Re: Definition of stress tensor in continuum mech

On Oct 5, 7:13 am, Edward Green <spamspamsp...@xxxxxxxxxxx> wrote:
On Oct 3, 7:23 am, GSS <gurcharn_san...@xxxxxxxxx> wrote:
On Oct 3, 2:06 am, Edward Green <spamspamsp...@xxxxxxxxxxx> wrote:
On Oct 2, 7:21 am, GSS <gurcharn_san...@xxxxxxxxx> wrote:

I invite all readers to give their opinion, if any, whether the
spacetime continuum as modeled in GR is assumed to be a rigid or a
deformable continuum?

If it is assumed to be a deformable continuum, then how come that the
strained state of this continuum has never been examined in terms of
associated strain tensors?

No answers, but a few platitudes:

Clearly "opinion" is not worth to much here: what is needed is some
definition of what it means for a continuum to be deformable or rigid.
Opinion would enter more into a choice of what definition might be a
reasonable one to adopt: there may be more than one.

Let me attempt to distinguish between a rigid and a deformable
continuum of points. Let P be *any* point in this continuum. Let P1,
P2, P3, .... Pn be n points in the neighborhood of P. Let ds_1 be the
separation distance between points P and P1, ds_2 be the separation
distance between points P and P2, ..., ds_n be the separation distance
between points P and Pn. If these separation distances ds_1, ds_2,
ds_3,...., ds_n from point P to all of its neighborhood points, remain
constant and invariant with time under all circumstances, then the
continuum under consideration can be regarded as rigid. If under
certain circumstances, these separation distances change to say ds'_1,
ds'_2, ds'_3,...., ds'_n then the continuum under consideration can
be regarded as deformable.

Yes, that sounds reasonable. Applying this to GR requires some more
work. For one thing, if "spacetime" is your continuum, the thing is
invariant for all time. We would have to resort to some other
consideration to say that one part of spacetime is distended wrt
another, than "changes in time".

Yes, for the spacetime continuum your point is valid. The problem
however is that when we assume the spacetime to be a physical entity,
we tend to visualize it as a sort of 4-D 'Block'. It is in this
'Block' view of spacetime that "nothing moves". Actually the notion of
spacetime is a mathematical abstract notion used to model the dynamic
phenomenon in 3-D physical space. The 3-D space with all its material
content does not physically exist for all past and future times. One
convenient view to visualize the situation could be to regard the 3-D
space, with all its interacting material content, as 'moving' along
the time coordinate just as a train moves along its tracks!!

I have referred to the spacetime continuum "as modeled in GR". As per
GR some regions of spacetime are *flat* and some are *curved*
depending on the matter-energy distribution in the vicinity. When we
shift our focus from a flat region to a curved region of spacetime we
can notice the *change* in spacetime characteristics. What I have
shown is that in the GR model of spacetime, when we shift our focus
from a *flat* region to a *curved* region of spacetime we can notice
the *change in separation distances between the neighborhood spacetime
points* thereby implying deformation of the spacetime continuum.

If we restrict ourselves to "space", then again we are faced with some
difficulties regarding "changing in time". Which time? How do we track
a "point in space" as it evolves? Do we paint it? And of course we
must give meaning to "distance".

Meaning to "distance" is given by the very definition of metric of
space. As far as time is concerned, we can always use the
internationally accepted standard notion of time as UTC or TAI. While
considering the deformation of space as per the GR model, I have only
considered 'static gravitational fields' produced by spherically
symmetric bodies of matter and for which the Schwarzschild solution is
considered valid.

Since the separation distance dS between two neighborhood points of
the spacetime continuum does change under the influence of
gravitational field (as per GR), obviously the spacetime continuum is
assumed to be deformable in GR.

I don't know what that means. Assuming by "separation" you mean
"under the spacetime metric", nothing "changes" under the influence of
the gravitational field, since (as someone once put it) "nothing moves
in spacetime".

As explained above, here "change" is from *flat* region to *curved*
region of spacetime continuum. Let me illustrate the notion of change
under the influence of gravitational field, through an example.

When a surface is represented in the parametric form by 2-d surface
coordinates, the intrinsic geometry of the surface is described by its
2-d metric tensor. The Riemann tensor composed from the 2-d metric
components is non-zero for a curved surface and zero for a plane or
flat surface.

Let us consider a large circular metal ring of radius R, filled inside
with a plane thin film membrane (rubber membrane or soap film). The
intrinsic geometry of any small region of this thin film can be
represented by a 2-d flat metric with zero Riemann tensor.

Let us now imagine that we exert a steady pressure over a small
localized region of this film (say by impinging an air jet) in such a
way that a small hemispherical bubble of radius r<<R is formed in this
local region. The 2-d surface of this hemispherical bubble can be
represented by a modified 2-d metric with non-zero Riemann tensor.
**Obviously it is not difficult to visualize that the localized
hemispherical bubble induced by a steady external pressure is actually
a deformed (elongated/stretched) membrane with a curved surface in
comparison to the undeformed plane membrane in the surrounding
region.**

By moving the impinging air jet sideways, location of the
hemispherical bubble on the large plane membrane can be easily
shifted. The state of deformation of the curved membrane in comparison
to the plane membrane can be studied in detail by comparing the
Riemannian metric of the curved surface with the Euclidean metric of
the plane surface. It can be easily shown that all displacements
produced on the curved surface of the membrane are continuous and
finite.

Let us now come back to our familiar 3-d space continuum and consider
the gravitational field of our solar system. Kindly imagine the
trajectory of the barycenter of our solar system within our galaxy.
Let S be the location of the barycenter on this trajectory at certain
time t. Let V be the volume of the solar system within which the
gravitation of the sun is significant. Let D be the diameter of this
volume. Now imagine that at certain different time t+T the barycenter
of the solar system is located at a different point S1 on the
trajectory such that the distance SS1 is greater than 2D. It is quite
obvious that at time t, the volume V of space around point S1 is free
from any gravitational influence and hence as per GR it can be
regarded as 'flat' or Euclidean space. But at time t+T the volume V of
space around S1 is occupied by the solar system and hence as per GR
the metric of this space will get modified to Riemannian metric to
produce physical deformations in the space continuum.

The essential point I am stressing here is that a plane membrane
surface with Euclidean metric does get deformed into a curved surface
with Riemannian metric under the influence of external pressure. It is
precisely in the same way it has been postulated in GR that 'flat'
space with Euclidean metric gets deformed to a 'curved' space with
Riemannian metric under the influence of a steady state gravitational
field. Accordingly the corresponding deformation characteristics have
been examined in detail.

I'm not ridiculing your idea, just pointing out we have to be very
methodical about giving it meaning. My opinion is, yes, it's probably
possible to make a well-motivated case that something is behaving
"elastically" in GR; there may be more than one mapping.

No, you have missed the point. I have not indicated anywhere that
'something' is behaving "elastically" in GR. What I have actually
shown is that the space *continuum* as modeled in GR gets *deformed*
under the influence of a gravitational field to the state of *dis-
continuum* leading to the *invalidity of GR*. You may like to have a
re-look at this inference.

http://www.geocities.com/gurcharn_sandhu/pdf_art/invalidity_gr.pdf

GSS


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