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{\bf The Relativistic Theory of the Asymmetric Field}\\
\hspace*{1.28cm}by Stephen Abbott, Isle of Wight
\bigskip
{\it Abstract\/}
The fundamental concepts of General Relativity have been reexamined. The
heuristic value of the
concept of general covariance in seeking new laws of nature is acknowledged,
in that any new
tensor, which arises naturally within the development, is deemed as a
candidate for representing a
physical quantity. The inclusion of the anti-symmetric part of the metric
connection is seen to lead
to field equations which do not require the ad hoc introduction of an energy
tensor of matter. The
natural appearance of this tensor shows that the field equations of General
Relativity may be derived
from a geometrical tensor identity. The theory indicates, even more than
ever, how the physics which
can occur in the Universe coincides with the geometry of space-time.
\bigskip
\bigskip
{\it Keywords\/}
Christoffel tensor, geodesic equation, transposition invariance, curvature
tensor,
cosmological constant, energy tensor, field theory.
\bigskip
\bigskip
2000 MSC 83C05
\newpage
\bigskip
{\it Introduction\/}
One of the earliest references on the foundations of General Relativity is
Einstein's book [1]. Here, as a continuation of his work,
the expansion of the equations of the gravitational field has led to new
possibilities for using a
4-dimensional field theory as a means of unification. Previously, systems
with more than four
dimensions have been given more emphasis since it seemed that
electromagnetism could not be
included in a natural way into the 4-dimensional system of gravitational
equations. Unification
methods which contract the metric connection to a trace alone will lose much
of the physical
structure which may be present. Within this field theory, however, the
unification of electromagnetism
and gravitation arises from the coupling of the metric connection and the
four-potential. The expanded
system of equations must eventually be shown to be consistent with the known
facts of physical cosmology [2].
\bigskip
{\it Remarks on the Displacement Field\/}
The `displacement field' is also known as the `metric connection' and the
anti-symmetric part of
the displacement field will be represented below by a capital `sigma'. This
anti-symmetric part of
the connection transforms as a tensor which suggests that it may have some
physical existence in
its own right, other than being just a part of the connection. This is the
basis of the current
work, however, it differs from Einstein's non-symmetric theory, in that the
symmetry of the
metric tensor itself is retained. The heading for this section is taken from
a sub-heading in the
main reference, where it is remarked that the lower indices of the
connection, in the equation for
the increment of a vector, `play quite different roles' [1]. This indicated
that the requirement of
symmetry for the metric connection may have been too restrictive.
\bigskip
{\it The Asymmetric Connection\/}
The symbol `gamma' has been retained for the symmetric part of the
displacement field and the
symbol `sigma' has been introduced for the anti-symmetric part. The word
`asymmetric' is used to
distinguish this connection from the `non-symmetric' connection used in
Appendix II of the main
reference. The theory of the `non-symmetric field' introduced a metric
tensor [3] which was also
non-symmetric. The metric tensor occurs in the separation formula for a pair
of events in space-time
and it is difficult to see why the quadratic terms should not be symmetric
in the coordinate
indices, so here the symmetry of the metric tensor itself is regarded as
essential. The covariant
derivative of a covariant vector in the asymmetric theory is now given by
\begin{equation}
B_{\mu ;\nu }=B_{\mu ,\nu }-\left(\Gamma ^{\alpha }_{\mu \nu }+\Sigma
^{\alpha }_{\mu \nu } \right)B_{\alpha }
\end{equation}
\vspace{0.30cm}
where we have used the semicolon to represent covariant differentiation and
comma subscript
`nu' to represent partial differentiation with respect to x subscript `nu'.
\bigskip
{\it The Natural Anti-Symmetric Tensor\/}
In order to generate the simplest
anti-symmetric tensor which
can be formed from the covariant derivative,
the symmetric part of the tensor
in equation (1) is removed by forming a difference.
\begin{equation}
\overline{\phi }_{\mu \nu }=B_{\mu ;\nu }-B_{\nu ;\mu }
\end{equation}
\vspace{0.30cm}
This, however, retains the anti-symmetric part of the connection, while the
symmetric part
cancels.
\begin{equation}
\overline{\phi }_{\mu \nu }=B_{\mu ,\nu }-B_{\nu ,\mu }-2\Sigma ^{\alpha
}_{\mu \nu }B_{\alpha }
\end{equation}
\vspace{0.30cm}
The first result for the asymmetric theory is to note the presence of the
third term in (3). This term
occurs in the same position as gauge theory extensions to the
electromagnetic field [4]. This
method of tensor formation from the four-potential was used to include the
electromagnetic field
into General Relativity. However, the third term is absent in the symmetric
theory and there is no
coupling between the metric connection and electromagnetism unless it is
introduced to the field
equations as an energy tensor. The expression will be rewritten using the
traditional notation for
the electromagnetic field.
