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.

Special Relativity’s “Newtonization” in Complex “Para-Space”: The Two Theories Equivalence Question

Complex model, say C3, of “para-space” as alternative to the real M4 Minkowski space-time for both relativistic and classical mechanics was shortly introduced as reference to our previous works on that subject. The actual aim, however, is an additional analysis of the physical and para-physical phenomena’ behavior as we formally transport observable mechanical phenomena [motion] to non-real interior of the complex domain. As it turns out, such procedure, when properly set, corresponds to transition from relativistic to more classic (or, possibly, just classic) kind of the motion. This procedure, we call the “Newtonization of relativistic physical quantities and phenomena”, first of all, includes the mechanical motion’s characteristics in the C3. The algebraic structure of vector spaces was imposed and analyzed on both: the set of all relativistic velocities and on the set of the corresponding to them “Galilean” velocities. The key point of the analysis is realization that, as a matter of fact, the relativistic theory and the classical are equivalent at least as for the kinematics. This conclusion follows the fact that the two defined structures of topological vector spaces i.e., the structure imposed on sets of all relativistic velocities and the structure on set of all “Galilean” velocities, are both diffeomorphic in their topological parts and are isomorphic as the vector spaces. As for the relativistic theory, the two approaches: the hyperbolic (“classical” SR) with its four-vector formalism and Euclidean, where SR is modeled by the complex para-space C3, were analyzed and compared.

On the Existence of a Minimum Universal Speed of Physical Transmissions Associated with Matter Wave in Special Relativity

In this work, we show that it is possible to establish coordinate transformations between inertial reference frames in the theory of special relativity with a minimum universal speed of physical transmissions. The established coordinate transformations, referred to as modified Lorentz transformations because they have almost identical form to the Lorentz transformations, also comply with the requirement of invariance of the Minkowski line element. Particularly, the minimum universal speed can be associated with the phase speed of de Broglie matter wave. As application, we also discuss the possibility to formulate relativistic classical and quantum mechanics for the special relativity associated with the modified Lorentz transformations, which describes physical processes that represent an expansion or a collapsing of massive quantum particles.

Does QM Embedded in 5th Dimensional Embedding Allow for Classical Black Hole Ideas Only in Early Universe, Whereas Corda Special Relativity Plus QM May Eliminate Event Horizons for Black Holes after Big Bang?

We first look at the possibility that the ideas of event horizons for black holes may have their application only in early universe conditions whereas Corda’s ground breaking work rejecting event horizons may be due to the formation of quantum mechanics free of an embedding in 5 dimensions allowing for a simpler more direct approach, which rejects the idea of a firewall. First, we present the idea of classical black hole physics applied only once as for the early universe, whereas in such a setting, there may be a way to present NLED and structure formation due to an initial entropy approach as outlined. Then the ideas of Corda’s breakthrough are presented for the reasons he illuminated in his recent work, due to QM being fully formed separate from higher dimensional embedding after the initial evolution of the universe.

Hamiltonian Formalism of de-Sitter Invariant Special Relativity

The Lagrangian of Einstein's special relativity with universal parameter c(SR_c)is invariant under Poincarétransformation,which preserves Lorentz metric η_μν.The SR_c has been extended to be one which is invariant underde Sitter transformation that preserves so-called Beltrami metric B_(μv).There are two universal parameters,c and R,inthis Special Relativity(denoted as SR_(cR)).The Lagrangian-Hamiltonian formulism of SR_(cR) is formulated in this paper.The canonic energy,canonic momenta,and 10 Noether charges corresponding to the space-time's,de Sitter symmetryare derived.The canonical quantization of the mechanics for SR_(CR)-free particle is performed.The physics related to itis discussed.

Three Kinds of Special Relativity via Inverse Wick Rotation

Since the special relativity can be viewed as the physics in an inverse Wick rotation of four-dimensional （4D） Euclid space, which is at almost equal footing with the 4D Riemann/Lobachevski space, there should be important physics in the inverse Wick rotation of 4D Riemann/Lobachevski space. Thus, there are three kinds of special relativity in de Sitter（dS）/Minkowski/anti-de Sitter（AdS） space at almost equal footing, respectively. There is an instanton tunnelling scenario in the Riemann-de Sitter case that may explain why A be positive and link with the multiverse.

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.

