   Next: Multipole Radiation Field Up: Full Scalar Radiation Field Previous: Full Scalar Radiation Field

### Source as a Sum of Multipoles

The circumstance of long wave lengths is expressed by the inequality where is the radius of the cylinder surrounding the source. This circumstance allows us to set (46)

throughout the integration region where the source is non-zero, and it allows us to introduce the st multipole moment (per unit length ) (47)

for the double integral on the right hand side of Eq.(45). This multipole density [#!factor_of_two!#] is complex. However, the reality of the master source implies and is implied by In terms of this multipole density the full scalar radiation field in is  (48)

The evaluation of the mode integral is now an easy two step task. First recall the th recursion relation (49)

where for negative one uses This recursion relation is a consequence of consolidating two familiar contiguity relations for the Bessel functions. Introduce Eq.(49) into the integrand of Eq.(48).

Second, use the standard expression for the Dirac delta function. Apply this equation to Eq.(48). Consequently, the full scalar radiation field in Rindler sector reduces to the following multipole expansion (50)

where     Doing the -integration yields   (51)

Here and mean that the source functions are evaluated in compliance with the Dirac delta functions at and respectively. Recall that is a strictly timelike coordinate in Rindler sector , while in the coordinate is strictly spacelike. Consequently, one should not be tempted to identify and with what in a static inertial frame corresponds to evaluations at advanced or retarded times. Instead, one should think of the observation event in and the source event in as lying on each other's light cones both of which cut across the future event horizons of and . More explicitly, one has which means that the source has been evaluated on the past light cone of at in Rindler sector . Similarly, which means that the source has been evaluated on the past light cone of at in Rindler sector . The expression, Eq.(51), for the full scalar radiation field is exact within the context of wavelengths large compared to the size of the source. Furthermore, one should note that even though there is only one -integral, and are source functions with distinct domains, namely, Rindler sectors and respectively.   Next: Multipole Radiation Field Up: Full Scalar Radiation Field Previous: Full Scalar Radiation Field
Ulrich Gerlach 2001-10-09