\hspace*{1.28cm}\vspace{0.30cm}
\begin{equation}
\overline{\phi }_{\mu \nu }=\phi _{\mu \nu }-2\Sigma ^{\alpha }_{\mu \nu
}B_{\alpha }
\end{equation}
\vspace{-0.30cm}\hspace*{1.28cm}
\vspace{0.30cm}
\begin{displaymath}
\phi _{\mu \nu }=B_{\mu ,\nu }-B_{\nu ,\mu }
\end{displaymath}
\bigskip
{\it Maxwell's Equations\/}
The second of Maxwell's equations can be written as
\begin{equation}
\frac{\partial \phi _{\mu \nu }}{\partial x_{\sigma }}+\frac{\partial \phi
_{\nu \sigma }}{\partial x_{\mu }}+\frac{\partial \phi _{\sigma \mu
}}{\partial x_{\nu }}=0
\end{equation}
\bigskip
With the usual identification of the electric and magnetic vector components
with elements of the
electromagnetic field tensor [4], this equation will still allow the `curl'
of the electric field to equal
the rate of change of the magnetic field. However, the other `curl' equation
is usually derived
from the divergence of the field. We may now speculate that, although
Maxwell's equations were
derived from empirical observations, the observable quantities may not have
been of a fundamental
nature. In the asymmetric theory, we now take equation (4) as an expression
for the natural field
tensor, and set its divergence to zero.
\begin{equation}
\frac{\partial \overline{\phi }_{\mu \nu }}{\partial x_{\nu
}}=\frac{\partial \phi _{\mu \nu }}{\partial x_{\nu }}-\frac{\partial
\left(2\Sigma ^{\alpha }_{\mu \nu }B_{\alpha } \right)}{\partial x_{\nu }}=0
\end{equation}
\bigskip
The above equation will allow the `curl' of the magnetic field to equal the
rate of change of the
electric field plus the current vector, if we now identify the current with
the term in `sigma'.
\begin{equation}
J_{\mu }=\frac{\partial \left(2\Sigma ^{\alpha }_{\mu \nu }B_{\alpha }
\right)}{\partial x_{\nu }}
\end{equation}
\vspace{0.30cm}
These equations begin the unification of electromagnetism and gravitation
which will be
completed if some deeper unification is possible with the `sigma' tensor. It
would be important to
show that the current can be represented as above in Minkowski space-time.
This is left as an
open question for now as we move on to consider gravitation.
\bigskip
{\it The Geodesic Equation\/}
In the symmetric theory this equation defines particle paths under the
influence of inertia and
gravitation alone [5]. The generalization to the asymmetric field is given
by
\begin{equation}
\frac{d^{2}x^{\mu }}{ds^{2}}+\left(\Gamma ^{\mu }_{\alpha \beta }+\Sigma
^{\mu }_{\alpha \beta } \right)\frac{dx^{\alpha }}{ds}\frac{dx^{\beta
}}{ds}=0
\end{equation}
\vspace{0.30cm}
When expanded, the term containing `sigma' is found to be equal to zero by
interchanging `alpha'
and `beta'. The geodesic equation retains its original form although, as
will be seen below, the
symmetric `gamma' is not the same `gamma' as in General Relativity.
\begin{equation}
\frac{d^{2}x^{\mu }}{ds^{2}}+\Gamma ^{\mu }_{\alpha \beta }\frac{dx^{\alpha
}}{ds}\frac{dx^{\beta }}{ds}=0
\end{equation}
\bigskip
\bigskip
{\it The Covariant Derivative of the Metric\/}
This is the starting point for attempts to express the `gamma' in terms of
the metric tensor. For
the symmetric theory the method has been described in many standard texts
where the covariant
derivative of the metric tensor is set to zero.
