Re: Problems of four-velocity
From: Ken S. Tucker (dynamics_at_vianet.on.ca)
Date: 02/21/05
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Date: 21 Feb 2005 13:00:42 -0800
Tom, pardon top post...I studied your reply.
There is an idea among some mathematical
physicists they may take the rules of static
manifolds, developed by Riemann et al, and
expect them to apply to relativity dynamics.
That's a major assumption, i.e. presumptious.
Tom Roberts wrote:
> Ken S. Tucker wrote:
> > Tom Roberts wrote:
> >>It is the path-independence of
> >>parallel transport that permits one to ignore the different tangent
> >> spaces in a flat manifold (like Minkowski spacetime).
> >
> > Tom, Minkowski spacetime is NOT flat,
> > it's NONORTHOGONAL.
>
> You are wrong on both counts: Minkowski spacetime is indeed flat, in
the
> sense that the Riemann curvature tensor is identically zero
everywhere.
> And a manifold (spacetime) cannot be said to be "nonorthogonal" or
> "orthogonal" -- those concepts simply do not apply to a manifold.
Yes they do.
> The COMPONENTS of the metric tensor of Minkowski spacetime,
> when projected onto Minkowski coordinates, are indeed
> orthogonal (i.e. the only nonzero terms are on the diagonal).
No you're mistaken get out Weinberg's "Grav and Cosmo"
provide a ref.
> > You'll arrive at ds^2 = dt^2 -dx^2 using
> > g_01= -v/c , that's nonorthogonal.
>
> You are confused. If indeed "ds^2 = dt^2 -dx^2" (which is a valid
line
> element on a 2-d Minkowski spacetime using Minkowski coordinates x
and
> t), then g_01=0 (not "-v/c"). Manifestly. And those metric components
> are orthogonal (i.e. the only nonzero terms are on the diagonal).
> Manifestly.
That's BS, and I'll prove it.
1) Only lengths perpendicular to motion in x,y,z
are invariant (in SR).
2) "s" (of ds) is invariant and is therefore
perpendicular to x,y,z
3) time "t" is NOT invariant, and is therefore
NOT perpendicular (orthogonal) to x,y,z.
4) You snipped without considering Minkowski's
work I refd and substituted your own deformed
ideas of dynamic relativity embodied within
"Dynamic nonorthogonal spacetime metrics" such as
g_0i.
Regards
Ken S. Tucker
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