Spacetime triangle diagram technique

In physics and mathematics, the spacetime triangle diagram (STTD) technique, also known as the Smirnov method of incomplete separation of variables, is the direct space-time domain method for electromagnetic and scalar wave motion.

The STTD technique belongs to the second among the two principal ansätze for theoretical treatment of waves — the frequency domain and the direct spacetime domain. The most well-established method for the inhomogeneous (source-related) descriptive equations of wave motion is one based on the Green's function technique. For the circumstances described in Section 6.4 and Chapter 14 of Jackson's Classical Electrodynamics, it can be reduced to calculation of the wave field via retarded potentials (in particular, the Liénard–Wiechert potentials).

Despite certain similarity between Green's and Riemann–Volterra methods (in some literature the Riemann function is called the Riemann–Green function ), their application to the problems of wave motion results in distinct situations:

and it was the Riemann–Volterra representation that Smirnov used in his Course of Higher Mathematics to prove the uniqueness of the solution to the above problem (see, item 143).

are invoked. The Riemann-Volterra approach presents the same or even more serious difficulties, especially when one deals with the bounded-support sources: here the actual limits of integration must be defined from the system of inequalities involving the space-time variables and parameters of the source term. However, this definition can be strictly formalized using the spacetime triangle diagrams. Playing the same role as the Feynman diagrams in particle physics, STTDs provide a strict and illustrative procedure for definition of areas with the same analytic representation of the integration domain in the 2D space spanned by the non-separated spatial variable and time.

Several efficient methods for scalarizing electromagnetic problems in the orthogonal coordinates ${\displaystyle x_{1},x_{2},x_{3}}$ were discussed by Borisov in Ref. The most important conditions of their applicability are ${\displaystyle h_{3}=1}$ and ${\displaystyle \partial _{3}(h_{1}/h_{2})=0}$, where ${\displaystyle h_{i},i=1,2,3}$ are the metric (Lamé) coefficients (so that the squared length element is ${\displaystyle ds^{2}=h_{1}^{2}dx_{1}^{2}+h_{2}^{2}dx_{2}^{2}+h_{3}^{2}dx_{3}^{2}}$). Remarkably, this condition is met for the majority of practically important coordinate systems, including the Cartesian, general-type cylindrical and spherical ones.

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