Re: general relativity question [Archive]

tessel@um.bot

Apr6-06, 04:00 AM

On Sat, 25 Mar 2006 markwh04@yahoo.com wrote (tangentially to zephir's
original question, IMHO, but OTH cartanian geometry is a suitable topic
for discussion here):

> As Cartan first showed back in the 1920's, the concepts of metric and
> connection are independent. In particular, you don't need to have a
> (non-degenerate) metric to have a connection. The example of Newtonian
> spacetime is a case in point.
>
> Curvature is defined fundamentally in terms of the connection, not the
> metric. So, it's meaningful to ask whether these spaces have curvature
> or not, even when they don't have a metric.

I just wanted to very quickly point out that Elie Cartan's enormously
important contributions along these lines include

*the well known "moving frames" (c.f. frames and coframes in discussing
"physical components" of tensorial quantities in metric theories of
gravitation),

*the formulation of the notion of Cartanian geometry, the common
generalization of Riemannian and Kleinian geometry, which is a natural
setting for much 20th century work in theoretical physics,

*beautiful examples of Kleinian and Cartanian geometries which arise in
the study of specific systems of ODEs, arising from Lie's symmetry
theory of differential equations.

(And of course he made many other fundamental contributions to
mathematics.)

A bit of background: Lie's symmetry theory of differential equations has
applications including

(a) justification of "dimensional analysis" approach to deducing the
form of the solutions of differential equations without actually solving
them,

(b) organizing almost all known methods for solving ODEs and PDEs,

(c) motivating and establishing Noether's theorem on
mechanical/dynamical symmetries (quite a bit more general than the
version presented in most textbooks),

(d) motivating and establishing the canonical energy-momentum-stress
tensor associated with a Lagrangian formulation of a field theory.

These applications don't always require the context of Lie's theory but
it sure helps! Another reason why Lie's symmetry theory should be
interesting is that Lie and Klein worked closely together from about
1969, and what we now call Lie algebras and Lie groups arose in Lie's
later work as the essential mathematical background for Lie's symmetry
theory of differential equations. Yet another is that modern digital
computers render his theory a practical tool for routine work with
differential equations and systems of differential equations, including
highly nonlinear ones.

Next, I should very quickly sketch the essential motivation for the
notion a Cartanian geometry.

Klein considered Lie groups G and closed subgroups H, and showed how G
acting on the homogeneous space X =G/H (or the right coset space H\G,
which is probably more convenient here) arises as the "transformation
group" defining a notion of "homogeneous geometry" on X. This was a
terribly important development in 19th century mathematics because it
organizes and unifies a plethora of "noneuclidean geometries" and
relates "relative strengths" of competing geometries on a particular
base space X (if we think of G as a fiber bundle over X with fibers
conjugate to H) to the Lie group structure of larger groups.

Some examples should clarify this:

1. G = SO(3), H = SO(2), X = G/H is the sphere with round metric
geometry.

2. G = SO+(1,3), H = Sim(2), the subgroup of Euclidean similarities, X =
G/H is the Riemann sphere with Moebius geometry;

3. G = E(2), H = SO(2), X = G/H is the real plane with euclidean metric
geometry.

4. G = Sim(2), H the stabilizer of the origin in the natural action on
the real plane (generated by rotations around and dilations from the
origin, a two dimensional closed Lie subgroup of the four dimensional
group of similitudes), X = G/H the real plane with euclidean similarity
geometry.

5. G = PSL(3), H stabilizer of origin in the natural action on the real
plane (a six dimensional closed Lie subgroup of the eight dimensional
projective group), X = G/H the real plane with projective geometry.

6. G = E(3), H = SO(3), X = G/H is R^3 with euclidean metric geometry.

7. G = E(1,3), H = SO(1,3), X = G/H is R^4 with Minkowski geometry.

In this list, 1>>2 (1 is more rigid than 2) and 3>>4>>5. Also 2 and 7
are closely related via the observations of Penrose about light cones in
Minkowski geometry. Note that the congugates of H correspond to the
fibers over X; for example in 6 we have rotations about any point.
Also, 1,3 can be regarded "locally competing metric geometries" on a
small region of the real plane, and they have the same isotropy group H
but different parents (and also different topology, when one attempts to
extend these geometries to a suitable action by our transformation group
on the largest possible manifold).

However, Klein left out the other great stream of thought in 19th
century geometry, Riemannian geometry. Cartan realized that he could
absorb this by considering sections of appropriate fiber bundles, and in
the process unify and greatly generalize the ideas of Riemann and Klein.
Cartan also recognized more clearly than had Klein the thorny local
versus global problems which arise even in Kleinian geometry (as in
sphere versus plane above).

Cartan started with the observation that in Riemannian geometry on a two
dimensional manifold (say), parallel transporting a vector in a closed
loop typically brings it back -rotated- wrt its original orientation at
the basepoint of the loop. IOW, we simply crawled up the fiber over X
by a certain amount, depending on the area of the loop in the base
space! Cartan subsumes "moving frames" into this picture and shows that
his "structure equations" (see the book by Flanders mentioned below)
still hold with essentially no alteration.

But we get much more: we can introduce "isotropy groups" H other than
SO(n+1) or SO(1,n), and we can introduce "model groups" G other than
E(n+1) or E(1,n). For example, if we use G = SO(3) and H = SO(2)
instead of G = E(3) and H = SO(3), we obtain an analog of Riemannian
geometry in which our curvature tensor measures deviation from a round
sphere rather than from a plane, so that a "zero curvature" manifold is
a round sphere.

