/expression/model of the measurable universe (all its forces from the Plancky, pointy ultrino to the BB and any and all strange, possibly stringy, curved, n-dimensional, & etc. in between or amidst) - given that the impetus for GUT from the start IS unity/completeness? Despite, I know a 2500+ year historical record with a rather strong attraction to the beautiful elegance of consistancy.

It is most counterintuitive that inconsistancy then becomes a marker for fundamental veracity/agreement the more complete the GUT (or for that matter the GEM) expression. Yet this surprising in-your-face inconsistancy, oddly, or evenly, may be a consistant balance of inconsistancies when placed next to wave/particle and the where/when of quantum subatomics.

I briefly apologize for being rather off-topic - Sorry!
But sometimes, I can just not resist, and have a weakness for the gravity of my own conceits. And groaners.

I dislike my own speculations. (Clearly NOT theories, nor even hypotheses.) I am very fond of consistancy.

the Big Bang - Everything from nothing! Is this not perfect elegance?

fondly,

Bill Snyder
aka
A_Wanderer

sweetser · Jul 7, 2006

Java applet

Carl:

Wow!

Sorry, just a gut reaction. It will take a few days to think about this, but thought I should put on record my initial reation.

Another reflex reaction...

>A flat space version of force (i.e. change in velocity per unit time) as a function of position and velocity (i.e. phase space).

Spacetime can be treated as flat. Force cannot be treated as m dv/dt. Instead, use the chain rule Luke.
F = m dv/dt + v c dm/dR = - G M m/R^2
If you get GR correct, you should end up with an animation that is identical.

Damn, need to get ready for work...

Kudos, kudos,
doug

CarlB · Jul 9, 2006

sweetser said:

Kudos, kudos, doug

Yes, the Cambridge geometric algebra guys are awesome.

I've uploaded the next version of the applet. This one draws multiple test bodies, which makes for a more pretty display. I've set the initial conditions so as to give a demonstration of the conservation of angular momentum in the Newtonian potential:

http://www.gaugegravity.com/testapplet/SweetGravity.html

I guess it's possible that I'll add the GR simulation within the next 24 hours, and if so, I'll update this note accordingly.

Carl

Norman Albers · Jul 9, 2006

I am running hard to catch up with you and am almost there. Forgive me for not yet having read this entire disucssion; I shall, but need to start communicating ideas with you. We need to let go of the zero current and charge densities. I have shown minimal, clear construction of electrons and photons by allowing infinitesimal divergence in the field. Now I am studying GR (last chapter in Adler, Bazin, Schiffer) to find the lay of the land. We must allow the vacuum to cook up this inhomogeneous stuff! I have found fundamental problems, in that my photon fields can express a Lagrangian with three different orders of time derivatives, not useful.

sweetser · Jul 10, 2006

Godel and speculations in physics

Hello Bill:

What I was suggesting was: Does not Godel's grand reductio establish that any GUT which expression is more complete and and so (inversely?) LESS consistant (in comparison to any theory on an opposing gradient) shall be closer to the truest description/expression/model of the measurable universe (all its forces from the Plancky, pointy ultrino to the BB and any and all strange, possibly stringy, curved, n-dimensional, & etc. in between or amidst) - given that the impetus for GUT from the start IS unity/completeness?

I am unfamiliar with any way of taking Godel's incompleteness theorem and translating that into either a field equation or an action. As far as I know, the theorem remains in the house of logic.

Did you know that logic can be translated directly into algebra? Scientific American had an article on this a number of years ago. The translation is particularly relevant for fuzzy logic. Here's the cannonical example. There is a card which says on one side: "The statement on the other side of this card is true." OK, call that x. On the other side, the card reads: "The statement on the other side of this card is false." That would then be 1-x. So what is x?
x=1-x
2x=1
x=1/2
Math I can solve :-) The card is full of half truthes. The point of this story is that it may become possible to translate something at the foundations of logic into algebra someday. What would be required is a way to translate numbers into differential equations. I hope that sounds like BS, because I am clueless as to how to do that.

...next to wave/particle and the where/when of quantum subatomics.

I'd have to start a new thread on this forum to treat this right, but my one liner reply is that if you do 4D calculus correctly, the reason causality is different between classical physics and quantum mechanics is clear. Check out the video "Why Quantum Mechanics is Weird" at TheStandUpPhysicist.com.

I dislike my own speculations. (Clearly NOT theories, nor even hypotheses.) I am very fond of consistancy.

