| Topic: |
Science > Physics |
| User: |
"GSS" |
| Date: |
25 Jan 2008 09:15:46 AM |
| Object: |
Space Dynamics to Particle Structures |
1. Space-time Distortions or Space Dynamics
--------------------------------------------------
1.1 An important mathematical notion of spacetime curvature had
dominated the fundamental physics during the last century.
Mathematically however, the term 'spacetime curvature' implies a non-
zero value of the Riemann tensor computed from the metric coefficients
of the 4D spacetime manifold. In any gravitation free region of
space, the metric of 4D spacetime manifold may be represented by
g_ij(x) such that the corresponding Riemann tensor is zero and an
infinitesimal separation distance ds between two neighborhood
points P(x^i) and Q(x^i+dx^i) is given by:
(ds)^2 = g_ij dx^i dx^j ..................... (1)
However, in the same region of space under the influence of a
gravitational field, the metric of 4D spacetime manifold will be
represented by h_ij(x) (as per EFE) such that the corresponding
Riemann tensor is non-zero and an infinitesimal separation distance
ds' between the neighborhood points P' and Q' is given by:
(ds')^2 = h_ij dx^i dx^j ..................... (2)
1.2 To distinguish between a rigid and a deformable continuum of
space points, let P be any point in this continuum and P1, P2, ....
Pn be n points in the neighborhood of P. Let ds1 be the separation
distance between points P and P1, ds2 be the separation distance
between points P and P2, ..., dsn be the separation distance between
points P and Pn. If these separation distances ds1, ds2, .... dsn from
point P to all of its neighborhood points, remain constant under all
circumstances, then the continuum under consideration can be regarded
as rigid. If under certain circumstances, these separation distances
change to say ds1', ds2', .... dsn' then the continuum under
consideration will 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),
the spacetime continuum is assumed to be deformable in GR.
Considering only the spatial components of the metric tensor, it can
be shown that the separation distance ds between two neighborhood
points of the space continuum also changes under the influence of
gravitational field (as per GR). Therefore the space continuum is also
assumed to be deformable in GR.
1.3 Obviously, whenever the separation distance between
neighboring points P and Q in the space continuum, changes from ds to
ds' it implies a relative shift in the original positions of P and Q
to the changed positions say P' and Q' such that arc element P'Q' =
ds'. This relative shift in positions of P and Q to the changed
positions P' and Q' may be referred as the relative displacement of
these points. Specifically, the vector PP' may be defined as the
displacement vector U and the corresponding displacement of Q to Q'
will then be represented by the incremented displacement vector U
+dU . Hence, whenever the separation distance ds between two
neighboring points P and Q changes to ds' as given by equations (1)
and (2), the associated displacement vector field U will be defined at
all points P of the space continuum.
The deformation of the continuum can be said to be fully determined
when the displacement of every point P in the continuum is known or
uniquely determined. The existence of displacement vector U at every
point P as a function of position coordinates of P will constitute a
displacement vector field U in the continuum. The displacement vector
from point P(r) to P'(r') is given by the relation,
U = r' - r
= u^i a_i ...................... (3)
where u^i are the components of vector U and r, r' are the position
vectors of P and P'.
1.4 In general the displacement vector field U in the space
continuum will be a function of space coordinates and time. As such
the time dependent deformations of physical space, that could be
represented through a time dependent displacement vector field U, may
be described as the space-time distortions. The space and time
derivatives of U will correspond to the strain tensor field in the
space continuum. Since the above referred time dependent deformations
in the space continuum are reversible, the physical space continuum
can be assumed to be elastic in nature. Hence logically the
displacement vector and strain tensor fields will also be accompanied
by the corresponding stress tensor field in the physical space
continuum. The study of space-time distortions through the detailed
study of corresponding displacement vector, strain tensor and stress
tensor fields in the space continuum may be termed as space
dynamics. It is hoped that a detailed analytical study of space
dynamics can provide us a valuable insight into the fundamental
structures of various elementary particles and their interactions.
..........................
For further details on this topic, kindly refer to :
http://www.geocities.com/gurcharn_sandhu/paradigm/space_dynamics.pdf
GSS
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