# General Relativity and Theory of Spinors

1. Feb 6, 2005

### Blackforest

Be kind with me I am just a dentist; nevertheless I am trying to understand the GR and the Theory of Spinors.
In this sense I tried to explore the fundations of the GR and made use of a Taylorisation of the variations of the basis vectors de (with subscript 0, 1, 2 or 3) until the second order insteed of only until the first order as in the traditional moving frame method and I reach a surprising result that seems to be interesting:
if the Schwarz's condition holds, then the Riemann tensor can be built with the terms of second order of the Taylorisation first and second the terms of the first order should be isotropic vectors thus generating a sub-space with dimension at most equal to 2, thus suggesting a correlation with EM field....

My question is: Does it really make a physical sense to work with this Taylorisation insteed with the usual moving frame method establishing a linear dependance via the Christoffel's symbols of second kind between the de and the e? Thank you

2. Feb 7, 2005

### pervect

Staff Emeritus
Can anyone else make any sense out of this?

3. Feb 7, 2005

Staff Emeritus
I think Taylorization means a Taylor series, and the things he is developing the series for are the base vectors of a frame or vierbein, moving frames of four orthogonal unit vectors you can use to develop Riemannian geometry, and he wants to take the series to two terms before cutting them off instead of to one term as he says the standard procedure implies. I am unable to judge these points but perhaps someone else can.

So then he proceeds with the standard moving frames analysis to buil the Riemann curvature tensor, only not he has two terms for each entry in the tensor, and he says they (the approximations to the vecotrs) will now be isotropic and span a 2-dimensional space. I am not sure but it looks like he may hav hit on the Newman-Penrose tetrad formalism.

See http://arxiv.org/PS_cache/gr-qc/pdf/9807/9807001.pdf for a discussion of this formalism and its use.

Last edited: Feb 7, 2005
4. Feb 7, 2005

### marlon

The spinor representation in QM is just a way of looking at atoms if you will. I mean, when you say "an atom", you are now (spinorrepresentation) envisioning atoms as little arrows (the spins). A spinor is basically any mathematical object with a specific way of symmetry. If you rotate it 360° you don't get the same object but the exact opposite (i mean : if the object is A, then the opposite is -A). If you rotate another 360° you get the exact same object (eg : A).

One can use this representation when talking about stuff like ferromagnetism...Every atom on the lattice as an arrow and one atom tells his neigbour to align its arrow in the same direction. Every arrow in this lattice is therefore directed along one same direction (in the case of ferromagnetism)

regards
marlon

5. Feb 8, 2005

### Blackforest

"I think Taylorization means a Taylor series,...(SelfAdjoint)": yes. And yes also for the rest of your resumé. Sorry for the bad English and for the delay (my work + ? 8h time difference between America and Germany); thanks for discussing and commenting my question. I had until now not enough time to read the proposed link but I will do it in the next days.
"The spinor representation in QM is just a way of looking at atoms if you will. I mean, when you say "an atom", you are now (spinorrepresentation) envisioning atoms as little arrows (the spins). (Marlon)"... OK and thanks for the explanation; I am not so far with my work; I mean I was happy to introduce isotropic vectors in my way of doing and only starting from the point of view of the GR and I wanted to know of this Taylor serie makes sense. But I don't have until now find a spinor accorded to my approach... I am working. Regards Blackforest

6. Feb 16, 2005

### Blackforest

I don't like to speak after myself but accordingly to the fact that I should like to progress in my approach (GR + spinors) and to the fact that this approach is an invitation to consider the 2-D isotropic subspaces as a possible representation of a front of an EM wave related to a photon (for example; but perhaps is it totally false: please tell me), I need help in my representation of moving EM waves in vacuum. Is it correct to think that the fields E (electric) and H (magnetic) are parallel transported in despite of the fact that these fields always hold in a direction orthogonal to the motion of the photon? Or is it better to say that these fields are in translation ? In the case of a photon, what is the difference between these two notions? Thanks for the help. Blackforest