\begin{equation}
\frac{\partial g_{\mu \nu }}{\partial x_{\alpha }}-g_{\mu \beta }\Gamma
^{\beta }_{\nu \alpha }-g_{\nu \beta }\Gamma ^{\beta }_{\mu \alpha }=0
\end{equation}
\vspace{0.30cm}
The covariant derivative is now given using the fully asymmetric connection.
\begin{equation}
\frac{\partial g_{\mu \nu }}{\partial x_{\alpha }}-g_{\mu \beta
}\left(\Gamma ^{\beta }_{\nu \alpha }+\Sigma ^{\beta }_{\nu \alpha }
\right)-g_{\nu \beta }\left(\Gamma ^{\beta }_{\mu \alpha }+\Sigma ^{\beta
}_{\mu \alpha } \right)=0
\end{equation}
\vspace{0.30cm}
In this case it is not possible to express the `gamma' or the `sigma' in
terms of the metric tensor alone.
However, if we follow the same method as before, using a cyclic interchange
of indices with some
addition and subtraction of equations we obtain,
\begin{equation}
\frac{\partial g_{\nu \alpha }}{\partial x_{\mu }}+\frac{\partial g_{\alpha
\mu }}{\partial x_{\nu }}-\frac{\partial g_{\mu \nu }}{\partial x_{\alpha
}}=+2g_{\alpha \beta }\Gamma ^{\beta }_{\mu \nu }+2g_{\nu \beta }\Sigma
^{\beta }_{\alpha \mu }+2g_{\mu \beta }\Sigma ^{\beta }_{\alpha \nu }
\end{equation}
\vspace{0.30cm}
This indicates that the `gamma' should now be expressed in terms of the
familiar derivatives of the
metric tensor and a new symmetric tensor derived from the `sigma' and the
metric.
\bigskip
{\it The Symmetric Connection and the Christoffel Tensor\/}
The symmetric part of the connection is the sum of the Christoffel tensor of
the second kind and
the new tensor.
\begin{equation}
\Gamma ^{\sigma }_{\mu \nu }=\frac{1}{2}g^{\sigma \alpha
}\left(\frac{\partial g_{\nu \alpha }}{\partial x_{\mu }}+\frac{\partial
g_{\alpha \mu }}{\partial x_{\nu }}-\frac{\partial g_{\mu \nu }}{\partial
x_{\alpha }} \right)+S^{\sigma }_{\mu \nu }
\end{equation}
\vspace{0.30cm}
where the additional contribution to the symmetric part of the connection is
given by
\begin{equation}
S^{\sigma }_{\mu \nu }=g^{\sigma \alpha }\left(g_{\nu \beta }\Sigma ^{\beta
}_{\mu \alpha }+g_{\mu \beta }\Sigma ^{\beta }_{\nu \alpha } \right)
\end{equation}
\vspace{0.30cm}
This is a significant result. It is telling us that, if we try to introduce
an anti-symmetric part to the
connection, while retaining a symmetric metric tensor, then there will be a
symmetric contribution
to the metric connection. This also allows the geodesic equation to retain
its original form so that,
although the anti-symmetric part of the connection cancels explicitly in
equation (8), there is still a
new contribution to the geometry of the geodesic paths [6].
\bigskip
{\it The Contracted Symmetric Connection\/}
Contracting equation (13) gives
\begin{equation}
\Gamma ^{\nu }_{\mu \nu }=\frac{1}{2}g^{\nu \alpha }\frac{\partial g_{\nu
\alpha }}{\partial x_{\mu }}+\delta ^{\alpha }_{\beta }\Sigma ^{\beta }_{\mu
\alpha }+g^{\nu \alpha }g_{\mu \beta }\Sigma ^{\beta }_{\nu \alpha }
\end{equation}
\vspace{0.30cm}
The third term on the right-hand side is found to be equal to zero by
interchanging `nu' and
`alpha'. The second term becomes what can be described as `the trace of the
sigma tensor'. It
appears to be related closely to the determinant of the metric.
\begin{equation}
\Gamma ^{\nu }_{\mu \nu }=\frac{1}{\sqrt{g}}\frac{\partial
\sqrt{g}}{\partial x_{\mu }}+\Sigma ^{\alpha }_{\mu \alpha }
\end{equation}
\vspace{0.30cm}
It is now confirmed that the dimensions of `sigma' are those of reciprocal
length.
