Poincaré’s Relativistic Physics: Its Origins and Nature

Henri Poincaré (1854–1912) developed a relativistic physics by elevating the empirical inability to detect absolute motion, or motion relative to the ether, to the principle of relativity, and its mathematics ensured that it would be compatible with that principle. Although Poincaré’s aim and theory were similar to those of Albert Einstein (1879–1955) in creating his special theory of relativity, Poincaré’s relativistic physics should not be seen as an attempt to achieve Einstein’s theory but as an independent endeavor. Poincaré was led to advance the principle of relativity as a consequence of his reflections on late nineteenth-century electrodynamics; of his conviction that physics should be formulated as a physics of principles; of his conventionalistic arguments on the nature of time and its measurement; and of his knowledge of the experimental failure to detect absolute motion. The nonrelativistic theory of electrodynamics of Hendrik A.Lorentz (1853–1928) of 1904 provided the means for Poincaré to elaborate a relativistic physics that embraced all known physical forces, including that of gravitation. Poincaré did not assume any dynamical explanation of the Lorentz transformation, which followed from the principle of relativity, and he did not seek to dismiss classical concepts, such as that of the ether, in his new relativistic physics. Shaul Katzir teaches in the Graduate Program in History and Philosophy of Science, Bar Ilan University.     

On the gravitodynamics of moving bodies

In the present work we propose a generalization of Newton’s gravitational theory from the original works of Heaviside and Sciama, that takes into account both approaches, and accomplishes the same result in a simpler way than the standard cosmological approach. The established formulation describes the local gravitational field related to the observables and effectively implements the Mach’s principle in a quantitative form that retakes Dirac’s large number hypothesis. As a consequence of the equivalence principle and the application of this formulation to the observable universe, we obtain, as an immediate result, a value of Ω = 2. We construct a dynamic model for a galaxy without dark matter, which fits well with recent observational data, in terms of a variable effective inertial mass that reflects the present dynamic state of the universe and that replicates from first principles, the phenomenology proposed in MOND. The remarkable aspect of these results is the connection of the effect dubbed dark matter with the dark energy field, which makes it possible for us to interpret it as longitudinal gravitational waves.     

Feynman’s Proof of Maxwell Equations: in?the?Context?of Quantum Gravity

Many of us are familiar with Feynman’s “proof” of 1948, as revealed by Dyson, which demonstrates that Maxwell equations of electromagnetism are a consequence of Newton’s laws of motion of classical mechanics and the commutation relations of coordinate and momentum of quantum mechanics. It was Feynman’s purpose to explore the universality of dynamics of particles while making the fewest assumptions. We re-examine this formulation in the context of quantum gravity and show how Feynman’s derivation can be extended to include quantum gravity.     

Infinity and Newton’s Three Laws of Motion

It is shown that the following three common understandings of Newton’s laws of motion do not hold for systems of infinitely many components. First, Newton’s third law, or the law of action and reaction, is universally believed to imply that the total sum of internal forces in a system is always zero. Several examples are presented to show that this belief fails to hold for infinite systems. Second, two of these examples are of an infinitely divisible continuous body with finite mass and volume such that the sum of all the internal forces in the body is not zero and the body accelerates due to this non-null net internal force. So the two examples also demonstrate the breakdown of the common understanding that according to Newton’s laws a body under no external force does not accelerate. Finally, these examples also make it clear that the expression ‘impressed force’ in Newton’s formulations of his first and second laws should be understood not as ‘external force’ but as ‘exerted force’ which is the sum of all the internal and external forces acting on a given body, if the body is infinitely divisible.     

The Maxwell Electromagnetic Equations and the Lorentz Type Force Derivation—The Feynman Approach Legacy

R. Feynman’s “heretical” approach (Dyson in Am. J. Phys. 58:209–211, 1990; Dyson in Phys. Today 42(2):32–38, 1989) to deriving the Lorentz force based Maxwell electromagnetic equations is revisited, the its complete legacy is argued both by means of the geometric considerations and its deep relation with the vacuum field theory approach devised (Prykarpatsky et al. in Int. J. Theor. Phys. 49:798–820, 2010; Prykarpatsky et al. in Preprint ICTP, 2008, ). Being completely classical, we reanalyze the Feynman’s derivation from the classical Lagrangian and Hamiltonian points of view and construct its nontrivial relativistic generalization compatible with the vacuum field theory approach.     

