Howard E. Brandt

MAXIMAL PROPER ACCELERATION AND THE STRUCTURE OF SPACETIME

A limiting proper acceleration in nature follows deductively from known physics and compels the union of spacetime and four-velocity space into a maximal-acceleration invariant phase space having an intrinsic Kaluza-Klein-type fiber-bundle structure with manifest gauge properties. The Riemann curvature scalar of the bundle manifold is determined, and a possible action principle is considered to serve as a basis for the generation of field equations.

Structure of spacetime tangent bundle

The universal upper limit on attainable proper acceleration relative to the vacuum imposes restrictions on possible structures in the spacetime tangent bundle. Various features of the differential geometry of the spacetime tangent bundle are presented here. Also, a modified Schwarzschild solution is obtained, and the associated gravitational red shift is calculated.

Differential geometry of spacetime tangent bundle

Conditions are investigated under which the Levi-Civita connection of the spacetime tangent bundle corresponds to that of a generic tangent bundle of a Finsler manifold. Also, requirements are specified for the spacetime tangent bundle to be almost complex or Kählerian.

Riemann curvature scalar of spacetime tangent bundle

The maximum possible proper acceleration relative to the vacuum determines much of the differential geometric structure of the space-time tangent bundle. By working in an anholonomic basis adapted to the spacetime affine connection, one derives a useful expression for the Riemann curvature scalar of the bundle manifold. The explicit documentation of the proof is important because of the central role of the curvature scalar in the formulation of an action with resulting field equations and associated solutions to physical problems.

Connections and geodesics in the spacetime tangent bundle

Recent interest in maximal proper acceleration as a possible principle generalizing the theory of relativity can draw on the differential geometry of tangent bundles, pioneered by K. Yano, E. T. Davies, and S. Ishihara. The differential equations of geodesics of the spacetime tangent bundle are reduced and investigated in the special case of a Riemannian spacetime base manifold. Simple relations are described between the natural lift of ordinary spacetime geodesics and geodesics in the spacetime tangent bundle.

Finsler-spacetime tangent bundle

The Levi-Civita connection coefficients of the spacetime tangent bundle, for the case of a Finsler spacetime, are reduced to the form given by Yano and Davies for a generic tangent bundle of a Finsler manifold. A useful expression is also obtained for the Riemann curvature scalar of a Finsler-spacetime tangent bundle.

Khler spacetime tangent bundle

Requirements are delineated for the spacetime tangent bundle to be Kählerian. In particlar, an almost complex structure is constructed in the case of a Finsler spacetime, and its covariant derivative in terms of the bundle connection is shown to be vanishing, provided the gauge curvature field is vanishing. The Levi-Civita connection coefficients and the Riemann curvature scalar are also specified for the Kähler spacetime tangent bundle.

Curved electromagnetic missiles

An interesting topic has been the possible behavior of transient fields in the limit of great distances from their sources. Under the physical restriction that the total energy radiated is finite, it has been shown that the energy reaching a distant receiver can decrease with distance much more slowly than the usual r−2. Such cases of slow decrease have been referred to as electromagnetic missiles. All of the wide variety of known missiles propagate in essentially straight lines. It is shown here that such a missile can follow a path that is strongly curved. An example of a curved electromagnetic missile is explicitly constructed and some of its simpler properties are discussed.

Fun with pulses

In 1887, 26 years after Maxwell first published his seminal equations that define electromagnetic theory, Hertz produced the first controlled radio wave in the laboratory using an oscillating dipole. Since then, the use of electromagnetic waves has permeated our daily life. In the earlier applications the wave forms were almost sinusoidal, prominent examples being radio (AM and FM), television and radar. In each of these cases, there is a relatively narrow bandwidth around a carrier frequency.

Cubic dispersion relation for a relativistic backward-wave oscillator

The cubic approximation to the dispersion relation for a relativistic backward-wave oscillator is obtained, and the utility and limits of the approximation are presented. The approximation is obtained by Taylor series expansion of the wave admittance in the dispersion relation for the transverse-magnetic and free-streaming modes of a relativistic, thin, hollow, cylindrical electron beam moving along the axis of a disc-loaded waveguide in a strong axial magnetic field. The resulting cubic dispersion relation yields instability growth rates and frequencies which fall off beyond their maximum more sharply with increasing wavenumber than for the complete dispersion relation. The approximation is found to be quite good near the operating points of contemporary high-power relativistic backward-wave oscillators, namely, for relatively long wavelength and small ratio of Budkers parameter to the relativistic gamma factor of the beam. >