Since Zeeman topology and other fine topologies defined in **special** and **general** **theory** of **relativity** in above works have many interesting properties, we discuss these properties and also discuss inter-relationships among these topologies. Most important and remarkable of these results are the results proved by R. Göbel and G. Dossena. Göbel proved that the group of all homeomorphisms of a space-time of **general** **relativity** with Zeeman-like topology is the group of all homothetic transformations. And Dossena proved that the first homo- topy group of Zeeman topology for Minkowski space is non-trivial and contains uncountably many subgroups isomorphic to Z. In particular, this topology is not simply connected. Lindstrom generalized the results of Göbel and gave a sequence of Zeeman-like topologies which are in the ascending order of fineness.Thus, in Section 2, we describe Zeeman topology and other fine topologies on Minkowski space and discuss their properties. We also discuss t-topology, s-topology and A-topology introduced by Nanda [3]-[5] and studied in details by G. Agrawal and S. Shrivastava [14] [15]. In Section 3, we describe path topology of Hawking-King-McCarty (HKM topology), and improvements by Malament [10], Fullwood [18] and D.H. Kim [13]. We also discuss properties of HKM topology proved recently by R. Low. In Section 4, we describe the work of Göbel on Zeeman-like topologies defined on space-time of **general** **relativity** and discuss the results proved by him. We also remark on the work of other researchers, especially that by Lindstrom [11] and Mashford [19].

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Using his principle of equivalence, Einstein took another giant step that led him to the **general** **theory** of **relativity**. Einstein reasoned that since acceleration (a space-time effect) can mimic gravity (a force), perhaps gravity is not a separate force after all.

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In the curved Riemannian space-time, operating with the components of five-dimensional metric tensor, one can obtain ten components of metric tensor of the Einstein’s **general** **theory** of **relativity**, four components of electromagnetic vector potential Ā of the Maxwell **theory**, and one component which theoretically can describe any new scalar field [8].The five-dimensional continuum proposed in the article, which includes two temporal coordinates and three spatial coordinates, absorbed all the advantages of the Kaluza five-dimensional world over the flat four-dimensional Minkowski continuum, revealed the connection of the macrocosm, including temporal representations, with microcosm, charge and mass of elementary particles, with the presence of the space environment (dark energy and dark matter), with the existence of vector and scalar fields. His predecessor can be considered the Eddington's Five-Dimensional Continuum (Uranoid), which includes, in addition to the four-dimensional continuum of Minkowski, the fifth time coordinate [9]. Eddington’s Uranoid is the object under study environment (the entire universe is composed of elementary particles). It contains, in addition to the four dimensions of the continuum Minkovsky (x1, x2, x3, t), the fifth - time t0. “The E-frame provides a fifth direction perpendicular to the axes x1, x2, x3, t; and the position vector can be extended t0:

After many years of research we came to the conclusion that previous authors did not get satisfactory solutions and they failed to build the most **general** transformation equations of **Special** **Relativity** even in one dimensional space, because they did not properly define the universal invariant velocity and did not fully deploy the properties of anisotropic time-space.

Annihilation refers to the process of objects dying out in the background of nature (ether). And as a concept of physics, it refers to a process of converting into the ether after a matter and its antimatter meet. That is to say, the most fundamental matter and its antimatter are ultimate particle and charge (whose essence is an electro-hole). Every one of electro-ultimate particles is a unified body made up of the two. In this article, the so-called pure ether, which is completely made up of the electro-ultimate particles moving at the highest speed in reality. Therefore, the annihilation of meaning in **general** is not an ideal conversion process. The above-mentioned the ultimate particles radiated from the surface of the positron to the ether, as well as they are sharing electric charges with electro-ultimate particles, all of which can be belonged to this process. Let's take another angle, it is also so. In terms of the above- mentioned positron which is annihilated layer by layer, because every one of electro-ultimate particles renders as the negative charge of one unit, after the positron is impacted by them, some ultimate particles maybe come into contact with each other and share charges. So, there is a condition for forming charge layers. When a certain charge layer is sufficient to restrict some of the ultimate particles inside it, a new high-density particle is come into being. If only a few of the ultimate particles are confined in the charge layer and displayed as electrically

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typically 0 (zero) and 1 (one) was published by Leibniz (Leibniz, 1703) himself in 1703. In the following, George Boole (1815-1864), an English mathematician, was able to develop in a very short time an impressive algebra of logic (Boole, 1854) as an mathematical extension of the traditional (Aristotelian) logic. According to Boole, “… the principle of contradiction … affirms that it is impossible for any being to possess a quality, and at the same time not to possess it … “ (Boole, 1854, p. 49). Accordingly, “Hence x(1 - x) will represent the class whose members are at once ‘men,’ and ‘not men,’ and the equation (1) thus express the principle, that a class whose members are at the same time men and not men does not exist. In other words, that it is impossible for the same individual to be at the same time a man and not a man.” (Boole, 1854, p. 49). Aristotle's earliest formal study of logic has had an unparalleled influence on science. While some authors (i. e. Kant) where of the opinion that Aristotle has discovered everything that is possible to know about logic other (Russell) pointed to many serious limitations of Aristotle’s logic. It is worth to mention that Lukasiewicz's allegations that Aristotle's law of contradiction has no logical worth (Lukasiewicz & Wedin, 1971) are unfounded (Seddon, 1981). In **general**, even if something like a many-valued or dialectical logic as a non-classical logic which does not restrict the number of truth values to only two, either true or false, usually denoted by “0” and “1”, is necessary, this does not falsify Aristotle’s logic completely. The relationship between Aristotle’s logic and a consistent multi-valued logic is similar to the relationship between Newtonian mechanics and Einstein’s **special** **theory** of **relativity**, the one passes over into the other and vice versa.

