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EUCLIDEAN RELATIVITY
Vu B Ho
Advanced Study, 9 Adela Court, Mulgrave, Victoria 3170, Australia
Email: vubho@bigpond.net.au
Abstract: In this work, we present in more details the formulation of Euclidean relativity. We
show that even though there are profound differences between Einstein special relativity and
Euclidean special relativity, general relativity with both pseudo-Euclidean metric and
Euclidean metric have many common features. For example, both forms of metric can be
used to describe the precession of planetary orbits around a gravitational mass and the
cosmological evolution.
In our work on the motion of quantum particles, when they are viewed as three-dimensional
Riemannian manifolds, we suggested that their motion could be described by extending the
isometric transformations in classical physics to the isometric embedding between smooth
manifolds [1]. According to the Whitney embedding theorem, in order to smoothly embed
three-dimensional Riemannian manifolds we would need an ambient six-dimensional
Euclidean space [2,3]. It has also been shown in our works on the temporal dynamics that a
six-dimensional Minkowski pseudo-Euclidean spacetime is obtained by extending one-
dimensional temporal continuum to three-dimensional temporal manifold [4]. While the
question of whether it is possible to smoothly embed three-dimensional Riemannian
manifolds in six-dimensional pseudo-Euclidean spacetime remains, we showed that it is
possible to apply the principle of relativity and the postulate of a universal speed to formulate
a special theory of relativity in which the geometry of spacetime has a positive definite metric
by modifying the Lorentz transformation. The modified Lorentz transformation gives rise to
new interesting features, such as there is no upper limit for the relative speed between inertial
reference frames, the assumed universal speed is not the speed of any physical object or
physical field but rather the common speed of expansion of the spatial space of all inertial
frames. Furthermore, we also showed that when the ratio of the relative speed and the
universal speed approaches infinite values, there will be a conversion between space and
time, therefore not only the concept of motion but the concepts of space and time themselves
are also relative [1]. In this work we will develop and present in more details the Euclidean
special relativity and extend it to the Euclidean general relativity.
In classical physics, the concept of a pseudo-Euclidean spacetime was introduced by
Minkowski in order to accommodate Einstein theory of special relativity in which the
coordinate transformation between the inertial frame with spacetime coordinates
 and the inertial frame with coordinates  are derived from the
principle of relativity and the postulate of a universal speed . The coordinate transformation
is the Lorentz transformation given by




where  
and  
[5]. It is shown that the Lorentz transformation given in
Equations (1-4) leaves the Minkowski spacetime interval  invariant.
Spacetime with this metric is a pseudo-Euclidean space. We now show that it is possible to
construct a special relativistic transformation that will make spacetime a Euclidean space
rather than a pseudo-Euclidean space as in the case of the Lorentz transformation. Consider
the following modified Lorentz transformation




where  
and will be determined from the principle of relativity and the postulate of
a universal speed. Instead of assuming the invariance of the Minkowski spacetime interval, if
we now assume the invariance of the Euclidean interval then from the
modified Lorentz transformation given in Equations (5-8), we obtain the following
expression for

It is seen from the expression of given in Equation (9) that there is no upper limit in the
relative speed between inertial frames. The value of at the universal speed is
. For the values of , the modified Lorentz transformation given in Equations
(5-8) also reduces to the Galilean transformation. However, it is interesting to observe that
when we have and , and in this case from Equations (5) and (8), we
obtain  and , respectively. This result shows that there is a conversion
between space and time when , therefore in Euclidean special relativity, not only the
concept of motion but the concepts of space and time themselves are also relative. It is also
worth mentioning here that the Euclidean relativity of space and time also provides a
profound foundation for the temporal dynamics that we have discussed in our other works
[6]. In the present situation, if in the inertial frame with spacetime coordinates 
the dynamics of a particle  law  
, then since
 and  in the inertial frame is
converted to a temporal law of dynamics  
in the inertial frame with
spacetime coordinates .
As in the case of the Lorentz transformation given in Equations (1-4), we can also derive the
relativistic kinematics and dynamics from the modified Lorentz transformation given in
Equations (5-8), such as the transformation of a length, the transformation of a time interval,
the transformation of velocities, and the transformation of accelerations. Let be the proper
length then the length transformation can be found as

It is observed from the length transformation given in Equation (10) that the length of a
moving object is expanding rather than contracting as in Einstein theory of special relativity.
Now if  is the proper time interval then the time interval transformation can also be found
to be given by the relation


It is also observed from the time interval transformation given in Equation (11) that the
proper time interval is longer than the same time interval measured by a moving observer.
With the modified Lorentz transformation given in Equations (5-8), the transformation of
velocities can be found as follows












