DifferentialGeometry/Tensor/RiemannInvariants

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Tensor[RiemannInvariants] - calculate the Riemann (curvature) invariants for a spacetime

Calling Sequences

RiemannInvariants(g, C, options)

RiemannInvariants(NTetrad, options)

RiemannInvariants(NPRSc, NPWSc, options)

RiemannInvariants(RSp, WSp, barWSp, options)

Parameters

g - a metric tensor with signature [1, -1, -1, -1]

C - (optional) the curvature tensor of the metric g

NTetrad - a null tetrad for the metric g

RSp - the Ricci spinor for the metric g

WSp - the Weyl spinor for the metric g

barWSp - the conjugateWeyl spinor

NPRSc - the Newman-Penrose Ricci scalars for the metric g

NPWSc - the Newman-Penrose Weyl scalars for the metric g

options - the (optional) keyword argument author = A, where A = "ZakharyMcIntosh" or A = "CarminatiMcLenaghan"

options - the (optional) keyword argument invariants = L, where L is a list denoting the invariants to be computed, or L= "AllComplex"

Description

•

A Riemann invariant is a scalar invariant constructed from the curvuture tensor of a metric g. These are second invariants in the sense that no covariant derivatives of the curvature are used. There are many lists of Riemann invariants listed in the literature. At present the command RiemannInvariants will compute those found in the papers:

–	J. Carminati and R. G. McLenaghan, Algebraic invariants of the Riemann tensor in a four dimensional Lorentzian spacetime, J. Math. Physics 32(11), 3135--314.

–	Zakhary and C. B. G. McIntosh, A complete set of Riemann Invariants, General Relativity and Gravitation 29(5) 1997, 539 -- 581.

•	The command RiemannInvariants with return a table T which contains the Riemann invariants as denoted in these paper.

–	If author = "ZakharyMcIntosh" and invariants = "AllComplex", then the indices of T are ["I", "J", "R", "I6", "I7", "I8", "K", "L", "M", "M1", "M2"].

–	If author = "ZakharyMcIntosh" and invariants = "AllReal", then the indices of T are [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].

–	If author = "CarminatiMcLenaghan", then the indices of T are ["r1", "r2", "r3", "w1", "w2", "m1", "m2", "m3", "m4", "m5"].

–	The default option is author = "ZakharyMcIntosh" and invariants = "AllComplex".

•	The invariants of ZakharyMcIntosh are calculated using either the spinor formalism or the Newman-Penrose formalism. Use either the 2nd, 3rd or 4th calling sequences. See Examples 1--4.

•	The invariants of CarminatiMcLenaghan are calculated using either the tensor or spinor formalism. Use the 1st or 4th calling sequence. See Examples 5--7.

•	This command is part of the DifferentialGeometry:-Tensor package, and so can be used in the form RiemannInvariants(...) only after executing the commands with(DifferentialGeometry); with(Tensor); in that order. It can always be used in the long form DifferentialGeometry:-Tensor:-RiemannInvariants.

Examples

Example 1.

Compute the ZakharyMcIntosh Riemann invariants directly from the metric. The tensor versions of these formulas have not yet been implemented and an empty table is returned.

(2.1)

M >

(2.2)

M >

(2.3)

Example 2.

Compute the ZakharyMcIntosh Riemann invariants from a null tetrad. Use the command DGGramSchmidt to first construct an orthonormal tetrad.

M >

(2.4)

Use the command NullTetrad to construct a null tetrad.

M >

NTetrad := [2^(1/2)*D_t/(2*x^2)+2^(1/2)*D_z/2, 2^(1/2)*D_t/(2*x^2)-2^(1/2)*D_z/2, 2^(1/2)*D_x/2+((1/2)*I)*2^(1/2)*D_y, 2^(1/2)*D_x/2-((1/2)*I)*2^(1/2)*D_y]

(2.5)

M >

Inv2 := TABLE(["I6" = 1/((3*x^4)), "R" = "R", "I7" = 0, "I" = 1/((9*x^4)), "M1" = 4/(9*x^8), "I8" = 0, "M2" = -4/(27*x^10), "L" = 2/(9*x^8), "K" = -2/(3*x^6), "J" = -1/((27*x^6)), "M" = 0])

(2.6)

M >

(2.7)

Example 3.

Compute the ZakharyMcIntosh Riemann invariants from the Newman-Penrose curvature scalars. Use the command NPCurvatureScalars to compute the Newman-Penrose curvature scalars.

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(2.8)

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Inv3 := TABLE(["I6" = 1/((3*x^4)), "R" = "R", "I7" = 0, "I" = 1/((9*x^4)), "M1" = 4/(9*x^8), "I8" = 0, "M2" = -4/(27*x^10), "L" = 2/(9*x^8), "K" = -2/(3*x^6), "J" = -1/((27*x^6)), "M" = 0])

(2.9)

Example 4.

Compute the ZakharyMcIntosh Riemann invariants from the Ricci spinor and the Weyl spinor. For this we need to extend our spacetime manifold to a rank 4 vector bundle.

