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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

[1] Vu B Ho, On the motion of quantum particles and Euclidean relativity (Preprint,

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.

[9] Vu B Ho, A Special Relativity of Circular Reference Frame (Preprint, ResearchGate,

20170, viXra 1708.0195v1.

[10] Vu B Ho, Spacetime Structures of Quantum Particles (Preprint, ResearchGate, 2017),

viXra 1708.0192v1.

[11] Lewis Ryder, Introduction to General Relativity, Cambridge University Press, New

York, 2009.

[12] Vu B Ho, Relativistic Spacetime Structures of a Hydrogen Atom (Preprint,

ResearchGate, 2017), viXra 1709.0321v1.

[13] Vu B Ho, A Quantum Structure of Spacetime (Preprint, ResearchGate, 2017).