2001/08/15 by Gerald Kaiser, Kaiser, Gerald
Mathematics · Physics and Astronomy · #31-XX #32-XX #35-XX #78-XX #Advanced Mathematical Physics Problems #Complex Variables (math.CV) #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Radio Astronomy Observations and Technology #math-ph #math.CV #math.MP #msc:31-XX #msc:32-XX #msc:35-XX #msc:78-XX
paper · pdf · doi:10.48550/arxiv.math-ph/0108006
10 pages, Invited paper, NATO Advanced Research Workshop on Clifford Analysis and its Applications, Prague, October 30 - November 3, 2000
arxiv created 2001/08/15 · openalex publication_date 2001/08/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G(xr-xe) be the causal Green function for the wave equation in four spacetime dimensions, representing the signal received at the spacetime point xr due to an impulse emitted at the spacetime point xe. Such emission and reception processes are highly idealized, since no signal can be emitted or received at a single (mathematical) point in space and time. We present a simple model for \sl extended \rm emitters and receivers by extending G analytically to a function G(zr- ze), where ze=xe+iye is a complex spacetime point representing a circular \sl pulsed-beam emitting antenna dish \rm centered at xe and emitting in the direction of ye, and zr=xr-iyr represents a circular \sl pulsed-beam receiving antenna dish \rm centered at xr and receiving from the direction of yr. The holomorphic Green function G(zr-ze) represents the \sl coupling \rm between the emission from ze and the reception at zr. To preserve causality and give nonsingular coupling, the orientation vectors ye and yr must belong to the \sl future cone \rm V+ in spacetime. Equivalently, ze and zr belong to the \sl future and past tubes \rm in complex spacetime, respectively. The space coordinates of ye and yr give the spatial orientations and radii of the dishes, while their time coordinates determine the \sl duration and focus \rm of the emission and reception processes. The \sl directivity \rm D(y) of the communication process is a convex function on V+, i.e., D(yr+ye)≤ D(yr)+D(yr). This shows that the efficiency of the communication can be no better than the sum of its emission and reception components.