Einstein's Field equations & quaternions / octonions

  • #1
You can rewrite Maxwell's equations using d'Alembertian operator on quaternions.
Can something similar be done for Einstein's Field equations and is there an advantage in doing so?
Will this help in finding solutions to the equations or e.g. calculation of proper time,proper distance?

Will octonions help?
Clifford analysis?
 

Answers and Replies

  • #2
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International Journal of Theoretical Physics Vol. 25, No. 6 (1986), 581{588.

Curvature Calculations with Spacetime Algebra

David Hestenes

Abstract. A new method for calculating the curvature tensor is
developed and applied to the Scharzschild case. The method employs
Clifford algebra and has definite advantages over conventional
methods using differential forms or tensor analysis.


--------------------------------------------------------------------------------------------
GRAVITY, GAUGE THEORIES AND GEOMETRIC ALGEBRA

Anthony Lasenby, Chris Doran and Stephen Gull

Astrophysics Group, Cavendish Laboratory, Madingley Road,
Cambridge CB3 0HE, UK.

arXiv:gr-qc/0405033v1

--------------------------------------------------------------------------------------------



There's more of this, lots more. Search for Lasenby and Hestenes.
 
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  • #3
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Shortly after the fantastic experimental success of Dirac's approach of using spinors to model the degrees of freedom of electron motion, Mayer and Einstein studied how they fit into and were required by the symmetry demands of relativistic covariance: A. Einstein and W. Mayer, Preuss, Akad. Wiss. Phys. Math. Klass. Sitz., 1932

They found that quaternions can be reduced to a pair of spinors and also that tensor representations of field equations, at least in the case of electrodynamics, contain extra degrees of freedom not warranted by the demands of symmetry. In particular, time or space reflections should not be part of a continuous set of transformations.

This effectively means that spinors and, through their composition, quaternions, provide the most primitive mathematical structures able to specify covariant transformations.

(It should be noted that Pauli pioneered the rudiments of spinor use, Uhlenbeck and Goudsmit showed how their use was necessary to model electron orbitals, Dirac discovered the covariant application for the electron which led to the discovery of the positron while Majorana extended their application to other particles to indicate the existence of neutrinos)

Some books (with very different approaches):

Introduction to 2-Spinors in General Relativity, Peter O'Donnell
General Relativity and Matter, Mendel Sachs (excellent explanation of spinor-to-quaternion relationships)
Quantum Mechanics and Gravity, Mendel Sachs
Spinors and Space-time, Penrose and Rindler
 
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  • #4
Great, Peter O'Donnels book seems interesting.
 
  • #5
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I've always wondered why quaternions aren't/haven't been taught very much.

Perhaps there has been more investigation into their use than I'm aware of, but it seems like the choice of Gibbs' vector calc (IIRC developed from his study of quaternions) was the rather arbitrary result of Gibbs' book being used by Yale (and Yale's reputation leading to its use by other universities) as much as its actual benefit over quaternions (I could be totally wrong on this, it's just my perspective on it).

I'm not an expert on quaternion/vector analysis history by any means, but I've always wondered if there is any real reason why it isn't emphasized (or even mentioned at all) in most physics courses?
 
  • #6
Very interesting question. It could be best answered by R.P.Feynmann but I'll give my try. My opinion, psychology lies to it bases. Many persons are lazy and take everything what exist in existing books... which they don't understand... Abstractions are oldschool, everything has to be concrete, meaning practical and easy understood... learning without understanding... Complex and imaginary numbers are no more complex than real numbers... And real numbers are no more real than complex numbers...
Human brain is last part in human evolution, it takes effort to use it. And as education is for everyone... this means the questions asked by education should be answerable by everyone... ... ... ... ... ... ... ...
 
  • #7
bcrowell
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Another person working on this kind of thing is John Baez of UC Riverside.
 
  • #8
Patrick. R. Girard has also a nice book. (including HXH over R)
 
  • #10
E.g. when for simplicity focussing first on Maxwell's equations before General relativity.
You could use complex potentials,quaternions,spinors,geometric algebra,Hilbert spaces,Riemann sphere?,...
But many problems in a certain field have a best choise as set of mathematical tools.
Which one is according to you the best?
By preference a dense notation,not hiding but revailing the underlying nature of nature,using mathematical properties with fysical meaning,allowing solutions in closed form...
 
  • #11
sweetser
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Maxwell field equations

Hello:

I am glad my book is being put to use, that was the goal. This is an ongoing research project. I wish I could keep the book up-to-date, but there is only so much time in a day (I have a full time job, wife, and 2 year old).

Here is the right way to derive the Maxwell equations. All that is required is to generate the Lagrange density. Here is the real valued quaternion expression:

[tex]\mathcal{L}_{EM}=scalar\left( \frac{1}{4} (\nabla A - (\nabla A)^*)(A \nabla - (A \nabla)^*)+ (\nabla A - (\nabla A)^*)^2 - (A \nabla - (A \nabla)^*)^2 \right)[/tex]

The first term generates B2 - E2 for the two Maxwell source equations. While not as well known, the other Lorentz invariant E.B can be used to generate the two Maxwell homogeneous equations. From there, the derivation is standard, using Euler-Lagrange.

