Relativistic velocity addition and the relativity of space and time intervals
J. H. Field
备注：
11 pages, 5 figures. v2 has a simplifed presentation using directly the Reciprocity Principle. References changed, typos removed. v3 has much improved presentation arriving at same conclusions as v1 and v2. References changed. Two additional figures. v3 Added Note explains errors in earlier versions. See also arXiv:0807.0158. References added
A thought experiment first proposed by Sartori is analysed using the parallel velocity addition formula of special relativity. The distances and proper-time intervals between some similarly defined spatial coincidence events are found to be widely different in different inertial frames. This relativity of space and time intervals is quite distinct from the well-known time-dilatation and length contraction effects of special relativity. Sartori's claimed derivation of the parallel velocity addition formula, assuming relativistic time dilatation, based on the thought experiment, is shown to be fortuitous.
We present a way of teaching Einstein's special relativity. It starts with Galileo's relativity, the learners know from previous lectures. The lecture underlines that we can have three transformation equations for the space-time coordinates of the same event, which lead to absolute clock readings, time intervals and lengths (Galileo's relativity), to absolute clock readings but to relative time intervals and lengths (up-to-date Galileo transformations) and to relative clock readings time intervals and lengths.
Assuming that the relativistic universe is homogeneous and isotropic, we can unambiguously determine its model and physical properties, which correspond with the Einstein general theory of relativity (and with its two special partial solutions: Einstein special theory of relativity and Newton gravitation theory), quantum mechanics, and observations, too.
We revisit the problem of the Lorentz transformation of time-intervals in special relativity. We base our discussion on the time-interval transformation formula $ cΔt' = γ(cΔt - \vecβ \cdot Δ\vec{r}) $ in which $ Δt'$ and $ Δt $ are the time-intervals between a given pair of events, in two inertial frames $ S $ and $ S'$ connected by an general boost. We observe that the Einstein time-dilation-formula, the Doppler formula and the relativity of simultaneity, all follow when one the frames in the time-interval transformation formula is chosen as the canonical frame of the underlying event-pair. …
Space-like and time-like invariant space-time intervals are used to analyse measurements of spatial and temporal distances. The former are found to be Lorentz invariant --there is no `relativistic length contraction', whereas the latter exhibit the experimentally confirmed `time dilatation' effect, without, however, `relativity of simultaneity'. It is shown how the spurious predictions of length contraction and relativity of simultaneity arise from the use of inconsistent synchronisation procedures for spatially separated clocks.
Let $r(φ)$ denote the orbit of Mercury. We compare the formulae obtained via general relativity for $r(φ)$ and for the corresponding perihelion precession angle $Δφ$, with the formulae obtained via the relativistic gravitational law, $F=GMmγ^6/r^2$. The latter law can be derived from Newton's gravitational law by employing the gravitational rather than the rest masses of the Sun and Mercury. Remarkably, it is found that the two expressions for the dominant part of $r(φ)$ are identical and that the two expressions for $Δφ$ are also identical.
After having shown that the corresponding components of a four vector transform via the same transformation factors as the space-time coordinates of the same event do, we design a relativistic diagram that displays in true values theirs components. One diagram works for events generated by tardyons whereas a second diagram works for events generated by light signals or photons. We consider both approaching and receding tardyons respectively photons in each case. We also show how the relativistic diagram for radar and photographic detections of moving profiles.
In previous work, the author extended the Poincare Lie algebra to include a four position operator as a natural extension to a large fifteen parameter Lie algebra of operators. We here propose to generalize the metric contained in those structure constants to be the Riemann metric as determined by Einstein's equations from the energy momentum tensor. This gives a new type of "Lie" algebra whose structure constants are space-time dependent. One obtains a new type of uncertainty principle in strong gravitational fields and an altered spectra for the hydrogen atom.
The relativistic time dilatation effect and Einstein's second postulate of special relativity are used to analyse train/embankment thought expriments, both Einstein's original one, and an extension where observers on two trains moving at different speeds, as well as on the embankment, are considered. Whereas the relativistic analysis of Einstein's experiment shows, in contradiction to Einstein's interpretation, no `relativity of simultaneity' effect, the latter is apparent for certain events in the two-train experiment. The importance of relativistic kinematics --embodied for photons in Einste…
It is shown that if we can define a physical quantity with proper character in a given inertial reference frame (kinematic, dynamic, electromagnetic in its nature) which transforms when detected from a reference frame relative to which it moves with velocity $u_x$ as $F=\f{F^o}{\sqrt{1-\f{u_x^2}{c^2}}}$ then we can derive for it transformation equations following one and the same procedure, which involves the addition law of relativistic velocities which can be derived without using the Lorentz transformations. The transformation equation derived that way, generates the physical quantities $u_…
Beside the rise of total cross sections or interaction radii of colliding high energetic particles and the shrinkage of mean-free-paths of ultra relativistic particles (nucleii) in material media (anomalons), which have been shown to be of special relativistic origin [1], still other phenomena in high energy physics may arise from relativistic kinematics. In particular this seems to be the case with the EMC-effect and the so called atmosperic neutrino anomaly.
