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Gerald Kaiser

Publications and source records attributed to Gerald Kaiser.

At least 19 recordsLinked to original sources
arXiv2023

Thermal Spacetime, Part I: Relativistic Bohmian Mechanics

Positive-energy solutions of the Klein-Gordon equation form a Hilbert space of holomorphic functions on the future tube. This domain is interpreted as an extended phase space for the associated classical particle, the extra dimensions being the time and a new variable related to the inverse equilibrium temperature in an associated canonical ensemble. This gives a compelling formulation of relativistic Bohmian mechanics where the "hidden variables" are simply the classical trajectories of the particle in phase space. Interactions may be included through "holomorphic gauge theory."

math-ph↗
arXiv2016

Active and Reactive Energy Balance Equations in Active and Reactive Time

Electrical networks, and physical systems in general, are known to satisfy a power balance equation which states that the rate of change of the energy in time equals the power at the port of the network minus the power dissipated. However, when complex power is considered, there does not seem to exist a similar statement for the imaginary power, either in the time-domain or the frequency-domain. Recently, in the context of electromagnetic fields, it has been shown by complexifying the time to t+js and interpreting s as reactive time, that it is possible to set up an imaginary power balance in terms of the rate of change of reactive energy in reactive time. Here these ideas are specialized to linear and time-invariant RLC networks. For non-sinusoidal waveforms it is shown that the rate of change of reactive energy in reactive time contains all the essential properties and features of the commonly accepted definition of reactive power under sinusoidal conditions. We believe that this provides an unambiguous and physically motivated resolution to the longstanding debate on how to generalize reactive power to non-sinusoidal waveforms.

math.DS↗
arXiv2016

Does a phase shift occur in an AC arc?

This is a translation of a classic paper in German showing that the apparent power in an AC arc is larger than the active power although no phase shift exists between the voltage and the current, indicating that the reactive power vanishes. The phenomenon studied in this paper gave rise to a variety of mutually conflicting "power triangle" models relating the active, reactive, and apparent powers P, Q, and S whose merits are still under debate today.

physics.hist-ph↗
arXiv2015

Conservation of reactive electromagnetic energy in reactive time

The complex Poynting theorem (CPT) is extended to a canonical time-scale domain $(t,s)$. Time-harmonic phasors are replaced by the positive-frequency parts of general fields, which extend analytically to complex time $t+is$, with $s>0$ interpreted as a time resolution scale. The real part of the extended CPT gives conservation in $t$ of a time-averaged field energy, and its imaginary part gives conservation in $s$ of a time-averaged reactive energy. In both cases, the averaging windows are determined by a Cauchy kernel of width $Δt\sim \pm s$. This completes the time-harmonic CPT, whose imaginary part is generally supposed to be vaguely `related to' reactive energy without giving a conservation law, or even an expression, for the latter. The interpretation of $s$ as reactive time, tracking the leads and lags associated with stored capacitative and inductive energy, gives a simple explanation of the volt-ampere reactive (var) unit measuring reactive power: a var is simply one Joule per reactive second. The related 'complex radiation impedance density' is introduced to represent the field's local reluctance to radiate.

physics.optics↗
arXiv2014

Completing the complex Poynting theorem: Conservation of reactive energy in reactive time

The complex Poynting theorem is extended canonically to a time-scale domain $(t, s)$ by replacing the phasors of time-harmonic fields by the analytic signals $X(r, t+is)$ of fields $X(r,t)$ with general time dependence. The imaginary time $s>0$ is shown to play the role of a time resolution scale, and the extended Poynting theorem splits into two conservation laws: its real part gives the conservation in $t$ of the scale-averaged active energy at fixed $s$, and its imaginary part gives the conservation in $s$ of the scale-averaged reactive energy at fixed $t$. At coarse scales (large $s$, slow time), where the system reduces to the circuit level, this may have applications to the theory of electric power transmission and conditioning. At fine scales (small $s$, fast time) it describes reactive energy dynamics in radiating systems.

math-ph↗
arXiv2012

Electromagnetic helicity wavelets: a model for quasar engines?

