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Symmetry
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Relativity Based on Symmetry
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Relativity Based on Symmetry

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Physics".

Deadline for manuscript submissions: closed (31 March 2022) | Viewed by 16297

Special Issue Editor

Prof. Dr. Yaakov Friedman

E-Mail Website
Guest Editor
Departments of Physics and Mathemetics, Jerusalem College of Technology, Jerusalem, Israel
Interests: special and general relativity; extending relativity; testing relativity; quantum behavior
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

“Based on this century of experience, it is generally supposed that a Final theory will rest on principles of symmetry." S. Weinberg 1979 Nobel Prize winner.

The space-time symmetry resulting from the Principle of Relativity is the basis for Special Relativity (SR). This symmetry and some minor assumptions uniquely define transformations between inertial systems and imply the existence of an invariant metric. The domain  of all admissible velocities in SR is a Bounded Symmetric Domain under affine maps on   but if we use an alternative way of describing the relative motion of objects, the symmetric velocities, the corresponding domain is symmetric with respect to the conformal transformations. This description leads to analytic solutions of the relativistic dynamics equation.

General Relativity (GR), also called the geometric theory of gravitation, is based on Einstein’s field equation, which is generally covariant. The predictions of this theory have successfully passed all tests. Most tests of GR are based on the Schwarzschild solution for a spherically symmetric, non-rotating body. The solution preserves all the symmetries of the problem, but also contains an additional symmetry, which can and should be tested.

To extend relativity to the microscopic region, one needs to extend the geometric description of GR to non-gravitational forces and to derive a mathematical model for motion in fast-changing force-fields propagating with constant speed.

Prof. Dr. Yaakov Friedman
Guest Editor

Manuscript Submission Information

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Keywords

  • Spacetime symmetry
  • Bounded symmetric domain
  • Spin-half representation
  • Geometric dynamics
  • Tests of general relativity
  • One-way speed of light
  • Schwarzschild solution
  • Relativity of space-time
  • Relativistic Newtonian dynamics.

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Published Papers (7 papers)

