RADIATED POWER

The electric and magnetic field components are obtained from the wave function $\psi_F(\xi,\tau,r,\theta)$ by taking the partial derivatives listed in the tables in Section II.C. With their help we shall now find the Poynting vector component along the $\tau$-direction, namely

$$\frac{1}{4\pi}(\hat B_r \hat E_\theta-\hat B_\theta \hat E_r)\xi \equiv T^\xi_{~\tau}~.$$

Its space integral,

$$\int_0^\infty \int_0^\infty \int _0^{2\pi} T^\xi_{~\tau}\,\xi d\tau \, rdr \,d\theta ~,$$ (57)

is the total radiated momentum (=radiant energy flow) into the $\tau$-direction. It is positive (resp. negative) whenever the source is confined to Rindler sector $I$ (resp. $II$). Furthermore, the $\tau$-momentum is independent of the synchronous time $\xi$ because $\tau$ is a cyclic coordinate. This $\tau$-momentum measures the energy radiated by the two accelerated sources, and it takes the place of what in a static inertial reference frame is the emitted energy.

Both the T.E. and the T.M. field have the same Poynting vector component along the $\tau$-direction. More precisely, reference to the table of T.E. and the T.M. field components (Section II) shows that in Rindler sector $F$ this Poynting object is

$$T^\xi_{~\tau}=\frac{\xi}{4\pi} \left[ \frac{\partial}{\part... ...ac{1}{r}\frac{\partial\psi}{\partial \theta}\right) \right] ~,$$ (58)

the same for both types of fields. Furthermore, the wave function $\psi$ is governed by a wave equation, which is also common to both fields. Consequently, the mathematical analysis which relates observations to the radiation sources is the same for both types of radiation fields. However, it is the difference in two types of sources which is important from the viewpoint of physics.

The only difference lies in the source and hence in the amplitude and phase of $\psi$ in $F$. Comparing the ensuing Eq.(60) with Eq.(63), one sees that T.E. and T.M. polarized radiation are caused by the densities of magnetic and electric dipole moment respectively.