Motion and Special Relativity in Complex Spaces

A natural extension of the Lorentz transformation to its complex version was constructed together with a parallel extension of the Minkowski M4 model for special relativity (SR) to complex C4 space-time. As the [signed] absolute values of complex coordinates of the underlying motion’s characterization in C4 one obtains a Newtonian-like type of motion whereas as the real parts of the complex motion’s description and of the complex Lorentz transformation, all the SR theory as modeled by M4 real space-time can be recovered. This means all the SR theory is preserved in the real subspace M4 of the space-time C4 while becoming simpler and clearer in the new complex model’s framework. Since velocities in the complex model can be determined geometrically, with no primary use of time, time turns out to be definable within the equivalent theory of the reduced complex C4 model to the C3 “para-space” model. That procedure allows us to separate time from the (para)space and consider all the SR theory as a theory of C3 alone. On the other hand, the complex time defined within the C3 theory is interpreted and modeled by the single separate C1 complex plane. The possibility for application of the C3 model to quantum mechanics is suggested. As such, the model C3 seems to have unifying abilities for application to different physical theories.

Developments of Space-Time, Complete Special Relativity, and Superluminal Entangled Communication

Time and space are two very important basic concepts in philosophy and nature.In special relativity we proved that there are necessarily two symmetrical topological structures separated by the light-cone,which includes the generalized Lorentz transformation(GLT)for the spacelike interval,in which phase velocity is superluminal.Based on quantum entanglement as new fifth interaction,we research a simple superluminal entangled communication,whose key is to establish two mutually entangled particles or devices Alice(A)and Bob(B).We observe and control the information of A position,and then can know the corresponding results of the other B.This is not to send directly information each other.It may be superluminal,and should agree and test GLT.Moreover,we research some new possible developments of time and space,such as the fractal dimension extended to the complex dimension,the higher dimensional time,and the arrow of time.A generalized Noether’s theorem is proposed.In quantum theory,we search the higher dimensional complex space in supersymmetry,and the space-time operators.

Thoroughly Testing Einstein’s Special Relativity Theory, and More

Einstein’s Special Relativity (ESR) has enjoyed spectacular success as a mathematical construct and in terms of the experiments to which it has been subjected. Possible vulnerabilities of ESR will be explored that break the symmetry of reciprocal observations of length, time, and mass. It is shown how Newton could also have derived length contraction . Einstein’s General Relativity (EGR) will also be discussed occasionally such as a changed perspective on gravitational waves due to a small change in ESR. Some additional questions addressed are: Did Einstein totally eliminate the Ether? Is the physical interpretation of ESR completely correct? Why should there be a maximum speed limit, and should it always be the same? The mass-energy equation is revisited to show that in 1717 Newton could have derived the modern , and not known that it violates the foundation of his mechanics. Tributes are paid to Einstein and others.

Non-Uniqueness of Einstein’s Special Relativity, and the Inconclusiveness of High Energy (Relativistic) Physics

In this paper, we present a new form of “special relativity” (BSR), which is isomorphic to Einstein’s “special relativity” (ESR). This in turn proves the non-uniqueness of Einstein’s “special relativity” and implies the inconclusiveness of so-called “relativistic physics”. This work presents new results of principal significance for the foundations of physics and practical results for high energy physics, deep space astrophysics, and cosmology as well. The entire exposition is done within the formalism of the Lorentz SL(2C) group acting via isometries on real 3-dimensional Lobachevskian (hyperbolic) spaces L3 regarded as quotients SL(2C)/SU(2). We show via direct calculations that both ESR and BSR are parametric maps from Lobachevskian into Euclidean space, namely a gnomonic (central) map in the case of ESR, and a stereographic map in the case of BSR. Such an identification allows us to link these maps to relevant models of Lobachevskian geometry. Thus, we identify ESR as the physical realization of the Beltrami-Klein (non-conformal) model, and BSR as the physical realization of the Poincare (conformal) model of Lobachevskian geometry. Although we focus our discussion on ball models of Lobachevskian geometry, our method is quite general, and for instance, may be applied to the half-space model of Lobachevskian geometry with appropriate “Lorentz group” acting via isometries on (positive) half space, resulting yet in another “special relativity” isomorphic with ESR and BSR. By using the notion of a homotopy of maps, the identification of “special relativities” as maps from Lobachevskian into Euclidean space allows us to justify the existence of an uncountable infinity of hybrid “special relativities” and consequently an uncountable infinity of “relativistic physics” built upon them. This is another new result in physics and it states that so called “relativistic physics” is unique only up to a homotopy. Finally, we show that “paradoxes” of “special relativities” in either ESR or BSR are simply common distortions of maps between non-isometric spaces. The entire exposition is kept at elementary level accessible to majority of students in physics and/or engineering.

