Re: Warped Vacuum?

Dr ***:
>"Bilge" <dubious@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> wrote in message

> General relativity says that there are no ``straight'' lines.
>dr
>I think that was a point I was trying to make {:-) and length was modified
>by gravity and velocity. So that straight became a purely subjective
>statement.

It's not a purely subjective statement.

>b
>Geodesics are the ``straightest'' lines possible. On the surface
>of a sphere, for example, the geodesics are great circles. These
>_are_ unambiguous and define extremal paths on the sphere, which
>is why pilots use them for navigation. Locally, the paths ``appear''
>straight, but since great circles intersect at exactly 2 points,
>those paths cannot satisfy euclid's parallel postulate.
>dr
>So circles could be defined as warped straight lines.?

You start with the definition of arc length:

s = \integral sqrt(g_ab dx^a dx^b)

In three dimensional euclidean space, that means,

s = \integral sqrt(dx^2 + dy^2 + dz^2)

You define an _affine_ parameter to characterize the arc length, call it
dl, so that you have (for the more general case):

s = \integral sqrt(g_ab (dx^a/dl) (dx^b/dl)) dl

You then find the variation of that integral that gives an
extremal length (max or min). You do that by making the replacements,
x^u -> x'^u = x^u + \delta x^u, g_uv -> g'_uv = g'_uv + \delta g'_uv
to obtain s'. Then you require that s - s' = 0 and solve for the
parameters. The result is the geodesic equation.

If you don't allow g_uv to vary, then you get what we call a straight
line. If you allow g_uv to vary, then the geodesics you get depend upon
the constraints on the metric.

>Perhaps if you cut a great circle in half along its length it would satisfy
>Euclid ?{:-)
>I think this was about your objection to the word warped in the title of
>this thread ? our perhaps we are debating about nothing or a vacuum which
>you conceptualise as nothing? but that I conceptualise as different
>vacuum/dielectric states but untimely as nothing.

My objection is to thinking that space and time have any pre-ordained
``way to be'' and that a departure from the pre-ordained implies the
manipulation of some ``thing.'' Space and time are what they are and
flat is nothing more than a very specific case of curved. The natural
state of affirs is the most general. Lots of physicists would really
like an excuse to assume the universe is flat so that they could construct
gravity like the other four forces, but no excuse for that assumption
exists. So, they don't assume it.



.

Relevant Pages

  • Re: Warped Vacuum?
    ... >>by gravity and velocity. ... So that straight became a purely subjective ... >>of a sphere, for example, the geodesics are great circles. ...
    (sci.physics.relativity)
  • Re: Warped Vacuum?
    ... So that straight became a purely subjective ... could be seen as exiting in a mind or set of minds. ... >of a sphere, for example, the geodesics are great circles. ...
    (sci.physics.relativity)
  • Three parallels do not cut - one should assume, shouldnt one?
    ... Discussing geometry with a pure mathematician is really difficult, ... At the other end of specialisation we find straight lines and circle- ... so in these cases geodesics are not straight lines. ... A plane has become very important to the scientists, ...
    (sci.math)
  • Thraed reinstation:Three parallels do not cut - one should assume, shouldnt one?
    ... Three parallels do not cut - one should assume, ... At the other end of specialisation we find straight lines and circle- ... so in these cases geodesics are not straight lines. ... A plane has become very important to the scientists, ...
    (sci.math)
  • Re: Cantor Confusion
    ... And maybe that's why they're taxicab drivers. ... you get these nonsensical definitions for squares and circles. ... Are circles composed of straight lines? ... Learn some mathematics for a change. ...
    (sci.math)