Einstein,de Sitter and the beginning of relativistic cosmology in 1917

In 1917, both Einstein and de Sitter proposed a new interpretation of the universe as a whole: the structure of the universe could be described in terms of relativistic field equations. Their contributions marked the beginning of the modern scientific comprehension of the origin and evolution of the universe. Our aim is to propose a critical review paper, based on references in primary sources, on the formulation in 1917 of Einstein’s and de Sitter’s models of the universe, which represents a fundamental chapter in the history of relativistic Cosmology.     

On the Electromagnetic Origin of Inertia and Inertial Mass

We address the problem of inertial property of matter through analysis of the motion of an extended charged particle. Our approach is based on the continuity equation for momentum (Newton’s second law) taking due account of the vector potential and its convective derivative. We obtain a development in terms of retarded potentials allowing an intuitive physical interpretation of its main terms. The inertial property of matter is then discussed in terms of a kind of induction law related to the extended charged particle’s own vector potential. Moreover, it is obtained a force term that represents a drag force acting on the charged particle when in motion relatively to its own vector potential field lines. The time rate of variation of the particle’s vector potential leads to the acceleration inertia reaction force, equivalent to the Schott term responsible for the source of the radiation field. We also show that the velocity dependent term of the particle’s vector potential is connected with the relativistic increase of mass with velocity and generates a longitudinal stress force that is the source of electric field lines deformation. In the framework of classical electrodynamics, we have shown that the electron mass has possibly a complete electromagnetic origin and the obtained covariant equation solves the “4/3 mass paradox” for a spherical charge distribution.     

Introducing Relativistic Quantum Mechanics to Energy Students

A method for introducing relativistic quantum mechanics to energy students is described. The method complements existing modern physics courses and relies on Feynman’s relativistic path integral approach to display a relationship between classical dynamics, quantum theory, and relativistic quantum theory.     

Gravity of monopole and string and the gravitational constant in 3He-A

We discuss the effective metric produced in superfluid ³He-A by such topological objects as the radial disgyration and monopole. In relativistic theories these metrics are similar to that of the local string and global monopole, respectively. But in ³He-A they have a negative angle deficit, which corresponds to a negative mass of the topological objects. The effective gravitational constant in superfluid ³He-A, deduced from a comparison with relativistic theories, is G∼Δ⁻², where the gap amplitude Δ plays the part of the Planck energy. G depends on temperature roughly as (1−T ²/T _c ² )⁻² and corresponds to a screening of Newton’s constant. Pis’ma Zh. éksp. Teor. Fiz. 67, No. 9, 666–671 (10 May 1998) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Relativistic perihelion precession of orbits of Venus and the Earth

Among all the theories proposed to explain the “anomalous” perihelion precession of Mercury’s orbit first 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 the 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 from amongst the three tests he proposed. With the advent of the space age, the level of observational accuracy has improved further and it is now possible to detect this precession for other planetary orbits of the solar system — viz., Venus and the Earth. This conclusively proved that the phenomenon of “anomalous” perihelion precession of planetary orbits is a relativistic effect. Our previous papers presented the mathematical model and the computed value of the relativistic perihelion precession of Mercury’s orbit using an alternate relativistic gravitational model, which is a remodeled form of Einstein’s relativity theories, and which retained only experimentally proven principles. In addition this model has the benefit of data from almost a century of relativity experimentation, including those that have become possible with the advent of the space age. Using this model, we present in this paper the computed values of the relativistic precession of Venus and the Earth, which compare well with the predictions of general relativity and are also in agreement with the observed values within the range of uncertainty.      

Why gravity experiments are so exciting

Being the most fundamental interaction gravity not only describes a particular interaction between matter, but also covers issues like the notion of space and time, the role of the observer and the relativistic measurement process. Gravity is geometry and, as a consequence, allows for the existence of black holes, non-trivial topologies, a cosmological big bang, time-travel, warp drive, and other phenomena not known from non-relativistic physics. Here we present the experimental basis of General Relativity, in particular its foundations encoded in the Einstein Equivalence Principle and its predictions in the weak and strong gravity regime. We discuss various routes to search for effects possibly signaling effects of the looked for quantum theory of gravity. We lay emphasis on assumptions to be tested which are only rarely discussed in the literature like tests of Newton’s axioms, tests of conservation laws, etc. We propose an experiment testing the order of time derivatives in the equation of motion.     

