Natural Science Forum / Physics / Research / July 2008

Oops. Wrong cite. It's
http://federation.g3z.com/Physics/index.htm#Solutions

Expanding on my previous reply: yes.

In fact, the fixes required resolve a mystery that's been in my mind
for a long time. You see, back when I first "learned" SR -- actually
BEFORE I learned it -- I came up with my own transformation which is
quite unlike the Lorentz transformation in that it effectively leaves
time intact, while doing something strange with the spatial
coordinates. In effect, I came up with 2 time-like axes; one which
remains absolute and the other which shifts "locally".

I never really understood what I did (in geometric terms), until
resolving the issue you're asking about. In fact, the extra coordinate
is required to match up with the 11th generator of the Galilei group.
In turn, this is needed to pass over to a consistent Galilean limit --
not just to the 10-parameter Galilean group, but its 11-parameter
central extension.

Without it, you can't resolve the correspondence limit! In that sense,
GR is not the correct inverse correspondence limit of the Newtonian
theory of gravity -- it only works with the 10 parameter group, not
the 11 parameter centrally extended group.

The 11th generator of the Galilei group is accommodated, which fixes
the "missing hole" left behind by the (1/c)^2 -> 0 limit. Without this
fix, then because you lose the ability to change the direction of the
equal-time planes in Galilean physics, (since simultaneity is absolute
in non-relativistic physics), you lose an important degree of freedom
in the corresponding gravitational field equations.

In fact, this has held up efforts by people in the past to come up
with a field law for Newtonian gravity in the same mould as Einstein's
equations that has a seamless limit with respect to it.

The fix means that the stress tensor becomes a 4x5 quantity T^{mu}_a;
mu = 0, 1, 2, 3; a = 1, 2, 3, 4, 5; representing the current density
for the P_a, the ath component of momentum; defining
  P_0 = E = total energy
  P = (P_1, P_2, P_3) = momentum
  P_4 = H = kinetic energy
  P_5 = M = relativistic mass.
where M = E/c^2, E - H = constant.

The mass shell P^2 - E^2/c^2 has to be changed to P^2 - 2MH + z H^2,
where z is the parameter that distinugishes Galilean spacetime (z = 0)
from Lorentzian space-time (z = (1/c)^2).

This leads to a 5-metric
  g^{ab} P_a P_b = P^2 - 2MH + z H^2.
The most general linear transformations that preserve both this and
the 1-form
  mu = M - z H
is the 11-parameter Galilei group, itself.

This is how the recent journal reference I cited also handles the
issue (it proceeds to write down a path integral form for the Galilean
Dirac equation).

The generalized Lorentz transformations consists of 5 translations and
a 6-parameter homogeneous group. The infinitesimal form of the latter
is:
  delta(P) = omega x P - u M
  delta(M) = z delta(H) = -z u.P
  delta(H) = -u.P
  delta(mu) = 0

A coordinate form can be erected by writing out the corresponding
canonical 1-form:
  theta = P.dr - H ds + M du = P.dr - H dt + mu du.
  t = s - z u
  dr = (dx, dy, dz)
The action of the Lorentz group on the coordinates is uniquely
determined by the requirement that it leave theta invariant. This
yields the following infinitesimal form:
  delta(dr) = omega x dr - u dt
  delta(du) = u.dr
  delta(dt) = -z u.dr
  delta(ds) = 0

You can see, here, that t is just the "real time"; while s is a
vestige of the "Galilean time" s that lingers on even when z > 0. The
coordinate u = (s - t)/z is conjugate to the mass. For the Galilean
limit, t -> s, and u loses its moorings from s and t and becomes
independent.

That's the non-trivial element that has to be added to GR to make it
have a Galilean limit. The coordinate s plays no direct role in the
dynamics ... in effect, space-time is foliated in 4-dimensional
sections along surfaces of constant s. However, as z -> 0, all the
variability you see on the right-hand side of
  delta(dt) = -z u.dr
is lost and passes over to u:
  delta(du) = u.dr.

This extra element has to remain in place when writing down the
Galilean form of GR.

So .. now you can write down the configuration variables for a
gravitational field dynamics. One has FIVE frame 1-forms, which I'll
call *x* = (theta^1, theta^2, theta^3); s = theta^4, u = theta^5;
along with t = theta^0 = s - z u.

One has FIVE torsion 1-forms, derived from these:
  *X* = (Theta^1, Theta^2, Theta^3); S = Theta^4, U = Theta^5
along with T = Theta^0 = S - z U.

The most general connection that resolves the metric and 1-form has 6
independent components, which can be labelled *sigma* = (sigma_1,
sigma_2, sigma_3); and *alpha* = (alpha_1, alpha_2, alpha_3). One can
then write down the equations for the torsion and 6 curvature 2-forms
(*Sigma* and *Alpha*), by rote-following the Cartan structure
equations
  d*x* - *sigma* x *x* + *alpha* t = *X*
  dt - z *alpha*.*x* = T = S - z U; (t = s - z u)
  du + *alpha*.*x* = U
  ds = S
  d*sigma* - 1/2 (*sigma* x *sigma*) + z/2 (*alpha* x *alpha*) =
*Sigma*
  d*alpha* - *sigma* x *alpha* = *Alpha*
where I use 3-vector notation in conjunction with the differential
forms' wedge product (e.g. *alpha* t = (alpha_1 ^ t, alpha_2 ^ t,
alpha_3 ^ t); and *sigma* x *alpha* = *alpha* x *sigma*).

One can formulate the question: what is the most general algebraic
combination of these configuration coordinates that is Lorentz
invariant (hence, independent of *sigma* and *alpha*) that yields 4-
forms suitable for a Lagrangian?

Without the extra element (u and s), there are 6 combinations. The
extra elements make for 2 more. This leads to a potentially richer
field law --- though I haven't yet looked at it in detail.

Most importantly, the laws corresponding to the u and U fields are
CRITICALLY involved in formulating a consistent set of laws for the
stress tensor (especially the energy density). Lose this, and you lose
the ability to consistently ramp down from z = (1/c)^2 > 0 -> z = 0.
Newtonian gravity would have to be written in "by hand" instead of
derived, in this way, as a z -> 0 limit.

An interesting feature: one of the invariants is linear in the torsion
and is not present in the ordinary Lorentz-based gravity theory (X.x +
Ts)u, I think it is. This can lead to a non-zero torsion, even in the
absence of external sources -- another issue I haven't looked at in
detail yet.