By using the general concepts of special relativity and the requirements of quantum mechanics, Dirac equation is derived and studied. Only elementary knowledge of spin and rotations in quantum mechanics and standard handlings of linear algebra are employed for the development of the present work.
The principles of the special theory of relativity are extremely simple. A knowledge of the Pythagorean theorem and an ability to perform the simplest algebraic operations are sufficient to be conversant with the kinematics of the special theory of relativity, as well as the time dilation and contraction of the longitudinal dimensions of moving bodies that are associated with relative motion. However, the simplicity of the fundamentals of the theory of relativity are in surprising contrast with the difficulty of the perception and at times the total nonacceptance of the consequences of the spe…
Among all the theories proposed to explain the 'anomalous' perihelion precession of Mercury's orbit announced in 1859 by Le Verrier, the general theory of relativity proposed by Einstein in November 1915, alone could calculate Mercury's 'anomalous' precession with a precision demanded by observational accuracy. Since Mercury's precession was a directly derived result of the full general theory, it was viewed by Einstein as the most critical test of general relativity, amongst the three tests proposed by him. With the advent of the space age, the observational accuracy level has improved furthe…
In this article, we argue that the theory of special relativity, as formulated by Einstein, is a philosophical rather than a scientific theory. What is scientific and experimentally supported is the formalism of the relativistic mechanics embedded in the Lorentz transformations and their direct mathematical, experimental and observational consequences. This is in parallel with the quantum mechanics where the scientific content and experimental support of this branch of physics is embedded in the formalism of quantum mechanics and not in its philosophical interpretations such as the Copenhagen …
In a previous paper published in this journal, we described a new relativistic wave equation that accounts for the propagation of light from a source to an observer in two different inertial frames. This equation, which is based on the primacy of the Doppler effect, can account for the relativity of simultaneity and the observation that charged particles cannot exceed the speed of light. In contrast to the Special Theory of Relativity, it does so without the necessity of introducing the relativity of space and time. Here we show that the new relativistic wave equation based on the primacy of t…
In non-relativistic as well as in special relativistic quantum theory, {\em mass} and {\em charge} are {\em pure numbers} appearing in various (quantum) operators and admit {\em any values}, {\it ie}, values for these quantities are to be prescribed {\em by hand}. This is, in a theory of probability, understandable since we need to {\em assume} some {\em intrinsic} properties of the object we are calculating the probability about. Then, if we {\em specify}, in some satisfactory manner, mass and charge for a point of the space in a suitable general-relativistic framework, the quantum theoretica…
The major extant relativity theories - Galileo's Relativity (GaR), Lorentz's Relativity (LR) and Einstein's Special Relativity (SR), with the latter much celebrated, while the LR is essentially ignored. Indeed it is often incorrectly claimed that SR and LR are experimentally indistinguishable. Here we show that (i) SR and LR are experimentally distinguishable, (ii) that comparison of gas-mode Michelson interferometer experiments with spacecraft earth-flyby Doppler shift data demonstrate that it is LR that is consistent with the data, while SR is in conflict with the data, (iii) SR is exactly d…
The axiomatic definition of mass in classical mechanics, outlined by Mach in the second half of 19th century and improved by several authors, is simplified and extended to the theory of special relativity. According to the extended definition presented here, the mass of a relativistic particle is independent of its velocity and coincides with the rest mass, i.e., with the mass defined in classical mechanics. Then, force is defined as the product of mass and acceleration, both in the classical and in the relativistic framework.
Fractional calculus represents a natural tool for describing relativistic phenomena in pseudo-Euclidean space-time. In this study, Fractional modified special relativity is presented. We obtain fractional generalized relation for the time dilation.