Form Equation (12), if we let then we obtain 
. Therefore in this case
only when the relative speed between two inertial frames vanishes, . In other
words, the universal speed is not the common speed of any moving physical object or
physical field in inertial reference frames. In order to specify the nature of the assumed
universal speed we observe that in Einstein theory of special relativity it is assumed that
spatial space of an inertial frame remains static and this assumption is contradicted to
Einstein theory of general relativity that shows that spatial space is actually expanding.
Therefore it seems reasonable to suggest that the universal speed in the modified Lorentz
transformation given in Equations (5-8) is the universal speed of expansion of the spatial
space of all inertial frames. The transformations of accelerations can be derived from the
modified Lorentz transformation and the transformations of velocities given in Equations
(12-14). The transformation of the accelerations can be found as









 











By carrying out the thought experiment of the collision of two identical masses in two inertial
frames that are moving relative to each other, we can derive the following relationship
between the rest mass observed in the rest frame and the mass observed from other
frame as [7,8]

It is seen from Equation (18) that when . However, when we also have
the conversion between space and time , therefore we may speculate that there may
also be a conversion between the spatial mass and the temporal mass of a particle when
[4]. Form Equation (18) we obtain

Since both and are variables, we obtain the following relation by differentiation

 
, we have



Using Equations (21), the change of kinetic energy  can be obtained as

Using Equations (21) and (22) we arrive at

Since , therefore we have . By integrating both sides of Equation (23)
 
  

we obtain the following expression for the kinetic energy

For , we have  and Equation (25) reduces to . However, we
have when . The relativistic momentum of a particle of mass with
velocity can also be defined by the following relation

Then we have when . In magnitudes, , where the
total energy is defined by the relation . From this definition, we
obtain when . Using the relations  and , we also obtain the
following Euclidean relativistic energy-momentum relationship

Now consider the rotating frames of reference in the form of concentric circles as shown in
the figure below [9]
Let r and t be the radius and time of a circular frame which is regarded as being stationary.
Let and be the radius and time of a circular frame which is rotating with respect to the
-frame with a constant angular speed about the common centre O. Denote s and the
arc-length positions of a particle in the -frame and -frame respectively. If we
assume , then from the figure above we obtain the following relations



From Equations (28) and (30), we obtain

Together with , Equation (31) can be seen as a form of kinematical Galilean
transformations of circular reference frames. In order to formulate a Euclidean special
relativity for circular reference frames, we assume that the relativistic transformations for the
rotating frames take the following forms





where is a quantity that will be determined and is an undetermined universal speed. The
quantity can be determined if we simply assume the following Euclidean identity

With the assumed relation given in Equation (34), we obtain

We finally obtain the following Euclidean special relativistic transformations for circular
reference frames




As in the case of linear inertial reference frames, there are new features considering the
special relativity of circular reference frames that have a Euclidean metric. For example, we
have and  
when , and from Equations (36) and (37) we have
and , respectively. There is also a conversion between space and time.
In the following we will extend our presentation of the Euclidean relativity to general
relativity. It is obvious that we can still assume that Einstein theory of general relativity can
also be applied to Riemannian spacetime manifolds which are endowed with positive definite
metrics. In the original Einstein theory of general relativity, the field equations of the
gravitational field are proposed to take the form [5]


where  is the covariant form of the energy-momentum tensor,  is the Ricci tensor
defined by the relation







and the metric connection 
is defined in terms of the metric tensor  as





and  is the Ricci scalar curvature. However, as discussed in our previous works
on the field equations of general relativity [10], if we rewrite Einstein field equations in the
following form



then Einstein field equations can be interpreted as a definition of an energy-momentum tensor
as that of Maxwell theory of the electromagnetic field. In this case, the basic equations of the
gravitational field can be proposed using the contracted Bianchi identities


Even though Equation (42) is purely geometrical, it has a form of Maxwell field equations of
the electromagnetic tensor,  . If the quantity
 can be perceived as a
physical entity, such as a four-current of gravitational matter, then Equation (42) has the
status of a dynamical law of a physical theory. With the assumption that the quantity
 to be identified with a four-current of gravitational matter then a four-current
 can be defined purely geometrical as follows

For a purely gravitational field, Equation (42) reduces to

Using the identity , Equation (44) implies

where is an undetermined constant. Using the identities  and , we
obtain  
, and the energy-momentum tensor given in Equation (41) reduces to


As shown in the appendix 1, the Schwarzschild vacuum solution with can be obtained
with a Riemannian positive definite metric for a centrally symmetric field given in the form