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formation of the **Special** **Theory** of **Relativity** [9,10], while they are under the exertion of the magnetomotive force of the external magnetic field, and while the round- ed electromagnetic fields are moving with the speed v close to the speed c of light in vacuum, because the re- striction of inertia is not applicable to the rounded elec- tromagnetic fields here, since they are massless. De- composing the vector radii of the rounded electromag- netic fields into three components in the Cartesian coor- dinate system and taking a **general** function f(r) for the amplitude of the magnetic vector, instead of restricting ourselves to the Coulomb law of the inverse of the square of the vector radius or the distance, one can write the expressions of the components of the magnetic vector of the rounded electromagnetic fields in their own rest frame or in the moving reference frame (with respect to an observer who is at rest and observes the motion of the rounded electromagnetic fields) in the z-direction with a speed v, using polar and Cartesian coordinates as follows

“action-at-a-distance” **theory** and Maxwell's field **theory**. In the case of **general** **relativity** he sought to resolve the clash between Newtonian **theory** and **special** **relativity**. In both cases Einstein formulated physical principles which are also methodological: the principle of **relativity** in the case of **special** **relativity** and that of equivalence in the case of **general** **relativity**. Einstein was also led to modify

A system of clock-synchronised stationary observers is an essential feature of **Special** **Relativity**. Einstein [3, §3] holds that the Lorentz Transformation associates coordinates x, y, z, t of his ‘stationary system’ K with the coordinates ξ, η, ζ, τ of his ‘moving system’ k. A system of clock-synchronised stationary observers and the Lorentz Transformation are the bases for Einstein’s time dilation and length contraction. It is regarded in **general** by physicists [4, §12.1] that a stationary system of observers k which are clock-synchronised when at rest are not synchronised when they all move together with respect to a clock-synchronised ‘stationary system’ K, as illustrated in figure 1.

e = is the transformation for physical time and its independence from the space dimensions, showing that time, interpreted as duration, is not directly related to space. It is very interesting to note that the transformations (28) and (30) don’t satisfy the invariance condition (20), giving further evidence to the conclusion that generally x 4 ≠ ict . We have seen that a procedure of clocks synchronization different than the Lorentz one leads to different classes of space and time transformations in which we cannot assume the validity of Minkowski metric invariance, as also occurring in the **general** case represented by transformations (14) when 2

Let us make a calculation where M is the mass of earth, while m is the mass of a satellite and assume that their common point of gravity is stationary. The result will be that v of Equation (3) will be different when the time dilation is measured from the earth or from the satellite. Our calculations are still based on the limits of SR. Thus, no influence of acceleration or gravity (as in **general** **relativity** [6]) is included in our results where v M is the velocity of earth and

In some of his writings in the early 1960s, Dirac shared his thoughts regarding **relativity** **theory** [2]. He pointed out that in gravitational **theory** as well as in some other areas of physics the application of quantum me- chanics led to an unforeseen predicament. When one looks at space-time sections (cuts) of Einstein’s **General** **Relativity** (EGR), some degrees of freedom drop out of the **theory**. The gravitational field is a tensor field with ten components of which only 6 are needed to describe the physical world. This is the heart of the quandary. When one picks out the relevant six, this destroys the four-dimensional symmetry of space-time that was origi- nally built into ESR and carried over into EGR. This led Dirac to question the basicness of the space-time 4-dimensional formalism of ESR in which the laws of physics must display four-dimensional symmetry. It is almost unbelievable that Dirac would come to such a conclusion, since the Dirac equation [3] for which he is the most famous, is formulated by putting space and time on an equal footing in quantum mechanics.

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The above-laid considerations reflect a completely dismal **general** physical picture of the world. If this picture is further accepted in the scientific community, then many countries will continue wasting their time and money in empty projects like the International Reactor for Thermonuclear Synthesis, Large Hadrons Collider and the like. The now existing army of “brothers’ talc-tellers” will depict for us more and more fantastic physical scenarios. Amazed people will listen to these breathtaking stories about parallel universes, worm holes, the teleportation of large objects, travelling in time, horizontal events and any other stuff like this, and demand more and more money from their Governments for putting up new shows. Leaders of states must remember that “the viability of any idea is determined by the quantity of people feeding on it”. We are confident that in reality our world is not like this [31].

