🚀 go-pugleaf

Tom Roberts wrote: > > GSS wrote:
> > You could say "deformed" relative to "undeformed" or relative to
> > "original". The Euclidean space may be considered undeformed.
>
> Not in GR, and not in geometry in general. In particular, it is not in
> general possible to map the points of a Euclidean manifold 1-to-1 to the
> points of a curved manifold, while preserving the local topological
> structure (i.e. the neighborhoods).
>
> This is so basic it is mind-boggling that you think this: try to
> describe the "deformation" you imagine from E^2 to S^2 (infinite 3-d
> Euclidean space to the surface of a sphere).... Then try a Klein
> bottle.... These examples don't begin to touch the difference between
> E^4 and E^(1,3), and are vastly simpler than the typical manifolds of GR....
>
> Trying to consider "deformations" like this is hopeless. The geometry of
> any manifold used as a physical model of the world must be understood
> INTRINSICALLY.
>
> IOW: in GR it is the distributions of matter and energy (plus boundary
> conditions) that determine the geometry (topology and metric) of the
> manifold; it is not possible to remove the matter and energy from the
> universe -- but that is what must happen to get your "undeformed space"
> (which isn't Euclidean, anyway). Removing all matter and energy from the
> universe -=-

$$ Vacuum is a MENTAL state or CONDiTiON ..at least, in Tom's case.
                                                          ```Brian.

> -=- yields a _DIFFERENT_ physical situation and a _DIFFERENT_
> universe -- the word "deformed" is woefully inadequate to describe that.
>
> IOW: you can only deform one thing to another if they are commensurate,
> and different manifolds in general are not.
>
> > You may
> > also view the term "deformed" in the sense of "altered".
>
> Hmmm. That's still woefully inadequate. <shrug>
>
> > Consider the
> > relative positions of N number of neighborhood points in the
> > Euclidean space. If the relative positions of all these points remain
> > "unchanged" or "unaltered" or the "same" for all time to come, that
> > is to say if the relative positions of all neighborhood points remain
> > "invariant" with time, we can say the space continuum is undeformed.
> > If on the other hand the relative positions of all these neighborhood
> > points get "changed" or "altered" at some later time due to 'some
> > reason', we can say the space continuum is deformed.
>
> But that is not sufficient to handle the general case of a curved
> manifold. In particular, this cannot alter the topology of the manifold,
> as you are only admitting continuous deformations.
>
> For instance, you cannot possibly obtain the Schwarzschild spacetime
> from E^4 this way. In fact, you cannot even get Minkowski spacetime from
> E^4 this way, and it _has_ the same topology as E^4.
>
> Moreover, you are discussing evolution over time, which cannot possibly
> obtain the manifolds of GR from Euclidean manifolds.
>
> Specifically: you are imposing a geometrical structure E^3xR on the
> world, where E^3 is "space" and R is "time". Such background structure
> is clearly unwarranted -- why assume this particular structure rather
> than some other?? GR, of course, avoids this by not having any
> background geometrical structure at all (geometry is a dynamical aspect
> of the model).
>
> IOW: you are essentially assuming a Galilean world.
>
> > In our sincere and most revolutionary attempt to represent the
> > gravitational phenomenon through geodesic equations in four
> > dimensional space-time manifold, we mainly focused on the geometry of
> > particle traces in the space-time continuum.
>
> That is not "revolutionary", that is quite standard in GR, which is ~90
> years old. And your claim simply is not true -- most people "focus" on
> the curvature and metric tensors, and the topological structure of the
> manifold, not "particle traces".
>
> > In the process we lost
> > focus on the geometry of space continuum itself which by definition
> > consists of a continuum of points.
>
> _YOU_ have indeed lost that (if you ever had it), but people
> knowledgeable about GR have not. <shrug>
>
> The geometry is wholly determined by the metric and the topology --
> that's why people concentrate on them. <shrug>
>
> > Once we focus on the *geometry* of
> > the space continuum, the very first unique feature we notice is that
> > the space continuum itself gets deformed in the presence of
> > gravitational field.
>
> I repeat: your usage of "deformed" is nonsense. The time evolutions you
> call "deformations" are not at all what happens in GR. In GR the many
> solutions to the field equation are each their own individual manifold,
> and there is no direct relationship between any pair of them (in
> general). None of them, of course, are Euclidean. <shrug>
>
> Like all too many people around here, you need to STUDY GR and
> differential geometry. Yes, to do that you will need to unlearn many
> misconceptions.
>
>         [You seem to be resonating with Ken Tucker. I caution against
>          that, as he is in the same boat (needing to unlearn many
>          misconceptions). The two of you will sail off into never-
>          never land, unrelated to any physics, the real world, or
>          the _real_ theory of GR.]
>
> Tom Roberts     tjroberts@lucent.com
Re: Invalidity of General Theory of Relativity.
Re: Vacuum is a MENTAL state or CONDiTiON ..at least, in Tom's case.