tensor(deprecated)/invars - Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : tensor(deprecated)/invars

tensor

  

invars

  

compute the scalar invariants of the Riemann tensor of a space-time, based on the Newman-Penrose curvature components

 

Calling Sequence

Parameters

Description

Examples

References

Calling Sequence

invars( 'flag', Curve, conj_pairs)

Parameters

flag

-

one of the following ten values: 'r1', 'r2', 'r3', 'w1', 'w2', 'm1', 'm2', 'm3', 'm4', or 'm5'

Curve

-

curve component table holding the Newman-Penrose curvature components

conj_pairs

-

optional parameter of a list of pairs (pair: list of two elements) of names that holds the variable names to be treated as complex conjugates in the calculations.

Description

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][RiemannInvariants] and Physics[Riemann] instead.

• 

This function calculates any of the ten invariants listed above of the Riemann tensor, as designated by their corresponding flags.  For detailed definitions and descriptions of these invariants, refer to the paper listed in the References section of this page.

• 

Simplification :

– 

tensor[invars] has two simplifiers, `tensor/invars/simp` and `tensor/invars/Msimp`.

– 

`tensor/invars/simp` is applied once after the invariant has been formally constructed.

– 

Due to lengths of the actual formulas for the invariants, when calculating r3, m2, m3, m4, and m5, an extra simplifier, `tensor/invars/Msimp`, is employed.  `tensor/invars/Msimp` is used to simplify the sum of every 15 terms in the formulas for the five invariants mentioned above.  And then `tensor/invars/simp` is applied on top of `tensor/invars/Msimp` to put the 15-term segments together.

– 

Note: that if the user finds it unnecessary, one of these simplifiers can actually be defined to perform no action.

• 

This function is part of the tensor package, and can be used in the form invars(..) only after performing the command with(tensor), or with(tensor, invars).  The function can always be accessed in the long form tensor[invars].

Examples

Important: The tensor package has been deprecated. Use the superseding commands DifferentialGeometry[Tensor][RiemannInvariants] and Physics[Riemann] instead.

withtensor:

Define the coordinate variables and the covariant natural basis metric :

coordt,r,θ,φ:

g_comptsarraysymmetric,1..4,1..4:

forito4doforjfromi+1to4dog_comptsi,j0enddoenddo:g_compts1,1ar:g_compts2,2br:g_compts3,3r2:g_compts4,4r2sinθ2:gtableindex_char=1,1,compts=opg_compts

gtableindex_char=−1,−1,compts=ar0000br0000r20000r2sinθ2

(1)

Now give a tetrad that transforms the above metric into the one in Debever's formalism :

h_comptsarraysparse,1..4,1..4:

h_compts1,112212ar12:

h_compts1,212212br12:

h_compts2,112212ar12:

h_compts2,212212br12:

h_compts3,312212r:

h_compts3,412I212rsinθ:

h_compts4,312212r:

h_compts4,412I212rsinθ:

hcreate1,1,oph_compts

htableindex_char=1,−1,compts=2ar22br2002ar22br200002r2I22rsinθ002r2I22rsinθ

(2)

Obtain the curvature components.

SPNnpspincoord,h,G,any:

CurvenpcurveSPN,any:

Specify the simplification wanted :

`tensor/invars/simp`:=proc(x) x end proc:

Now you are ready to compute any of the ten invariants.  For example,

R1invarsr1,Curve

R1ⅆⅆrbrar+ⅆⅆrarbr28br4r2ar2+2ⅆ2ⅆr2arbrarr2ⅆⅆrbrⅆⅆrararr2brⅆⅆrar2r2+4ar2br24ar2br264ar4br4r4

(3)

Repeat with a different simplification :

`tensor/invars/simp`:=proc(x) simplify(factor(x)) end proc:

R1_invarsr1,Curve

R1_4ⅆ2ⅆr2ar2br2ar2r44arbrr2brⅆⅆrar2r2+ⅆⅆrbrⅆⅆrararr24ar2brbr1ⅆ2ⅆr2ar+br2ⅆⅆrar4r4+2ⅆⅆrbrbrⅆⅆrar3arr4+r2ar2r2ⅆⅆrbr28br3+16br2ⅆⅆrar28r2ar3brⅆⅆrbrbr3ⅆⅆrar+8ar4r2ⅆⅆrbr2+2br2br1264ar4br4r4

