An Exceptionally Technical Discussion of AESToE

[MTd2]

Sorry... But I don't get where you are trying to go.

 

[Lawrence B. Crowell]

Sorry... But I don't get where you are trying to go.

I have this idea cooking in my head, so I am bouncing it off here. It concerns a general approach to framing fields. This is just the preliminary parts here, and I am slamming out some of the notation --- I hope I get the indices etc more or less right.

A differential form and its dual vector $\vec v$, $\underline f$ is seen in the product

$$\vec{v} \underline{f}~=~v^i\vec{\partial_i}\underline{dx}^j f_j~=~v^if_i$$

What I did was to assume that the differential form had the form ${\underline f}~=~e^{V}{\underline dx}$. The contraction is then

$$v^i\vec{\partial_i}\underline{dx}^je^{V_j}~=~v^ie^{2V_j}{\partial_i}\underline{dx}^j~+~v_i(\partial_iV_j){\underline dx}^j.$$

Then consider a transformation $e^{V}~\rightarrow~e^{-i\chi^\dagger}e^{V}e^{i\chi}$ which gives a variation in V as

$$\delta V~=~i(\chi~-~\chi^\dagger)~-~{i\over 4}[(\chi~+~\chi^\dagger),~V]$$

and the deviation in the contraction is

$$\langle v,~F\rangle~\rightarrow~v_ie^{V_i}~+~e^{-i\chi^\dagger}\big(\partial_iV~+~i\partial_i(\chi~-~\chi^\dagger)\big)e^{i\chi}$$

We might now want a form of this contraction which is gauge covariant. So to do this we back track and consider $y^i~=~x^i~+~\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}\lambda^{\dot\beta}$. We then have that

$$dy^i~=~dx^i~+~ d\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}\lambda^{\dot\beta}~+~\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}d\lambda^{\dot\beta}$$

The differential operators dual to this system are

$$\partial_i,~D_\alpha~=~\partial_\alpha~+~i\sigma^i_{\alpha{\dot\beta}}{\bar\lambda}^{\dot\beta}\partial_i$$

Then for the vector ${\underline f}~=~e^{V}{\underline dy}$ there exists a differential form contraction will result in

$$\omega_A~=~\sigma^i_{\alpha{\dot\beta}}\partial_i V d\lambda^{\dot\beta}~+~d\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}\lambda^{\dot\beta}~+~\lambda^\alpha\sigma^i_{\alpha{\dot\beta}}d\lambda^{\dot\beta},$$

where the left index A runs over i and $\alpha$. This is analogous to the differential forms $\omega^0~=~\gamma(dx^0~-~u^0 dx)$ and $\omega^i~=~\gamma(dx^i~-~u^idt)$ corresponding to special relativity. This is a Finsler bundle, and from this a generalized lifting condition (an Euler-Lagrange equation constraint) will be derived for the framing of fields.

An invariant vector for the contraction $v^\alpha (D_\alpha\Phi) {\underline d\lambda})$ for $\Phi~=~(1/4) {\bar D}{\bar D}V$ will define under the contraction

$$\Phi_\alpha~=~D_\alpha\Phi~=~\frac{1}{4}{\bar D}{\bar D}D_\alpha V$$

which if we impose the holomorphic condition $D_\alpha\chi^\dagger~=~0$ then this is gauge invariant for $V~\rightarrow~V~+~\chi~+~\chi^\dagger$. In general

$$\Phi_\alpha~=~\frac{1}{4}{\bar D}{\bar D}e^{-V}D_\alpha e^V$$

which is also gauge covariant as $\Phi_\alpha~\rightarrow~e^{-i\chi}\Phi_\alpha e^{i\chi}$

 

[MTd2]

How does it relate to this thread?

 

[Lawrence B. Crowell]

How does it relate to this thread?

I think that the whole framing of fields should be generalized. I think that Garrett's work is in some ways just the beginning.

Lawrence B. Crowell

 

[MTd2]

Ok, but I would you mind giving an intuitive reason to understand your choice for this generalization?