\bigskip
{\it The Transposition Invariant Derivative\/}
We now introduce transposition of the lower indices which will be required
later for evaluation of
curvature. Firstly, we write the partial derivative of equation (16) with
respect to x, using the
comma notation for partial derivatives.
\begin{equation}
\Gamma ^{s}_{is,k}=\frac{\partial }{\partial
x_{k}}\left(\frac{1}{\sqrt{g}}\frac{\partial \sqrt{g}}{\partial
x_{i}}+\Sigma ^{s}_{is} \right)
\end{equation}
\vspace{0.30cm}
Transposing indices i and k, gives `gamma' tilde.
\begin{equation}
\tilde{\Gamma }^{s}_{is,k}=\frac{\partial }{\partial
x_{i}}\left(\frac{1}{\sqrt{g}}\frac{\partial \sqrt{g}}{\partial
x_{k}}+\Sigma ^{s}_{ks} \right)
\end{equation}
\vspace{0.30cm}
A transposition invariant derivative of the contracted symmetric connection
is given by
\begin{equation}
\Gamma ^{s}_{is,k}+\tilde{\Gamma }^{s}_{is,k}=2G_{ik}+H_{ik}
\end{equation}
\vspace{0.30cm}
where the two tensors on the right hand side are given by equations (20) and
(21)
\begin{equation}
G_{ik}=\frac{\partial }{\partial
x_{i}}\left(\frac{1}{\sqrt{g}}\frac{\partial \sqrt{g}}{\partial x_{k}}
\right)
\end{equation}
\begin{equation}
H_{ik}=\frac{\partial \Sigma ^{s}_{is}}{\partial x_{k}}+\frac{\partial
\Sigma ^{s}_{ks}}{\partial x_{i}}
\end{equation}
\vspace{0.30cm}
The latter quantity is a transposition invariant derivative of the sigma
tensor trace.
\bigskip
{\it The Curvature Tensor\/}
The derivation of the contracted curvature tensor is independent of the
symmetry properties of the
connection. For the asymmetric connection we write the negative as
\begin{equation}
-K_{ik}=\left(\Gamma ^{s}_{ik}+\Sigma ^{s}_{ik} \right)_{,s}-\left(\Gamma
^{s}_{is}+\Sigma ^{s}_{is} \right)_{,k}-\left(\Gamma ^{s}_{it}+\Sigma
^{s}_{it} \right)\left(\Gamma ^{t}_{sk}+\Sigma ^{t}_{sk}
\right)+\left(\Gamma ^{s}_{ik}+\Sigma ^{s}_{ik} \right)\left(\Gamma
^{t}_{st}+\Sigma ^{t}_{st} \right)
\end{equation}
\bigskip
{\it Transposition Invariance\/}
We now appeal to Einstein's comments on transposition invariance [1]. This
is the parting of the
ways for the non-symmetric theory and the asymmetric theory. By persisting
with the calculation
in terms of the `gamma' and `sigma' we will find a transposition symmetric
tensor which may be
used in field equations. Since the connection has both symmetric and
anti-symmetric parts, to
form a transposition symmetric tensor from the contracted curvature tensor
we take
\begin{equation}
-2M_{ik}=-\left(K_{ik}+\tilde{K}_{ik} \right)
\end{equation}
\bigskip
Rather than give the explicit version of this in terms of `gamma' and
`sigma', the method of
evaluation of some of the terms which occur in the expansion will be given.
\newpage
To evaluate some of the terms containing products, we consider various
values of a parametric
tensor `pi' and its transpose `pi' tilde.
\bigskip
If
$ \Pi _{ik}=\Gamma ^{s}_{it}\Gamma ^{t}_{sk}$
\bigskip
then
$ \tilde{\Pi }_{ik}=\Gamma^{s}_{kt}\Gamma ^{t}_{si}=\Gamma ^{s}_{ti}\Gamma
^{t}_{ks}=\Gamma ^{s}_{it}\Gamma ^{t}_{sk}$
\bigskip
and
$ \tilde{\Pi }_{ik}=\Pi_{ik} $
\vspace{-0.21cm}
\bigskip
This product is transposition symmetric, so the value is doubled in the
expansion of equation (23).
\bigskip
Also,
$ \Pi _{ik}=\Gamma ^{s}_{ik}\Gamma ^{t}_{st}$
is transposition symmetric since `gamma' is symmetric.