The complex distance function $ζ$, which plays a prominent role in the definition of scalar (acoustic) wavelets, is found to determine a complex extension of the spherical coordinate system that is ideally suited for the construction of highly focused electromagnetic beams with helicities conforming to the oblate spheroidal geometry of $ζ$. This is used to build a basis of electromagnetic wavelets $F^m$ radiated or absorbed by the branch disk $D$ of $ζ$. $F^m$ has integer angular momentum $m$ around the z axis and definite spheroidal helicity. We use a regularization method to compute its singular charge-current density and show that the total charge vanishes. Hence $F^m$ is due solely to electric and magnetic polarization currents. $D$ acts as a magnetic dipole antenna, and its axis as a coupled electric dipole antenna. We propose this as an idealized electromagnetic model for quasars (in flat spacetime, without gravity), with $D$ representing the accretion disk and the vortex singularities along its axis representing the jets. In the regularized version, the accretion disk is represented by a solid flat oblate spheroid and the jets by two solid narrow semi-hyperboloids.

gr-qc↗
arXiv2011

Electromagnetic inertia, reactive energy, and energy flow velocity

In a recent paper titled "Coherent electromagnetic wavelets and their twisting null congruences," I defined the local inertia density (I), reactive energy density (R), and energy flow velocity (v) of an electromagnetic field. These are the field equivalents of the mass, rest energy, and velocity of a relativistic particle. Thus R and I are Lorentz-invariant and |v|<=c, with equality if and only if R=0. The exceptional fields with |v|=c were called "coherent" because their energy moves in complete harmony with the field, leaving no inertia or reactive energy behind. Generic electromagnetic fields become coherent only in the far zone. Elsewhere, their energy flows at speeds |v|<c. The purpose of this paper is to confirm and clarify this statement by studying the local energy flow in several common systems: a time-harmonic electric dipole field, a time-dependent electric dipole field, and a standing plane wave. For these fields, the energy current (Poynting vector) is too weak to carry away all of the energy, thus leaving reactive energy in its wake. For the time-dependent dipole field, we find that the energy can flow both transversally and inwards, back to the source. Neither of these phenomena show up in the usual computation of the energy transport velocity which considers only averages over one period in the time-harmonic case.

physics.class-ph↗
arXiv2011

Huygens' principle in classical electrodynamics: a distributional approach

We derive Huygens' principle for electrodynamics in terms of 4-vector potentials defined as distributions supported on a surface surrounding the charge-current density. By combining the Pauli algebra with distribution theory, a compact and conceptually simple derivation of the Stratton-Chu and Kottler-Franz equations is obtained. These are extended to freely moving integration surfaces, so that the fields due to charge distributions in arbitrary motion are represented. A further generalization is obtained to multiple surfaces, which can be used to enclose clusters of transmitters, scatterers and receivers.

math-ph↗
arXiv2011

Coherent electromagnetic wavelets and their twisting null congruences

We construct an electromagnetic field whose scalar potential is a pulsed-beam wavelet Psi (an analytic continuation of a classical Huygens wavelet). The vector potential A is determined up to three complex parameters by requiring that (a) it satisfies the Lorenz gauge condition with Psi, (b) its current density is supported on the same disk D as the charge density of Psi, (c) it is axisymmetric, and (d) it has the same retarded time dependence as Psi. By choosing one of the parameters in A appropriately, the electromagnetic field generated by the four-potential (A, Psi) can be made null, meaning that E^2=B^2 and E dot B=0. We call such fields coherent because upon being radiated, they do not loiter around the source, generating electromagnetic inertia (a new concept related to reactive energy) but immediately propagate out at the speed of light. The coherent EM wavelets define a twisting null congruence of light rays in Minkowski space, which we show to be identical to the Kerr congruence associated with the Kerr-Newman metric. The latter represents a black hole due to a time-independent charge-current density on a massive disk D spinning at the angular velocity c/a, where a is the radius of D. By contrast, our coherent wavelets are electromagnetic pulsed beams radiated by pulsed charge-current distributions on D, still spinning at the uniform rate c/a.

