Natural Science Forum / Physics / Particle Physics / February 2006

      In the General Theory of Relativity, the mathematical notion of
space-time continuum is implied to be an entity that can get physically
deformed and curved under the influence of gravitational field.  Let us
first examine the notion of space-time continuum.

Space-Time Continuum
--------------------
       Consider a very simple example of a particle motion along
X-coordinate. This motion can be represented through a distance-time
curve or trace on an X-T coordinate plane.  The velocity and
acceleration of the particle at any point along the X-axis will be
represented by the slope and curvature of the trace at that point. Let
us now consider a particle moving in a circular orbit in XY plane.  The
motion of this particle can be represented as a helical trace in a XY-T
coordinate space or manifold.  The velocity and acceleration
characteristics of this particle will be represented by the geometry of
helical trace in the XY-T manifold.  An important point to be noted
here is that the helical trace does not physically exist anywhere at
any time; it is just a mathematical or graphical representation of the
motion of a particle over a period of time.

       Similarly the motion of various particles in three-dimensional
physical space can be represented through suitable traces in a
four-dimensional XYZ-T space-time manifold.  An important point to be
noted here too is that four-dimensional traces of particles do not
physically exist anywhere at any time; these are just mathematical
representations of the motion of particles in three dimensional space
over a period of time. The 4-D geometry of the particle traces is just
a mathematical representation.  In the same way, a four-dimensional
space-time manifold XYZ-T does not physically exist anywhere at any
time; it is just a mathematical notion even if a logically convenient
one.  However, due to the logical convenience and common usage, the
mathematical notion of space-time manifold got assigned a more
sophisticated name of space-time continuum.  In geometrical terms of
course we may regard the space-time continuum as composed of a
continuum of 3-D space points and an orthogonal fourth dimension of
time continuum.  Whenever certain phenomenon is not varying with time,
we can study that phenomenon in 3-D space continuum independently.

Main Postulate of General Relativity
------------------------------------
       The main postulate of General Relativity is that the
gravitational phenomenon can be satisfactorily represented by 'suitably
adjusting' the metric properties of the space-time manifold.   For this
the metric coefficients of the space time manifold are required to
satisfy a set of partial differential equations (EFE) involving
energy-momentum tensor, whereby the non-rectilinear trajectories of
mass particles will transform into geodesics. Thus the study of
dynamical trajectories of mass particles in a gravitational field will
reduce to the study of geodesics in the space time manifold defined by
specified metric coefficients. After doing so, the main postulate could
be extended to imply that the gravitational field itself 'somehow'
modifies the metric coefficients of space-time manifold such that the
trajectories of mass particles 'naturally' turn out to be geodesics.

Line element 'ds' and the Space Metric
--------------------------------------
        The most fundamental concept underlying all the basic notions
of space is that of the absolute invariance of space points.  Say, in
any particular coordinate system X with origin at point O, let us
consider a particular space point P with coordinates (x^i).  If Q is
another space point in the neighborhood of P, then an infinitesimal
separation distance  ds  between the points  P  and  Q  is  given  by:

(ds)^2 = g_ij dx^i dx^j
..................... (1)

where g_ij are the metric coefficients in coordinate system x^i.  The
invariance of space points P, Q etc. implies that ds will remain
constant, even when the coordinate system is transformed from X to say
Y.  Such transformations that ensure the invariance of the line element
ds and hence the invariance of space points in general, are said to be
admissible transformations.  Obviously, if the coefficients g_ij
constitute the metric of Euclidean space, then all other coefficients
obtained through any admissible transformation of coordinates will also
represent the metric of same Euclidean space.

       On the other hand let us consider an arbitrary change in metric
coefficients   g_ij(x) in equation (1) to h_ij(x).   Here, by an
arbitrary change we mean a change that is brought about on any
considerations other than through an admissible transformation of
coordinates.  From equation (1), it can be easily seen that an
arbitrary change in metric coefficients  g_ij(x)  to say  h_ij(x),
without any corresponding admissible transformation of coordinate
parameters x^i, will lead to a change in separation distance ds between
points P  and Q  to say  ds'.  Therefore,

(ds')^2 = h_ij dx^i dx^j         ....... (2)

Obviously, whenever the separation distance between neighboring points
P and Q 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. The arbitrarily changed
coefficients h_ij(x) can be associated with the metric of a Riemannian
space, whereas the original coefficients g_ij(x) represented the metric
of Euclidean space.  Hence, whenever the metric [g_ij(x)] of Euclidean
space is changed to the metric [h_ij(x)] of Riemannian space, the
separation distance ds between two neighboring points P and Q will
change to the separation distance ds' as given by equations (1) and (2)
with the associated displacement vector field U defined at all points P
of the space continuum.    The displacement vector field U will fully
determine the deformed or strained state of the continuum as already
discussed in the previous article titled 'Infinitesimal Strain in
Continuous Media'.