From De Sitter Special Relativity and Snyder's Model to Complete Yang Model

The de Sitter special relativity on the Beltrami-de Sitter-spacetime and Snyder＇s model in the momentum space can be combined together with an IR-UV duality to get the complete Yang model at both classical and quantum levels, which are related by the proposed Killing quantization. It is actually a special relativity based on the principle of relativity of three universal constants （c, ρp, R）.

From the Complete Yang Model to Snyder＇s Model, de Sitter Special Relativity and Their Duality

By means of the Dirac procedure, we re-examine Yang＇s quantized space-time model, its relation to Snyder＇s model, the dS special relativity and their UV IR duality. Starting from a dimensionless dS3-space in a （5＋1）- dimensional Mink-space a complete Yang model at both classical and quantum level can be presented and there really exists Snyder＇s model, the dS special relativity and the duality.

Determination of Photon and Elementary Particles Rest Masses Using Maxwell’s Equations and Generalized Potential Dependent Special Relativity

The nature and origin of the photon and elementary rest masses are some of the challeng-ing problems that physics face. The approaches used to solve these problems are complex and time-consuming. Specifically, the photon rest mass pays attention to theoretical physi-cists. Many experimental works show that the photon rest mass is non zero. This problem can be solved using generalized potential dependent special relativity, which has been de-rived using simple arguments, and Maxwell’s equations, besides the conventional Einstein energy-momentum relation. The results obtained show that the rest mass of photons and elementary particles are strongly dependent on the vacuum energy and a universal con-stant. This result conforms with the models that predict time decaying vacuum energy as-sociated with production of smaller rest mass particles followed by larger masses. The two potential dependent mass expressions conform with the cosmological models that suggest the photon is generated first by assuming the universe consisting of total constant vacuum with decaying cosmological part and mass generating part. Using Maxwell’s equations, beside plank and De Broglie hypothesis together with special relativity energy-momentum relation the photon rest mass is estimated. It was shown that the photon rest mass is ex-tremely small compared to the electron mass.

Analysis of a Stochastic Emission Theory Regarding Its Ability to Explain the Effects of Special Relativity

In this article, we investigate the physical consequences that would result if electromagnetic field quanta were emitted at random speeds by a source and if the receiver could only perceive the fraction of the quantum field that is slower than the speed of light in its individual rest frame. The analysis shows that this plausible hypothesis eliminates the weak points of conventional emission theories and that both postulates of special relativity are fulfilled. Furthermore, the results demonstrate that this theory can explain numerous experiments that are usually interpreted using different aspects of special relativity. However, the resulting quantum field theory is not equivalent to the special theory of relativity and requires neither spacetime nor Lorentz transformation. Furthermore, this approach offers a starting point for interpreting quantum effects and effects that contradict the special theory of relativity.

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.

Simple Test of Special Relativity Involving Fresnel Drag and Light Propagation in a Uniformly Moving Medium

In this paper, a simple test of special relativity involving light transmission through a uniformly moving medium and Fresnel’s drag coefficient is discussed. It is shown using Fresnel drag that there is a difference in propagation time of two light beams travelling in opposite directions in such a medium and that this time difference is independent of the index of refraction of the medium. This result has been experimentally confirmed by Wang et al. but cannot be explained by special relativity. Fresnel drag in the context of ether theory provides a full and accurate explanation that is consistent with the existence of a preferred frame.

Effective Mass Model Reduced to Ordinary Mass Using Newton’s, Quantum and Generalized Special Relativity

The concept of the effective mass in crystals shows that the electron mass is affected by the crystal field and was experimentally verified. A useful expression for effective mass was obtained. Unfortunately this expression showed that the effective mass vanishes in the ab-sence of the external field. This is in conflict with observations which show that it reduces to the ordinary mass. To cure this defect a new model is developed assuming the existence of vacuum force as verified experimentally as shown by Casimir effect. Using Newton’s second law and the quantum expression of momentum, useful expressions were found. The same expression was found using generalized special relativity. Strikingly the three models reduced to the conventional one in the absence of vacuum, they also reduced to the ordinary electron mass in the absence of all forces.