The Embedding of Schwarzschild in Braneworld

The braneworlds models were inspired partly by Kaluza-Klein’s theory, where both the gravitational and the gauge fields are obtained from the geometry of a higher dimensional space. The positive aspects of these models consist in perspectives of modifications it could bring in to particle physics, such as: unification in a TeV scale, quantum gravity in this scale and deviation of Newton’s law for small distances. One of the principles of these models is to suppose that all space-times can be embedded in a bulk of higher dimension. The main result in these notes is a theorem showing a mathematical inconsistency of the Randall-Sundrum braneworld model, namely that the Schwarzschild space-time cannot be embedded locally and isometrically in a five dimensional bulk with constant curvature (for example AdS-5). From the point of view of semi-Riemannian geometry this last result represents a serious restriction to the Randall-Sundrum’s braneworld model.     

Aberration and the Fundamental Speed of Gravity in the Jovian Deflection Experiment

We describe our explicit Lorentz-invariant solution of the Einstein and null geodesic equations for the deflection experiment of 2002 September 8 when a massive moving body, Jupiter, passed within 3.7’ of a line-of-sight to a distant quasar. We develop a general relativistic framework which shows that our measurement of the retarded position of a moving light-ray deflecting body (Jupiter) by making use of the gravitational time delay of quasar’s radio wave is equivalent to comparison of the relativistic laws of the Lorentz transformation for gravity and light. Because, according to Einstein, the Lorentz transformation of gravity field variables must depend on a fundamental speed c, its measurement through the retarded position of Jupiter in the gravitational time delay allows us to study the causal nature of gravity and to set an upper limit on the speed of propagation of gravity in the near zone of the solar system as contrasted to the speed of the radio waves. In particular, the v/c term beyond of the standard Einstein’s deflection, which we measured to 20% accuracy, is associated with the aberration of the null direction of the gravity force (“aberration of gravity”) caused by the Lorentz transformation of the Christoffel symbols from the static frame of Jupiter to the moving frame of observer. General relativistic formulation of the experiment identifies the aberration of gravity with the retardation of gravity because the speed of gravitational waves in Einstein’s theory is equal to the speed of propagation of the gravity force. We discuss the misconceptions which have inhibited the acceptance of this interpretation of the experiment. We also comment on other interpretations of this experiment by Asada, Will, Samuel, Pascual–Sánchez, and Carlip and show that their “speed of light” interpretations confuse the Lorentz transformation for gravity with that for light, and the fundamental speed of gravity with the physical speed of light from the quasar. For this reason, the “speed of light” interpretations are not entirely consistent with a retarded Liénard–Wiechert solution of the Einstein equations, and do not properly incorporate how the phase of the radio waves from the quasar is perturbed by the retarded gravitational field of Jupiter. Although all of the formulations predict the same deflection to the order of v/c, our formulation shows that the underlying cause of this deflection term is associated with the aberration of gravity and not of light, and that the interpretations predict different deflections at higher orders of v/c beyond the Shapiro delay, thus, making their measurement highly desirable for deeper testing of general relativity in future astrometric experiments like Gaia, SIM, and SKA.     

Cartan–Weyl Dirac and Laplacian Operators,Brownian Motions: The Quantum Potential and Scalar Curvature,Maxwell’s and Dirac-Hestenes Equations,and Supersymmetric Systems

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincaré–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q     

Variational wave equations of two fermions interacting via scalar,pseudoscalar, vector,pseudovector and tensor fields

We consider a method for deriving relativistic two-body wave equations for fermions in the coordinate representation. The Lagrangian of the theory is reformulated by eliminating the mediating fields by means of covariant Green's functions. Then, the nonlocal interaction terms in the Lagrangian are reduced to local expressions which take into account retardation effects approximately. We construct the Hamiltonian and two-fermion states of the quantized theory, employing an unconventional “empty” vacuum state, and derive relativistic two-fermion wave equations. These equations are a generalization of the Breit equation for systems with scalar, pseudoscalar, vector, pseudovector and tensor coupling.     

Relativistic dynamics of spherical timelike shells

The dynamics of general relativistic timelike spherically symmetric thin shells of matter is considered, putting special emphasis on the physical interpretation of the models and, therefore, on the dependence of the dynamical behavior upon then the choice of the equation of state. From this point of view the general formalism is reviewed both in the Israel’s and in the canonical (Lagrangian and Hamiltonian) approach. Known exact solutions corresponding to closed equations of state are reviewed as well, and a new, wide class of nonlinear barotropic solutions is introduced and discussed in detail.     

The low-energy compton scattering and polarizability of spin-0 Hadrons in a Gaugeinvariant approach

In this work, the amplitude of the interaction of an electromagnetic field with π-mesons has been determined in the context of a relativistic gauge-invariant approach by solving electrodynamic equations with the use of the covariant Green’s function method. Based on this approach, the effective Lagrangian of the two-photon interaction with spin-0 hadrons taking into account the electric and magnetic polarizabilities has been obtained.