Re: tensor help

On May 28, 8:21 pm, hetware <massl...@xxxxxxxxxxxx> wrote:
Eric Gisse wrote:
On May 28, 5:35 am, hetware <massl...@xxxxxxxxxxxx> wrote:
Eric Gisse wrote:
On May 27, 11:11 pm, hetware <massl...@xxxxxxxxxxxx> wrote:
Well, not completely. GR really /is/ related to elasticity.
Einstein's field equation(s) are closely analogous to the relationship
between stress and strain.

The technology GR uses is related to elasticity - stress tensors and
such, and mixed with a healthy dose of differential geometry.

The Einstein field equations are closely analogous to the generalized
Hooke's law (which gives the "field equations" relating stress and strain.
In elasticity these equations relate geometry to stress-energy. In GR the
field equations related geometry to stress-energy.

Generalized to _what_ though? The only guess that makes sense to me
[always a dangerous concept] would be F = K.x where K is a constant
tensor and x is a column vector. No matter what, force will be
proportional to displacement in some way - which isn't even remotely
close to how GR behaves.

Yes, geometry is related to stress energy - but I simply don't see the
parallel between elasticity, or Hooke's law, and general relativity
outside of some common mathematical technology.


But I completely disagree with the statement that GR is related to
elasticity. GR is no more related to elasticity than it is related to
electrodynamics.

Yes, the electromagnetic stress-energy tensor is the component of the GR
momenergy tensor contributed by the electromagnetic field.

Electrodynamics uses tensors too - the Maxwell stress
tensor in the non-covariant formalism or the Faraday tensor in the
covariant formalism.

That is simply not a meaningful statement.

Well, it's kind of a chicken and egg thing. Many authors define tensors
in terms of how their components behave under coordinate transformations,
so
it makes sense to assume that we are talking about such transformations.
I will say, many authors also talk in terms of covariant, contravariant
and
mixed tensors, which IMO is quite wrong. Though your intent may have
been
consistent with common usage, I stand by my critique. As for
Potter, "Lunar Society" sums it up nicely.

There is a reason for that

Yes there is.

- look at Symon for example. He makes _no_
distinction between covariant/contravariant tensors

That would be the correct thing to do.

Only with tensors defined on the R^n manifold - it simply renders the
distinction unnecessary, it doesn't make it truly go away.

You will only make mistakes if you try that in Minkowski space.


because the metric
which is used to raise/lower indicies is the identity tensor.

For the wrong reason.

What makes it wrong?


I can't
recall whether he made that explicit though, but I wouldn't be
surprised if it was swept under the rug or relegated to a one line
explanation.

Covariant and contravariant are not characteristics of invariant geometric
objects such as tensors. Components can be either covariant or
contravariant, and the covariant expression of an invariant tensor can be
changed to the contravariant form without effecting the tensor.

This is close enough to how I understand it.


That seems to be a frequent occurance when tensors are used in
classical mechanics. However, such behavior cannot be gotten away with
when the metric is something else - the distinction between co and
contravariant is always [should be] made in such cases.

That has nothing to do with whether the invariant tensors are covariant or
contravariant.

Yes it does.

In classical mechanics, there is no distinction made between covariant
or contravariant anything. I claim this is because the metric tensor,
which is used to change an object's expression to covariant or
contravariant form, is the identity tensor.


While I have seen tensors defined in many ways, it always ends up
based upon how they transform under coordinate transformations.

A tensor is a geometric object which is independent of any specific
coordinate system used to represent it, and whose components transform
according to stated transformation rules - typically multiplication by the
matrix of partial derivatives relating coordinate systems.

That's how I have seen tensors defined - through how they transform.


--http://www.vho.org/GB/c/DC/gcgvcole.htmlhttp://www.vho.org/GB/Books/dth/http://www.germarrudolf.com/http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm


.

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