 

[Lawrence B. Crowell]

Ok, but I would you mind giving an intuitive reason to understand your choice for this generalization?

I think that what is required is a general framing system for fields, which remove the sharp distinctions between inertial and noninertial frames. The radiation reaction of an accelerated charge is derived from the Lamour radiation formula for an oscillating charge. This is an important key: The radiation reaction is due to a changing acceleration, such as due to the orbit of a charged particle in a bound state. Another example of a quantum physics on an accelerated from is Unruh radiation, which is calculated for a constant acceleration.

The radiation reaction of an accelerated charge is due to the interaction of a charge with its own field. This of course leads to some conundrums. The Coulomb attraction between two charges diverges as the distance between the charges goes to zero. Hence the self-energy due to the charge of an electron will be infinite. It might be simply declared that an electron does not interact with its own field, but this leads to open questions, in particular a charged particle will resist changes in its momentum more than an uncharged particle --- the field contributes to inertia. So we might then say that even if the electron interacts with its field it does so in a spherically symmetric manner and the region "at infinity" pushes back in a way as to cancel out this self-repulsion. But a naive interpretation of this will mean that a charged particle under a small perturbation will have its "self-force" canceled in one direction more than another, so the electron can accelerate to v ---> c by riding on its own field. Clearly that does not happen, and one way out of this was the Wheeler-Feynman absorber theory with positied advanced potentials coming in from the future that in a Lorentz invariant manner prevented these spurious solutions.

The Unruh radiation is due to the interaction of a body with the vacuum state. The motion of a uniformly accelerating object is defined within a Rindler wedge which partitions off a corner of spacetime as what the body on the accelerated from can have complete causal contact with. An instanteous surface of simultaneity on the accelerated frame will have field amplitudes on it which are unique within the Rindler wedge, but across the split horizon beyond the wedge spatial surfaces intersect, which means that Cauchy data is not uniquely specified there. So fields or quanta within the Rindler wedge which are nonlocally correlated on any spatial surface of simultaneity will be entangled with quanta across the horizon and in a way which breaks up the time ordering of fields. This ambiguity or "scrambling" means that fields accelerated on the frame will exhibit a thermal distribution of quanta. A body on an accelerated frame will then come to a thermal equilibrium with a temperature corresponding to this "Unruh radiation."

Are these two related? Probably, but in ways not currently well understood. This may point to a central problem with the foundations of physics. Theories are formulated on a frame bundle --- something which has persisted since the days of Newton. Newton's first law dictates that physics is to be observed from inertial frames. The Einstein equivalence principle operates to give a sharp distinction between inertial frames and accelerated frames. Yet recent developments at least suggests that a comprehensive theory of quantum gravity, and one which can account for the behavior of gauge fields in accelerated frames, may well require a more general formulation of physics on a more general frame construction which places inertial and noninertial frames on a commensurate basis.

Lawrence B. Crowell

 

[MTd2]

Larence, have you seen STVG? You might look at its equations and think very seriously about it. The preditive power is superior to any dark matter model, and can also finely describe the Train Wreck Cluster (Abel 520), which isn´t very well modeled by dark matter, and the Bullet Cluster.

http://arxiv.org/PS_cache/gr-qc/pdf/0506/0506021v7.pdf

And its predictive tests:

http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.0364v4.pdf

http://arxiv.org/PS_cache/arxiv/pdf/0708/0708.1264v3.pdf

http://arxiv.org/PS_cache/astro-ph/pdf/0702/0702146v3.pdf

I think it has some relation to your ideas. Check this one for a sanity test: http://arxiv.org/PS_cache/arxiv/pdf/0710/0710.3415v2.pdf

 

[Lawrence B. Crowell]

Thanks for the references. I have heard of these theories by Moffat. It is similar to my idea that gravitation involves a hierarchy of spins.

Unfortunately I have become a bit tied up with some things so I have not been able to continue here much. I hope to break out more on my idea about the bundle structure of framing this weekend.