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Research

10 pages, 320 KiB  
Article
A Physically Meaningful Relativistic Description of the Spin State of an Electron
by Yaakov Friedman
Symmetry 2021, 13(10), 1853; https://doi.org/10.3390/sym13101853 - 3 Oct 2021
Cited by 4 | Viewed by 2528
Abstract
We introduced a new model to present the states of a two-state quantum system. The space is the complexified Minkowski space. The Lorentz group acts by the linear extension of its action on the four-vectors. We applied this model to represent the spin [...] Read more.
We introduced a new model to present the states of a two-state quantum system. The space is the complexified Minkowski space. The Lorentz group acts by the linear extension of its action on the four-vectors. We applied this model to represent the spin state of an electron or any relativistic spin 1/2 particle. The spin state of such particle is of the form U+iS, where U is the four-velocity of the particle in the lab frame, and S is the 4D spin in this frame. Under this description, the transition probability between two pure spin states ϱ1 and ϱ2 of particles moving with the same velocity are defined by use of Minkowski dot product as 12<ϱ2|ϱ1>. This transition probability is Lorentz invariant, coincide with the quantum mechanics prediction and thus agree with the experimental results testing quantum mechanics predictions based on Bell’s inequality. For a a particle of mass m and charge q with the spin state ϱ, the total momentum is mcϱ and the electromagnetic momentum is qϱ. This imply that the Landé g factor for such particles must be g=2. We obtain an evolution equation of the spin state in an electromagnetic field which defines correctly the anomalous Zeeman effect and the fine structure splitting. Full article
(This article belongs to the Special Issue Relativity Based on Symmetry)
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9 pages, 270 KiB  
Article
Testing Relativistic Time Dilation beyond the Weak-Field Post-Newtonian Approximation
by Yaakov Friedman and Esra Yudkin
Symmetry 2020, 12(4), 500; https://doi.org/10.3390/sym12040500 - 1 Apr 2020
Viewed by 2794
Abstract
In General Relativity, the gravitational field of a spherically symmetric non-rotating body is described by the Schwarzschild metric. This metric is invariant under time reversal, which implies that the power series expansion of the time dilation contains only even powers of [...] Read more.
In General Relativity, the gravitational field of a spherically symmetric non-rotating body is described by the Schwarzschild metric. This metric is invariant under time reversal, which implies that the power series expansion of the time dilation contains only even powers of v / c . The weak-field post-Newtonian approximation defines the relativistic time dilation of order ϵ (or of order ( v / c ) 2 ) of the small parameter. The next non-zero term of the time dilation is expected to be of order ϵ 2 , which is impossible to measure with current technology. The new model presented here, called Relativistic Newtonian Dynamics, describes the field with respect to the coordinate system of a far-removed observer. The resulting metric preserves the symmetries of the problem and satisfies Einstein’s field equations, but predicts an additional term of order ϵ 3 / 2 for the time dilation. This term will cause an additional periodic time delay for clocks in eccentric orbits. The analysis of the gravitational redshift data from the Galileo satellites in eccentric orbits indicates that, by performing an improved satellite mission, it would be possible to test this additional time delay. This would reveal which of the coordinate systems and which of the above metrics are real. In addition to the increase of accuracy of the time dilation predictions, such an experiment could determine whether the metric of a spherically symmetric body is time reversible and whether the speed of light propagating toward the gravitating body is the same as the speed propagating away from it. More accurate time dilation and one-way speed of light formulas are important for astronomical research and for global positioning systems. Full article
(This article belongs to the Special Issue Relativity Based on Symmetry)
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17 pages, 304 KiB  
Article
Symmetry of the Relativistic Two-Body Bound State
by L.P. Horwitz and R.I. Arshansky
Symmetry 2020, 12(2), 313; https://doi.org/10.3390/sym12020313 - 22 Feb 2020
Viewed by 2217
Abstract
We show that in a relativistically covariant formulation of the two-body problem, the bound state spectrum is in agreement, up to relativistic corrections, with the non-relativistic bound-state spectrum. This solution is achieved by solving the problem with support of the wave functions in [...] Read more.
We show that in a relativistically covariant formulation of the two-body problem, the bound state spectrum is in agreement, up to relativistic corrections, with the non-relativistic bound-state spectrum. This solution is achieved by solving the problem with support of the wave functions in an O ( 2 , 1 ) invariant submanifold of the Minkowski spacetime. The O ( 3 , 1 ) invariance of the differential equation requires, however, that the solutions provide a representation of O ( 3 , 1 ) . Such solutions are obtained by means of the method of induced representations, providing a basic insight into the subject of the symmetries of relativistic dynamics. Full article
(This article belongs to the Special Issue Relativity Based on Symmetry)
18 pages, 411 KiB  
Article
Relativistic Gravitation Based on Symmetry
by Yaakov Friedman
Symmetry 2020, 12(1), 183; https://doi.org/10.3390/sym12010183 - 20 Jan 2020
Cited by 2 | Viewed by 3947
Abstract
We present a Relativistic Newtonian Dynamics ( R N D ) for motion of objects in a gravitational field generated by a moving source. As in General Relativity ( G R ), we assume that objects move by a geodesic with respect to [...] Read more.
