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HELP FOR: invars
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SYNOPSIS:
- GRTensorII includes a library of 18 scalar polynomial invariants of the
Riemann tensor:
{ Ricciscalar, R1, R2, R3, M3, M4 } [Real]
{ W1, W2, M1, M2, M5, M6 } [Complex]
- The set includes those defined by
[CM]: J. Carminati and R. G. McLenaghan (1991), J. Math. Phys., 32, 3135.
This set has been shown to be complete for perfect fluid, and null and
non-null Maxwell Ricci tensors.
- An additional invariant, M6, is included. It corresponds to that given by
E. Zakhary and C. B. G. McIntosh (1996) (preprint)
who suggest that the set including this invariant is complete for
all Ricci and Weyl tensor types.
- To refer to the full set of invariants the name `invars' is used,
as in `grcalc ( invars ):'.
- Individual invariants can be referred to by their names outlined in
the table below and the following paragraphs.
- Four invariants are constructed from the Ricci tensor (or spinor)
alone. The GRTensorII names for these invariants corresponds to that
of [CM]. These are { Ricciscalar, R1, R2, R3 }. They are all real
valued. They can be referred to as a group using the name Rinvars,
as in `grcalc ( Rinvars ):'.
- Four invariants are constructed from the Weyl tensor (or spinor) alone.
The GRTensor names for these invariants corresponds to that of [CM],
These are { W1, W2 }. They are both complex valued, and the
individual real and imaginary parts can be accessed via the names
W1R, W1I, W2R, W1I. They can also be referred to as a group using the
name Winvars, as in `grcalc ( Winvars ):'.
- Ten invariants can be constructed from combinations of Ricci and Weyl
tensors. The GRTensorII names for these invariants corresponds to those of
[CM] except for the additional invariant which is named M6. The names are
{ M1, M2, M3, M4, M5, M6 }. The real and imaginary parts of the complex
invariants can be accessed via the names M1R, M1I, M2R, M2I, M5R, M5I,
M6R, M6I. They can also be referred to as a group using the name Minvars,
as in `grcalc ( Minvars ):'.
- The invariants listed in [CM] can be calculated as a group using the
names CM (for the full set), CMR (for the Ricci invariants), CMW (for
the Weyl set) and CMM (for the `mixed' invariants).
These are summarized in the following table (for definitions in terms of
the Ricci and Weyl tensors, see ?cmscalar):
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GRTensorII name Spinor definition
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A B A'B'
R1 Phi Phi
ABA'B'
A A' B B' C C'
R2 Phi Phi Phi
B B' C C' A A'
A A' B B' C C' D D'
R3 Phi Phi Phi Phi
B B' C C' D D' A A'
ABCD
W1 Psi Psi
ABCD
AB CD EF
W2 Psi Psi Psi
CD EF AB
ABA'B' CD
M1 Psi Phi Phi
ABCD A'B'
AB CDA'B' EF
M2 Psi Psi Phi Phi
ABCD EF A'B'
___ ABA'B' CDC'D'
M3 Psi Psi Phi Phi
ABCD A'B'C'D'
___ ABC'E' CEA'B' D D'
M4 Psi Psi Phi Phi Phi
ABCD A'B'C'D' E E'
CDEF A'B'E'F' AB
M5 Psi Psi Psi Phi Phi
ABCD A'B' EFE'F'
ABA'B' CDC'D' EF
M6 Psi Phi Phi Phi Phi
ABCD EFA'B' C'D'
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W1R, W2R, M1R, M2R, Re(W1), Re(W2), Re(M1), Re(M2), Re(M5), Re(M6)
M5R, M6R
W1I, W2I, M1I, M2I, Im(W1), Im(W2), Im(M1), Im(M2), Im(M5), Im(M6)
M5I, M6I
invars { Ricciscalar, R1, R2, R3, W1, W2, M1, ... , M6 }
Rinvars { Ricciscalar, R1, R2, R3 }
Winvars { W1, W2 }
Minvars { M1, M2, M3, M4, M5, M6 }
CMinvars { Ricciscalar, R1, R2, R3, W1, W2, M1, ... , M5 }
SSinvars (*) { Ricciscalar, R1, R2, W2R}
CM CMinvars
CMR Rinvars
CMW Winvars
CMM { M1, M2, M3, M4, M5 }
CMB SSinvars
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(*) In type B warped product spaces (e.g. spherical symmetry),
only a subset of the invariants need be calculated.
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Notes:
- It can be shown for general spacetimes that any invariant, I, of order
five or less can be written as a polynomial, P, in this set of 18
invariants,
I := P (Ricciscalar, R1, ... , M6)
- It is not difficult to show that the 18 listed invariants are not
related by polynomials of the above form. However, it is not known if
relationships of the form
P ( Ricciscalar, R1, ... , M6 ) = 0
exist between these invariants.
Calculation algorithms:
i. If the spacetime has been specified by a null tetrad, spinor polynomials
in Weyl and Ricci curvature components (of the form given in the appendix
of [CM]) are used in the calculation by default.
ii. If the spacetime has been specified by a non-null basis, the basis
components of Ricci and Weyl are used by default. For the definitions
of these invariants in terms of the Ricci and Weyl tensors, see
?cmscalar.
iii. Otherwise the regular metric components of the Ricci and Weyl tensor are
used.
- In cases (ii) and (iii) intermediate tensors C2(dn,dn,up,up), CS(up,dn) and
CSstar(up,dn) are calculated in the course of calculating the invariants.
Intermediate simplification of these objects (via gralter) may help if a
MapleV `Object to large' error is encountered.
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EXAMPLE:
> qload ( schw ):
> grcalc ( Winvars ):
> grdisplay ( Winvars ):
For the schw spacetime:
CM invariant Re(W1)
2
m
W1R = 6 ----
6
r
CM invariant Im(W1)
W1I = 0
CM invariant Re(W2)
3
m
W2R = - 6 ----
9
r
CM invariant Im(W2)
W2I = 0
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SEE ALSO: cmscalar, dinvar, grt_objects, basis, grcalc, gralter.
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