Lawrence B. Crowell

 

[John G]

rntsai: "Now a3 also has a d2 subalgebra, under that the 15 breaks down as 3+3+4+4+1; the 3+3 is d2 itself (rotations/boosts), the 4 is translation, the other 4 works out to special conformal operations; the 1 is scaling or dilation."

Tony Smith: "one d4 includes a conformal d3 = a3 = SU(2,2) = Spin(2,4) Conformal MacDowell-Mansouri gravity that acts on the 4-dim physical spacetime part of the 8-dim Kaluza-Klein"

Lawrence B. Crowell: "Spin(8) embeds spin(6) as well. So it appears possible to define a conformal theory of gravity. Conformal gravity is a "good thing."

Garrett: "Regarding conformal gravity: in this E8 theory the frame and Higgs are literally multiplied in the frame-Higgs part of the connection. Because of this, the scale of the Higgs and the scale of the frame are a shared degree of freedom -- the conformal degree of freedom is described redundantly by the frame and Higgs scale."

http://www.space.com/scienceastronomy/080229- spacecraft -anomaly.html

Seems like a job for conformal gravity or as I said it in another forum, "conformal dark energy unimodular relativity". Are you listening, David Finkelstein? Grumble, grumble.

 

[MTd2]

This anomaly was first noticed in 1990, in a Galileo flyby, his is anomaly is said to be descbided http://en.wikipedia.org/wiki/Flyby_anomaly . It's not something really new, so there should be something more to that article. Besides, near Earth asteroids also suffer this effect.

 

[CarlB]

I've typed up a new preon model that should be natural to get E8 style quantum numbers here:
http://carlbrannen.wordpress.com/2008/03/03/pascals

 

[Lawrence B. Crowell]

I am not sure if this is evidence of conformal gravity. These anomalous flyby trajectories may be due to unknown aspects of the mass distribution in the Earth. At least that has to be ruled out first. Also I don't consider the Pioneer anomaly as anything worth considering. If this were evidence of new physics it should be impacting the Voyager crafts as well. This sounds like an "instrument error," since it depends on two identical systems. A recent pop-science report on these is at

http://www.space.com/scienceastronomy/080229- spacecraft -anomaly.html

The SU(2,2) or spin(2, 4) has 15 elements (3+3) + (4+4)+ 1 where the 3+3 are boosts and rotations and the other elements are the dilaton fields. The system exists in the space $R^{4,2}$ with the metric

$$ds^2~=~-du^2~-~dt^2~+~dx^2~+~dy^2~+~dz^2~+~dv^2,$$

where this "light cone" is $PR^{4,2}$ of five dimensions. Then if we consider the plane $v~-~u~=~F$, then for fixed F this defines a hyperboloid which is the spacetime with the standard metric. This spacetime is determined by the intersection of the lightcone and the plane ${\cal P}\cap PR^{4,2}$Now we can "fiddle" F as we see fit, and define a different hyperboloid which are all identical under a rescaling, so long as the plane does not go through the origin of the 5 dimensional lightcone. The plane which goes through the origin defines conformal "infinity."

In four dimensions the action for such a theory with a dilaton field $\phi$is

$$S~\simeq~\int d^4x\Big(\sqrt{-g}\kappa\phi^2R~+~\frac{1}{2}|\nabla\phi|^2~-~V(\phi)\Big),$$

is reduced the problem to a scalar field on a four-manifold. The fifth dimension defines $\phi$ as a dilaton field that acts on the embedded four dimensional spacetime or D4-brane. For the scalar potential $V(\phi)~=~-(1/2)m^2\phi^2~+~(\sigma^2/4)\phi^4 F_{ab}F^{ab}$ defines a Klein-Gordon wave equation for $\phi$

$$\partial_a\partial^a\phi~=~-m^2\phi~+~\sigma^2\phi^3 F_{ab}F^{ab},$$

where $F_{ab}$ is a Kaluza-Klien U(1) field tensor.

Now the dilaton field presumably after inflation "settles" down to a configuration so that $\nabla^2\phi~=~0$. But this field is multicomponent and maybe a "Goldstone" part is removed and a component remains with a small field amplitude. So for 9 components, eight of which might be identified as the Higgs fields, the remaining field might have a residual component with some small amplitude.