7. Mar 21, 2005

### Blackforest

I am a little bit disappointed. This is a forum where discussion about independant research is not allowed: ok, I agree. I am not a professional: ok it's true. Where can I find someone or some place to really get an opinion concerning my approach? I make the choice to post my thread here because it is one of the few place where I had the chance to receive a commentary (I thank you for this). All that I can read here on the different forums is very helpfull, I must say, but I though that a forum could offer a little corner for a more precise discussion; is it a dream? What is the way in general for an amateur to access to such discussions? I hope you get a little bit time for an answer. Best regards

8. Mar 25, 2005

### Blackforest

It looks like if I am speaking in the desert. Within the GR any form of energy has an influence on the geometry; it must be true for the EM energy. Conversely is it realistic to think that some modifications of the geometry could generate polarisations ? This is what my calculations seem to demonstrate. Is it realistic to imagine that fluctuations of the geometry at Planck's scala generate "quanto-polarisations" ? Is it realistic to think that the geometry in vacuum is an average flat geometry in reality resulting of fluctuations around the Minkowski's metric? Are all these question at the boarder of the science or still science-fiction?

9. Mar 25, 2005

### pervect

Staff Emeritus
If you find out where to get good feedback from a professional for free, please tell the rest of us! As far as practical possibilities go, this forum, sci.physics.research are the most obvious. If you have enough interest and time and money, you could consider trying to take some graduate level courses and "hook up" with a compatible professor. (Not that there's any guarantee of succces).

As far as your quesiton goes, I don't really follow exactly what you're trying to do, but a light beam should indeed follow a geodesic through space-time, which means that the light beam and it's associated fields should be parallel transported (aka Fermi-Walker transported) along it's geodesic path.

Furthermore, because of the defintion of a geodesic, the tangent vector (the direction of travel) when parallel transported should still be the tangent vector - i.e. a geodesic is a curve which parallel transports a tangent vector along the curve.

10. Mar 26, 2005

### Blackforest

"If you find out where to get good feedback from a professional for free, please tell the rest of us! "

... I am afraid... I think you are right: I don't know any professional that works for free and so it is for me in my own profession... (except sometimes for very poor people, but it is not the subject). I am sorry because I didn't realize immediatly that these forums were made for professionals only. As I understand it now clearly this first point eliminates any conversation with professionals for amateur like me, at least here. I think that I can't progress alone or in speaking about everything or nothing with other amateurs. So, the only logical consequence of this kind of logic is very simple: I beg my pardon to have been sometimes disturbing your activities herer and I wish you all a good continuation in your job; I give back to Cesar what to Cesar belongs and I give up with my curiosity for physics. Thanks. Have all a nice day.

Oh, do you know what? When a patient ask me what I am doing for him, what technic I am involving to save his tooth, I don't refuse to answer and I give some explanations, even if it is obviously not his job... and even if he does not always understand everything from my explainations...

I think everybody has not only the right but the duty to know more about himself and his neighbourrough.

Best regards

11. Apr 21, 2005

### Blackforest

You indirectly asked me in your last post what I am doing concerning the physics: “… I don’t know what you are doing but…” Even if it sounds a little bit surprising for professionals that an amateur like me is “loosing” his time with such complicated stuff, I don’t see why it would be forbidden to be curious about the world (nature) and to have the ambition to learn with the goal to obtain a part of the answer to the great question: “How does it work?”. I suppose it is a common point between all people visiting these forums.
Concerning what I am exactly doing: I belong to the category of laymen who don’t believe that we absolutely need a BB to explain everything but that creation-annihilation mechanisms are certainly permanent in the nature. In accordance with this point of view (politically incorrect, I agree) and to the evidence that our universe is a gigantic bath full of “vacuum”, I am trying to develop a theory to explain the origin of particles… starting from the energy that would be contained in these regions that we call “vacuum”. This is naturally leading me in the direction of the ZPF considerations and so and … but not only. I try to win progressively a general and complete overview on the subject. I have read a part of the fantastic work of Mark D. Roberts (11 April 2004) with the title: “Vacuum Energy”.
My own (and very modest calculations) were starting with the Maxwell’s Laws and a strange demonstration that seemed to allow the existence of a force in vacuum without source … (if one accepts to introduce the mathematical question: “When and how can we write such equations: u x w = [P]. w + k?”). That sounds stupid except, may be, if one thinks that the local geometry around a wave is not always totally flat and that the answers to the mathematical question is depending on this local geometry. This mathematical question (apparently easy) can be, for a part but not always, connected with the Poincaré work when and if u x w is understood as the new formulation (in another frame) of w that was written in a first one. These personal considerations are naturally leading me to learn more about an eventual connections between motions and underground geometry; including spins; particularly applied to EM waves.
That is my answer to your indirect question; that’s why I asked about parallel transport. My question was bad formulated and was: “Is the spin of a given particle (following a geodesic) parallel transported? Can the direction of the spin be different of the direction of the motion (there is an angle between speed and spin)? ” Thanks for attention and the answers.