(4)

Verify the two results are identical :

simplifyR1R1_

0

(5)

Specify the "inner" simplification, namely `tensor/invars/Msimp`:

`tensor/invars/Msimp`:=proc(x) x end proc:

M3invarsm3,Curve

M3ⅆ2ⅆr2ar2br2ar2r4arbrr2brⅆⅆrar2r2+ⅆⅆrbrⅆⅆrararr24ar2brbr1ⅆ2ⅆr2ar+br2ⅆⅆrar4r44+ⅆⅆrbrbrⅆⅆrar3arr42+r2ar2r2ⅆⅆrbr28br3+10br2ⅆⅆrar242ar3br32brr2ⅆⅆrbrⅆⅆrar+ar4r2ⅆⅆrbr2+8br2br122ⅆ2ⅆr2arbrarr2brⅆⅆrar2r22rarrⅆⅆrbr+2brⅆⅆrar2+ar2rⅆⅆrbr2br2+2br2576ar8br8r8

(6)

Repeat with a different "outer" simplifier :

`tensor/invars/simp`:=proc(x) x end proc:

M3_invarsm3,Curve

M3_2ⅆ2ⅆr2arbrarr2ⅆⅆrbrⅆⅆrararr2brⅆⅆrar2r2+2ⅆⅆrbrar2r4ar2br22brⅆⅆrararr+4ar2br2ⅆⅆrbrar+ⅆⅆrarbr24608ar6br8r6+2ⅆ2ⅆr2arbrarr2ⅆⅆrbrⅆⅆrararr2brⅆⅆrar2r2+2ⅆⅆrbrar2r4ar2br22brⅆⅆrararr+4ar2br22ⅆ2ⅆr2arbrarr2ⅆⅆrbrⅆⅆrararr2brⅆⅆrar2r2+4ar2br24ar2br29216ar8br8r8

(7)

Verify the two results are identical :

simplifyM3M3_

0

(8)

Demonstrate the use of the conj_pairs parameter :

M3__invarsm3,Curve,r,rBAR,θ,thetaBAR

M3__2ⅆ2ⅆr2arbrarr2ⅆⅆrbrⅆⅆrararr2brⅆⅆrar2r2+2ⅆⅆrbrar2r4ar2br22brⅆⅆrararr+4ar2br2ⅆ2ⅆrBAR2arBARbrBARarBARrBAR2ⅆⅆrBARbrBARⅆⅆrBARarBARarBARrBAR2brBARⅆⅆrBARarBAR2rBAR2+2ⅆⅆrBARbrBARarBAR2rBAR4arBAR2brBAR22brBARⅆⅆrBARarBARarBARrBAR+4arBAR2brBARⅆⅆrbrar+ⅆⅆrarbr24608ar4br6r4arBAR2brBAR2rBAR2+2ⅆ2ⅆr2arbrarr2ⅆⅆrbrⅆⅆrararr2brⅆⅆrar2r2+2ⅆⅆrbrar2r4ar2br22brⅆⅆrararr+4ar2br2ⅆ2ⅆrBAR2arBARbrBARarBARrBAR2ⅆⅆrBARbrBARⅆⅆrBARarBARarBARrBAR2brBARⅆⅆrBARarBAR2rBAR2+2ⅆⅆrBARbrBARarBAR2rBAR4arBAR2brBAR22brBARⅆⅆrBARarBARarBARrBAR+4arBAR2brBAR2ⅆ2ⅆr2arbrarr2ⅆⅆrbrⅆⅆrararr2brⅆⅆrar2r2+4ar2br24ar2br29216ar6br6r6arBAR2brBAR2rBAR2

(9)

References

  

Carminati, J., and McLenaghan, R.G. "Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space." Journal of Mathematical Physics, Vol. 32 No. 11. (Nov. 1991).

See Also

DifferentialGeometry[Tensor][RiemannInvariants]

Physics[Riemann]

tensor(deprecated)

tensor(deprecated)[conj]

tensor(deprecated)[frame]

tensor(deprecated)[npcurve]

tensor(deprecated)[npspin]

tensor(deprecated)[simp]