Diagrammatic Approach for Investigating Two Dimensional Elastic Collisions in Momentum Space II: Special Relativity

The diagrammatic approach to the collision problems in Newtonian mechanics is useful. We show in this article that the same technique can be applied to the case of the special relativity. The two circles play an important role in Newtonian mechanics, while in the special relativity, we need one circle and one ellipse. The circle shows the collision in the center-of-mass system. And the ellipse shows the collision in the laboratory system. These two figures give all information on two dimensional elastic collisions in the special relativity.

A New Version of Special Relativity Absorbed the Uncertainty Principle: Its Content as Well as Application and Experimental Test

Based on the space spherical symmetry of 3-dimensional and the translational symmetry of time and the uncertainty principle, a 4-dimensional space-time cylinder model of quarks and leptons is established. With this model, equations of the special relativity can be extended more perfectly, thereby achieving a unity of the special relativity and quantum mechanics in deeper level. New equations can not only interpret issues explained by old equations but also solve several important pending problems. For example, a formula to strictly calculate the coefficient ξ of Lorentz invariance violation (LIV) is derived, to above 4 × 1019 eV UHECR protons the calculated |ξ| -30, although there is the LIV effect it is too weak to change the GZK cutoff, which is consistent with observations of HiRes and Auger;Also, a relation formula between the Hubble constant and several basic constants is derived, thus theoretically calculated H0 = 70.937 km·s-1·Mpc-1, which is well consistent with the final observation result of HST Key Project. In addition, an unusual effect predicted by new equations can be experimentally tested in the electron storage ring;a preliminary experiment result has hinted its signs of existence.

Is the kinematics of special relativity incomplete?

A thorough analysis of composite inertial motion (relativistic sum) within the framework of special relativity leads to the conclusion that every translational motion must be the symmetrically composite relativistic sum of a finite number of quanta of velocity. It is shown that the resulting spacetime geometry is Gaussian and the four-vector calculus to have its roots in the complex-number algebra. Furthermore, this results in superluminality of signals travelling at or nearly at the canonical velocity of light between rest frames even if resting to each other.

The Space Dilation: Analyzing Progressive Speed Reduction and Its Relativistic Parallels

This paper presents a conceptual exploration that draws an intriguing parallelbetween a hypothetical travel scenario and the principles of special relativity. The scenario involves a traveler who reduces their speed by an amount proportional to the distance traveled. Despite initially traveling at a high speed towards a given destination, the continual reduction in speed results in an asymptotic approach to the goal, analogous to the unattainable speed of light in relativity. Mathematically, the scenario is expressed through the Harmonic Series, demonstrating that the total travel time increases without bound, making the destination theoretically unreachable within a finite timeframe. This exploration mirrors the relativistic velocity addition and time dilation effects, providing a compelling analogy for understanding asymptotic limits. By highlighting the profound implications of diminishing returns and unattainable goals, this paper aims to stimulate further discussion and exploration of these fascinating parallels.

A NEW INVARIANT QUANTITY OF SPECIAL RELATIVITY AND THE MINIMAL MOVING MASS SPACE-TIME REFERENCE SYSTEM

Ⅰ. PROBLEM AND RESULTS In physics, in order to describe a motion mathematically one needs a space-time reference system ∑(x, y z; t). From the mathematical point of view, any two reference systems are equivalent, one only needs a transformation between the coordinates of the two systems.

One Electron Atom in Special Relativity with de Sitter Space-Time Symmetry

The de Sitter invariant Special Relativity （dS-SR） is SR with constant curvature, and a natural extension of usual Einstein SR （E-SR）. In this paper, we solve the dS-SR Dirac equation of Hydrogen by means of the adiabatic approach and the quasi-stationary perturbation calculations of QM. Hydrogen atom is located in the light cone of the Universe. FRW metric and ACDM cosmological model are used to discuss this issue. To the atom, effects of de Sitter space-time geometry described by Beltrami metric are taken into account. The dS-SR Dirac equation turns out to be a time dependent quantum Hamiltonian system. We reveal that： （i） The fundamental physics constants me, h, e variate adiabatically along with cosmologic time in dS-SR QM framework. But the fine-structure constant α≡ - e^2/（hc） keeps to be invariant; （ii） （2s^1/2 - 2p^1/2）-splitting due to dS-SR QM effects： By means of perturbation theory, that splitting △E（z） are calculated analytically, which belongs to O（1/R^2）-physics of dS-SR QM. Numerically, we find that when ｜R｜ = {103 Gly, 104 Gly, 105 Gly}, and z = {1, or 2}, the AE（z） 〉〉 1 （Lamb shift）. This indicates that for these cases the hyperfine structure effects due to QED could be ignored, and the dS-SR fine structure effects are dominant. This effect could be used to determine the universal constant R in dS-SR, and be thought as a new physics beyond E-SR.

