On the Quark Scaling Theorem and the Polarisable Dipole of the Quark in a Scalar Field

In this article the possible impact on the present state of particle physics theory is discussed of two unrecognized theoretical elements. These elements are the awareness that (a) the quark is a Dirac particle with a polarisable dipole moment in a scalar field and that (b) Dirac’s wave equation for fermions, if derived from Einstein’s geodesic equation, reveals a scaling theorem for quarks. It is shown that recognition of these elements proves by theory quite some relationships that are up to now only empirically assessed, such as for instance, the mass relationships between the elementary quarks, the relationship between the bare mass and the constituent mass of quarks, the mass spectrum of hadrons and the mass values of the Z boson and the Higgs boson.

Relativistic Maxwell Theory and Einstein Field Equation derived from Variational Principle

In this article, I'll be reviewing relativistic mechanisms using the calculus of variation in the classical limit. The variational principle is considered to be one of the most important mechanisms to build a theory. Newton's second law of motion is a consequence of Euler-Lagrange equations which gives the least (or stationary) trajectory of a particle between any two arbitrary points. I'll the use action principle by deriving the relativistic Maxwell's field equation, geodesic equation, and Einstein's field equation.

Parallel transport and geodesics

The mathematics of parallel transport and of affine and metric geodesics is presented. The geodesic equation is obtained in several different ways, bringing out its role both as a geometric statement and as an equation of motion. The Euler-Lagrange method to find metric geodesics, and hence Christoffel symbols, is explained. The role of conserved quantities is discussed. Killing’s equation and Killing vectors are introduced. Fermi-Walker transport (the non-rotating freely falling cabin) is defined and discussed. Gravitational redshift is calculated, first in general and then in specific cases.

Forces in Schwarzschild, Vaidya and generalized Vaidya spacetimes

In this paper we investigate how the leading term in the geodesic equation in Schwarzschild spacetime changes under the coordinate transformation to Eddington-Finkelstein coordinates. This term corresponds to the Newton force of attraction. Also we consider this term when we add the energy-momentum tensor of the form of the null dust and the null perfect fluid into right-hand side of the Einstein equation. We estimate the value of this force in Vaidya spacetime when the naked singularity formation occurs. Also we give conditions in generalized Vaidya spacetime when this force of attraction is replaced by the force of repulsion.

Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

The classical uncertainty principle inequalities are imposed over the general relativity geodesic equation as a mathematical constraint. In this way, the uncertainty principle is reformulated in terms of proper space–time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression for the uncertainty principle (GeUP). This re-formulation confirms the need for a minimum length of space–time line element in the geodesic, which depends on a Lorentz-covariant geodesic-derived scalar. In agreement with quantum gravity theories, GeUP imposes a perturbation over the background Minkowski metric unrelated to classical gravity. When applied to the Schwarzschild metric, a geodesic exclusion zone is found around the singularity where uncertainty in space-time diverged to infinity.

Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

The classical uncertainty principle inequalities were imposed as a mathematical constraint over the general relativity geodesic equation. In this way, the uncertainty principle was reformulated in terms of the proper space-time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression for the uncertainty principle (GeUP). This re-formulation confirmed the necessity for a minimum length for the space-time line element in the geodesic, dependent on a geodesic-derived scalar which made the expression Lorentz-covariant. In agreement with quantum gravity theories, GeUP required the imposition of a perturbation over the background Minkowski metric unrelated to classical gravity. When applied to the Schwarzschild metric, a geodesic exclusion zone was found around the singularity where uncertainty in space-time diverged to infinity.

Divergent reflections around the photon sphere of a black hole

From any location outside the event horizon of a black hole there are an infinite number of trajectories for light to an observer. Each of these paths differ in the number of orbits revolved around the black hole and in their proximity to the last photon orbit. With simple numerical and a perturbed analytical solution to the null-geodesic equation of the Schwarzschild black hole we will reaffirm how each additional orbit is a factor $$e^{2 \pi }$$ e 2 π closer to the black hole’s optical edge. Consequently, the surface of the black hole and any background light will be mirrored infinitely in exponentially thinner slices around the last photon orbit. Furthermore, the introduced formalism proves how the entire trajectories of light in the strong field limit is prescribed by a diverging and a converging exponential. Lastly, the existence of the exponential family is generalized to the equatorial plane of the Kerr black hole with the exponentials dependence on spin derived. Thereby, proving that the distance between subsequent images increases and decreases for respectively retrograde and prograde images. In the limit of an extremely rotating Kerr black hole no logarithmic divergence exists for prograde trajectories.

Differential Geometry and Acoustics: A Survey

A connection between acoustic rays in a moving inhomogeneous fluid medium and the null geodesic of a pseudo-Riemannian manifold provides a mechanism to derive several well-known results commonly used in acoustic ray theory. Among these include ray integrals for depth dependent sound speed and current profiles commonly used in ocean and aero acoustic modelling. In this new paradigm these are derived by application of a symmetry of the effective metric tensor known as isometry. In addition to deriving well-known results, the application of the full machinery of differential geometry offers a unified approach to modelling acoustic fields in three dimensional random environments with time dependence by, (1) using conformal symmetry to simplify the geodesic equation, and (2) application of geodesic deviation as a generalization of geometric spread.

Deflection of light by a Coulomb charge in Born–Infeld electrodynamics

We study the propagation of light under a strong electric field in Born–Infeld electrodynamics. The nonlinear effect can be described by the effective indices of refraction. Because the effective indices of refraction depend on the background electric field, the path of light can be bent when the background field is non-uniform. We compute the bending angle of light by a Born–Infeld-type Coulomb charge in the weak lensing limit using the trajectory equation based on geometric optics. We also compute the deflection angle of light by the Einstein–Born–Infeld black hole using the geodesic equation and confirm that the contribution of the electric charge to the total bending angle agree.