Deformed State of the Space Continuum
-------------------------------------
       However, as per the fundamental postulate of General
Relativity, gravitational field of a gravitating body changes the
metric [g_ij(x)] of the surrounding  Euclidean space to the metric
[h_ij(x)] of Riemannian space in accordance with Einstein's Field
Equations.  With this change in the metric of space continuum, the
separation distance ds between any pair of neighboring space points P
and Q will also change to ds' as given by equations (1) and (2).   That
means all space points within the region of gravitational field will
experience relative displacements with the displacement vector field U
producing the strained state of the space in that region.  The strained
state is fully represented by the strain tensor E with its mixed
components e^i_j given by the covariant derivatives of the displacement
vector U as,

e^i_j  = (u^i_j +u^j_i)/2        .......(3)

In fact, subtracting (1) from (2)  we get an important relation between
the covariant components of the strain tensor e_ij and the metric
coefficients as,

(ds')^2 - (ds)^2 = {h_ij - g_ij} dx^i dx^j = 2.e_ij dx^i dx^j   ....
(4)

with   2.e_ij =  h_ij - g_ij      .....  (5)

where e_ij represent the covariant components of strain tensor E
expressed as functions of coordinate parameters x^i.

Gravitation Induced Metric and Associated Strain Tensor
-------------------------------------------------------
       Let us consider a spherical polar coordinate system with origin
at point O and the coordinate parameters r, theta and phi. Further for
the sake of convenience let us represent theta by letter q and phi by
letter p. The metric coefficients for this coordinate system in the
un-deformed or gravitation free space continuum are given as,

 g_rr = 1  ;  g_qq = r^2 ;  g_pp= r^2.Sin^2 (q)  ..... (6)

The arc element or the separation distance ds between two neighboring
space points P and Q in this region will be given by:

(ds)^2 =  g_rr (dr)^2 + g_qq (dq)^2 + g_pp (dp)^2
       = 1.(dr)^2 + r^2.(dq)^2 + r^2.Sin^2(q).(dp)^2    ..... (7)

Now let us assume that a spherically symmetric body of mass M and
radius r0, is located at the origin O of this coordinate system.  Due
to the gravitational field in its vicinity (i.e. r > r0 > 0), the
modified metric coefficients h_ij are given by the Schwarzschild
solution as:

h_rr = 1/(1 - 2GM/c^2r) ; h_qq = r^2 ;  h_pp= r^2.Sin^2 (q)  .... (8)

Thus the modified radial metric coefficient h_rr at any particular
space point P(r, q, p) can be taken as a function of M and its value in
the region under consideration is always greater than unity for M>0.
The arc element or the modified separation distance ds' between two
neighboring space point positions P' and Q' in this region will be
given by:

(ds')^2 = h_rr (dr)^2 + h_qq (dq)^2 + h_pp (dp)^2
       = (1/(1 - 2GM/c^2 r)).(dr)^2 + r^2.(dq)^2 + r^2.Sin^2
(q).(dp)^2   .... (9)

Therefore, using equation (5) we can compute the induced strain tensor
components e_ij from the modified metric coefficients h_ij as,
2 e_rr = h_rr - g_rr = (1/(1 - 2GM/c^2 r)) - 1     ...... (10)

with the factor 2GM/c^2 r <<1, equation (10) will get simplified to,

e_rr  =  GM/c^2 r           .......  (11)
and   e_qq = h_qq - g_qq = 0  ;  e_pp = h_pp - g_pp = 0   .....  (12)
and   e_qp =  e_pq = e_rq = e_qr = e_rp = e_pr = 0       ......  (13)

This set of strain tensor components constitutes the strain field
induced in the region of space continuum where the gravitational field
of M has modified the metric coefficients to h_ij.

