Re: Tensors
From: Ken S. Tucker (dynamics_at_uniserve.com)
Date: 12/16/04
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Date: Wed, 15 Dec 2004 22:42:20 -0800
Anthony Smales <arse@hotmail.com> wrote in message
news:gtacnQZT9IDQIF3cSa8jmA@karoo.co.uk...
> This is what I understand tensors to be, please correct me if I am getting
> this wrong:
>
> I imagine a contravariant tensor to be a vector in a 3D space, hovering
> above a manifold - like a straight line representing the path 'as the crow
> flies' being shown above the curves of the mountains and valleys below.
>
> Question: Does the contravariant tensor actually touch the 'mountain
peaks'
> by default?
>
> As we move across the vector, we want to know information about the
surface
> below. We do this by generalising the vector into an object which gives
the
> (coordinate) information of the surface below as we move along it. This
> object is a contravariant tensor.
>
> The difference between the contravariant and the surface below is given by
> another tensor - the metric tensor, which varies as we move along the
> contravariant tensor.
>
> As we move along the contravariant tensor, the metric tensor traces out
> another tensor across the curved surfsace (or 'manifold'). This is a
> covariant tensor - it is actually in the curved surface.
>
> Of course, when applied to general relativity, the surface is the 4D
surface
> of space-time and the curves are due to gravitational warping of
spacetime.
>
> The tensors can be expanded to a field of such tensors - here the
> contravariant tensor field is a tangent space to the manifold (which is
the
> dual space of the contravariant tensor).
>
> Well, that's as I understand it. I may well have it all wrong, please let
me
> know :)
It's important to use precise language and have a good knowledge of
curvilinear CS's. Have a peek at,
http://arxiv.org/abs/gr-qc/9807044
it's a bit basic, but a start.
Ken S. Tucker
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