My level of discomfort is dictated by how much of it can be translated directly into math. I have wondered about Godel, and am totally uncomfortable with it since zero can be translated into algebra. I am very comfortable saying I have a proposal to achieve unification of gravity and EM because it is so specific, I can feed it directly into Mathematica, a fact that so far has failed to make an impression with professional physicists (a busy lot).

I do have speculations in between. Let me give you the details of one of them. The standard model has the gauge symmetry U(1)xSU(2)xSU(3). There are obvious questions to ask: why three groups, why these in particular? The answer we have is clear: we have no idea. This is the kind of problem that is beyond hard. There is nothing you can do to "work" on it. All you can do is be aware of the clarity of our ignorance about the standard model.

What kind of symmetry characterizes the 4D wave equation at the heart of my unified field proposal? Because my background involved much work with quaternions, one of the things I know is that the group SU(2) are the unit quaternions. How does one make a unit quaterion? Easy, just make sure the diagonal is zero, A-A^*. The unit quaternion has three of the four degrees of freedom available to a quaternion. What should the fourth one do? The group U(1) is usually represented as the complex numbers with a norm of one. The group is Abelian, that is to say it commutes with other members of the group. Quaternions are well know as not being Abelian. Under special circumstances, quaternions can behave like an Abelian group. The prime example is if all the quaternions point in the same darn direction. I realized a normalized quaternion would commute with its SU(2) sidekick, like so:
\frac{A}{|A|} (A-A^{*}) = (A-A^{*}) \frac{A}{|A|}
This is the electroweak symmetry. If I write my GEM field equations like so,
J_q - J_m = \square^* \square \frac{A}{|A|} (A-A^*)
this has U(1)xSU(2) symmetry! Now the unified field theory contains gravity through the diffeomorphism symmetry, EM through the U(1) symmetry which is not perfect due to massive particles, and the SU(2) symmetry of the weak force which is behind nuclear radiation.

Now we come to the part that I do not understand: SU(3) symmetry. I know its Lie algebra has to have 8 parts to it. The multiplication table has to be different that the standard quaternion multiplication table. It is possible that the \square^{*} \square part of the GEM field equations have what it takes. There are 8 parts, that is easy. What I (and only I) call the Euclidean product, a^{*} b, is not assiociative, because (a b)^{*} c \not= a^{*} (b c). The Hamilton product, a b is associative because (a b) c = a (b c). Quaternions under the Euclidean product still are a group: there is an identiy (1, 0, 0, 0), there is always an inverse, and all products are still quaternions. I am not enough of a professional math guy to prove SU(3) connection. The one course I failed, got an outright E, was a Harvard class on group theory and particle physics. As for the E, it is so a person failing at Harvard remains a cut above the rest.

Now both questions about the standard model have the chance of being answered. The symmetry of the GEM unified field equations is Diff(M)xU(1)xSU(2)xSU(3). The four forces of Nature, gravity (Diff(M)), EM (U(1)), the weak force (SU(2)) and the strong force (SU(3)) all fit comfortably in the same home. This is the sort of specific speculation that keeps me crazy.

doug

sweetser · Jul 10, 2006

Studying GR, studying GEM

Hello Norman:

Sean Carroll's lecture notes on General Relativity are freely available on the web. I printed them out, and for each chapter, transcribed them into my own form. That process took two months, and now I think I have a concrete handle on the math behind GR.

This forum is more a call and response venue: a question gets asked, I reply, and we go back and forth a few times. It is not organized in a logical way :-) For that, I recommend either downloading the half hour shows at TheStandUpPhysicist.com, or ordering the DVD (1 sale in the world so far, to a physicist friend). Although each show is only a half hour, it would take more time if you tried to confirm that the math makes sense.

doug

CarlB · Jul 10, 2006

sweetser said:

Now both questions about the standard model have the chance of being answered. The symmetry of the GEM unified field equations is Diff(M)xU(1)xSU(2)xSU(3). The four forces of Nature, gravity (Diff(M)), EM (U(1)), the weak force (SU(2)) and the strong force (SU(3)) all fit comfortably in the same home. This is the sort of specific speculation that keeps me crazy.

I think that trying to get SU(3) in there is just one symmetry too far.

You do not have an explanation for why there are exactly three generations of particles. These kinds of things, that is, the representations of these symmetries that happen to correspond to the elementary particles, are at least as important as the symmetries themselves.

