arXiv subjects

Jonathan Gratus

Publications and source records attributed to Jonathan Gratus.

At least 19 recordsLinked to original sources

arXiv2025

The stress-energy distributional multipole for both uncharged and charged dust

In this paper, we formulate the distributional uncharged and charged stress-energy tensors. These are integrals, along a worldline, of derivatives of the delta-function. These distributions are also multipoles and they are prescribed to any order. They represent an extended region of non-self-interacting uncharged or charged dust, shrunken to a single point in space. We show that the uncharged dust stress-energy multipole is divergence-free, while the divergence of the charged dust stress-energy multipole is given by the current and the external electromagnetic field. We show that they can be obtained by squeezing a regular dust stress-energy tensor onto the worldine. We discuss the aforementioned calculations in a coordinate-free manner.

gr-qc↗

arXiv2025

Electromagnetic Boundary Conditions for Space--time Interfaces

We give a general family of electromagnetic boundary conditions applicable to arbitrary space--time interfaces between electromagnetic media, which include the known space--only and time--only boundary conditions as special cases. These boundary conditions describe a broad class of electromagnetic interfaces, including surfaces in arbitrary motion, ultra-thin (metasurface) media, and cases where the media on one or both sides of the boundary can be both spatially and temporally dispersive. Our approach utilizes 4-dimensional spacetime and addresses the question of how, and in what ways, an electromagnetic field may be connected across a 3-dimensional hypersurface. We show that our proposed boundary conditions are the most general conditions consistent with causality and linearity.

physics.optics↗

arXiv2025

The Vlasov Bivector: A Parameter-Free Approach to Vlasov Kinematics

Plasma kinetics, for both flat and curved spacetime, is conventionally performed on the mass shell, a 7--dimensional time-phase space with a Vlasov vector field, also known as the Liouville vector field. The choice of this time-phase space encodes the parameterisation of the underling 2nd order ordinary differential equations. By replacing the Vlasov vector on time-phase space with a bivector on an 8--dimensional sub-bundle of the tangent bundle, we create a parameterisation free version of Vlasov theory. This has a number of advantages, which include working for lightlike and ultra-relativistic particles, non metric connections, and metric-free and premetric theories. It also works for theories where no time-phase space can exist for topological topological reasons. An example of this is when we wish to consider all geodesics, including spacelike geodesics. We extend the particle density function to a 6--form on the subbundle of the tangent space, and define the transport equations, which correspond to the Vlasov equation. We then show how to define the corresponding 3--current on spacetime. We discuss the stress-energy tensor needed for the Einstein-Vlasov system. This theory can be generalised to create parameterisation invariant Vlasov theories for many 2nd order theories, on arbitrary manifolds. The relationship to sprays and semi-sprays is given and examples from Finsler geometry are also given.

math-ph↗

arXiv2023

Broadband Terahertz Generation in a Corrugated Waveguide with matched Phase and Group Velocities

\begin{abstract} We show that it is possible to design corrugated waveguides where phase and group velocities coincide at an inflection point of the dispersion relation, allowing an extended regime of interaction with a charge particle beam. This provides a basis for designing travelling slow-wave structures with a broadband interaction between relativistic charged particle beams and propagating terahertz waves allowing an energy exchange between beam and wave, amplifying terahertz radiation. We employ Fourier-Mathieu expansion, which gives approximate analytic solutions to Maxwell equations in a corrugated waveguide with periodically undulating cross-section. Being analytic, this enables quick design of corrugated waveguides, determined from desirable dispersion relations. We design a three dimensional waveguide with the desired dispersion and confirm the analytical predictions of the wave profile, using numerical simulations. Madey's theorem is used to analyse the strength of the wave-beam interaction, showing that there is a broad frequency interaction region.

physics.app-ph↗

arXiv2023

Moment tracking and their coordinate transformations for macroparticles with an application to plasmas around black holes

Particle-in-cell codes usually represent large groups of particles as a single macroparticle. These codes are computationally efficient but lose information about the internal structure of the macroparticle. To improve the accuracy of these codes, this work presents a method in which, as well as tracking the macroparticle, the moments of the macroparticle are also tracked. Although the equations needed to track these moments are known, the coordinate transformations for moments where the space and time coordinates are mixed cannot be calculated using the standard method for representing moments. These coordinate transformations are important in astrophysical plasma, where there is no preferred coordinate system. This work uses the language of Schwartz distributions to calculate the coordinate transformations of moments. Both the moment tracking and coordinate transformation equations are tested by modelling the motion of uncharged particles in a circular orbit around a black hole in both Schwarzschild and Kruskal-Szekeres coordinates. Numerical testing shows that the error in tracking moments is small, and scales quadratically. This error can be improved by including higher order moments. By choosing an appropriate method for using these moments to deposit the charge back onto the grid, a full particle-in-cell code can be developed.

