represents and computes a tetrad (vierbein) and the corresponding null vectors of the Newman-Penrose formalism - Maple Programming Help

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Physics[Tetrads][e_] - represents and computes a tetrad (vierbein) and the corresponding null vectors of the Newman-Penrose formalism

Physics[Tetrads][eta_] - represents the (tetrad) metric of a local system of references

Calling Sequence

e_[a, mu]

e_[a, mu, keyword]

e_[keyword]

eta_[a, b]

eta_[a, b, keyword]

eta_[keyword]

Parameters

_mu

-

a spacetime index related to a global system of references, these are names representing integer numbers between 0 and the spacetime dimension, they can also be the numbers themselves

_a, b_

-

the tetrad indices related to a local system of references, as names representing integer numbers the same way as the spacetime indices

keyword

-

optional, it can be definition, matrix, nonzero, and can be given alone or together with convariant or contravariant indices.

Description

• 

The e_[a, mu] and eta[a, b] commands respectively represent the tetrad (also vierbein; by default, this is an orthonormal tetrad) and the tetrad metric, that is, the metric of the local frame - which by default is inertial, of Minkowski type. These two tensors are defined in terms of each other by 𝔢a,μ𝔢bμ=ηa,b.

• 

Both e_ and eta_ accept the keywords accepted by the other tensors of the Physics package, these are definition, matrix and nonzero, that can be given with or without indices. If given with indices, the corresponding output takes their character (covariant or contravariant) into account. In the case of e_, you can also use the keyword nullvectors, to see the null vectors corresponding to a given tetrad. Note anyway that these null vectors are available as commands of the Tetrads package; these are the l_, n_, m_ and mb_ commands.

Examples

withPhysics:withTetrads

Setting lowercaselatin letters to represent tetrad indices

Defined as tetrad tensors see ?Physics,tetrads,e_a,μ,eta_a,b,gamma_a,b,c,lambda_a,b,c

Defined as spacetime tensors representing the NP null vectors of the tetrad formalism see ?Physics,tetrads,l_μ,n_μ,m_μ,mb_μ

IsTetrad,NullTetrad,OrthonormalTetrad,SimplifyTetrad,TransformTetrad,e_,eta_,gamma_,l_,lambda_,m_,mb_,n_

(1)

Setupmathematicalnotation=true

mathematicalnotation=true

(2)

In a flat space, the spacetime gmu,nu and tetrad etaa,b metrics are the same, so the orthonormal tetrad 𝔢a,μ is just the identity

g_[]

gμ,ν=1000010000100001

(3)

eta_[]

ηa,b=1000010000100001

(4)

e_[]

𝔢a,μ=1000010000100001

(5)

In a curved spacetime, for instance, set a Local Rotational Symmetry metric metric:

g_13,7,5

Systems of spacetime Coordinates are: X=t,x,y,z

Default differentiation variables for d_, D_ and dAlembertian are: X=t,x,y,z

The metric in coordinates t,x,y,z

Parameters: ε,At,Bt,A1

Comments: _ⅇpsⅈlon=1 or _ⅇpsⅈlon=-1

Resetting the signature of spacetime from "+ - - -" to `- + + +` in order to match the signature in the database of metrics:

gμ,ν=ε0000εAt20000Bt2ⅇ2A1x2coshx212Bt2ⅇ2A1xcoshxsinhx002Bt2ⅇ2A1xcoshxsinhxBt2ⅇ2A1x2coshx21

(6)

PDEtools:-declareAt,Bt

Atwill now be displayed asA

Btwill now be displayed asB

(7)

The default orthonormal tetrad is now

e_[]

𝔢a,μ=ε0000Aε0000BⅇA1xcosh2xBⅇA1xsinh2xcosh2x000BⅇA1xcosh2x

(8)

The following null vectors correspond to this tetrad:

e_nullvectors

lμ=2ε22Aε200,nμ=2ε22Aε200,mμ=002BⅇA1xcosh2x22BⅇA1xsinh2x+I2cosh2x,m&conjugate0;μ=002BⅇA1xcosh2x22BⅇA1xsinh2x+I2cosh2x

(9)

You can compute these null vectors directly since these are also part of the Tetrads package:

l_μ2,l_μn_μ,l_μm_μ,l_μmb_μ

lμlμμ,lμnμμ,lμmμμ,m&conjugate0;μlμμ

(10)

mapu→u=SumOverRepeatedIndicesu,

lμlμμ=0,lμnμμ=−1,lμmμμ=0,m&conjugate0;μlμμ=0

(11)

You can query about the definition of any of these tensors in the same way you can now query any other tensor:

m_definition

mμlμμ=0,mμnμμ=0,mμmμμ=0,mμm&conjugate0;μμ=1,gμ,ν=lμnνlνnμ+mμm&conjugate0;ν+mνm&conjugate0;μ

(12)

eta_definition

ηa,b=𝔢a,μ𝔢bμbμ

(13)

e_definition

𝔢a,μ𝔢bμbμ=ηa,b

(14)

Verify the definition of the tetrad 𝔢a,μ given above

TensorArray,simplifier=simplify

−1=−10=00=00=00=01=10=00=00=00=01=10=00=00=00=01=1

(15)

See Also

d_, D_, g_, gamma_, IsTetrad, l_, lambda_, m_, mb_, n_, NullTetrad, OrthonormalTetrad, Physics, Physics conventions, Physics examples, SimplifyTetrad, Tetrads,, TransformTetrad