Other physically important examples of Yang-Mills theories include the theories of the electroweak force, the strong force, and various “grand unified theories”; these correspond to principal bundles with structure groups SU(2) × U (1), SU (3), and (usually) SU (n) or SO(n) for larger n. (Here SU (n) refers to the **special** unitary group of degree n, which is isomorphic to the group of unitary n × n matrices with determinant 1; SO(n), meanwhile, is the **special** orthogonal group of degree n, which is isomorphic to the group of orthogonal n ×n matrices with determinant 1.) Matter represented by sections of vector bundles associated to these principal bundles include quark and neutrino fields. These examples differ from electromagnetism in that their structure groups are generally non-Abelian, which has a number of consequences. For instance, the relationship between the principal connection and curvature given in Eq. (1) does not simplify as in Eq. (3); likewise, the form of the Yang-Mills equation relative to a choice of section of the relevant principal bundle, given for electromagnetism in Eq. (5), becomes more complicated. Another important difference is that in the non-Abelian case, horizontal and equivariant Lie algebra valued forms on a principal bundle are not invariant under vertical principal bundle automorphisms. Thus, unlike in electromagnetism, the field strength and charge-current density associated with other Yang-Mills theories will generally depend on the choice of section used to represent them as fields on spacetime.

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In fact, to particle wave, its phase velocity is not equal with its group velocity. This phenomenon will take a physical paradox. In Quantum Mechanics, particle and wave is accompanying. Particle velocity is particle movement speed. Phase velocity is wave movement speed. To particle wave duality, this property will force that the two speeds must be equal at anywhere. They must have the same pace. This is particle-wave **special** property. Other wave not has this **special** requirement, because other wave not has the **special** particle wave duality. To particle wave, if its phase velocity is not equal with its group velocity, not have same pace. This will lead to a bad result. Particle move to a point, but wave not move to that point. On the other hand, wave move to a point, but particle not move to that point. We compute a wave function, get the chance finding the particle at a point, but can’t find any particle at that point. Because particle movement speed is not equal with wave speed, particle can’t move to that point. So this phenomenon will take a physical paradox. So, In order to avoid this paradox, phase velocity must be equal with group velocity. But in

In this paper we consider the more **general** case of a Universe as a distribution of ‘intrinsic’ matter characterized by time- space coordinates, and curved in a system of other coordinates by‘extrinsic’ matter distributions, with other coordinates, of mass and other possible charges. We believe that such a

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being able to select the ‘relevant level’ at will (see Section 5.2.3 below) does dis- solve the distinction between selection and drift, but is not problematic. Imagine there is some change in a human population that could be considered an exam- ple of drift, such as a rise in blue eye color. While eye color has no effect on the rate of phenotypic change of humans, if we look at the gene level, the situa- tion is different. According to gene-level accounts (Dawkins, 2006; Sterelny and Kitcher, 1988; Gardner and Welch, 2011), the gene for blue eyes has increased in replication and hence has an increased fitness. Likewise, the relativistic ac- count would view the increased number of genes for blue eyes as an increase in their (the population of genes for blue eyes) resistance to change, as described in Section 5.1.1 above. Hence what looks like an unimportant change at the human level is relevant at the gene level. This result is **general** to the relativistic position: all biological change is due to Natural Selection on some level or another. As the example shows, though, explaining Natural Selection is a matter of carefully understanding which population is relevant to the change.

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suffered from the assumption that the 5-dimensional metric does not depend on the extra coordinate (the cylinder condition). Hence the proliferation in recent years of various versions of Kaluza-Klein **theory**, supergravity and superstrings. In the last years number of authors Wesson (1992), chatterjee et. al(1994a), Chakraborty and Roy (1994) have considered multi dimensional cosmological model. Kaluza-Klein achievements is shown that five dimensional **general** rela- tivity contains both Einsteins four-dimensional **theory** of gravity and Maxwells **theory** of electromagnetism.

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Some scientists [1, 2, 3] were concerned with this problem and they came to the sim- ilar conclusions as ours, but they applied in their works the Cartan’s approach to the connection in the principal bundle [2, 13, 14]. This approach is not well known among geometrists and relativists. We have used only the standard **theory** of connection in the principal bundle which was created by Ehresmann - Cartan’s student [4, 8]. His approach is commonly used in differential geometry and in **relativity**. We would like to emphasize that the formulation of the EP action in the form (46) can be important for quantizing of **general** **relativity** (because gauge fields are quantized).

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The possibility that Lorentz invariance can be violated in nature has currently become a subject of interest. People often doubt if the **special** **relativity** (SR) is only an approximate symmetry of nature [1, 2]. To give a quantitative measure of Lorentz invariance violation (LIV), one can build up a test **theory** where the Lagrangian of electrodynamics can be slightly deformed by adding to it a tiny Lorentz violating term. One such deformation considered by the authors of Ref. [1] (see also [3]) following standard practice causes the speed of light c to differ from the maximum attainable speed c 0 (which hereafter, unless stated