Incompatibility of the Induced Strain Components
------------------------------------------------
        As discussed earlier, for complete description of the strained
state of the space continuum, we must be able to uniquely determine the
displacement vector field U from the specified strain tensor
components. For this the strain tensor components are required to
satisfy Saint Venant's integrability or compatibility conditions. The
displacement vector components obtained from the integration of partial
differential equations of the type (3), must be single valued, finite
and continuous functions of coordinates and must satisfy physical
constraints over the boundary of the region of space under
consideration.

       It can be easily seen that the radial strain components e_rr
given by equation (11), with all other components being zero, cannot
satisfy the required compatibility conditions.  In order to illustrate
and highlight this problem, let us consider the relative displacement
vector U that gives rise to the strain components  e_rr, e_qq and e_pp
.   If u^r is the only non-zero component of the displacement vector U,
then the strain components dependent on u^r are given by,

e_rr =  Du^r/Dr   ;   e_qq = u^r/r  and  e_pp = u^r/r    ....  (14)

Obviously, if the radial strain component e_rr is non-zero, the radial
displacement component u^r must be non-zero.  But once the radial
displacement component u^r is non-zero, the tangential strain
components e_qq and e_pp  cannot be zero. This precisely is the
incompatibility of the strain components  e_rr, e_qq and e_pp induced
by the static gravitational field of a spherically symmetric
gravitating body of mass M.  This incompatibility is not limited to the
strain components induced by the Schwarzschild metric of spherically
symmetric, static gravitational fields but is applicable to all strain
components induced by the Riemannian metric obtained from EFE.  In fact
one of the essential requirements imposed by the standard compatibility
conditions on strain components e_ij is that  the Riemann tensor
composed from e_ij must be a zero tensor.  This can be true only if
both metrics of equation (5), namely g_ij  and h_ij are Euclidean which
however contradicts the basic postulate of General Relativity.
Therefore, the specification of metric coefficients (8) as per the
Schwarzschild solution is physically invalid and unacceptable.

Further, even if we overlook the compatibility conditions for a while,
from equations (11) and (14) we get,

 e_rr = du^r/dr =  GM/c^2 r     ....... (15)

which appears to be easily integrable.   A simple integration of
equation (15) yields,

 u^r = (GM/c^2).Ln(r/r0)     .......... (16)

Apparently we seem to have obtained a nice looking solution for the
radial displacement component u^r in spite of the fact that the
compatibility conditions were not satisfied by the strain components.
Well, a closer look at equation (16) will show that this finite value
of u^r will give rise to a finite value of tangential strain components
e_qq and e_pp? (equation (14)) whereas these components are required
to be zero as per the modified metric h_ij (equation (12)) of the space
continuum. This accounts for the incompatibility.  Further the radial
displacement u^r given by equation (16) tends to infinity as r tends to
infinity.  This is invalid since as per physical constraints, u^r must
tend to zero when the gravitational field tends to zero at infinitely
large r.

Physical Invalidity of General Theory of Relativity
---------------------------------------------------
       In our sincere and most revolutionary attempt to represent the
gravitational phenomenon through geodesic equations in four dimensional
space-time manifold, we mainly focused on the geometry of particle
traces in the space-time continuum.  In the process we lost focus on
the geometry of space continuum itself which by definition consists of
a continuum of points.

        In GR, the 'structure' of physical space continuum has been
tampered with imprudently by hypothesizing that the metric coefficients
of space are affected by the presence of gravitational field as per the
Einstein's Field Equations. The Einstein's Field Equations require
the metric of space under gravitational influence to be inherently
Riemannian. However, as seen above, when the metric of space is
'changed' from Euclidean to Riemannian, the 'induced' deformation of
the space continuum gives rise to a mutually incompatible set of strain
components leading to discontinuities and voids in the locations of
neighboring space points. Hence the induced Riemannian metric
transforms the space continuum to space dis-continuum that is
physically invalid and unacceptable. The compatibility conditions of
strain components require the metric of space to be Euclidean even
under the influence of a gravitational field.

      Therefore, the main postulate of General Theory of Relativity is
found to be logically invalid, mainly on account of the 'incompatible
deformation of space' induced by the Riemannian metric associated with
gravitational field.  Secondly on account of the violation of the
essential compatibility conditions for the induced strain components,
the space continuum is implied to deform in to a dis-continuum of
points.  The General Theory of Relativity is thus physically invalid,
null and void, irrespective of any claimed utility, application or
validation of this 'theory'.

GSS
http://www.geocities.com/gurcharn_sandhu/index.html