We
now address unjustified concerns regarding elements that are present
in this theory. Specifically, we have in mind objections to torsion, to
teleparallelism, to classical geometry (i.e. to non-gauge geometry)
and to Finsler geometry in particular. Nay SayersObjections to Torsion 1. Torsion is related to spin, not to the electromagnetic field or any other fields for that matter. This is a tremendous error, which has been criticized in no uncertain terms by Pommaret
in the introduction to one of his books, where he states: “… a new
concept of torsion was added to that of curvature, but the choice of
such word has deeply mislead mechanicians who tried to relate it to
couple-stress, a nonsense… , as couple-stress is
related to rotations through the principle of virtual moves while
torsion is a 2-form with value in the Lie algebra in the subgroup of
translations” 19 (emphasis in original). This criticism applies also to
the association of torsion with spin, which is a form of what Pommaret
calls couple-stress.2.
Torsion cannot be related to the electromagnetic field because there
is no way to match a tensor with three indices to a tensor with two
indices. There are weak and strong answers to this objection.
The weak answer is that the superscript is a gauge index and the
subscript is a spacetime index, and only the spacetime indexes need to
be matched with those of the electromagnetic field (see Andrade et al.
PRL 84, 4533 (2000)). We do not wish to insist
on this explanation because it may be right for the wrong reasons, as
the authors invoke general covariance, thus invariance under spacetime
transformations (See number ten
in this section). The authors of this web site advocate that the
appropriate arena for this program of unification is the Finsler bundle
and beyond (specifically a related Kaluza-Klein type of space). In that
bundle, the superscript has to do with reduced tangent spaces of
dimension 4; the subscripts have to do with differentiable manifolds of
dimension 7.3. Nobody has seen torsion of spacetime anywhere. We
are lucky that Einstein did not listen if anybody told him in 1914
that nobody had seen curvature and that, therefore, his effort to
geometrize the gravitational interaction was bound to fail.4.
What difference does it make whether one says that a theory has
torsion, or that the affine connection is the Christoffel symbol but
happens to be accompanied by a certain tensor. The first alternative
may offer the opportunity of a different geometrical interpretation of
the theory, but it is still the same theory. These closing
statements by Weinberg in his reply to comments by R. Becker in
Physics Today (April 2006, p. 16) overlook the fact that Becker had not
advocated torsion per se, but teleparallelism (TP), which of course
involves torsion but is a far more meaningful postulate. Let us,
however, assume that Becker had only advocated torsion. What difference
does it make? If there is torsion, physics can employ additional “…
resources of pure mathematics in attempts to perfect and generalize the
mathematical formalism that forms the existing basis of theoretical
physics, and after each success…” (emphasis in
original by Dirac in “Quantized Singularities in the Electromagnetic
Field”). Or, as Einstein put it “The theoretical scientist is compelled
in an increasing degree to be guided by purely mathematical, formal
considerations in his search for a theory …” and “The creative principle
resides in the mathematics”. Or, to listen to Feynman: “Until now it
appears that where our logic is the most abstract it always gives
correct results –it agrees with experiment”. But, since the proof is in
the pudding, let us cook.The autoparallels are more relevant than the extremals (with which they do not coincide when the torsion is not zero) since extremals depend only on the metric, and the autoparallels depend also on the torsion, and can even be defined in the absence of metric. One then realizes that one could fit the Lorentz force in the autoparallels, but that the connection has to be Finslerian. One then further realizes that only the zeroeth component of Finslerian torsions contributes to the autoparallels, where we find the geodesic equation together with additional terms that take the form of the Lorentz force (among other terms for more complicated torsions). In principle, this means unification at the level of the equations of motion with gravitational and electromagnetic forces. This in turn means that the non-contributing components of the torsion have an SO(3) symmetry, which in quantum mechanical equations means a SU(2) symmetry (continues in #5). Of course, that may be just a coincidence, but it is one which has to be viewed in the context that spacetime has necessarily an affine connection and, therefore, the property of having torsion in a general sense, which may happen to be zero. Nobody has proved that spacetime is torsionless. At most one can prove that such and such torsion theory is not viable. Objections to Teleparallelism 5.
What difference does it make whether one says that spacetime is
teleparallel, or that the affine connection is the Christoffel symbol
but happens to be accompanied by a certain tensor (S.