\bigskip
If
$ \Pi _{ik}=\Sigma ^{s}_{it}\Gamma ^{t}_{sk}+\Sigma ^{t}_{sk}\Gamma
^{s}_{it}$
then
$ \tilde{\Pi }_{ik}=\Sigma ^{t}_{ks}\Gamma ^{s}_{ti}+\Sigma ^{s}_{ti}\Gamma
^{t}_{ks}$
\bigskip
$ \tilde{\Pi} _{ik}=-\Sigma ^{t}_{sk}\Gamma ^{s}_{it}-\Sigma ^{s}_{it}\Gamma
^{t}_{sk}$
and
$ \tilde{\Pi }_{ik}=-\Pi _{ik}$
so the terms cancel in the expansion of equation (23).
\bigskip
If
$ \Pi _{ik}=\Sigma ^{s}_{ik}\Gamma ^{t}_{st}+\Sigma ^{t}_{st}\Gamma
^{s}_{ik}$
then \vspace{0.21cm}
$ \Pi _{ik}+\tilde{\Pi }_{ik}=2\Sigma ^{t}_{st}\Gamma ^{s}_{ik}$
\bigskip
If
$ \Pi _{ik}=\Sigma ^{s}_{it}\Sigma ^{t}_{sk}$
then
$ \tilde{\Pi} _{ik}=\Sigma ^{s}_{kt}\Sigma ^{t}_{si}=\Sigma ^{s}_{ti}\Sigma
^{t}_{ks}$
and
$ \tilde{\Pi }_{ik}=\Pi _{ik}$
\bigskip
If
$ \Pi _{ik}=\Sigma ^{s}_{ik}\Sigma ^{t}_{st}$
then
$ \tilde{\Pi }_{ik}=\Sigma ^{s}_{ki}\Sigma ^{t}_{st}=-\Sigma ^{s}_{ik}\Sigma
^{t}_{st}$
$ \tilde{\Pi }_{ik}=-\Pi _{ik}$
\bigskip
To evaluate some of the differential terms, we consider various values of a
parametric tensor
`delta' and its transpose `delta' tilde.
\bigskip
If
$ \Delta _{ik}=\Gamma ^{s}_{ik,s}$
then
$ \tilde{\Delta }_{ik}=\Gamma ^{s}_{ki,s}=\Gamma ^{s}_{ik,s}$
and
$ \tilde{\Delta }_{ik}=\Delta _{ik}$
so the value is doubled in the
expansion of equation (23).
\bigskip
If
$ \Delta _{ik}=\Gamma ^{s}_{is,k}$
then
$ \tilde{\Delta} _{ik}=\Gamma ^{s}_{ks,i}$
and using equation (19),
$ \Delta _{ik}+\tilde{\Delta} _{ik}=2G_{ik}+H_{ik}$
\bigskip
If
$ \Delta _{ik}=\Sigma ^{s}_{ik,s}$
then
$ \tilde{\Delta }_{ik}=-\Delta _{ik}$
\bigskip
If
$ \Delta _{ik}=\Sigma ^{s}_{is,k}$
then, using equation (21),
$ \Delta _{ik}+\tilde{\Delta} _{ik}=H_{ik}$
a tensor which will play a part in
the field equations.
\newpage
{\it The Transposition Symmetric Curvature Tensor\/}
We now return to the expansion of equation (23) of the previous section, and
using the results of
that section we simplify,
\begin{displaymath}
-2M_{ik}=2\Gamma ^{s}_{ik,s}-\left(2G_{ik}+H_{ik}\right)-2\Gamma
^{s}_{it}\Gamma ^{t}_{sk}+2\Gamma ^{s}_{ik}\Gamma ^{t}_{st}-H_{ik}-2\Sigma
^{s}_{it}\Sigma ^{t}_{sk}+2\Sigma ^{t}_{st}\Gamma ^{s}_{ik}
\end{displaymath}
\vspace{0.30cm}
Collecting some of the familiar terms containing the `gamma',
\begin{equation}
M_{ik}=\overline{R}_{ik}+H_{ik}+P_{ik}-Q_{ik}
\end{equation}
\begin{equation}
\overline{R}_{ik}=-\Gamma^{s}_{ik,s}+G_{ik}+\Gamma ^{s}_{it}\Gamma
^{t}_{sk}-\Gamma ^{s}_{ik}\Gamma ^{t}_{st}
\end{equation}
\begin{equation}
H_{ik}=\Sigma ^{s}_{is,k}+\Sigma ^{s}_{ks,i}
\end{equation}
\begin{equation}
P_{ik}=\Sigma ^{s}_{it}\Sigma ^{t}_{sk}
\end{equation}
\begin{equation}
Q_{ik}=\Sigma ^{t}_{st}\Gamma ^{s}_{ik}
\end{equation}
\bigskip
Equation (25) is recognized as the contracted curvature tensor
found in the field equations of General Relativity if we use equation (20),
however the `gamma'
are now defined by equation (13).