The Schwarzschild vacuum solution is found as

where is a constant of integration that can be identified with the mass of the source of a
physical field. It should be mentioned here that if the line element given in Equation (47) is
the description of the physical field of a three-dimensional physical object which is
isometrically embedded in a six-dimensional Euclidean space then the time can be
assumed to be the temporal arclength of a temporal curve. In order to investigate the nature of
the constant we examine the motion in this spacetime that is described by the geodesic
equation



 
With  and , the geodesic equation for can be found to satisfy the
relation [11]


where is a constant of integration. For  we obtain following the relations
















On the other hand, if we divide the line element given in Equation (48) by , we
obtain the equation







For a planar motion with  
, Equation (53) reduces to


where is a constant of integration. Using Equations (50) and (55), Equation (54) is reduced
to the equation





Using the identity



, Equation (56) is simplified to



By differentiating Equation (57) with respect to , we have





From Equation (58), we obtain the following two equations




It is seen that as in the case of Schwarzschild solution with the Minkowski pseudo-
Riemannian metric, Equation (59) describes a circle and Equation (60) can be used to
describe the precession of planetary orbits around a gravitational mass if the constant is
identified with the gravitational mass as  
and the constant is defined in
terms of the semi-latus rectum of an ellipse as .
It is noted that if the field endowed with the Riemannian metric given in Equation (47) is still
spherically symmetric but now time-dependent then the metric can be shown to be written in
the form [11]

Similar to the case of time-independent spherically symmetric metric as shown in the
appendix 1, the time-dependent metric given in Equation (61) can be reduced to the time-
independent Schwarzschild metric given in Equation (48) if the following condition is
assumed

where can be shown to be time-independent from the condition 
  
.
For the case of a gravitational field, the constant of integration can be identified as

. Therefore, if is time-independent then the mass of a gravitational source

symmetric vacuum solution of the field equations of general relativity is necessary static. It
         orem, it
cannot practically applied to physical reality because there is no physical object which has a
constant mass can be a physical star. In fact, most of the stars that we are observing at the
moment had already turned into other forms of energy many billion years ago. Therefore, the
  s can shine brightly in a mathematical universe but it must
completely stay dim in the physical universe that we are living in. And definitely the
Birkhoffs theorem cannot be applied to spacetime structures of quantum particles even
though for convenience a spherically symmetric spacetime line element may be assumed.
However, it may be speculated that due to the conversion between space and time as well as
the conversion between the spatial mass and the temporal mass, the Birkhoffs theorem could
be applied instead to the total conservation of spatial-temporal mass of a physical system that
is defined entirely in terms of geometrical objects. This total symmetry of spacetime needs a
sophisticated mathematical formulation that requires further investigation. In the mean time,
as has been shown in our previous works that we can derive equations that can be used to
construct line elements to describe the spacetime structures of quantum particles for given
Ricci scalar curvatures [10,12]. For example, if we assume a quantum particle to have a Ricci
scalar given by the equation of the form

 
then, as shown in the appendix 2, by seeking a line element of the form

where is constant, the quantity  satisfies the following differential
equation






 
Similarly, the spacetime structure of a quantum particle can be described by the equation







 



where is a wavefunction that satisfies the Schrödinger wave equation


We can extend the above discussions to the case when we consider not only space but time to
be a three-dimensional manifold. The infinitesimal distance  between two neighbouring
points and is defined by the relation , where
. If we consider the case when both spatial and temporal manifolds
are centrally symmetric then a general spacetime line element of the six-dimensional
spacetime manifold endowed with positive definite metric can be written as

where the infinitesimal distance has been chosen to have a spatial dimension. We now
consider the case when we can arrange the  directions of both the spatial manifold and
the temporal manifold so that

then the line element given in Equation (68) becomes

In the following, we show that there are profound differences in the structure of space-time
that arise from the line element given in Equation (70). First, we show that the line element
given in Equation (70) can lead to the conventional structure of space-time in which,
effectively, space has three dimensions and time has one dimension. The line element in
Equation (70) can be re-written in the form


where we have defined the new quantity that has the dimension of speed as  
. The
meaning of the speed can be interpreted as follows. As discussed in our previous works
[13], spacetime can be considered as being composed of spatial-temporal quanta that have a
very short lifespan. Each of these quanta of spacetime has its own spacetime structure after
having been created, which can be described by the line element given by Equation (70). In
order for a quantum of spacetime to disintegrate it simply expands rapidly. Therefore the
speed in the line element given by Equation (71) is the speed at which a quantum of
spacetime is expanding. Overall, the structure of spacetime at any given moment is a
collection of expanding spatial-temporal quanta. When , then the only observable
structure of spacetime is that of the form of the positive definite Schwarzschild metric
. Instead of the form given in Equation
(71), the line element given in Equation (70) can also be re-written in a different form as
follows