But if you assume that the leptons and quarks are composites of three particles, then the SU(3) becomes natural at the same time as you get three generations automtically. And the masses of the leptons become understandable by the Koide relation.

So long as your algebra contains the 3x3 complex matrices, it is of course possible to force SU(3) into it. But this does not imply anything interesting. One could fit any of a large number of symmetries into 3x3 complex matrices. To show a real derivation, you would need to not only pick out SU(3) and all that, but you also need to show why the particular representations that are seen in the elementary particles show up.

As an example, consider the electroweak symmetry. It is not enough to show SU(2) x U(1). What you nee to show is that one ends up with an SU(2) doublet and two SU(2) singlets for each flavor.

If you take a look at the algebra, you will find that this form, a doublet and two singlets, arises naturally. But the same cannot be said of SU(3). SU(3) arises much more naturally as the result of assuming that the elementary particles are composite.

But hey, if you want to put SU(3)xSU(2)xU(1) into complex 4x4 matrices (as is more natural for a Clifford algebra or quaternions, I suspect), then you should read this paper:
http://arxiv.org/abs/math.GM/0307165

Carl

bda · Jul 11, 2006

Hi Doug,

Pardon me for hopping in like this but I thought you could possibly use one little piece of information about the exponential metric. You may or may not know that the exponential metric seems to have been first proposed by Houssein Yilmaz in "H. Yilmaz, New approach to general relativity, Phys. Rev. 111(5), pp. 1417-1426 (1958)". You can do a search on "Yilmaz metric" and a lot of references come up.

In its Euclidean form the exponential metric has been used by Hans Montanus and myself; among other references see for instance: "J.M.C. Montanus, Proper-time formulation of relativistic dynamics, Found. Phys. 31(9), pp. 1357-1400 (2001)" and "J.B. Almeida, Geometric drive of the Universe's expansion, http://www.arxiv.org/abs/physics/0507102 ."
Jose

bda · Jul 11, 2006

CarlB said:

But hey, if you want to put SU(3)xSU(2)xU(1) into complex 4x4 matrices (as is more natural for a Clifford algebra or quaternions, I suspect), then you should read this paper:
http://arxiv.org/abs/math.GM/0307165

Carl

As author of that paper may I suggest also http://www.arxiv.org/abs/quant-ph/0606123

Jose

sweetser · Jul 11, 2006

Exponential metric

Hello Jose:

The exponential metric is elegant, in an almost measurable way. The number of strokes of the pen required to write it down is small. If the exponent goes to zero, the terms run to one. Exponentials appear over and over again in critical physics equations. There is a good reason why. The exponential is a small step away from unity that embodies a simple harmonic oscillator.

I am not the best student of the literature, but I do have a copy of the Yilmaz paper. I was able to find 4 papers that had the exponential metric:

N. Rosen. (note: this appears like the first reference)
A bi-metric theory of gravitation.
General Relativity Gravitation, 4(6):435-447, 1973.

H. Yilmaz.
Physical foundations of the new theory of gravitation.
Annals Phys., 101:413-432, 1976.

S. Kaniel and Y. Itin.
Gravity on parallelizable manifolds.
113 B(3):393-400, 1998.

Keith Watt and Charles W. Misner.
Relativistic scalar gravity: A laboratory for numerical relativity.
1999.

The way these four papers generated that metric were not elegant. Misner says he is doing this just so numerical calculations go faster, an argument of convenience over conviction. Rosen tosses in another metric field, and because that can store energy and momentum, it means that for an isolated source, gravity waves can have a dipole mode of emission, which disagrees with experiment. I don't recall the details of the other two, other than I did not get excited by said details. I believe they were variations off of the rank 2 field theory of GR, whereas I am trying to show a rank 1 field theory is a different way to implement a metric theory for gravity via symmetry.

The field equations I have were called beautiful by Feynman (in reference to the EM equations in the Lorentz gauge, not as a unified field theory). The equation that generates the exponential metric is also elegant in its directness - it is the divergence of the connection:
 \rho_{m}=2\partial_{\mu}\Gamma_{\nu}{}^{\:0\mu}A^{\nu} 
I remain stuck in 4D where few people do gravity research. Retro man!

doug

sweetser · Jul 12, 2006

Alterations for Newton's classical force law

Hello Carl:

