Date: Feb 14, 2012 1:54 PM
Author: Rock Brentwood
Subject: Re: Acoustic Metrics, and the OPERA neutrino result
On Feb 12, 3:51 pm, "Androcles" <H...@Hgwrts.phscs.Feb.2012> wrote:
> What we really need is to create a super-theory ("Quantum Gravity")
> that incorporates and reconciles the best bits of general relativity
> and quantum mechanics, or as the tailor said to the emperor buying
> an invisible new suit, "Never mind the quality, feel the width!"
But how do you propose to do that? It would be a lot like getting
partisans on the Conservative and Liberal side of the aisle to quietly
shack up together.
The basic problem is that the two paradigms totally disagree on what
the nature of time is. And that disagreement cannot be reconciled
(which ironically may lead to a solution I briefly describe below).
GR, or any other formulation of dynamics that uses space-time geometry
(like Newton-Cartan) treats time as a dimension that's "all there";
while QM wants time to be the arena of change, a place in which change
takes place; particularly non-deterministic change.
The only simple way to get these two views of time to mesh
consistently is to treat them as *different*. That means, basically,
you have not just "space flowing in time", but "*spacetime* flowing in
time". The "time" in "spacetime" is the one that goes with GR or any
other geometric theory. The "time" in "flowing with time" is the one
that goes with the Schroedinger or Heisenberg equation.
The key may lie in better understanding just what non-relativistic
theory actually is, and how it relates to Relativity.
Whereas in Relativity, there is a natural affinity for a 4-dimensional
geometry, in non-relativistic the geometry naturally associated with
it is *5* dimensional. This is seen directly in the transformation law
for mass (m), kinetic energy (H) and momentum (p):
m -> m, p -> p - m (delta v), H -> H - p.(delta v) + m(delta v)^2/2
They transform as a *5*-vector. There are 2 invariants you can
construct out of this:
(1) the linear invariant m, (2) the quadratic invariant, p^2 - 2mH.
The quadratic invariant is a metric in disguise. The corresponding
metric for the coordinates comes out of the association:
p <-> del ... momentum associated with spatial translations
H <-> -d_s ... kinetic energy associated with translations of time
but requires that the mass be associated a symmetry of its own
m <-> d_u ... mass associated with a symmetry for an extra
Then the quadratic invariant becomes the 5-D "laplacian"
del^2 + 2 d_t d_u
and associated with this is the metric
dx^2 + dy^2 + dz^2 - 2 dt du.
All of this remains the same when going over to relativity, except for
the inclusion of a relativistic correction:
p^2 - 2mH becomes p^2 - 2mH - H^2/c^2.
So the Laplacian becomes
del^2 + 2d_t d_u - 1/c^2 d_t^2
the corresponding metric becomes
dx^2 + dy^2 + dz^2 - 2 dt du - du^2/c^2.
and -- the punchline -- the time (t) is replaced by
a non-invariant time t
an invariant time s, given by ds = dt + du/c^2.
By setting the factor (1/c)^2 to 0, you get all the formulae for non-
relatvistic theory; while by setting it to a positive value you get
relativity. By considering the two in concert, you end up seeing the
appearance of an extra feature that was not present in either
relativity or non-relativistic theory when considered alone: the
distinction between a coordinate time (t) and absolute time (s).
All of this can be done for curved space-time geometries as well. It
leads to BOTH the Newtonian Law of gravity (when setting the factor (1/
c)^2 to 0) via the Newton-Cartan equation, AND to Einstein Field
Equations of GR (when keeping the (1/c)^2 positive). But there is now
an extra feature in both: (1) the appearance of a 5th coordinate (u),
(2) an invariant vector field given by the invariant derivative
operator (d_u), and (3) an "absolute time" s given by the invariant
So, the idea is to just like with the irreconcilability of GR and QM
by using s for the Schroedinger equation in quantum theory and t for
the field equations in GR.