So this sort of physics certainly can be tested. Maybe this residual scalar field amplitude gives a Brans-Dicke type of gravity on the classical level. But of course we have to eliminate mass distribution effects first.

Lawrence B. Crowell

 

[Tony Smith]

Lawrence B. Crowell said that he does not "... consider the Pioneer anomaly as anything worth considering. If this were evidence of new physics it should be impacting the Voyager crafts as well. ...".

No. That is not true.
According to a google cache of an Independent UK 23 September 2002 article by Marcus Chown:
"... The Pioneers are "spin-stabilised", making them a particularly simple platform to understand. Later probes ... such as the Voyagers and the Cassini probe ... were stabilised about three axes by intermittent rocket boosts. The unpredictable accelerations caused by these are at least 10 times bigger than a small effect like the Pioneer acceleration, so they completely cloak it. ...".

Tony Smith

 

[Kea]

These anomalous flyby trajectories may be due to unknown aspects of the mass distribution in the Earth.

Yes, Louise Riofrio has suggested that the black hole at the centre of the Earth explains the anomalies.

 

[Lawrence B. Crowell]

"... The Pioneers are "spin-stabilised", making them a particularly simple platform to understand. Later probes ... such as the Voyagers and the Cassini probe ... were stabilised about three axes by intermittent rocket boosts. The unpredictable accelerations caused by these are at least 10 times bigger than a small effect like the Pioneer acceleration, so they completely cloak it. ...".

Tony Smith

There might possibly be something with new physics going on, but this acceleration is not measured by a controlled experiment. We have two old spaceprobes heading out of the Oort region which have a small anomalous acceleration. Two other spacecraft of a different design do not exhibit this. So there could be a range of plausible explanations for this which can't be eliminated easily. It also again appears to be device dependent. There might be a range of possible reasons for this: a tiny leaks from a tank, asymmetry in how optical or IR radiation is effecting off and heating surfaces, and ... . In an experiment you have to isolate variables to remove competing influences. We are not able to do this with the Pioneer craft, which means we simply can't determine much of anything about the source of this anomalous acceleration.

Lawrence B. Crowell

 

[Lawrence B. Crowell]

Yes, Louise Riofrio has suggested that the black hole at the centre of the Earth explains the anomalies.

I hope not. A mini black hole eating away the core of the Earth is not comforting. If the thing were at the center of the Earth I would suspect its gravitational influence would be radial. Then again the thing might be weaving a path through the Earth's interior eating it out like a worm in an apple. In that case there might be a time dependent gravitational anomaly.

Black holes are thermodynamically unstable. Consider a black hole has a mass $M$ that determines a temperature $T~\sim~A/4$, for the area $A~=~2\pi r^2$ and $r~=~2M$. Further consider the black hole as having the same temperature as its environment, which is easily modeled as the CMB. Then if the black hole absorbs a quanta with a mass $\delta m$ or emits by Hawking radiation a quanta of the same mass the black hole temperature deviates from that of its environment. Black hole thermodynamics does not permit equilibrium in the standard sense. The effective heat capacity of spacetime is negative.

This means a black hole in the Earth's interior would either emit radiation and explode, or it would eat away at the planet and turn it into a black hole the size of a ball bearing, or in the case of a Jovian giant about the size of a basketball. I don't remember what the Earth's interior temperature is, I think it is around 5000deg C, so the black hole would probably either quantum decay quickly or devour the planet. That temperature zone has a narrow range where a black hole could be quasi-stable. I suppose we will have to wait and see if any of these extrasolar planets identified by Doppler wobbling turn out to be little black holes.

Lawrence B. Crowell

 

[MTd2]

Two other spacecraft of a different design do not exhibit this.

You can´t just cut voyager's propellers for this experiment. They are still functioning and returning useful scientific results where they are, the heliopause. There are much better ways to make this experiment, such as attaching probes to comets and studying their trajectory with high precision telemetry.