\bigskip
{\it Field Equations\/}
There is some freedom of choice in selecting the form of field
equations. We could follow Einstein's heuristic approach [1], and introduce
the `energy tensor of
matter'. Alternatively, we can just form a generally covariant equation from
the transposition
symmetric curvature tensor, and explore the consequences.
\begin{equation}
M_{ik}-\frac{1}{2}g_{ik}M=0
\end{equation}
\vspace{0.30cm}
where the scalar of the symmetric curvature is given by
\begin{equation}
M=\overline{R}+H+P-Q
\end{equation}
\vspace{0.30cm}
and the four scalars on the right-hand side are found by contracting
equations (25) to (28) with
the contravariant metric tensor.
\vspace{0.30cm}
Any new field equations must reduce to those of General Relativity, as a
zero order test, when
the metric connection is completely symmetric so we attempt to recover the
cosmological equations
by setting
\begin{equation}
\overline{R}_{ik}-\frac{1}{2}g_{ik}\overline{R}+\Lambda g_{ik}=-\kappa
T_{ik}
\end{equation}
\vspace{0.30cm}
where the bar indicates that the functions depend upon a symmetric part of
the connection which
has been augmented, as in equation (13). The cosmological constant is then
given by
\begin{equation}
\Lambda =-\frac{1}{2}\left(H+P-Q \right)
\end{equation}
\vspace{0.30cm}
This equation shows how the cosmological constant will be related to the
fundamental constants
of nature once the scalars for the energy tensor have been identified. The
algebraic sum of scalars
with opposite signs implies that the cosmological constant may change in
sign as the Universe
evolves.
\vspace{0.30cm}
The energy tensor has been included in equation (31) to show where it must
be introduced within
the equations of General Relativity in order to represent the sources of
gravitation. Here, the term
represents forms of energy which have not been introduced explicitly within
the unification.
\begin{equation}
\kappa T_{ik}=H_{ik}+P_{ik}-Q_{ik}
\end{equation}
\vspace{0.30cm}
We have identified equation (33) as an expression for the energy tensor of
the asymmetric field. It
may yet transpire that we have the liberty to introduce particular forms of
the energy tensor in
order to study specific problems.
\bigskip
{\it Conclusion\/}
Here we have a relativistic field theory which does not require the ad hoc
introduction of either
the current vector or the energy tensor of matter. The natural appearance of
the energy tensor confirms that the asymmetric field has non-zero energy.
The derivation of transposition symmetric field equations which
include the anti-symmetric part of the metric connection is complete. The
field equations of General
Relativity are contained within the asymmetric theory as a special case of
complete symmetry.
\newpage
\bigskip
{\it References\/}
\bigskip
{}[1] Albert Einstein, {\it The Meaning of Relativity,\/} Chapman and
Hall, London, 1956.
\bigskip
{}[2] Fang Li Zhi and Li Shu Xian, {\it Creation of the Universe,\/}
World Scientific, Singapore, 1989.
\bigskip
{}[3] G.Stephenson and P.M.Radmore, {\it Advanced Mathematical Methods
for Engineering and
Science Students, \/}Cambridge University Press, Cambridge, 1990, p19.
\bigskip
{}[4] Tsou Sheung Tsun, Why are Physicists Excited by the W-Boson, {\it
Bulletin\/}, Vol 19, 1983,
p165.
\bigskip
{}[5] A.D.Osborne, Gravitation and the Problem of Space-time
Singularities, {\it Bulletin\/}, Vol 21,
1985, p123.
\bigskip
{}[6] M.V.Berry, {\it Principles of Cosmology and Gravitation\/},
Institute of Physics Publishing,
Bristol, 1976, p55.
\end{document}