math-ph↗
arXiv2009

Quantum Physics, Relativity, and Complex Spacetime: Towards a New Synthesis

The positivity of the energy in relativistic quantum mechanics implies that wave functions can be continued analytically to the forward tube T in complex spacetime. For Klein-Gordon particles, we interpret T as an extended (8D) classical phase space containing all 6D classical phase spaces as symplectic submanifolds. The evaluation maps $e_z: f\to f(z)$ of wave functions on T are relativistic coherent states reducing to the Gaussian coherent states in the nonrelativistic limit. It is known that no covariant probability interpretation exists for Klein-Gordon particles in real spacetime because the time component of the conserved "probability current" can attain negative values even for positive-energy solutions. We show that this problem is solved very naturally in complex spacetime, where $|f(x-iy)|^2$ is interpreted as a probability density on all 6D phase spaces in T which, when integrated over the "momentum" variables y, gives a conserved spacetime probability current whose time component is a positive regularization of the usual one. Similar results are obtained for Dirac particles, where the evaluation maps $e_z$ are spinor-valued relativistic coherent states. For free quantized Klein-Gordon and Dirac fields, the above formalism extends to n-particle/antiparticle coherent states whose scalar products are Wightman functions. The 2-point function plays the role of a reproducing kernel for the one-particle and antiparticle subspaces.

math-ph↗
arXiv2009

Generalized Huygens principle with pulsed-beam wavelets

Huygens' principle has a well-known problem with back-propagation due to the spherical nature of the secondary wavelets. We solve this by analytically continuing the surface of integration. If the surface is a sphere of radius $R$, this is done by complexifying $R$ to $R+ia$. The resulting complex sphere is shown to be a real bundle of disks with radius $a$ tangent to the sphere. Huygens' "secondary source points" are thus replaced by disks, and his spherical wavelets by well-focused pulsed beams propagating outward. This solves the back-propagation problem. The extended Huygens principle is a completeness relation for pulsed beams, giving a representation of a general radiation field as a superposition of such beams. Furthermore, it naturally yields a very efficient way to compute radiation fields because all pulsed beams missing a given observer can be ignored. Increasing $a$ sharpens the focus of the pulsed beams, which in turn raises the compression of the representation.

math-ph↗
arXiv2006

Extended wave propagators as pulsed-beam communication channels

Let P_0(x_r-x_e) be the causal propagator for the wave equation, representing the signal received at the spacetime point x_r due to an impulse emitted at the spacetime point x_e. Such processes are highly idealized since no signal can be emitted or received at a precise point in space and at a precise time. We propose a simple and compact model for extended emitters and receivers by continuing P_0 to an analytic function P(z_r - z_e), where z_e=x_e+iy_e represents a circular pulsed-beam emitting antenna centered at x_e and radiating in the spatial direction of y_e while z_r=x_r-iy_r represents a circular pulsed-beam receiving antenna centered at x_r and receiving from the spatial direction of y_r. The space components of y_e and y_r give the spatial orientations and radii of the antennas, while their time components s_e, s_r represent the time a signal takes to propagate along the antennas between the center and the boundary. The analytic propagator P(z_r - z_e) represents the transmission amplitude, forming a communication channel. Causality requires that the extension/orientation 4-vectors y_e and y_r belong to the future cone V, so that z_e and z_r belong to the future tube and the past tube in complex spacetime, respectively. The invariance of P(z_r - z_e) under imaginary spacetime translations has nontrivial consequences.