Interaction between an Accelerated Mass in Straight Motion and a Hidden Energy Reservoir as a Strict Mathematical Consequence of Special Relativity

A. Einstein and H.A. Lorentz had found that the mass of an accelerated body traveling at relativistic velocity appears to depend on whether the acceleration is performed in the direction of motion or in a transverse direction. E.P. Epstein rejected this result in the “Annalen der Physik”;he rather postulated an additional force that turns up when the body is accelerated in the longitudinal direction. It can be shown that the concept of an increased longitudinal mass is based on a simple mathematical error. When correcting this error, it turns out that Epstein’s additional, hidden force is indispensable in order to avoid an inner inconsistency of Special Relativity. It does most of the total work absorbed by the moving object, and is thus responsible for most of the increase in its energy (=mass), given the speed attained is relativistic. In other words: While the total force on the body needed to maintain a constant acceleration a0 is “(1-v2/c2)-1ma0=m0(1-v2/c2)-3/2a0”, the technical force needed to maintain that acceleration amounts only to “ma0=m0(1 - v2/c2)-1/2a0”. The total energy of two objects that undergo a symmetrical, elastic head-on collision is therefore not conserved during the collision, thus requiring the involvement of a hidden reservoir of energy. This result is confirmed by calculations that use the concept of momenergy. The phenomenon of an apparent disappearance of energy has been noticed in particle physics already (target-experiment), but its consequences have been ignored. Instead, an explanation has been given (reduced “energy of the center of mass”) which is inconsistent and violates the relativity principle.

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.

Hydrodynamic Analogy to Special Relativity

By studying of a slender body moving in a fluid wave-medium, e.g., in air or in shallow water, it was found that the hydrodynamic momentum mass and the total energy of the fluid field can be expressed in forms of and E=mc2, where v is the body moving speed, c is the wave speed and is the hydrodynamic mass at the zero speed. Thus a hydrodynamic analogy to the relativistic particle motion in vacuum can be traced. The velocity dependence of mass and the mass-energy equivalence are universal for any wave medium, which should not be regarded as a consequence of relative Lorentz time-space, but one of the existence of wave in the medium. Its further inference leads to an even more significant physical picture. If the mass particle moves in an unbounded space at a supercritical speed, i.e. , waves are generated and radiated from it, like the Mach waves by the supersonic plane, and the particle itself experiences a resistance as reaction from the wave radiation. By an extension of this analogy, it can be interred from a hydrodynamic superconductive phenomenon that particles or waves can move possibly at a superluminal speed without experiencing any resistance through a tunnel (a bounded space) under certain conditions. Therefore the speed of light is not the limit of our physical world and superluminal phenomena are possible.

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.

Quantum Mechanics and Special Relativity:The Origin of Momentum Operator

The free relativistic particle, by definition, has to move in an inertial reference frame with uniform velocity less than the speed of light. The corresponding movement of a material quantum particle describes a wave packet, composed of matter waves—solutions of the Schr?dinger equation. The maximum of packet, corresponding to the largest probability to find the particle, has to move with the same uniform velocity, defined by the initial condition. It has been shown that the traditional definition of the quantum momentum operator i.e. taking it to correspond to the special relativity theory, relativistic momentum, cannot produce precise description of a relativistic matter particle. Different definitions are investigated and one that solves this issue is found. Obtained original expression of relativistic kinetic energy operator creates new possibilities for relativistic quantum systems theory.

Einstein’s Concept of Clock Synchronization Conflicts with the Second Relativity Postulate

Einstein defined clock synchronization whenever photon pulses with timetags traverse a fixed distance between two clocks with equal time spans ineither direction. Using the second relativity postulate, he found clocksmounted on a rod uniformly moving parallel with the rod’s length cannot besynchronized, but clocks attached to a stationary rod can. He dismissed thisdiscrepancy by claiming simultaneity and clock synchronization were not commonbetween inertial frames, but this paper proves with both Galilean and Lorentztransformations that simultaneity and clock synchronization are preservedbetween inertial frames. His derivation means moving clocks can never besynchronized in a “resting” inertial frame. Ultraprecise atomic clocks intimekeeping labs daily contradict his results. No algebraic error occurred inEinstein’s derivations. The two cases of clocksattached to a rod reveal three major conflicts with the currentsecond postulate. The net velocity between a photon source and detector plusthe “universal” velocity c is mathematically equivalent toEinstein’s clock synchronization method. As the ultraprecise timekeepingcommunity daily synchronizes atomic clocks on the moving Earth withultraprecise time uncertainty well below Einstein’s lowest limit ofsynchronization, the theoretical resolution of the apparent conflict isaccomplished by expanding the second relativity postulate to incorporate thenet velocity between the photon source and detector with the emitted velocity c as components of the total velocity c. This means the magnitudeof the total photon velocity can exceed the speed limit (299792458 m/s) set by the standard velocity c. .