physics.plasm-ph↗

arXiv2022

The tensorial representation of the distributional stress-energy quadrupole and its dynamics

We investigate stress-energy tensors constructed from the covariant derivatives of delta functions on a worldline. Since covariant derivatives are used all the components transform as tensors. We derive the dynamical equations for the components, up to quadrupole order. The components do, however, depend in a non-tensorial way, on a choice of a vector along the worldline. We also derive a number of important results about general multipoles, including that their components are unique, and all multipoles can be written using covariant derivatives. We show how the components of a multipole are related to standard moments of a tensor field, by parallelly transporting that tensor field.

math-ph↗

arXiv2021

Maxwell-Lorentz without self interactions: Conservation of energy and momentum

Since a classical charged point particle radiates energy and momentum it is argued that there must be a radiation reaction force. Here we present an action for the Maxwell-Lorentz without self interactions model, where each particle only responds to the fields of the other charged particles. The corresponding stress-energy tensor automatically conserves energy and momentum in Minkowski and other appropriate spacetimes.

physics.class-ph↗

arXiv2021

Temporal boundaries in electromagnetic materials

Temporally modulated optical media are important in both abstract and applied applications, such as spacetime transformation optics, relativistic laser-plasma interactions, and dynamic metamaterials. Here we investigate the behaviour of temporal boundaries, and show that traditional approaches that assume constant dielectric properties, with loss incorporated as an imaginary part, necessarily lead to unphysical solutions. Further, although physically reasonable predictions can be recovered with a narrowband approximation, we show that appropriate models should use materials with a temporal response and dispersive behaviour.

physics.optics↗

arXiv2020

The Distributional Stress-Energy Quadrupole

We investigate stress-energy tensors constructed from the delta function on a worldline. We concentrate on quadrupoles as they make an excellent model for the dominant source of gravitational waves and have significant novel features. Unlike the dipole, we show that the quadrupole has 20 free components which are not determined by the properties of the stress-energy tensor. These need to be derived from an underlying model and we give an example motivated from a divergent-free dust. We show that the components corresponding to the partial derivatives representation of the quadrupole, have a gauge like freedom. We give the change of coordinate formula which involves second derivatives and two integrals. We also show how to define the quadrupole without reference to a coordinate systems or a metric. For the representation using covariant derivatives, we show how to split a quadrupole into a pure monopole, pure dipole and pure quadrupole in a coordinate free way.

gr-qc↗

arXiv2020

Temporary Singularities and Axions: an analytic solution that challenges charge conservation

We construct an analytic solution to Maxwell's equations that violates global charge conservation, by building on the possibilities demonstrated in [Foundations of Physics 49, 330 (2019)]. The construction is valid for a spacetime containing a temporary singularity and a Maxwellian electrodynamics containing a proposed "topological" axion field. As such, it demonstrates that although "physics breaks down" at a singularity, the physical laws on the spacetime still impose constraints on what can happen. Further, the concepts of transformation optics can be applied to show that our specific mathematical solution has a much wider applicability.

gr-qc↗

arXiv2019

Electromagnetism, Axions, and Topology: a first-order operator approach to constitutive responses provides greater freedom

We show how the standard constitutive assumptions for the macroscopic Maxwell equations can be relaxed. This is done by arguing that the Maxwellian excitation fields (D,H) should be dispensed with, on the grounds that they (a) cannot be measured, and (b) act solely as gauge potentials for the charge and current. In the resulting theory, it is only the links between the fields (E,B) and the charge and current (\rho,J) that matter; and so we introduce appropriate linear operator equations that combine the Gauss and Maxwell-Ampere equations with the constitutive relations, eliminating (D,H). The result is that we can admit more types of electromagnetic media -- notably, the new relations can allow coupling in the bulk to a homogeneous axionic material; in contrast to standard EM where any homogeneous axion-like field is completely decoupled in the bulk, and only accessible at boundaries. We also consider a wider context, including the role of topology, extended non-axionic constitutive parameters, and treatment of Ohmic currents. A range of examples including an axonic response material is presented, including static electromagnetic scenarios, a possible metamaterial implementation, and how the transformation optics paradigm would be modified. Notably, these examples include one where topological considerations make it impossible to model using (D,H).

physics.optics↗

arXiv2019

Evaporating black-holes, wormholes, and vacuum polarisation: must they always conserve charge?