M >

(2.10)

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(2.11)

N >

(2.12)

Use the command SolderForm to compute the Newman-Penrose curvature scalars.

N >

S := x^2*2^(1/2)*dt*D_z1*D_w1/2+x^2*2^(1/2)*dt*D_z2*D_w2/2+2^(1/2)*dx*D_z1*D_w2/2+2^(1/2)*dx*D_z2*D_w1/2-((1/2)*I)*2^(1/2)*dy*D_z1*D_w2+((1/2)*I)*2^(1/2)*dy*D_z2*D_w1+2^(1/2)*dz*D_z1*D_w1/2-2^(1/2)*dz*D_z2*D_w2/2

(2.13)

Use the command RicciSpinor to compute the Newman-Penrose curvature scalars.

N >

(2.14)

Use the command WeylSpinor to compute the Newman-Penrose curvature scalars.

M >

WeylSpin := -dz1*dz1*dz1*dz1/(2*x^2)+dz1*dz1*dz2*dz2/(6*x^2)+dz1*dz2*dz1*dz2/(6*x^2)+dz1*dz2*dz2*dz1/(6*x^2)+dz2*dz1*dz1*dz2/(6*x^2)+dz2*dz1*dz2*dz1/(6*x^2)+dz2*dz2*dz1*dz1/(6*x^2)-dz2*dz2*dz2*dz2/(2*x^2)

(2.15)

Use the command ConjugateSpinor to compute the Newman-Penrose curvature scalars.

N >

barWeylSpin := -dw1*dw1*dw1*dw1/(2*x^2)+dw1*dw1*dw2*dw2/(6*x^2)+dw1*dw2*dw1*dw2/(6*x^2)+dw1*dw2*dw2*dw1/(6*x^2)+dw2*dw1*dw1*dw2/(6*x^2)+dw2*dw1*dw2*dw1/(6*x^2)+dw2*dw2*dw1*dw1/(6*x^2)-dw2*dw2*dw2*dw2/(2*x^2)

(2.16)

Use the command NPCurvatureScalars to compute the Newman-Penrose curvature scalars.

N >

Inv4 := TABLE(["I6" = 1/((3*x^4)), "R" = "R", "I7" = 0, "I" = 1/((9*x^4)), "M1" = 4/(9*x^8), "I8" = 0, "M2" = 4/(27*x^10), "L" = 2/(9*x^8), "K" = 2/(3*x^6), "J" = 1/((27*x^6)), "M" = 0])

(2.17)

Example 5.

Compute the CarminatiMcLenaghan Riemann invariants directly from the original metric.

N >

(2.18)

M >

Inv5 := TABLE(["r2" = 0, "w1" = 2/(3*x^4), "r3" = 1/((4*x^8)), "r1" = 1/((x^4)), "m2" = 4/(9*x^8), "w2" = 2/(9*x^6), "m1" = 2/(3*x^6), "m5" = 8/(27*x^10), "m4" = 0, "m3" = 4/(9*x^8)])

(2.19)

The curvature tensor can be given as an optional argument.

M >

(2.20)

M >

Inv5 := TABLE(["r2" = 0, "w1" = 2/(3*x^4), "r3" = 1/((4*x^8)), "r1" = 1/((x^4)), "m2" = 4/(9*x^8), "w2" = 2/(9*x^6), "m1" = 2/(3*x^6), "m5" = 8/(27*x^10), "m4" = 0, "m3" = 4/(9*x^8)])

(2.21)

Example 6.

Compute the CarminatiMcLenaghan Riemann invariants directly from a null tetrad. The Newman-Penrose versions of these invariants have not yet been implemented and an empty table is returned.

M >

(2.22)

Example 7.

Compute the CarminatiMcLenaghan Riemann invariants from the Ricci spinor and the Weyl spinor.

M >

Inv6 := TABLE(["r2" = 0, "w1" = 2/(3*x^4), "r3" = 1/((4*x^8)), "r1" = 1/((x^4)), "m2" = 4/(9*x^8), "w2" = 2/(9*x^6), "m1" = 2/(3*x^6), "m5" = (4/27)*((abs(x)^12+x^12)/(x^10*abs(x)^12)), "m4" = 0, "m3" = 4/(9*x^8)])

(2.23)

N >

TABLE(["r2" = 0, "w1" = 2/(3*x^4), "r3" = 1/((4*x^8)), "r1" = 1/((x^4)), "m2" = 4/(9*x^8), "w2" = 2/(9*x^6), "m1" = 2/(3*x^6), "m5" = 8/(27*x^10), "m4" = 0, "m3" = 4/(9*x^8)])

(2.24)

	See Also
	DifferentialGeometry, Tensor, assuming, ConjugateSpinor, CurvatureTensor, Physics[Riemann], NullTetrad, NPCurvatureScalars, SolderForm, RicciSpinor, Physics[Ricci], WeylSpinor, Physics[Weyl]