math-ph↗
arXiv2004

Eigenwavelets of the Wave equation

We study a class of localized solutions of the wave equation, called eigenwavelets, obtained by extending its fundamental solutions to complex spacetime in the sense of hyperfunctions. The imaginary spacetime variables y, which form a timelike vector, act as scale parameters generalizing the scale variable of wavelets in one dimension. They determine the shape of the wavelets in spacetime, making them pulsed beams that can be focused as tightly as desired around a single ray by letting y approach the light cone. Furthermore, the absence of any sidelobes makes them especially attractive for communications, remote sensing and other applications using acoustic waves. (A similar set of "electromagnetic eigenwavelets" exists for Maxwell's equations.) I review the basic ideas in Minkowski space, then compute sources whose realization should make it possible to radiate and absorb such wavelets. This motivates an extension of Huygens' principle allowing equivalent sources to be represented on shells instead of surfaces surrounding a bounded source.

math-ph↗
arXiv2004

Making electromagnetic wavelets II: Spheroidal shell antennas

In the companion paper, a coherent charge-current distribution for radiating electromagnetic wavelets was constructed on an oblate spheroidal surface S_a. Its main drawback was the necessity of including magnetic along with electric charges, making the sources impossible to realize. Here we show how this difficulty can be overcome by using Hertz potentials to generate a charge-current distribution due solely to bound electric charges. However, this distribution still appears difficult to realize because it consists of multiple surface layers on S_a. We show how it can be replaced by a simple volume distribution on a spheroidal shell. Our method generalizes the usual construction of equivalent Huygens sources, based on boundary conditions on an interface between electromagnetic media, by allowing the transition to be gradual without incurring addition complexity.

math-ph↗
arXiv2004

Energy-momentum conservation in pre-metric electrodynamics with magnetic charges

A necessary and sufficient condition for energy-momentum conservation is proved within a topological, pre-metric approach to classical electrodynamics including magnetic as well as electric charges. The extended Lorentz force, consisting of mutual actions by F=(E, B) on the electric current and G=(H, D) on the magnetic current, can be derived from an energy-momentum "potential" if and only if the constitutive relation G=G(F) satisfies a certain vanishing condition. The electric-magnetic reciprocity introduced by Hehl and Obukhov is seen to define a complex structure on the tensor product of 2-form pairs (F,G) which is independent of but consistent with the Hodge star operator defined by any Lorentzian metric. Contrary to a recent claim in the literature, it does not define a complex structure on the space of 2-forms itself.

math-ph↗
arXiv2004

Making electromagnetic wavelets

Electromagnetic wavelets are constructed using scalar wavelets as superpotentials, together with an appropriate polarization. It is shown that oblate spheroidal antennas, which are ideal for their production and reception, can be made by deforming and merging two branch cuts. This determines a unique field on the interior of the spheroid which gives the boundary conditions for the surface charge-current density necessary to radiate the wavelets. These sources are computed, including the impulse response of the antenna.

math-ph↗
arXiv2004

Helicity, polarization, and Riemann-Silberstein vortices

Riemann-Silberstein (RS) vortices have been defined as surfaces in spacetime where the complex form of a free electromagnetic field given by F=E+iB is null (F.F=0), and they can indeed be interpreted as the collective history swept out by moving vortex lines of the field. Formally, the nullity condition is similar to the definition of "C-lines" associated with a monochromatic electric or magnetic field, which are curves in space where the polarization ellipses degenerate to circles. However, it was noted that RS vortices of monochromatic fields generally oscillate at optical frequencies and are therefore unobservable while electric and magnetic C-lines are steady. Here I show that under the additional assumption of having definite helicity, RS vortices are not only steady but they coincide with both sets of C-lines, electric and magnetic. The two concepts therefore become one for waves of definite frequency and helicity. Since the definition of RS vortices is relativistically invariant while that of C-lines is not, it may be useful to regard the vortices as a wideband generalization of C-lines for waves of definite helicity.

math-ph↗