To make a direct connect to the physics literature, we need a different kind of applet. The issue is to generate a velocity profile for a disk galaxy given the mass distribution function as a function of the radius. Let me do this in crude ASCII graphics. We start with a mass/area function that has an exponential decay:

m/area
|.
|.
| .
| ...
-------...
R

From that we calculate the velocity profile:

vel. calculated
| .
| . .
|. .
|.
-------------
R

This has the "Keplarian decline", which is what Newton's theory should generate if the applet is written correctly. Test out GR in the flat spacetime if you like, overlay them, and a difference should not be visable at this resolution. What is seen in Nature is this:

vel. observed
| . . .
| .
|.
|.
-------------
R

When programming, there are so many ways to do things, it helps to be specific. Let's use the mass distribution profile for the galaxy NGC3198. It is a galaxy that is too faint for you or I to ever see with our eyes, no matter what size telescope we used (something like a magnitude of 23). The velocity ramps up to 150,000 m/s and stays there. The total mass is 1x10^{40} kg. The mass per area as a function of the radius is m/area = 37 Exp(-R'/2.23') solar masses/pc^2.

The velocity profile is not the only problem with Newton's law. The solution is unstable, so disk galaxies should collapse, but they don't. That's unreasonable, because galaxies last a long time.

Let's consider the forces: it is gravity versus the centrifugal acceleration:
m \frac{V(R)^2}{R} - \frac{G m M}{R^2} = \frac{d m V}{d t}
To see the proverbial Keplarian decline, the centrifugal forces and gravity are in balance, so solve for V:
V=\sqrt{\frac{G M}{R}}
If the mass drops off as the square root of 1/R, the velocity can stay constant. The observed light curve instead makes it look like the mass drops off exponentially, much faster than the square root of 1/R.

Algebraically, there are three things that can be done. The first is to "Stuff the Mass box", which goes under the name of the dark matter hypothesis. Folks cannot figure out what dark matter is, but they do know we need more of it than the stuff we know huge amounts about. We cannot see the stuff directly yet despite the extraordinary care astronomers use to analyze light. Since we cannot see it, and don't know what it is made of, folks at computer terminals make up a dark matter distribution that can generate the velocity profile and lead to a stable visible mass distribution. I hope this area of study sounds suspicious to you.

The second approach is "Switch the Equation". It goes under the name of MOND, or Modification Of Newtonian Dynamics. It is claimed that when gravity gets super wimpy, then it becomes a 1/R force law:
\frac{V(R)^2}{R} - \frac{G m M}{R^2} = \frac{d m V}{d t} iff \frac{G M}{R^2} > 10^{-10}
\frac{V(R)^2}{R} - m \sqrt{\frac{a_0 G M}{R^2}} = \frac{d m V}{d t} iff \frac{G M}{R^2} < 10^{-10}, a_0 = 10^{-10}
MOND does a good job with real data. That is is strength. The theory is a weakness. Recently someone figured out the Lagrangian required to get this sort of thing to work out, and I heard it apparently is not a pretty site. Suspicious? You should be.

And the third possibility is "The relativistic chain rule for a distributed mass source". That probably is not familiar. Well, the chain rule should be. What does a force do? If you say it is mA, technically you are wrong. Force is a change in momentum, so F = m\frac{d V}{d t}+ V \frac{d m}{d t}. The second term is literally the stuff of rocket science. For a spiral galaxy that doesn't change its mass over eons, that term can safely be assumed to contribute nothing.

Technically, my expression for force was also wrong. The relativistic force is F^{\mu}=\frac{d m V^{\mu}}{d \tau}. The important thing to focus on is the d \tau. That Greek letter is for a change with respect to the spacetime interval, not exclusively the time interval. This means that formally, it could be a change in space that the rocket term describes. The classical force would then be:
\frac{V(R)^2}{R} - \frac{G m M}{R^2} = m \frac{d V}{d t}+ V \frac{d m}{d R/c}
It is like the rocket force term, but applies to space. There is not a label for it, so I'll call it the rocket-space term, a flip of space rocket because this is a flippy idea. What a rocket-space term says can be described. What does a force do? It is a change in momentum. There is the familiar sort of change in momentum, when something changes its velocity. Well, in the outer reaches of a spiral galaxy, there is ZERO change in velocity, even though there still is a gravitational force changing momentum. That's a real puzzle. The force must be changing something else: where mass is in space. That is exactly the kind of curve I wrote earlier - the amount of mass per area drops exponentially in th outer reaches. The change in momentum as one moves away from the center is seen as the change in mass times a constant velocity. In words, it is an exact match.

This is really a new idea. That is rare in physics. It is worth a try, so see if it is consistent with data from a specific galaxy. I should say I have a way to derive this expression, but it takes longer to do so.