 

[Lawrence B. Crowell]

I've typed up a new preon model that should be natural to get E8 style quantum numbers here:
http://carlbrannen.wordpress.com/2008/03/03/pascals

I suspect that systems of inequivalent Hadamard matrices will give a triality on the Jordan algebras $J^3(V)$ of 27 dimensions. There is then a fundamental representation of 27-dimensions, where three copies of these modulo three dimension for the $Z_3$ gives the 78 dimensions of $E_6$. Might the MUB's be a way of constructing a triality condition on the Jordan algebra to define the 3-cyclicity on the E_6?

From there this might be extended to $E_8$ lattices and sporadic groups. The Hamming code $H_8$ can be used to construct $E_8$, with the Weyl group matrix $W(E_8)$. This construction involves the Kissing number (minimal sphere packing condition in 8-dim) with the icosian pairing of quaterions $diag[Q_4,~Q_4]$. This gives [itex]2^4\times 14~=~224[/tex] minimal vectors $((\pm 1/2)^4,~0^4)$, and just as in the Grossett polytope these give the 240 lattice with the 8 additional weights for 248. A triality structure on the Hadamard matrices might be one natural way to construct a $\Lambda_{24}$.<br /> <br /> I am not sure what bearing this has on the preon models you are working up here. The Pascal triangle construction you are arguing for look like some polynomial system you are constructing form the MUB system. I will have to look at this more closely. I am still doing a bit to get a working knowledge of Bengtsson papers.<br /> <br /> Lawrence B. Crowell[/itex]

 

[Kea]

Black holes are thermodynamically unstable.

Given that the justification for black holes in the Earth is a quantum gravitational cosmology, in which cosmic time is directly linked to the emergent classical thermodynamics of black holes, these semiclassical arguments do not necessarily hold. Admittedly, much has yet to be done.

P.S. Hadamard matrices are cool.

 

[Lawrence B. Crowell]

You can´t just cut voyager's propellers for this experiment. They are still functioning and returning useful scientific results where they are, the heliopause. There are much better ways to make this experiment, such as attaching probes to comets and studying their trajectory with high precision telemetry.

Voyager's PGU (propulsion guidance unity) does not run continuously and was only used to tweak the trajectory of the craft for insertions through orbital windows. I am not sure if the system is even usable any more.

The problem with the whole Pioneer anomaly is that this is an unintended unconcontrolled experiment that measures some sort of accelerating force. The problem is that I see no way to conclusively eliminate device dependencies in order to conclude that there is some sort of real physics here.

Lawrence B. Crowell

 

[MTd2]

Voyager's PGU (propulsion guidance unity) does not run continuously and was only used to tweak the trajectory of the craft for insertions through orbital windows.

"The identical Voyager spacecraft are three-axis stabilized systems that use celestial or gyro referenced attitude control to maintain pointing of the high-gain antennas toward Earth."

http://voyager.jpl.nasa.gov/ spacecraft /index.html

It is still working too keep the antenna aimed at earth. That small accelaration always change the orbit enough so much that it invalidates any possible attempt to measure the anomaly.

 

[Lawrence B. Crowell]

Given that the justification for black holes in the Earth is a quantum gravitational cosmology, in which cosmic time is directly linked to the emergent classical thermodynamics of black holes, these semiclassical arguments do not necessarily hold. Admittedly, much has yet to be done.

P.S. Hadamard matrices are cool.

The Hawking result with black hole radiation requires a classical or semi-classical metric back reaction. So the theory is approximate. Yet I imagine that it is a pretty good theory for the quantum radiance of black holes with a mass $M~>>~M_p$.

