a) Though in places, e.g., in Section 2.4, the symbols, ,
,
…, i.e., with an
can be thought of as the abstract representation of a null tetrad (i.e., Penrose’s
abstract index notation [66
]), in general, our intention is to describe vectors in a coordinate
representation.
b) The symbols, ,
most often represent the coordinate versions of different null
geodesic tangent fields, e.g., one-leg of a Bondi tetrad field or some rotated version.
c) The symbol, , (with hat) has a very different meaning from the others. It is used to represent
the Minkowski components of a normalized null vector giving the null directions on an arbitrary light
cone:
The Bondi time, , is closely related to the retarded time,
. The use of the retarded
time,
, is important in order to obtain the correct numerical factors in the expressions for
the final physical results. Derivatives with respect to these variables are represented by
Frequently, in this work, we use terms that are not in standard use. It seems useful for clarity to have some of these terms defined from the outset:
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where the indices, represent three-dimensional Euclidean indices. To avoid extra notation
and symbols we write scalar products and cross-products without the use of an explicit Euclidean
metric, leading to awkward expressions like
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appears as the harmonics in the harmonic expansions. Thus, care must be used when
lowering or raising the relativistic index, i.e.,
.
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The conventions used here were chosen so that the numerical coefficient of in
was equal to
one; this re-scaling can simply be viewed as choosing a different (perhaps less natural)
identification for the electromagnetic quadrupole moment, or as a sort of gauge choice for our
results.
http://www.livingreviews.org/lrr-2012-1 |
Living Rev. Relativity 15, (2012), 1
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