We can also obtain geometries in which transporting a vector in a closed
loop brings it back with the same orientation but a different length.
Clearly, manifolds with this kind of Cartan connection cannot be metric
geometries! (And you might recognize the basic idea of Weyl's flawed
original example of a gauge theory.)

Playing around, typically we obtain geometries (Cartan connections) with
torsion in addition to curvature. Indeed, while Riemmannian and
semi-Riemmanian manifolds feature connections having zero torsion but
nonzero curvature (and also uniquely defined by additional requirements
formulated in terms of an added structure, a metric tensor), there are
Cartanian manifolds which feature connections having zero curvature but
nonzero torsion. This seems to involve a notion of "isotropy" more
related to a notion of "translation" (or better, "dislocation", as in a
crystal) than a notion of "rotation". In past discussions in this
newsgroup, we started to explore how this relates to "teleparallel
gravitation", but didn't get very far. And in other past discussions,
we explored how Cartan's formulation of "geometry" relates to de Rham
cohomology.

References:

Unsatisfactory in many ways but which is also the only extant book
devoted to Cartanian geometry of which I am aware:

Richard Sharpe, Differential geometry Springer-Verlag, GTM 166, 1997

Don't be misled by the bizarrely inappropriate title; this book concerns
Cartanian geometry, not differential geometry in general! Unfortunately,
Sharpe doesn't even attempt to discuss physical applications, and the
somewhat unmotivated (but important) examples he discusses are much too
limited to display the full scope of the theory.

The classic book

Harley Flanders, Differential Forms with Applications to the Physical
Sciences, Dover reprint 1989 (org. publ. 1963)

contains an introduction to Riemannian geometry via "moving frames",
from which the generalization to Cartanian geometry can be inferred,
perhaps with a little help from previous discussions in this newsgroup
of the basics of Cartanian geometry.

A new sketch of dimensional analysis, without the context of Lie's
ideas, but with many nice examples for those who are familiar with
hydrodynamics:

Hans G. Hornung,
Dimensional Analysis
Dover, 2004 (not a reprint!!)

A readable, well-balanced introduction to Lie's theory of DEs and later
developments such as Noether's theorem and completely integrable equations
such as the KdV:

Peter J. Olver,
Applications of Lie Groups to Differential Equations
Second Edition
Springer-Verlag, 1993

Another excellent introduction to Lie's theory, very beautifully
organized (and beautifully translated by Malcom MacCallum):

Has Stephani,
Differential equations: their solutions using symmetries
Cambridge University Press, 1989

Less clear, somewhat disorganized, and with a distracting notation
(adopting Mathematica style square brackets for functions!), but offering
invaluable detailed discussion of the application to boundary layers in
hydrodynamics:

Brian J. Cantwell,
Introduction to Symmetry Analysis
Cambridge University Press, 2002

The main goal of the next book is to explain Cartan's beautiful and
far-reaching solution of a vast generalization of the basic problem of
local equivalence in semi-Riemannian geometry (given two examples, are
they locally isometric?), and to put this in the historical context of
Lie's theory, Kleinian geometry, and Noether's theorem:

Peter J. Olver,
Equivalence, Invariants, and Symmetry
Cambridge University Press, 1995

There are several other valuable books explaining Lie's ideas regarding
the symmetries of DEs; I have only mentioned a selection here. Enjoy!

"T. Essel"

Oh No

Apr8-06, 04:00 AM

Thus spake tessel@um.bot
>Playing around, typically we obtain geometries (Cartan connections)
>with torsion in addition to curvature. Indeed, while Riemmannian and
>semi-Riemmanian manifolds feature connections having zero torsion but
>nonzero curvature (and also uniquely defined by additional requirements
>formulated in terms of an added structure, a metric tensor), there are
>Cartanian manifolds which feature connections having zero curvature but
>nonzero torsion. This seems to involve a notion of "isotropy" more
>related to a notion of "translation" (or better, "dislocation", as in a
>crystal) than a notion of "rotation". In past discussions in this
>newsgroup, we started to explore how this relates to "teleparallel
>gravitation", but didn't get very far. And in other past discussions,
>we explored how Cartan's formulation of "geometry" relates to de Rham
>cohomology.

Hi,

On the subject of teleparallelism, at one point you were going to try
and read a paper I had written, but it had too much dependency on
foundations of quantum theory. Things have come on a long way since
then, and I have taken out all of that stuff. In my latest paper I only
require that there is a reference frame in which it is possible to
define plane wave states, which is, of course, where teleparallelism
comes in.

You were right, however. Although I do get back to classical gtr in the
classical correspondence, the method I was using, generalising the k-
calculus, only holds when dealing with wave theory, which is part of the
quantum treatment. The interesting thing is that leads to a radical
difference in predictions wrt to cosmological redshift. My latest paper
concentrates on the empirical evidence, which I think stands up
extremely well. Is there any chance you would care to give it another
shot?

The main difference is that velocity of recession is half that indicated
in the standard model, so the universe (for given cosmological
parameters) is twice as old and requires only 1/4 of the critical
density for closure. I get an extra factor sqrt(1+z) in the distance-
redshift relation. Reanalysing the supernova data shows this is
consistent with a universe of just over critical mass and no
cosmological constant (nice graphs from Bob Day). I have the Pioneer
blue-shift and MOND as predictions, though both are quantum effects on
redshift and not evidence of a change in Newtonian dynamics. The most
conclusive test, I think, comes from galactic profiles from rotation
curves, lensing, and evolutionary models. There is a huge literature on
this in astro-ph, and what it amounts to is that the standard CDM model
is inconsistent.

Regards

--
Charles Francis
substitute charles for NotI to email