New Approach to Synchronize General Relativity and Quantum Mechanics with Constant “K”-Resulting Dark Matter as a New Fundamental Force Particle

Planck scale plays a vital role in describing fundamental forces. Space time describes strength of fundamental force. In this paper, Einstein’s general relativity equation has been described in terms of contraction and expansion forces of space time. According to this, the space time with Planck diameter is a flat space time. This is the only diameter of space time that can be used as signal transformation in special relativity. This space time diameter defines the fundamental force which belongs to that space time. In quantum mechanics, this space time diameter is only the quantum of space which belongs to that particular fundamental force. Einstein’s general relativity equation and Planck parameters of quantum mechanics have been written in terms of equations containing a constant “K”, thus found a new equation for transformation of general relativity space time in to quantum space time. In this process of synchronization, there is a possibility of a new fundamental force between electromagnetic and gravitational forces with Planck length as its space time diameter. It is proposed that dark matter is that fundamental force carrying particle. By grand unification equation with space-time diameter, we found a coupling constant as per standard model “αs” for that fundamental force is 1.08 × 10-23. Its energy calculated as 113 MeV. A group of experimental scientists reported the energy of dark matter particle as 17 MeV. Thorough review may advance science further.

Evidence for special relativity with de Sitter space-time symmetry

I show the formulation of de Sitter Special Relativity （dS-SR） based on Dirac-Lu-Zou-Guo’s discussions. dS-SR quantum mechanics is formulated, and the dS-SR Dirac equation for hydrogen is suggested. The equation in the earth-QSO framework reference is solved by means of the adiabatic approach. It’s found that the fine-structure ＂constant＂ α in dS-SR varies with time. By means of the t-z relation of the ΛCDM model, α’s time-dependency becomes redshift z-dependent. The dS-SR’s predictions of △α/α agree with data of spectra of 143 quasar absorption systems, the dS-space-time symmetry is SO（3, 2） （i.e., anti-dS group） and the universal parameter R （de Sitter ratio） in dS-SR is estimated to be R ≈ 2.73×1012 ly. The effects of dS-SR become visible at the cosmic space-time scale （i.e., the distance 109 ly）. At that scale, dS-SR is more reliable than Einstein SR. The α-variation with time is evidence of SR with de Sitter symmetry.

One Electron Atom in Special Relativity with de Sitter Space-Time Symmetry(Ⅱ): ——Higher Order Contributions

This is a follow of previous work entitled "One Electron Atom in Special Relativity with de Sitter SpaceTime Symmetry" [Commun. Theor. Phys. 57(2012) 930]. In this paper, we consider the higher order calculations and contributions in the previous framework to solve one electron atoms in de Sitter invariant relativistic quantum mechanics. The next-to-leading-order calculations in 1/R2-expansions show that the fine-structure constant α is variant with cosmologic time going by in the de Sitter invariant special relativistic quantum mechanics with standard FRW cosmologic model.

Extra Time Dimension: Deriving Relativistic Space-Time Transformations, Kinematics, and Example of Dimensional Compactification Using Time-Dependent Non-Relativistic Quantum Mechanics

We consider a five-dimensional Minkowski space with two time dimensions characterized by distinct speeds of causality and three space dimensions. Formulas for relativistic coordinate and velocity transformations are derived, leading to a new expression for the speed limit. Extending the ideas of Einstein’s Theory of Special Relativity, concepts of five-velocity and five-momenta are introduced. We get a new formula for the rest energy of a massive object. Based on a non-relativistic limit, a two-time dependent Schrödinger-like equation for infinite square-well potential is developed and solved. The extra time dimension is compactified on a closed loop topology with a period matching the Planck time. It generates interference of additional quantum states with an ultra-small period of oscillation. Some cosmological implications of the concept of four-dimensional versus five-dimensional masses are briefly discussed, too.