****
So this is what I am thinking about programming-wise. Presume a mass/area function of m/area = 37 Exp(-R'/2.23') solar masses/pc^2. Use the Newtonian equation to calculate the velocity for R over a range of say .1 to 30 arcseconds. Plot velocity versus R. I know it is easier to skip the units, but try not to as a check that the max velocity is in fact 150,000 m/s, and the total mass is 10^{40}. Newton's theory gets the peak right. Then include the rocket-space term and see what happens to the curve.

doug

bda · Jul 12, 2006

Hi Doug

sweetser said:

Hello Carl:

And the third possibility is "The relativistic chain rule for a distributed mass source".
...

This is really a new idea. That is rare in physics. It is worth a try, so see if it is consistent with data from a specific galaxy. I should say I have a way to derive this expression, but it takes longer to do so.

Actually I wrote about this a few years ago with someone who has since passed away. We never agreed that the paper was fit for even placing in arxiv, so it remaind in my hard disk until now.

Note that I had not yet got the exponential metric quite right at that time, so now the paper would have to be revised in that aspect; this should not produce significant alterations to the main conclusions.

Jose

bda · Jul 12, 2006

Doug,

bda said:

Note that I had not yet got the exponential metric quite right at that time, so now the paper would have to be revised in that aspect; this should not produce significant alterations to the main conclusions.

Jose

I had a quick look at the paper and I now realize there are a few mistakes, none of them serious and all easy to correct. The main difference after correction will consist on doubling the exponent in Eq. (7), but that will have no consequences for the discussion and conclusions.

Jose

sweetser · Jul 12, 2006

Hello Jose:

The exponential metric does not lead obviously to the space-rocket term. There is an error of omission in this express:
\frac{V(R)^2}{R} - \frac{G m M}{R^2} = m \frac{d V}{d t}+ V \frac{d m}{d R/c}
Force is a vector equation. One must include them. You'll notice that the space-rocket term points along \vec{V}, not along \vec{R}! The correct vector expression is:
\frac{V(R)^2}{R}\hat{R} - \frac{G m M}{R^2}(\hat{R}+\hat{V}) = m \frac{d V}{d t}\hat{R}+ V \frac{d m}{d R/c}\hat{V}
The rocket-space term suggests gravity works classically in a new direction. This is ONLY relevant for masses that are distributed over a significant amount of space. It is the passive mass small m whose distribution is changed. Of course the sum of the passive small mass is the active mass M. If you feel it is a bit confusing to have the mass in different part of the same equation, that is the way rocket science works!

doug

CarlB · Jul 12, 2006

sweetser said:

V = \sqrt{GM/R}
If the mass drops off as the square root of 1/R, the velocity can stay constant.

Please forgive a naive amateur, but to get the velocity constant, don't you have to increase the mass proportional to R^2, that is the mass per unit area has to be constant? [Edit: Okay, now I see it. Integral of mass has to be proportional to R, but mass is proportional to area. Nevermind.]

Were I asked to do the simulation you're talking about, I would do it with a large number (maybe 1000) sample points. It would run amazingly slowly in Java. Amazingly slowly. But you could eventually get a result out of it. That's a second stage operation. For now, you still need to tell me what the dv/dt equation is for just a test mass orbiting around a point mass. It turns out that extracting this information from flat space gravity theorists is harder than I expected.

I've now added the logic to throw three types of balls in the air, Newton and two others. I believe I have the correct equations for general relativity, and for the Cambridge geometric algebra relativity, and I should upload the applet later tonight. I've also changed the colors, and added bright white test masses so that you can see them retracing their orbits.

Carl

bda · Jul 13, 2006

Doug

sweetser said:

Hello Jose:

The exponential metric does not lead obviously to the space-rocket term.

I agree; no matter which metric one uses the main thing is that velocity is basically determined by

\frac{v^2}{r} = -\frac{\mathrm{d}V}{\mathrm{d} r},

where V is the gravitational potential; the metric introduces a correction which is significant only for very small r and large M.
You then plug in V =G M(r)/r and apply the the derivation rules
this then results in

v^2 = \frac{G }{r}}\left(M - r \frac{\mathrm{d} M} {\mathrm{d} r} \right);

it is then obvious that one can get constant velocity.

In my paper I apply this to measured velocity profiles of real galaxies and derive the respective mass distributions; I then compare those with observed light intensity and H1 profiles.

Jose

Unifying Gravity and EM

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