If you have never gone down this path (Lagrangain -> field equations), I recorded a 1.5 hr talk and all the pdfs aimed at an MIT undergrad audience (available at http://visualphysics.org/Talks [Broken]).

In my own research, I have concluded that quaternions cannot do gravity. We know gravity involves changes in symmetric metrics. Those changes are also symmetric. Quaternions are asymmetric, being a sum of both symmetric and antisymmetric parts. To be completely symmetric, I have begun using what I am calling "California numbers", which have exactly the same rules of multiplication as quaternions, but without a single minus sign. I basically put the same J's, Del's, and A's plus a few conjugate operators through the Lagrange/Euler-Lagrange machine to get my field equations for gravity. I am blogging weekly at http://Science20.com/standup_physicist if you want to follow along.

One more plug: quaternions lost out because people don't know what they mean visually. http://Visualphysics.org is a site whose goal is to animate quaternion expressions. It can be done, but the results are often odd. We will have to see if it wins people over.

Doug
 
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  • #12
642
15
E.g. when for simplicity focussing first on Maxwell's equations before General relativity.
You could use complex potentials,quaternions,spinors,geometric algebra,Hilbert spaces,Riemann sphere?,...
But many problems in a certain field have a best choise as set of mathematical tools.
Which one is according to you the best?
By preference a dense notation,not hiding but revailing the underlying nature of nature,using mathematical properties with fysical meaning,allowing solutions in closed form...
There's probably a certain amount of personal preference and prior familiarity with the notation that would lead a person down a certain path as well how important the connection is to experimental results. One potential problem with the stereographic projection-Riemann Sphere approach to spinor representation is that one point on the sphere's pole becomes a singularity and the math breaks down there. It's also a non-linear type of projection so that inner product space and hence Hilbert space would not be preserved in any inverse projection back to the problem domain (which presumably is the experimental domain).

A secondary goal to a book in progress is the full development of a spinor-quaternion program which should be comprehensible and usable by most anyone who understands the Maxwell equations and wave equations. The idea is to formulate integrated solutions to potentials, fields, forces, waves and media.

The post here https://www.physicsforums.com/showpost.php?p=3127444&postcount=26 has 2 different spinor-quaternion representations of the EM field equations. A suggested starting point would be to prove whether each is form compliant with the Maxwell equations or not. Form compliant meaning that any solution of the spinor-quaternion equations is also a solution of the Maxwell equations in vector form. Those are "naive" equations in the sense that the exact representation specifics of the spinors haven't been worked out or specified and are intended to simply map into vector space as a first step.
 
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  • #13
sweetser
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Solid technical reason to skip quaternions: representing the Lorentz group

The Lorentz group is vital to modern physics. Minkowski figured out that the Lorentz boost was like a rotation in spacetime. Since quaternions do space rotations, it should be a simple step to do spacetime rotations, right?

The answer is no. The answer should be no. Rotations can be done with a connected group while the Lorentz group is not connected. That is an important difference. All the way back in 1911 and 1912, two different people figured out how to get biquaternions to do spacetime rotations, but not real-valued quaternions.

So it was, at least up to 1995. A fellow named De Leo figured out how to use quaternion triple products to do a Lorentz boost. It can be done. I read the paper a few times, but I didn't quite get it, my bad. Where are the hypercomplex sines and cosines?

Jump ahead 15 years. I am telling this poor student in Indonesia to read and implement a paper I didn't understand well enough myself. I felt guilty. I think there MUST be a way to do a boost with quaternions. Here is a regular spatial rotation:

[tex] R \rightarrow R' = (\cos(\alpha), \vec{I} ~ \sin(\alpha)) ~R~ (\cos(\alpha), -\vec{I} ~ \sin(\alpha))[/tex]

So drop in a pair of hypercomplex functions in the same place:

[tex] R \rightarrow R' = (\cosh(\alpha), \vec{I}~ \sinh(\alpha)) ~R~ (\cosh(\alpha), -\vec{I} ~ \sinh(\alpha))[/tex]

This gets 2 things right! It also gets two things wrong by omission, and two other things wrong by bein where they should not be. Bummer. If the right thing could be added in, while the bad stuff got subtracted away... Yup, it can and has been done. Here is the answer:

[tex] R \rightarrow R'= h^* b h + \frac{1}{2}((h^* h^* b)^* - (h h b)^*)[/tex]

where the h is the hyperbolic cosine/sine quaternion, the star is the conjugate operator. Here you can see that a Lorentz group is not connected.

This is all it takes to get over the technical objection about quaternions and the Lorentz group. A http://visualphysics.org/preprints/qmn10091026". In other words, this will remain obscure.
 
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