RADIATION: MATHEMATICAL RELATION TO THE SOURCE

Any Maxwell field $F_{\mu\nu}$ can be obtained from a single Klein-Gordon scalar field $\psi$, a solution to the scalar wave Eq.(5). This is done with the help of the T.E. and T.M. tables of derivatives. Similarly, any K-G field $\psi$ can be obtained from the source function $S$. This is done with the help of the unit impulse response $\mathcal G$ (Green's function), the solution to

$$\left( -\frac{\partial ^2}{\partial t^2} + \frac{\partial^... ...-t')\delta(z-z') \frac{\delta(r-r')}{r} \delta(\theta-\theta')$$ (34)

In terms of $\mathcal G$ the solution to the inhomogeneous wave equation, Eq.(27) is

$$\psi(t,z,r,\theta) = \int_{-\infty}^\infty \int_0^\infty \int_0^\infty \int_0^{2\pi} {... ...au',\xi',r',\theta') ~4\pi S(t',z',r',\theta')~~dt'\, dz'\, r' dr'\, d\theta'~,$$ (35)

Here $S$ is the scalar source, which is non-zero only in Rindler sectors $I$ and $II$.