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HELP FOR: Basis/tetrad Object Library
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SYNOPSIS:
- a set of object definitions for describing spacetimes in terms of
non-holonomic bases and the Newman-Penrose formalism.
- In order to use the objects defined in this library, a n-basis must be
defined using makeg, or loaded using grload (or qload).
- The covariant derivative indices for spacetimes specified by a basis
are `cbdn' and `cbup', as in
grcalc ( R(bdn,bdn,cbdn):
This form of covariant derivative uses the rotation coefficients
rot(bdn,bdn,bdn) defined for the basis, rather than the Christoffel
symbols Chr(dn,dn,dn) which must be translated into a basis and are
thus usually less efficient.
Ordinary partial derivatives along the basis vectors can be specified
using the indices `pbdn' and `pbup', as in
grcalc ( Psi2(pbdn) ):
For null tetrads, the derivative operators corresponding to these
derivatives can also be used (see ?grt_operators).
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GRTensor name Description
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e(bdn,up) - basis vectors (l,n,m,mbar).
e(bdn,dn) - basis 1-forms.
nullt(up), nullt(dn) - fancy display for null tetrads.
basisv(up), basisv(dn) - fancy display for general bases.
l(up), n(up), m(up), - individual basis vectors (null tetrad only).
mbar(up)
e1(up), ... , e4(up) - individual basis vectors (general 4d basis).
w1(dn), ... , w1(dn) - individual basis 1-forms (general 4d basis).
testNP(bdn,bdn) - check that a set of basis vectors satisfy an
NP inner product by calculating e{(a)c}*e{(b)^c}
eta(bdn,bdn) - defines the inner product of two basis vectors.
lambda(bdn,bdn,bdn) - intermediate objects for calculation of rotation
coefficients.
rot(bdn,bdn,bdn) - Ricci rotation coefficients.
str(bdn,bdn,bdn) - structure constants.
NPkappa, NPsigma,
NPlambda, NPmu, NPrho
NPnu, NPtau, NPpi,
NPepsilon, NPgamma,
NPalpha, NPbeta - Newman-Penrose spin coefficients kappa ... beta.
NPSpin - collection of all NP spin coefficients.
NPkappabar, NPsigmabar,
NPlambdabar, NPmubar,
NPrhobar, NPnubar,
NPtaubar, NPpibar,
NPepsilonbar, NPgammabar,
NPalphabar, NPbetabar - complex conjugates of the Newman-Penrose spin
coefficients.
NPSpinbar - collection of complex conjugates of all NP spin
coefficients.
Psi0, Psi1, Psi2
Psi3, Psi4 - Weyl curvature coefficients
WeylSc - collection of Weyl coefficients coefficients
Phi00, Phi01, Phi10,
Phi02, Phi20, Phi12,
Phi21, Phi11, Phi22,
Lambda - Ricci curvature coefficients
RicciSc - collection of Ricci curvature coefficients
Ckappa ... Cbeta - NP spin coefficients using alternate formulations
(see below).
Ckappabar ... Cbetabar - complex conjugates of NP spin coefficients
using alternate formulations
CPsi0 ... CPsi4 - Weyl curvature coefficients using altenate
formulations
CPhi00 ... CPhi22, - Ricci curvature coefficients using alternate
CLambda formulations
CSpin, CSpinbar - collection of spin coefficients, alternate
formulations.
CRicciSc, CWeylSc - collection of Ricci and Weyl coefficients using
alternate formulations.
Petrov - the Petrov type of a NP basis.
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- to refer to the basis components of a standard GRTensor object, use the index
place-holders bdn and bup just as you would use dn and up (eg. to use the
basis components of g(dn,dn), type g(bdn,bdn) ).
- when using a null tetrad, the objects corresponding to the basis vectors
are:
l(up) ---> e(1,up),
n(up) ---> e(2,up),
m(up) ---> e(3,up),
mbar(up) ---> e(4,up).
To display tetrads, one can also use the objects
nullt(up), nullt(dn)
for null tetrads. For general 4d contravariant bases, the individual vectors
can be accessed as e1(up), e2(up), e3(up), e4(up) and together as basisv(up).
Contravariant basis vectors can be referenced as w1(dn), w2(dn), w3(dn),
w4(dn), and together as basisv(dn).
[Note, there is redundancy between these definitions, ie. e1(dn) = w1(dn),
etc. The form which is assigned by default depends on what form was
specified in the original spacetime file created by makeg().
These definitions are intended for display purposes, as well as for
use in grdef(). All subsequent calculations make use of the equivalent
objects e(bdn,up) and e(bdn,dn). Alteration to any of the nullt(up) or
basisv(up) vectors via gralter() will not affect subsequent calculations
from the basis.]
- the `C' objects listed above use the alternate formulations of
S. Allen, G. J. Fee, A. T. Kachura, F. W. Letniowski, R. G. McLenaghan,
(1994) Gen. Rel. Grav., 26, p. 26.
These definitions avoid the use of contravariant components of the basis
forms, thus avoiding inversion of the basis. If the basis vectors are
initially specified in covariant form, this may provide some time benefit.
See
D. Pollney, P. Musgrave, K. Lake, Class. Quant. Grav., 13, 2289.
for a discussion of this point.
- two GRTensorII commands relate specifically to null tetrads:
nptetrad - defines a null tetrad given a metric.
nprotate - performs rotations on a null tetrad.
See the individual help pages for these functions for more information.
- the result of a calculation of the Petrov type can be analyzed using
the PetrovReport() function. See ?PetrovReport for more information.
- the definitions of these objects are concisely reviewed in
S. Chandrasekhar, The Mathematical Theory of Black Holes, 2nd ed.,
Oxford University Press, 1992.
The NP objects are defined in
Ezra Newman and Roger Penrose (1962), J. Math. Phys., 3, 566;
and errata (1962), 4, 998.
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EXAMPLES:
> qload ( npschw ):
> grcalc ( WeylSc ):
> grdisplay ( _ ):
For the npschw spacetime:
Weyl Scalar, NP Psi0
Psi0 = 0
Weyl Scalar, NP Psi1
Psi1 = 0
Weyl Scalar, NP Psi2
m
Psi2 = - ----
3
r
Weyl Scalar, NP Psi3
Psi3 = 0
Weyl Scalar, NP Psi4
Psi4 = 0
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SEE ALSO: grt_objects, grt_operators, PetrovReport(), nptetrad(), nprotate(),
grdef().
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