We present a Relativistic Newtonian Dynamics ( R N D ) for motion of objects in a gravitational field generated by a moving source. As in General Relativity ( G R ), we assume that objects move by a geodesic with respect to some metric, which is defined by the field. This metric is defined on flat lab spacetime and is derived using only symmetry, the fact that the field propagates with the speed of light, and the Newtonian limit. For a field of a single source, the influenced direction of the field at spacetime point x is defined as the direction from x to the to the position of the source at the retarded time. The metric depends only on this direction and the strength of the field at x. We show that for a static source, the R N D metric is of the same form as the Whitehead metric, and the Schwarzschild metric in Eddington–Finkelstein coordinates. Motion predicted under this model passes all classical tests of G R . Moreover, in this model, the total time for a round trip of light is as predicted by G R , but velocities of light and object and time dilation differ from the G R predictions. For example, light rays propagating toward the massive object do not slow down. The new time dilation prediction could be observed by measuring the relativistic redshift for stars near a black hole and for sungrazing comets. Terrestrial experiments to test speed of light predictions and the relativistic redshift are proposed. The R N D model is similar to Whitehead’s gravitation model for a static field, but its proposed extension to the non-static case is different. This extension uses a complex four-potential description of fields propagating with the speed of light. Full article
(This article belongs to the Special Issue Relativity Based on Symmetry)
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18 pages, 343 KiB  
Article
The Pre-Potential of a Field Propagating with the Speed of Light and Its Dual Symmetry
by Yaakov Friedman, David Hai Gootvilig and Tzvi Scarr
Symmetry 2019, 11(12), 1430; https://doi.org/10.3390/sym11121430 - 20 Nov 2019
Cited by 5 | Viewed by 2417
Abstract
Relativity theory assumes that force fields propagate with the speed of light. We show that such force fields generated by a single source can be described by a pre-potential, which is a complex-valued function on spacetime outside the worldline of the source. The [...] Read more.
Relativity theory assumes that force fields propagate with the speed of light. We show that such force fields generated by a single source can be described by a pre-potential, which is a complex-valued function on spacetime outside the worldline of the source. The pre-potential is invariant under a spin-half representation of the Lorentz group acting on complexified spacetime. The complex four-potential of such a field is defined and calculated explicitly from the pre-potential without assuming any particular force law for the field. The real part of the obtained four-potential coincides with the known Liénard–Wiechert potential. The symmetry of the four-potential is described herein. The pre-potential satisfies the wave equation. The single source electromagnetic field derived from this four-potential is self-dual or anti-self-dual. The pre-potential and the four-potential are extended to a field with several sources. Full article
(This article belongs to the Special Issue Relativity Based on Symmetry)
15 pages, 465 KiB  
Article
Symmetry and Special Relativity
by Yaakov Friedman and Tzvi Scarr
Symmetry 2019, 11(10), 1235; https://doi.org/10.3390/sym11101235 - 3 Oct 2019
Cited by 7 | Viewed by 4408
Abstract
We explore the role of symmetry in the theory of Special Relativity. Using the symmetry of the principle of relativity and eliminating the Galilean transformations, we obtain a universally preserved speed and an invariant metric, without assuming the constancy of the speed of [...] Read more.
We explore the role of symmetry in the theory of Special Relativity. Using the symmetry of the principle of relativity and eliminating the Galilean transformations, we obtain a universally preserved speed and an invariant metric, without assuming the constancy of the speed of light. We also obtain the spacetime transformations between inertial frames depending on this speed. From experimental evidence, this universally preserved speed is c, the speed of light, and the transformations are the usual Lorentz transformations. The ball of relativistically admissible velocities is a bounded symmetric domain with respect to the group of affine automorphisms. The generators of velocity addition lead to a relativistic dynamics equation. To obtain explicit solutions for the important case of the motion of a charged particle in constant, uniform, and perpendicular electric and magnetic fields, one can take advantage of an additional symmetry—the symmetric velocities. The corresponding bounded domain is symmetric with respect to the conformal maps. This leads to explicit analytic solutions for the motion of the charged particle. Full article
(This article belongs to the Special Issue Relativity Based on Symmetry)
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9 pages, 245 KiB  
Article
Conformal Symmetry, Accelerated Observers, and Nonlocality
by Bahram Mashhoon
Symmetry 2019, 11(8), 978; https://doi.org/10.3390/sym11080978 - 2 Aug 2019
Cited by 3 | Viewed by 2383
Abstract
The acceleration transformations form a 4-parameter Abelian subgroup of the conformal group of Minkowski spacetime. The passive interpretation of acceleration transformations leads to a congruence of uniformly accelerated observers in Minkowski spacetime. The properties of this congruence are studied in order to illustrate [...] Read more.
The acceleration transformations form a 4-parameter Abelian subgroup of the conformal group of Minkowski spacetime. The passive interpretation of acceleration transformations leads to a congruence of uniformly accelerated observers in Minkowski spacetime. The properties of this congruence are studied in order to illustrate the kinematics of accelerated observers in relativistic physics. The generalization of this approach under conformal rescaling of the spacetime metric is examined. Full article
(This article belongs to the Special Issue Relativity Based on Symmetry)
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