sci.physics

On Oct 3, 2:06 am, Edward Green wrote:
> On Oct 2, 7:21 am, GSS 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.

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.

> Given such a definition, then yours is a technical question, not a
> matter of opinion.
>
> In terms of what might be a reasonable definition, we might wonder
> whether we want to ask the question of a four dimensional something,
> or a three dimensional something. There may be no natural definition
> of universal time, but I'm not convinced there might not be a natural
> idea of a three dimensional continuum which does its thing locally,
> and dynamically, and doesn't give a figo whether or not we want to
> search for a universal t coordinate.
>
Yes, instead of starting with a 4-D spacetime continuum, it is quite
appropriate to consider its subset, the 3-D space continuum, as
modeled in GR, to examine its deformation characteristics.

> Secondly, even given a sense of "deformation", by asking for an
> accompanying "stress" you are taking things a notch further. Stress
> and strain go together in ordinary material: it's plausible that if
> there is something like strain in space then there is also something
> like stress also, and something like a constitutive relation, but
> merely plausible (whether the whole analogical package can be put
> through might be a useful test of "reasonableness" in the
> identification).
>
No, I didn't ask for an accompanying "stress". I only mentioned strain
tensors associated with deformation of the continuum. We could take
the things a notch further at a later stage.

> All these something-likes are things to be searched for in the math of
> General Relativity, or in the class of maths which are sufficiently
> something-like General Relativity to reproduce its success. I guess we
> may offer our "opinion" about whether this might be a worthwhile
> search. I think it would.

In this regard, you may kindly examine the article on the strained
state of a continuum at,

/gurcharn_sandhu/pdf_art/

Last paragraph of this article is reproduced below for information.
"As shown above, the notion of deformed or strained state of the
continuum under study is derived from the variability or invariance of
arc element ds. Whenever the arc element ds changes over to ds' under
certain situations, the changed state of the continuum will be termed
the deformed or strained state. The strained state can be considered
fully defined or fully determined once we know or uniquely determine
the displacement vector field at all points of the continuum. The
strained state can also be defined through specification of strain
tensor components provided these components satisfy Saint Venant's
compatibility equations. Finally the strained state can also be
defined through specification of modified metric coefficients from
which the required strain tensor components can be computed subject to
the compatibility conditions. However, the compatibility conditions
require that the modified metric must be Euclidean to ensure that the
resulting strained state of the continuum corresponds to smooth,
finite and continuous displacement components and to avoid
discontinuities within the continuum. This fact is of crucial
importance for examining the validity of the current mathematical
model of General Relativity."

Further the validity of the current mathematical model of GR has also
been examined at,

/gurcharn_sandhu/pdf_art/

GSS