When , then the line element given in Equation (72) is reduced to the line element for a
spacetime manifold in which, effectively, time has three dimensions and space has one
dimension, namely, . This line element is
that of a quantum cell of spacetime and this gives the reason why a three-dimensional time
could not be observed at the macroscopic scale and the microscopic objects that occupy these
quantum elements of spacetime can be described as string-like objects. It is aslo seen from
Equations (71) and (72) that there is a conversion between space-like quantum cell and time-
like quantum cell when .
We would like to give a remark here on the formulation of Robertson-Walker metric to
describe the dynamical structure of the observable universe in modern cosmology. The
Robertson-Walker metric can be written in the following form

In the cosmological line element given in Equation (73), the time is a universal time and the
factor is an expansion factor. However, since the metric is conformally flat in order for
the spatial section of spacetime to be described as a curved space it must be embedded into a
four-dimensional Euclidean space . Since a flat space does not exist in Einstein general
relativity, a fictitious flat space must be introduced so that a three-dimensional
hypersurface can be embedded. However, as has been discussed above, within the framework
of the Euclidean relativity, a six-dimensional Euclidean space , which is regarded as a
natural setting, must exist in order to isometrically embed physical objects, which are
considered as three-dimensional Riemannian manifolds, including the observable universe as
a whole. For example, if we describe the dynamics of a physical object which is viewed as a
three-dimensional spatial manifold with respect to the time , which can be taken as the
temporal arclength of a temporal curve in the three-dimensional temporal manifold, then the
Robertson-Walker metric can be modified to take the following form

Similar to the case when the polar coordinates  are introduced to
describe a circle in the three-dimensional Euclidean space , the three-dimensional spatial
section can be described by introducing the spherical coordinates [7,11]




With the spherical coordinates given in Equations (75-78), the line element given in Equation
(74) can be expressed in the form


where the Gaussian curvature can take values  
 
.
Appendix 1
With the line element given in Equation (47), the tensor metric  and its inverse are given
as
  
   
 
 

  
   
 


The non-zero components of the affine connections are




















The non-zero components of the Ricci curvature tensor are
















For the vacuum solution, from  and , we obtain the identity



On integration Equation (5) we have

where is an undetermined constant. However, with the assumption that the metric given in
Equation (1) will approach the Euclidean metric as , we have . Therefore we
have

With the condition given in Equation (7), the component  can be rewritten as

 
From Equation (8) by integration we obtain

where is a constant of integration.
Appendix 2
With the line element given in Equation (64), we obtain the following non-zero components
of the affine connection

























































From the components of the affine connection given in Equation (5), we obtain





















































Using the relation  we obtain





References
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ResearchGate, 2017), viXra 1710.0253v1.
[2] Bang Yen Chen, Riemannian Submanifolds: A Survey, arXiv:1307.1875v1 [math.DG] 7
Jul 2013.
[3] Qing Han and Jia Xing Hong, Isometric Embedding of Riemannian Manifolds in
Euclidean Spaces (American Mathematical Society, 2006).
[4] Vu B Ho, A Temporal Dynamics: A Generalised Newtonian and Wave Mechanics (Pre-
print, ResearchGate, 2016), viXra 1708.0198v1.
[5] Einstein, A. et al, The Principle of Relativity, Dover, New York, 1952.
[6] Vu B Ho, A Temporal Dynamics: A Generalised Newtonian and Wave Mechanics (Pre-
print, ResearchGate, 2016), viXra 1708.0198v1.
[7] Ray DInverno, Introducing Einstein Relativity, Clarendon Press, Oxford, 1992.
[8] Anderson, Elmer E., Introduction to Modern Physics, Saunders College Publishing, New
York, 1982.
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viXra 1708.0192v1.
[11] Lewis Ryder, Introduction to General Relativity, Cambridge University Press, New
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ResearchGate has not been able to resolve any citations for this publication.
Chapter
Full-text available
Submanifold theory is a very active vast research field which plays an important role in the development of modern differential geometry. This branch of differential geometry is still so far from being exhausted; only a small portion of an exceedingly fruitful field has been cultivated, much more remains to be discovered. In this article, the author provides a broad review of Riemannian submanifolds in differential geometry.
  • Chen Bang Yen
Bang Yen Chen, Riemannian Submanifolds: A Survey, arXiv:1307.1875v1 [math.DG] 7 Jul 2013.