In 1909 Born studied the "relativistic undeformable body" but made the mistake of calling it "rigid". The "rigid body" as one can find in Relativity books is, in fact, this Born "undeformable body". In Relativity it is necessary to distinguish between "rigid" and "undeformable". The "undeformable" body (in the sense of the most rigid possible) must be the "deformable" body where schock waves propagate with maximum speed c. We present in this text the elastic laws for rigid bodies. We think that these laws, which are ignored by the majority of relativists, should be taught in the elementary rel…
A novel perspective on relativistic transformation recently-proposed provides an insight into the very meaning of the principle of relativity. With this novel perspective and Bell's theorem, we argue that special relativity, instead of quantum theory, should be radically reformulated to resolve inconsistencies between the two theories. A new theory of relativistic quantum mechanics is formulated upon this novel perspective. This new relativistic quantum mechanics is free from such anomalies as the negative probability density, the negative-energy states, Zitterbewegung, and the Klein paradox d…
Contemporary relativity theory is restricted in two points: (1) a use of the Riemannian space-time geometry and (2) a use of inadequate (nonrelativistic) concepts. Reasons of these restrictions are analysed in [1]. Eliminating these restrictions the relativity theory is generalized on the case of non-Riemannian (nonaxiomatizable) space-time geometry. Taking into account a progress of a geometry and introducing adequate relativistic concepts, the elementary particle dynamics is generalized on the case of arbitrary space-time geometry. A use of adequate relativistic concepts admits one to formul…
An 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 space-time geometry is Gaussian and the four-vector calculus has its roots in the complex-number algebra, furthermore, that Einstein's "relativity of simultaneity" is based on a misinterpretation of the principle of relativity. Among others predictions of the experimentally verified rise of the interaction-radii o…
In special relativity theory the physical quantities are generally expressed as function of the velocity. In the particular case of an extended object, the factor 1/gamma of Lorentz contraction of its length in the direction of motion is written as a function of the velocity of the extended object. The same happens with the Lorentz factor gamma of time interval dilation of a moving clock. In this paper we show that when gamma is written as a function of the relativistic energy of the considered object the resulting expressions for the length and for the time interval suggest that the contracti…
It is shown that the well-known relativistic correction of quantum Hamiltonian that is present in textbooks appears after quantization of oversimplified relativistic kinetic energy decomposition. Using the proper expression one obtains the kinetic energy operator which relation with the ususal one is consistent with corresponding relation of kinetic energies in special relativity theory. The Schrödinger equation with this operator gives new interesting possibilities for quantum phenomena description.
Klein-Gordon and Dirac equations are the motion equations for relativistic particles with spin 0 (so-called scalar particles) and 1/2 (electron/positron) respectively. For a free particle, the Dirac equation is derived from the Klein-Gordon equation by taking its square root in a bi-quaternionic formalism fully justified by the first principles of the scale relativity theory. This is no more true when an external electro-magnetic field comes into play. If one tries to derive the electro-magnetic Dirac equation in a manner analogous to the one used when this field is absent, one obtains an addi…
The present work shows that through a suitable change of variables relativistic dynamics can be mapped to light propagation in a non-homogeneous medium. A particle's trajectory through the modified space-time is thus formally equivalent to a light ray and can be derived from a mechanical equivalent of Fermat's principle. The similarities between light propagation and mechanics are then extended to quantum mechanics, showing that relativistic quantum mechanics can be derived from a wave equation in modified space-time. Non-relativistic results, such as de Broglie's wavelength, Schroedinger equa…
Relativistic thermodynamics is generalized to accommodate four dimensional rotation in a flat spacetime. An extended body can be in equilibrium when its each element moves along a Killing flow. There are three types of basic Killing flows in a flat spacetime, each of which corresponds to translational motion, spatial rotation, and constant linear acceleration; spatial rotation and constant linear acceleration are regarded as four dimensional rotation. Translational motion has been mainly investigated in the past literature of relativistic thermodynamics. Thermodynamics of the other two is deri…
The theories of quantum mechanics and relativity dramatically altered our understanding of the universe ushering in the era of modern physics. Quantum theory deals with objects probabilistically at small scales, whereas relativity deals classically with motion in space and time. We show here that the mathematical structures of quantum theory and of relativity follow together from pure thought, defined and uniquely constrained by the same elementary "combining and sequencing" symmetries that underlie standard arithmetic and probability. The key is uncertainty, which inevitably accompanies obser…
Some known relativistic paradoxes are reconsidered for closed spaces, using a simple geometric model. For two twins in a closed space, a real paradox seems to emerge when the traveling twin is moving uniformly along a geodesic and returns to the starting point without turning back. Accordingly, the reference frames (RF) of both twins seem to be equivalent, which makes the twin paradox irresolvable: each twin can claim to be at rest and therefore to have aged more than the partner upon their reunion. In reality, the paradox has the resolution in this case as well. Apart from distinction between…
NASA's Jet Propulsion Laboratory put on You Tube a problem that has been baffling the scientists for sometime. It involves an unexpected force acting on the space probes. The author proves that NASA'S satellite orbit anomaly problem can be solved in the frame of Einstein's Special Theory of Relativity. The anomaly confirms that gravity fields propagate with velocity of light as Einstein predicted. The proof is based on the authors discovery of the relativistic version of Newton's gravity field. The author provides formulas for relativistic equation of motion for a spacecraft in the joint gravi…
Evaluation of gravity control concepts should be examined with respect to currently known physical theories. In this work we study the hypothetical conversion of gravitational potential energy into kinetic energy using the formalism of general relativity. We show that the energy involved in the process greatly exceeds the Newtonian estimate, given the nature of general relativity. We conclude that the impact of any gravity manipulation for propulsion greatly depends fundamentally on its exact definition.