The Fundamental Definitions in Radar Measurement Principle and Theory of Space and Time

Fundamental definitions of distance and velocity in radar measurement principle are examined and revised from strict theoretical point of view. Synchronization scheme - for clocks in uniform, translatory relative motion is introduced as theoretical foundation for GPS and GLONASS type navigation and positioning technology. Traditional definitions of two-way radar measurement, based on arithmetic mean vlaue concept, turn out to be special cases of revised definitions for one-way radar measurement, based on geometric mean concept, derived from synchronization of moving clocks in accordance with the principle of relativity. The essential physical meaning of Lorentz transformation is interpreted in terms of radar measured parameters. Invariance or absoluteness of four dimensional interval turns out to be invariance or absoluteness of geometric mean time interval. The Lorentz factor turns out to be ratio of geometric mean and arithmetic mean time intervals in terms of radar measured parameters. Theoretical results are illustrated transparently by numerical examples. A crucial experiment for direct testing of the second postulate of special relativity by means of GPS of GLONASS type technology is proposed in this paper.

Space, Time, and Their Transformations

We construct a set T of space-the transformations between inertial systems that are completely equivalent for explaining the experimental evidence, by starting from two empirically based assumptions: (1) The two-way velocity of light is c in all inertial systems and in all directions; (2) Time dilation effects take place with the usual relativistic factor. The Lorentz transformation is an element of T and any two elements of T differ only as to a convention regarding clock synchronization. The simplest choice is to transform time independently of space coordinates (absolute synchronization) and can be called 'inerital transformation.' When accelerations are considered the equivalence is broken and the inertial transformation emerges as closest to physical reality.

Physical Space Is Finite

Metrological analysis shows that any clock in inertial motion in infinite space shall not have time dilation, due to relativity of such motion in such space. On the other hand, atomic clock in inertial motion in finite space shall exhibit time dilation, due to alteration of momentum of clock-defining particle caused by nonzero curvature of trajectory of such motion in such space. Therefore, time dilation experiment of atomic clock in inertial motion in physical space provides a direct and decisive way of determining geometry of physical space in real-time. Phenomenon of time dilation of atomic clock in inertial motion in physical space has long been observed and confirmed experimentally. Therefore, extent of physical space has to be finite, consistent with result of high precision experiment of free particle in high-speed motion conducted a decade ago.Keywords Geometry of Physical Space, Time Dilation, Atomic Clock, Special Relativity Theory.

Effects of Compactification on Free Massive Scalar Fields in Five-Dimensional Space-Time with an Extra Time Dimension: Analysing Some Results

This paper deals with some aspects of two-time physics (i.e., 2T + 3S five-dimensional space) for a Minkowski-like space with distinct speeds of causality for the time dimensions. Detailed calculations are provided to obtain results of Kaluza-Klein type compactification for free massive scalar fields and abelian free gauge fields. As already indicated in the literature, a tower of massive fields results from the compactification with mass terms having signs opposite to those of the ones appearing in other five-dimensional theories with an extra space dimension. We perform elaborate numerical calculations to highlight the magnitude of the imaginary masses and ask if we need to explore alternative compactification techniques.

Euclidean Model of Space and Time

The aim of this work is to show that the currently widely accepted geometrical model of space and time based on the works of Einstein and Minkowski is not unique. The work presents an alternative geometrical model of space and time, a model which, unlike the current one, is based solely on Euclidean geometry. In the new model, the pseudo-Euclidean spacetime is replaced with a specific subset of four-dimensional Euclidean space. The work shows that four-dimensional Euclidean space allows explanation of known relativistic effects that are now explained in pseudo-Euclidean spacetime by Einstein’s Special Theory of Relativity (STR). It also shows simple geometric-kinematical nature of known relativistic phenomena and among others explains why we cannot travel backward in time. The new solution is named the Euclidean Model of Space and Time (EMST).

Joint DOA and time delay estimation method for space-time coherent distributed signals based on search

Under dense urban fading environment, performance of joint multi-path parameter estimation method based on traditional point signal model degrades seriously. In this paper, a new space and time signal model based on multipath distribution function is given after new space and time manifold is reconstructed. Then joint spacetime signal subspace is obtained by converting acquired channel from time domain to frequency domain .Then space and time spectrum is formulated by the space sub-matrix and time sub-matrix taken out of joint space-time signal subspace, and parameters are estimated by searching the minimum eigenvalues of the space matrix and the time matrix. Lastly, A space and time parameters matching process is performed by using the orthogonal property between joint noise subspace and the space-time manifold. In contrast with tradition MUSIC, the algorithm we present here only need two 1- dimension searching and was not sensitive to different distribution function.