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HELP FOR: Carminati-McLenaghan Scalars
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- The invars library contains a set of scalar invariants polynomial
in the Riemann tensor listed by
[CM] J. Carminati and R. G. McLenaghan (1991), J. Math. Phys., 32, 3135.
Definitions of these invariants in terms of the Weyl and trace-free
Ricci tensors are provided below.
- The set contains four real invariants polynomial in the Ricci tensor
(Ricciscalar, R1, R2, R3 or collectively CMR), four complex invariants
polynomial in the Weyl tensor and its dual (W1R, W1I, W2R, W2I or
collectively W1, W2 or CMW). Finally there are eight mixed invariants
(M1R, M1I, M2R, M2I, M3, M4, M5R, M5I or collectively CMM). The invariants
can be referred to as a group using the name CM.
- See ?grt_invars for more information and spinor definitions of these
invariants.
a
Ricciscalar := R
a
b a
R2 := (1/4) S S
a b
b c a
R3 := (-1/8) S S S
a b c
b c d a
R4 := (1/16) S S S S
a b c d
abcd
W1R := (1/8) C C
abcd
abcd
W1I := (1/8) C* C
abcd
cd ef ab
W2R := (-1/16) C C C
ab cd ef
cd ef ab
W2I := (-1/16) C* C C
ab cd ef
ab cd
M1R := (1/8) S S C
acdb
ab cd
M1I := (1/8) S S C*
acdb
cd aefb
M2a := (1/16) S S C C
ef acdb
cd aefb
M2b := (1/16) S S C* C*
ef acdb
M2R := M2a - M2b
bc aefd
M2I := (1/8) S S C* C
ef abcd
M3 := M2a + M2b
ag ef c db
M4a := (-1/32) S S S C C
d ac befg
ag ef c db
M4b := (-1/32) S S S C* C*
d ac befg
M4 := M4a + M4b
cd ef aghb
M5a := (1/32) S S C C C
acdb gefh
cd ef aghb
M5b := (1/32) S S C C* C*
acdb gefh
cd ef aghb
M5c := (1/32) S S C* C C
acdb gefh
cd ef aghb
M5d := (1/32) S S C* C* C*
acdb gefh
M5R := M5a + M5b
M5I := M5c + M5d
In addition, the following invariant has been added to the set specifed in
[CM]:
ab e c f d
M6R := (1/32) C S S S S
cd a e b f
ab e c f d
M6I := (1/32) C* S S S S
cd a e b f
In GRTensorII, this invariant is not calculated as part of the CM set (ie.
grcalc(CM) will not calculate this object), however all of the invariants
including M6 can be calculated using the object name `invars', as in:
grcalc(invars).
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SEE ALSO: grt_invars, grt_objects.
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