A careful examination of the fundamentals of electromagnetic theory shows that due to the underlying mathematical assumptions required for Stokes' Theorem, global charge conservation cannot be guaranteed in topologically non-trivial spacetimes. However, in order to break the charge conservation mechanism we must also allow the electromagnetic excitation fields D, H to possess a gauge freedom, just as the electromagnetic scalar and vector potentials phi and A do. This has implications for the treatment of electromagnetism in spacetimes where black holes both form and then evaporate, as well as extending the possibilities for treating vacuum polarisation. Using this gauge freedom of D, H we also propose an alternative to the accepted notion that a charge passing through a wormhole necessarily leads to an additional (effective) charge on the wormhole's mouth.

gr-qc↗

arXiv2019

Maxwell's (D, H ) excitation fields: lessons from permanent magnets

Macroscopic Maxwellian electrodynamics consists of four field quantities along with electric charges and electric currents. The fields occur in pairs, the primary ones being the electric and magnetic fields (E,B), and the other the excitation fields (D,H). The link between the two pairs of field is provided by constitutive relations, which specify (D,H) in terms of (E,B); this last connection enabling Maxwell's (differential) equations to be combined in a way that supports waves. In this paper we examine the role played by the excitation fields (D,H), showing that they can be regarded as not having a physical existence, and are merely playing a mathematically convenient role. This point of view is made particularly relevant when we consider competing constitutive models of permanent magnets, which although having the same measurable magnetic properties, have startlingly different behaviours for the magnetic excitation field H.

physics.class-ph↗

arXiv2018

On $k$-jet field approximations to geodesic deviation equations

Let $M$ be a smooth manifold and $\mathcal{S}$ a semi-spray defined on a sub-bundle $\mathcal{C}$ of the tangent bundle $TM$. In this work it is proved that the only non-trivial $k$-jet approximation to the exact geodesic deviation equation of $\mathcal{S}$, linear on the deviation functions and invariant under an specific class of local coordinate transformations is the Jacobi equation. However, if the linearity property on the dependence in the deviation functions is not imposed, then there are differential equations whose solutions admit $k$-jet approximations and are invariant under arbitrary coordinate transformations. As an example of higher order geodesic deviation equations we study the first and second order geodesic deviation equations for a Finsler spray.

math-ph↗

arXiv2018

Mode profile shaping in wire media: Towards an experimental verification

We show that an experimentally plausible system consisting of a modulated wire medium hosted in a metal cavity can preserve the longitudinal field profile shaping predicted by Boyd et al. (2018) on the basis of a perfectly periodic wire-only structure. These new frequency domain numerical results are a significant step towards justifying the construction of an experimental apparatus to test the field profile shaping in practise.

physics.app-ph↗

arXiv2018

The correct and unusual coordinate transformation rules for electromagnetic quadrupoles

Despite being studied for over a century, the use of quadrupoles have been limited to Cartesian coordinates in flat spacetime due to the incorrect transformation rules used to define them. Here the correct transformation rules are derived, which are particularly unusual as they involve second derivatives of the coordinate transformation and an integral. Transformations involving integrals have not been seen before. This is significantly different from the familiar transformation rules for a dipole, where the components transform as tensors. It enables quadrupoles to be correctly defined in general relativity and to prescribe the equations of motion for a quadrupole in a coordinate system adapted to its motion and then transform them to the laboratory coordinates. An example is given of another unusual feature: a quadrupole which is free of dipole terms in polar coordinates has dipole terms in Cartesian coordinates. It is shown that dipoles, electric dipoles, quadrupoles and electric quadrupoles can be defined without reference to a metric and in a coordinates free manner. This is particularly useful given their complicated coordinate transformation.

math-ph↗

arXiv2018

Customized longitudinal electric field profiles: using spatial dispersion in dielectric wire arrays

We show how spatial dispersion can be used as a mechanism to customize the longitudinal profiles of electric fields inside modulated wire media, using a fast and remarkably accurate 1D inhomogeneous model. This customization gives fine control of the sub-wavelength behaviour of the field, as has been achieved recently for transverse fields in simpler slotted-slab media. Our scheme avoids any necessity to run a long series of computationally intensive 3D simulations of specific structures, in order to iteratively converge (or brute-force search) to an empirical `best-performance' design according to an abstract figure-of-merit. Instead, if supplied with an `ideal waveform' profile, we could now calculate how to construct it directly. Notably, and unlike most work on photonic crystal structures, our focus is specifically on the field profiles because of their potential utility, rather than other issues such as band-gap control, and/or transmission and reflection coefficients.

physics.app-ph↗

arXiv2017

A pictorial introduction to differential geometry, leading to Maxwell's equations as three pictures

In this article we present pictorially the foundation of differential geometry which is a crucial tool for multiple areas of physics, notably general and special relativity, but also mechanics, thermodynamics and solving differential equations. As all the concepts are presented as pictures, there are no equations in this article. As such this article may be read by pre-university students who enjoy physics, mathematics and geometry. However it will also greatly aid the intuition of an undergraduate and masters students, learning general relativity and similar courses. It concentrates on the tools needed to understand Maxwell's equations thus leading to the goal of presenting Maxwell's equations as 3 pictures.

math.DG↗