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HELP FOR: nptetrad
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CALLING SEQUENCE: nptetrad ( [lnSpace] )
PARAMETERS: lnSpace -- (optional) specifies a pair of coordinates
whose corresponding 1-forms will be used
to construct the l and n vectors of the
tetrad, eg. [t,r].
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SYNOPSIS:
- The nptetrad() command finds a null NP tetrad corresponding to a
given metric, g(dn,dn). The forms are saved as the object e(bdn,dn).
- Basis vectors are initially chosen as column vectors of the covariant
metric. From these vectors, an orthonormalization is carried out to
create a set of null vectors that satisfy the NP inner product,
c [ 0, 1, 0, 0 ]
e e = [ 1, 0, 0, 0 ].
(a)c (b) [ 0, 0, 0, -1 ]
[ 0, 0, -1, 0 ]
- If the optional `lnSpace' argument is supplied, then the 1-forms
corresponding to the given columns of g(dn,dn) are use preferentially
to construct the l and n vectors. The `lnSpace' argument is specified
as a MapleV list of two coordinate names (eg. [t,r]), or numbers
(eg. [4,1]). At least one of these coordinates should be timelike or
null.
- Because of inconsistencies in MapleV's application of the imaginary
number I=sqrt{-1} in symbolic expressions, incorrectly formed tetrads can
result. The output of an nptetrad() operation should be checked as
follows:
- the vectors l and n should have no imaginary components
(l and n correspond to the first and second rows of e(bdn,dn),
respectively).
- the vectors m and mbar should be complex conjugates of each other
(m and mbar correspond to the third and fourth rows of e(bdn,dn),
respectively).
- the inner product eta(bup,bup) should be of the NP form given above.
If any of these criteria can not be met, then the session should be
restarted and nptetrad() run again, possibly with a different choice
of lnSpace argument. [Note, the location of I in MapleV expressions
is strongly dependent on MapleV's memory state which can vary from
one run to the next. Thus different tetrads can result between two runs
of nptetrad() even if the input is identical.]
- The appearance of large terms in the components of the basis can
impair later calculations. To ensure that future calculations are
optimized, apply the relevant gralter() commands to the basis
vectors until they seem to be fully simplified. See ?gralter.
- The implementation of the Newman-Penrose formalism in GRTensorII
follows the original specification of Newman and Penrose (1962).
As such, it requires that the spacetime have a -2 signature. If
this is not the case, the program will ask the user if the metric
signature is to be reversed in the creation of the tetrad. If the
user answers `no' to this prompt, the tetrad can not be created and
the program aborts. A `yes' answer will cause the signature of
g(dn,dn) and g(up,up) to be negated.
- Tetrads created using nptetrad() can be saved using the grsaveg()
command.
- The results of the calculation are placed in the GRTensorII objects
e(bdn,dn) and eta(bup,bup), the basis vectors and basis inner product,
respectively.
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EXAMPLES:
> qload ( schw );
Calculating ds for schw ... Done. (0.010000 sec.)
Default spacetime = schw
For the schw spacetime:
Coordinates
a
x = [ r, theta, phi, t ]
Line element
2
ds =
2
d r 2 2 2 2 2 2
--------- + r d theta + r sin(theta) d phi + (- 1 + 2 m/r) d t
1 - 2 m/r
The Classical Schwarzschild metric
> nptetrad ( [t,r] );
The metric signature of the schw spacetime is +2.
In order to create an NP-tetrad, the signature of g(dn,dn) will be
changed to -2.
Continue? (1=yes):
> 1
The signature of the schw spacetime is now -2.
For the schw spacetime:
Basis inner product
[ 0 1 0 0 ]
[ ]
(a) (b) [ 1 0 0 0 ]
eta = [ ]
[ 0 0 0 -1 ]
[ ]
[ 0 0 -1 0 ]
Null tetrad (covariant components)
- r + 2 m
l[a] = [ 1, 0, 0, - --------- ]
r
r
n[a] = [ 1/2 ---------, 0, 0, 1/2 ]
- r + 2 m
1/2 1/2
m[a] = [ 0, - 1/2 2 r, - 1/2 I 2 r sin(theta), 0 ]
1/2 1/2
mbar[a] = [ 0, - 1/2 2 r, 1/2 I 2 r sin(theta), 0 ]
The null tetrad has been stored as e(bdn,dn).
>
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SEE ALSO: makeg, grsaveg, grt_basis, nprotate.
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