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Re: Acoustic Metrics, and the OPERA neutrino result
Posted:
Feb 14, 2012 1:54 PM


On Feb 12, 3:51 pm, "Androcles" <H...@Hgwrts.phscs.Feb.2012> wrote: > What we really need is to create a supertheory ("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 spacetime geometry (like NewtonCartan) 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 nondeterministic 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 nonrelativistic theory actually is, and how it relates to Relativity.
Whereas in Relativity, there is a natural affinity for a 4dimensional geometry, in nonrelativistic 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 (t) but requires that the mass be associated a symmetry of its own m <> d_u ... mass associated with a symmetry for an extra coordinate (u). Then the quadratic invariant becomes the 5D "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 noninvariant 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 nonrelativistic theory when considered alone: the distinction between a coordinate time (t) and absolute time (s).
All of this can be done for curved spacetime geometries as well. It leads to BOTH the Newtonian Law of gravity (when setting the factor (1/ c)^2 to 0) via the NewtonCartan 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 ds.
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.