Entries in Hadamard matrices can be defined according to sets of group homomorphisms. You can then define a "tower" or succession of such matrices in a recursive system of these homomorphisms. For the $n^{th}$ map matrix elements $M_n$ define the Hadamard matrix as

[tex] <br /> F_{2^n}~=~M^\dagger_n M_n.<br /> <br /> [/itex] <br /> <br /> So Hadamard matrices can be constructed in a manner similar to partial isometries used in some scattering theory. This also defines the Hadamard matrix as an error correction code of rank n. <br /> <br /> Hadamard matrices are also used to describe the non-unitary equivalency of vacua in black hole radiation and the Unruh effect. So error correction codes, Hadamard matrices, etc are I think required tools in one's tool kit to look at quantum gravity. I started a thread here on frame bundles, which is centered to a degree around these issues. Life suddenly got a bit busy with other things.<br /> <br /> Lawrence B. Crowell[/tex]

 

[Lawrence B. Crowell]

"The identical Voyager spacecraft are three-axis stabilized systems that use celestial or gyro referenced attitude control to maintain pointing of the high-gain antennas toward Earth."

http://voyager.jpl.nasa.gov/ spacecraft /index.html

It is still working too keep the antenna aimed at earth. That small accelaration always change the orbit enough so much that it invalidates any possible attempt to measure the anomaly.

The use of gyroscopic aiming is an internal force or torque on the craft. Assuming that Newton holds then this should not provide an external force on the craft. I might have to look this up, but the PGU on the Voyagers must be either used up or nearly so. They are not used to maintain any constant thrusting or used to aim the craft. The fuel and oxidant tanks on these things are like the pressurized cans of products we buy in the store --- use them enough and they eventually run out.

Lawrence B. Crowell

 

[MTd2]

I might have to look this up, but the PGU on the Voyagers must be either used up or nearly so.

According to the article on that page it ends between 2010-2012, if I am not mistaken.

Changing the subjet, really, between dark matter, superstrings, black holes everywhere, I think that Mosfat theory is nice, mathematicaly conservative, and fits nicely just too many different phenomena. Myabe that´s a better starting point, in the macrocosmic sense, to quantum gravity than GR. At least, I will try to persuit that path.

 

[Lawrence B. Crowell]

According to the article on that page it ends between 2010-2012, if I am not mistaken.

I think that Mosfat theory is nice,

So the craft is still alive in a sense. I suppose they must have some method for maintaining a telemetered connection with Pioneer as well.

As for Mosfat, what came to mind was MOSFET which is a transister. I tried looking this up but didn't find anything. In fact in a general search it appears there is some version of electronics called a Mosfat.

Lawrence B. Crowell

 

[MTd2]

As for Mosfat, what came to mind was MOSFET which is a transister. I tried looking this up but didn't find anything. In fact in a general search it appears there is some version of electronics called a Mosfat.

LOL! SORRY! :D

I meant Moffat! http://en.wikipedia.org/wiki/John_Moffat_(physicist)

And I referred to this: http://en.wikipedia.org/wiki/Scalar-tensor-vector_gravity

 

[MTd2]

Oh, I forgot to say! He works at PI institute, where Smolin also works!

 

[karnten07]

Yes, Louise Riofrio has suggested that the black hole at the centre of the Earth explains the anomalies.

I saw a news item on newscientist the other day: http://space.newscientist.com/artic...ion-may-account-for-wayward- spacecraft .html

They are suggesting that the anomalies are due to the Earth's rotation and how this effects space-time around it. I'm inclined to say that it sounds more plausible than black holes inside earth.

 

[Lawrence B. Crowell]

Scalar-tensor gravity, such as Brans-Dicke theory, serves two purposes. The origin was as a way of formulating some possible departures from Einstein's general relativity which might be testable. It is also possible that in conformal gravity that a component the dilaton field at low energy or the physical vacuum has some small oscillation about its $\phi^4$ minimal configuration. It will be very hard to test this IMO.

As for black holes in the Earth, clearly it is likely in the case of small anomalous accelerations of flyby spacecraft it is more likely due to some unknown mass distribution in the Earth, or maybe a moment of inertia that is not taken into account. Science is a bit similar to medicine --- always eliminate the most probable or basic explanation before working on more extreme diagnoses or explanations.

Lawrence B. Crowell

 

[MTd2]

The origin was as a way of formulating some possible departures from Einstein's general relativity which might be testable.

That was not the purpose of this theory. And it aparently fits better than GR on galactic scale.