Double Slit Interference

The third property of the radiation process is that it highlights the interference between the waves coming from Rindler sectors $I$ and $II$. The interference pattern, which is recorded on a hypersurface of synchronous time $\xi=constant$, has fringes whose separation yields the separation between the two localized in $I$ and $II$. Let these sources be located symmetrically at

$$\xi'_I=\xi'_{II}\equiv \xi'_0~,$$

and let them have equal proper frequency $\omega_0$ and hence (in compliance with the first term of the wave Eq.(27) equal Rindler coordinate frequency

$$\omega=\omega_0\xi'_0~.$$

Consequently, they are characterized by their amplitudes and their phases. Indeed, their form is

$$S^m_{I}(\tau +\sinh^{-1} u_I) = A^0_I \cos [\omega_0\xi'_0(\tau+\sinh^{-1}u_I)+\delta^m_I]$$

$$S^m_{II}(\tau -\sinh^{-1} u_{II}) = A^0_{II} \cos [\omega_0\xi'_0(\tau-\sinh^{-1}u_I)+\delta^m_{II}]~.$$ (55)

Thus the full scalar field, Eq.(54), expresses two waves. Both propagate in the expanding inertial frame, which is coordinatized by $(\xi,\tau,r,\theta)$. Their respective wave crests are located in compliance with the constant phase conditions $\tau\pm \sinh^{-1}u_I=const.$ Consequently, one wave travels into the $+\tau$-direction with amplitude $A^m_{I}$, the other into the $-\tau$-direction with amplitude $A^m_{II}$. They have well-determined phase velocities. Together, these two waves form an interference pattern of standing waves,

$$\psi_F(\xi,\tau,r,\theta) = \frac{1}{\sqrt{(\xi^2-\xi_0'^2-r^2)^2+(2\xi\xi_0')^2 }} \left[ \b... ...ay}(A^0_I-A^0_{II}) \cos [\omega_0\xi'_0(\tau-\sinh^{-1}u_I)+\delta^0_I]\right.$$

$$~ \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad... ...omega_0\xi'_0\sinh^{-1}u_I -\frac{\delta^0_{II}-\delta^0_{I}}{2}\right) \right]$$

$$+ \quad \textrm{higher~multipole~terms~of~order~}m=1,2,3,\cdots \quad .$$ (56)

The amplitude of this interference pattern is $A^0_{II}>0$, and there is a uniform background of amplitude $(A^0_I-A^0_{II})>0$. At synchronous time $\xi$ the interference fringes along the $\tau$-direction can be read off the factor

$$\sin (\omega_0\xi'_0\tau +\frac{\delta^0_{II}+\delta^0_{I}}{2})$$

in Eq.(56). Consequently, the fringes are spaced by the amount

$$\left( \begin{array}{c} \textrm{proper}\\ \textrm{fringe~... ...frequency}\\ \textrm{of~ the~source} \end{array} \right)}~.$$

This means that, analogous to a standard optical interference pattern, the fringe spacing is inversely proportional to the distance $2\xi'_0$ between the two sources. Furthermore, the position of this interference pattern depends on the phase of source $I$ relative to source $II$[#!phase!#]. It is difficult to find a more welcome way than the four Rindler sectors for double slit interference.

These observations lead to the conclusion that (i) the four Rindler sectors quite naturally accommodate a double slit interferometer, and that (ii) the spatial as well as the temporal properties of the interference fringes, together with the magnitude of the travelling background wave, are enough to reconstruct every aspect of the two sources, Eq.(55).