Weinberg, see #4). Cartan gave the answer to that question: if there is
TP, the parallel transport is integrable. There is equality of vectors
at a distance. One can integrate vector-valued integrands, which is not
legal if the affine connection is given by the Christoffel symbols.
See more on this in point number 6. Let us again take the pudding of
the previous section and transform into an absolute delicacy. In TP,
the second equation of structure gives the Einstein tensor in terms of
quadratic terms, like the traditional ones, plus terms which integrated
over all of space give zero, but which now have to be taken seriously
because of their gravitational implications. Hence there is new
physics, which is obvious but which authors usually choose to ignore.
It is a very relevant new physics since, as Feynman points out in his Lectures
(book 2, chapter 27, section 4) one would welcome a delicate
experiment to test through gravitational effects the contribution of
terms like the new ones which TP uncovers (see our preprint VII). Also,
if we set to zero the non-contributing terms of the torsion to the
equations of motion, the first equation of structure for the simplest
Finslerian torsion give the homogeneous pair of Maxwell’s equations.
Also, the invariant forms of a Finsler bundle can be separated into two
pieces as follows: on the one hand and on the other (see point #8).
If there is TP, the are the left invariant forms of the SO(3) group
and can thus be separated (They will be recovered later through a
product structure with the Kaluza-Klein type space that one can
construct with the invariants . So, again, SU(2), etc, etc, etc.6.
Teleparallelism (TP) cannot be the answer to the problem of
unification because GR has been shown to work with great precision. The
first problem with that statement is that GR is a fuzzy set of
statements. For instance, there was no concept of affine curvature in
Einstein’s 1915 GR theory because the concept of affine connection and
affine curvature did not even exist until two years later. The eventual
adoption of the Levi-Civita connection for GR is a step backwards. The
following Pommaret statement is worth citing because of its
forcefulness: “Indeed it is an apparent nonsense to
relate stress (or impulsion-energy in the 4-dimensional formalism)
—which by definition has to do with translations through the principle
of virtual work— with curvature that can be considered as a 2-form with
value in the Lie algebra in the subgroup of rotations” (emphasis in
original: Lie Pseudogroups and Mechanics). Notice that Pommaret is referring to affine curvature and of the linear group in n
dimensions, not metric curvature (in the original sense of set of
quantities which solve a problem of equivalence). Related to this,
Pommaret overlooks the fact that his criticism does not apply to GR
before physicists made it adopt an affine connection.Most importantly in GR with Levi-Civita connection (1917) understood, there is not equality of vectors at a distance; there is only path dependent parallel
transport. Which means: cosmologists cannot integrate distributions of
energy-momentum, which is vector-valued. They would have to first put
all energy-momentum vectors contributed by all points in space in the
same vector space. Only TP allows that. Einstein understood this.
However, they do integrate, and it appears that it works; after all, the
cosmological model may be lacking, but it is a step in the right
direction. Well, the reason why it works is because TP justifies it. The
Levi-Civita connection is canonically determined by a metric. The TP
connections are determined by a metric and a preferred frame field.
Cosmologists comoving frame of matter plays that role of preferred
frame. Are they doing 1915-GR cum Levi-Civita connection or are they
doing 1915-GR with TP connection? It is only in the second case that
their integratins are justified. Hence, they are doing TP, whether they
know it or not.7.
Teleparallelism (TP) cannot be the answer to the problem of
unification because of the topological restrictions it imposes on the
availability of geometries. This objection appeals to
geometric facts like, for instance, that a 2-dimensional and
teleparallel spacetime cannot be a sphere, since it is impossible to put
a constant vector field all over it, unless one removes at least one
point. Cartan showed why this argument fails. In discussing some
particular system (call it (1)) whose solving required first solving
some algebraic system, he stated that “every singularity-free solution of system (1) creates from the topological point of view, the continuum in which it exists” (emphasis in the original, “Letters on Absolute Teleparallelism”
Edited by R. Debever, p. 103). Consider the plane, where it is easy to
put different teleparallel connections. The Levi-Civita connection of
the plane is teleparallel in this case and the integration of the
system of equations of structure would yield the whole plane. But there
are other teleparallel connections on the plane (which leave out one
or more points), giving rise to other topologies There are other
reasons why this objection can fail, like if the connection were
schostastic, which we expect it to be, but which is a development far
in the future.Objections to Classical Geometry 8. Clasical geometry is not the answer to the problem of unification because of Quantum Mechanics (QM).
