2012, Vol.15, No.1, pp.84-94

It is shown that the generally covariant extended method of Riemann --Silberstein - Majorana - Oppenheime in electrodynamics, specified in Schwarzschild metrics, after separating the variables reduces the problems of electromagnetic solutions to a differential equation similar to that arising in the case of a scalar filed in the Schwarzschild space-time. This differential equation is recognized as a confluent Heun equation. Also, the electromagnetic field is treated on the base of 10-dimensional Duffin - Kemmer approach, when in addition to six components of the strength tensor one uses a 4-component electromagnetic potential. Corresponding system of 10 radial equations is simplified by the use of additional constraints steaming from eigenvalue equation for the spatial parity operator ; the radial system is divided into two subsystems of 4 and 6 equations respectively. In this second approach the problem of electromagnetic field reduces to the confluent Heun differential equation as well. In particular, we show explicitly how solutions found in complex form are embedded into the 10-dimensional formalism. Besides we determine radial functions that are responsible for gauge degrees of freedom.

Key words: Maxwell equations, Schwarzschild black hole, Heun equation

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