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 [50
]), 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. Their derivatives are represented by
Note: At this point we are taking the velocity of light as and omitting it; later, when we want
the correct units to appear explicitly, we restore the
. This entails, via
,
,
changing the prime derivatives to include the
, i.e.,
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 early.
![]() |
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
![]() |
appears as the harmonics in the harmonic expansions. Thus, care must be used when
lowering or raising the relativistic index, i.e.,
.
Symbol/Acronym |
Definition |
![]() ![]() |
Future conformal null infinity, Complex future conformal null infinity |
![]() ![]() |
Future, Past conformal timelike infinity, Conformal spacelike infinity |
![]() ![]() |
Minkowski space, Complex Minkowski space |
![]() ![]() |
Bondi time coordinate, Retarded Bondi time
( |
![]() |
Derivation with respect to |
![]() |
Derivation with respect to |
![]() |
Affine parameter along null geodesics |
![]() |
|
![]() |
Tensorial spin- |
![]() ![]() |
|
![]() |
Metric function on |
![]() |
Application of |
![]() |
Null tetrad system; |
NGC |
Null Geodesic Congruence |
NP/SC |
Newman–Penrose/Spin-Coefficient Formalism |
![]() |
Metric coefficients in the Newman–Penrose formalism |
![]() |
Weyl tensor components in the Newman-Penrose formalism |
![]() |
Maxwell tensor components in the Newman–Penrose formalism |
![]() |
Complex divergence of a null geodesic congruence |
![]() |
Twist of a null geodesic congruence |
![]() ![]() |
Complex shear, Asymptotic complex shear of a NGC |
![]() |
|
![]() |
Cut function on |
![]() |
Complex auxiliary (CR) potential function |
![]() |
Derivation with respect to |
![]() |
Good-Cut Equation, describing asymptotically shear-free NGCs |
GCF |
Good-Cut Function |
![]() |
Stereographic angle field for an asymptotically
shear-free NGC at |
![]() |
CR equation, describing the embedding of |
three-dimensional CR Structure |
A class of one-forms describing a real three-manifold
of |
![]() |
Complex four-dimensional solution space to the Good-Cut Equation |
![]() |
Complex electromagnetic dipole |
![]() |
Complex center-of-charge world line, lives in
|
![]() |
|
![]() |
Complex gravitational dipole |
![]() |
Complex center-of-mass world line, lives in |
UCF |
Universal Cut Function |
![]() |
UCF; corresponding to the complex center-of-charge world line |
![]() |
Bondi Mass Aspect |
![]() |
Bondi mass |
![]() |
Bondi linear three-momentum |
![]() |
Vacuum linear theory identification of angular momentum |
http://www.livingreviews.org/lrr-2009-6 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |