REFLECTIONS
Reflections on my article "Canonical form of
self-adjoint endomorphisms and associated stress-energy tensors"
by F. Sánchez M.
My
article "Canonical form of self-adjoint endomorphisms and
associated stress-energy tensors," is based a part of the publication of GS
Hall, Physical Structure of the Energy-Momentum Tensor in General
Relativity (Internat. Journal of Theor. Physics, Vol.25 No 4 1986) and
other authors.
This publication is from my point of view particularly interesting to
better understand the nature of the energy from your drive tensor
invariant elements.
Development
of this work is done from a rather simple in its mathematical context
but in my opinion an important physical content.
The simplicity of the mathematical development does not imply lack of
value or significance in its physical content.
In
some places I use classic words as eigenvalues or eigenvectors.
They are compatible with a more mathematical terminology as invariant
subspaces etc. ..
In the cases studied
I think I have to use this nomenclature because is more
suitane for the
nonspecialist reader.
Some bibliography related to "Canonical form of self-adjoint
endomorphisms and
associated stress-energy tensors
by F. Sánchez M.
I
add some citations that I discussed important to understand the topic
before us.
It is not complete because it is a classic developed and used more or
less since the early twentieth century by many authors.
I limit myself only to some relevant quotes where you can find the
necessary information.
See bibliography
here.
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COMMENTS
Feedback:
There have been some comments, including several as anonymous.
The transcribed below:
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Comment
In the third paragraph of section 2.1.2.
speaks of two or more different isotropic eigenvectors.
I do not understand respect.
Could I make some clarification?
Anonymous
Received December 4, 2009
Answer:
I try to express that two or more isotropic eigenvectors, having to be
orthogonal are to be in the same isotropic straight line.
Therefore we have a single isotropic direction orthogonal to the rest of
the eigenvectors is not isotropic, ie a 2D hypersurface (which contains
the eigenvectors not isotropic), or simply have, in the simplest case, a
single line does not isotropic.
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Comment
I read your publication and see that using words related to mathematical
elements such as "eigenvector," "intrinsic value", which from my point
of view are not relevant in a modern mathematical nomenclature,
although used in classical texts and still, even in
many modern texts.
I also see that using single or multiple words as referring to the
eigenvalues.
Could you comment further on the meaning of such words and words used?
Anonymous
Received May 6, 2010
Answer:
The words referred to I have used because this is more understandable by
most people.
However, it is appropriate to make some clarification on them: Let an
endomorphism M
eigen-direction,eigen- straight
line or eigensubspace , in an endomorphism M is an invariant subspace of dimension 1.
Eigenvector is a vector of eigensubspace .
In the case of the minimal polynomial, the subspace itself is linear : M-λI =
0.
λ is the eigenvalue.
(I is the identity endomorphism.)
Obviously the above is a particular case (the simplest).
My article "Canonical form of adjoint endomorphisms and Associates Tensors" cases are analyzed starting with the simplest and
continuing thereafter for the most complex,like radical subspaces of
index higher than one, and minimal polynomials of degree higher than one
without real roots.
The vectors and values cited can not be defined in a subspace whose
minimal polynomial is of degree higher than 1, and real roots.
The word simple is used when all invariant subspaces of M are of
dimension 1
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