Certainly, one can always define classical geometry in such a
restrictive way that the objection holds. Consider, however, the
development by Kähler of a language which is the same for general relativity and quantum mechanics. Consider also our work on Clifford algebra (our paper
listed as #19, JMP 2002). The title speaks by itself: Quantum Clifford
Algebra from Classical Differential Geometry. What does that objection
become in view of these facts? Riemannian geometry was born as
anything that can be derived canonically from a metric. If one defines
classical geometry as anything that can be derived canonically from a
set of invariant differential forms that define a Finsler bundle or a
subset thereof (for instance, the subset defines a Riemannian
geometry) one can formulate a physical theory with sectors like gravity
and quantum mechanics, which are more sophisticated than, and which
contain the theories by the same name in the paradigm.9. Classical Geometry is not the answer to the problem of unification because it has been shown that gauge theories are.
The success of a theory does not mean that another theory could not do
even better. That comment is particularly significant if one takes
into account that gauge theories are theories on auxiliary
bundles, not directly related to the main or tangent bundle. We cannot
imagine that an astronaut at a pit stop in a voyage to another galaxy
would not prefer a meal of asparragus, lobster, filet mignon and
truffles (real food) instead of the plastic-like food served at the spacecraft (auxiliary food).
The first gauge theory was Weyl’s. As Cartan showed, Weyl connections
are particular cases of affine connections directly related to the
tangent bundle. May be there is unidentified classical geometry behind
our gauge theories. Thus, the fact that we may have an apparent gauge
geometry working in physics and we fail to recognize that it is a
disguised form of some sophisticated classical differential geometry (if
such is the case) will one day speak of our incompetence at this point
in history rather than of the nature of the world. History is being
made even when we do nothing. It will be judged as very bad history if
we fail to see what retrospectively may one day be viewed as obvious
and fail to check it.10. Classical Geometry cannot
the answer to the problem of unification because gravitation, unlike
gauge theories, has to do with the infinite group of diffeomorphisms.
Cartan already objected to the premise that gravitation has to do with
the infinite group of diffeomorphisms. But nobody seems to care, since
that statement still pervades the literature. In a paper of 1936
titled “The role of Lie group theory in the evolution of modern
geometry”, he had this to say: “… Riemannian geometry has been regarded
simply as the theory of invariants of a quadratic differential form in
n variables with respect to the infinite group of
analytic transformations on those variables”. One page later he would
state that “… the point of view from which we have looked at Riemannian
geometry does not make evident, and one can even say that completely
masks that which is geometric in it, in the intuitive sense of the
word.” See paper 24 and preprint V.11. Classical Geometry cannot be the answer to the problem of unification because of the non-universality of “electromagnetic fall”. Clearly,
we are talking about the q/m factor in the formula for electromagnetic
acceleration. First of all, free fall is not universal in gravitation
theory either: the “fall” of a body B of mass m comparable to that of other bodies interacting with it gravitationally depends on m,
since it affects how those “other bodies” are going to move, which
reflects back on how B moves. The so called universality is just a bad
way of saying that there is a test particle limit for the gravitational
interaction, which is not the case for the electromagnetic interaction.
But this “limitation” is just a reflection of the fact that torsion
(which is very naturally associated with the electromagnetic
interaction) is in some sense more fundamental than gravitation in
determining that matter is what it actually is. At the very small region
of space where a particle is said to be, the torsion that gives its
identity to the particle is the dominant feature of the geometry, not
the torsion of the external field. Add to that the fact that the
equations of the geometry are not linear, and the conclusion follows
that the particle sees an effective torsion, which differs from particle
to particle.Objections to Finsler Geometry 12.
Finsler geometry cannot be a main constituent of the solution to the
problem of unification because the metric of spacetime is
pseudo-Riemannian. One does not need to even discuss that since Finsler connections exist on Riemannian metrics (See our paper #7: Canonical connections … and Finslerian Connections on Riemannian metrics ). |
AuthorsJose Vargas and Doug Torr will make every effort to respond to questions on their papers and the subject of Cartan-Kähler Unification. |