Einstein-Cartan Gravity Needs 11 Generators (Mass Is Gauged)
mark...@yahoo.com
> 4. Special relativity 1905 n = 10, i.e. RIGID Poincare group P(10)
> includes T4 & O(3) as subgroups.
Newtonian Physics; Galilei group -- 11 parameters.
This raises an interesting issue: define the Galilean limit (that is,
the limit as c -> infinity) of Poincare (with 10) to Galilei (with
11).
> 9. Einstein-Cartan theory with torsion 4D world crystal dislocation gap
> fields in addition to curvature disclination fields has n = 10, i.e.
> localize rigid P10.
Since Poincare' has 10 degrees of freedom, while Galilei has 11; then
what's the Galilei limit of Einstein-Cartan gravity?
The spoler to the first question leads to a resolution of the second
Reference:
http://federation.g3z.com/Physics/index.htm#GeneralizedWigner
You actually need an 11th degree of freedom to define the Galilei
limit. The 11th parameter (in Galilei) is the "central charge"
associated with mass. To make Poincare' suitable for taking the limit
to Galilei (that is: to implement the correspondence principle with
respect to non-relativistic physics!), one needs to split the energy
generator E into two parameters -- kinetic energy H and "relativistic
mass" M. E does not have a Galilean limit. The mass shell condition (E/
c)^2 - P^2 = (mc)^2 (where P is the momentum and m the rest mass)
needs to be generalized to P^2 - 2MH + (1/c)^2 H^2 = constant. An
additional invariant emerges: M - (1/c)^2 H = constant.
The rest mass exists only for those symmetry group orbits where the
invariant
M^2 - (1/c)^2 P^2 = (M - (1/c)^2 H)^2 - (1/c)^2 (P^2 - 2MH + (1/
c)^2 H^2)
is positive. One can transform these orbits to the rest state (P = 0),
at which point M -> m. Thus, the invariant is
M^2 - (1/c)^2 P^2 = m^2, if M^2 > (1/c)^2 P^2.
However, the generalized invariant allows for well-defined mass/energy/
momentum relations even in the absence of this condition. In
particular, in Galilean relativity (where (1/c)^2 = 0), one has the
sector M = 0, P != 0, where M^2 - (1/c)^2 P^2 = 0. (The "synchrons").
These are "action-at-a-distance" modes with the invariant
P^2 - 2MH + (1/c)^2 H^2 = p^2
giving you the momentum associated with the action-at-a-distance
momentum transfer.
The analogue exists in Poincare' relativity. Here, one can have
M^2 - (1/c)^2 P^2 = 0 -- "luxons" or light-like modes
M^2 - (1/c)^2 P^2 < 0 -- "tachyons"
Both have well-defined mass/energy/momentum relations; the latter of
the form
M = p/sqrt(v^2 - c^2)
P = p v/sqrt(v^2 - c^2)
H = U + p c^2/sqrt(v^2 - c^2)
where U is the internal energy (which is where the 11th parameter
ultimately goes to).
The "generalized" Einstein-Cartan theory has an 11th mode
corresponding to this split of total energy into kinetic energy and
relativistic mass. In effect, it gauges mass.
Reference:
The Wigner Classification for Galilei/Poincare/Euclid
http://federation.g3z.com/Physics/index.htm#GeneralizedWigner
mark...@yahoo.com
> The mass shell condition ... needs to be generalized to
> P^2 - 2MH + (1/c)^2 H^2 = constant.
A dual 5-metric. Taking the 1-form
P_a dx^a - H dx^0 + M dx^4, (a = 1, 2, 3)
as the "canonical form" for co-tangent space this works out to
eta^{ab} = delta^{ab}
eta^{a0} = 0 = eta^{a4}
eta^{00} = (1/c)^2
eta^{04} = 1
eta^{44} = 0
as a, b range over 1, 2, 3.
> The "generalized" Einstein-Cartan theory has an 11th mode
> corresponding to this split of total energy into kinetic energy and
> relativistic mass. In effect, it gauges mass.
This looks like a resurrection of the old projective 5-geometry
unified field theory with Weyl's scale generator added in as either a
line or phase bundle (The central charge is treated as part of a phase
bundle structure). The signature is the same independent of the value
(and sign) of the parameter (1/c)^2.
The Galilean group without the central charge, along with the
Poincare' and Euclidean groups are all subgroups of the general affine
group GA(4), respecting the generalized 4-metric and dual 4-metric
(which have to be treated independently so that a Galilean limit can
be taken)
g_{mn} = diag(1, -(1/c)^2, -(1/c)^2, -(1/c)^2); g^{mn} = diag(-(1/
c)^2, 1, 1, 1).
So, where does the Galilean central charge fit in? This looks a lot
like something related to the conformal or projective group. The
quadratic form for the mass shell, in the limit (1/c)^2 -> 0 is the
quadratic form associated with the projective space P(R^5). Compare to
http://en.wikipedia.org/wiki/Conformal_group
(the "Projective model" section). The line bundle structure appears
there.