12. Apr 21, 2005

Staff Emeritus
I think if you tried to parallel transport the spin axis of of some particle's intrinsic spin around a closed path you would get a Berry phase shift. Does anybody know?

13. Apr 21, 2005

### Blackforest

Thank you for this helpful information. After have been looking for Berry phase shift in Google, I have seen (for example): March 19, 1993 by Phillip F. Schewe and Ben Stein

THE BERRY PHASE , a phase shift acquired by quantum systems solely from topological effects, can create noticeable effects in even simple chemical reactions. Chemists once thought that reaction rates for simple exchange reactions such as H+H2H2+H could be determined from first principles. Mark Wu and Aron Kupperman of Caltech have found that theoretical calculations for the D+H2DH+H reaction fit experimental data better when they included this esoteric phase effect, named after the University of Bristol's Michael Berry, who discovered it in quantum systems in 1983. (Physics Today, March 1993.)

14. Apr 26, 2005

### Blackforest

Wow... as I am reading, one is really reaching the top of the research with such a subject; it looks a little bit too "great" for me... unless I get enough time to learn a lot of things including some modern approaches on the quantum gravity (Wilson loop, Aha...-Bohm effect,...). The interesting thing for me is the formalism of the Berry connection (geometrical potential) used in cerebosi starting from the Born-Hoppenheimer formalism; it appears to be a complementary term only concerning the electrons around the atomic kern in the kinematic momentum. I am just thinking (and dreaming) if I could connect this formalism (by extrapollation) with my own approach in vacuum (NSEgb.pdf) where I am obliged to propose a special formalism to the speed field for particles(-waves). This would force me to interprete the wave functions as resulting from the variations of a scalar field apparented to a gravitational one... Does it make sense? Sorry for disturbing with my questions.

15. May 1, 2005

### Blackforest

At the end of the 19th century, Maxwell write a synthesis of experimental results concerning electricity and magnetism. He shows the imbrications between the two fields. At this time questions concerning alternative geometries and topologies are still developed or in development but nobody is in a correct mental and technical situation to connect both subjects. In a first step Einstein demonstrates the relativity of electricity and magnetism; in a second one he strongly suggests the importance of the background, that is of the geometry and, eventually, of the geometro-dynamics. Despite of this, Maxwell’s equations, even if they do not precise in any way the geometry where they hold, suggest the existence of forces, even in vacuum… without the presence of any source…

To have a hope to get a part of the answer concerning this apparently illogical suggestion, one is obliged to scrutinize a mathematical question that I have called the (E) question*.

Amazingly, systematic solutions of this question* in 3D space (3 spatial dimensions) have a formalism that I can connect with some equation of the ADM Approach (versus 1: time + 3: space) giving the relation between local extrinsic tensor on one hand and canonical momenta and metric tensor on the other hand (see attachment; I hope it is ok with; etgb25.pdf or word).

This fact gives a relief to my approach; the logical consequence of my theory is to believe that some forces arising from the considerations of the GR could generate a similar problematic than the Maxwell’s equation for the “vacuum”, that is a problematic related to the (E) question.

My next step will be to look for the holonomy generated by this (E) question when I consider it as a transformation.

Thanks for attention

Last edited: Feb 2, 2006