represents and computes the Ricci rotation coefficients - Maple Programming Help

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Physics[Tetrads][gamma_] - represents and computes the Ricci rotation coefficients

Physics[Tetrads][lambda_] - represents a linear combination of the Ricci rotation coefficients - see reference

Calling Sequence

gamma_[a, b, c]

gamma_[a, b, c, matrix]

gamma_[keyword]

lambda_[a, b, c]

lambda_[a, b, c, matrix]

lambda_[keyword]

Parameters

_mu, nu_

-

the spacetime indices 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, c_

-

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

Description

• 

The components of the gamma_[a, b, c] tensor are the Ricci rotation coefficients and the components of the lambda_[a, b, c] tensor are linear combinations of them, according to the definitions in the Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2 (definitions (98.9) and (98.10)). Thus,

γa,b,c=𝔢bμ𝔢cν𝒟ν𝔢a,μ

  

where 𝒟ν𝔢a,μ, represented in Maple by D_[nu](e_[a,mu]), is the covariant derivative of the tetrad, and

λa,b,c=𝔢bμ𝔢cν𝒟ν𝔢a,μ𝒟μ𝔢a,ν

  

From these definitions it follows that gammaa,b,c is antisymmetric in its first pair of indices, and the covariant derivatives entering the definition of lambdaa,b,c can be replaced by non-covariant ones using the d_ operator. You can retrieve these definitions directly in the worksheet entering gamma_[definition] and lambda_[definition].

• 

Both gamma_ and lambda_ 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.

Examples

withPhysics:withTetrads

_______________________________________________________

Setting lowercaselatin_ah letters to represent tetrad indices

Defined as tetrad tensors see ?Physics,tetrads, 𝔢a,μ, ηa,b, γa,b,c, λa,b,c

Defined as spacetime tensors representing the NP null vectors of the tetrad formalism see ?Physics,tetrads: see ?Physics,tetrads, lμ, nμ, mμ, m&conjugate0;μ

_______________________________________________________

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

(1)

Setupmathematicalnotation=true

mathematicalnotation=true

(2)

Set the spacetime to be curved, for instance set the Schwarzschild metric in spherical coordinates (see g_):

g_sc

_______________________________________________________

Systems of spacetime coordinates are:X=r,θ,φ,t

Default differentiation variables for d_, D_ and dAlembertian are:X=r,θ,φ,t

The Schwarzschild metric in coordinates r,θ,φ,t

Parameters: m

_______________________________________________________

g_μ,ν=rr+2m0000r20000r2sinθ20000r2mr

(3)

The definition of the Ricci rotation coefficients and related λa,b,c

gamma_definition

γa,b,c=ν𝔢a,μ𝔢bμbμ𝔢cνcν

(4)

lambda_definition

λa,b,c=ν𝔢a,μμ𝔢a,ν𝔢bμbμ𝔢cνcν

(5)

Isolating the covariant derivative that appears on the right-hand side, of the definitions of γa,b,c, we have

e_b,αe_c,β

γa,b,c𝔢bαbα𝔢cβcβ=ν𝔢a,μ𝔢b,α𝔢c,β𝔢b,μb,μ𝔢c,νc,ν

(6)

Simplifyrhs=lhs

β𝔢a,α=γa,b,c𝔢bαbα𝔢cβcβ

(7)

The γa,b,c and λa,b,c tensors are related by formulas (98.10) and (98.11) of Landau's book referenced at the end, this relationship can be obtained as follows

SubstituteTensor,

λa,b,c=γa,d,e𝔢dνdν𝔢eμeμ+𝔢fμfμ𝔢gνgνγa,f,g𝔢bμbμ𝔢cνcν

(8)

Simplifying the repeated indices,

Simplify

λa,b,c=γa,c,b+γa,b,c

(9)

Substituting the free indices in this expression and adding as follows, then isolating gammaa,b,c, we obtain the inverse of this relationship

+subsa=b,b=c,c=a,subsa=c,b=a,c=b,

λa,b,cλb,a,cλc,a,b=2γa,b,c

(10)

isolate,gamma_a,b,c

γa,b,c=λa,b,c2λb,a,c2λc,a,b2

(11)

The nonzero Ricci coefficients and related λa,b,c components for the Schwarzschild metric

gamma_nonzero

γa,b,c=1,2,2=Ir+2mr32,1,3,3=Ir+2mr32,1,4,4=Imr32r+2m,2,1,2=−Ir+2mr32,2,3,3=cosθrsinθ,3,1,3=−Ir+2mr32,3,2,3=cosθrsinθ,4,1,4=−Imr32r+2m

(12)

lambda_nonzero

λa,b,c=2,1,2=−Ir+2mr32,2,2,1=Ir+2mr32,3,1,3=−Ir+2mr32,3,2,3=cosθrsinθ,3,3,1=Ir+2mr32,3,3,2=cosθrsinθ,4,1,4=−Imr32r+2m,4,4,1=Imr32r+2m

(13)

The value of γ1,11

gamma_`~1`,1,1

0

(14)

The components of γa,b1

gamma_`~1`,a,b,matrix

gamma_~1,a,b=00000Ir+2mr3/20000Ir+2mr3/20000Imr3/2r+2m

(15)

The Riemann tensor in the tetrad system of coordinates can be expressed in terms of the Ricci rotation coefficients γa,b,c using formula (98.14) of Landau's book

Riemanna,b,c,d=d_dgamma_a,b,cd_cgamma_a,b,d+gamma_a,b,fgamma_f,c,dgamma_f,d,c+gamma_a,f,cgamma_f,b,dgamma_a,f,dgamma_f,b,c

Ra,b,c,d=dγa,b,ccγa,b,d+γc,f,d+γd,f,cγa,bfa,bfγa,f,cγbfdbfd+γa,f,dγbfcbfc

(16)

You can verify identities like this one by taking left-hand side minus right-hand side and computing an Array of components of this tensorial expression using TensorArray, then get its elements (components) using ArrayElems; recalling, ArrayElems returns a set with the elements of an Array, omitting all those elements equal to zero, so we expect an empty set here:

ArrayElemsTensorArraylhsrhs,simplifier=simplify

(17)

Let's verify in the same way the expression of the covariant derivative in terms of the Ricci rotation coefficients

β𝔢a,α=γa,b,c𝔢bαbα𝔢cβcβ

(18)

ArrayElemsTensorArraylhsrhs,simplifier=simplify

(19)

See Also

Array, ArrayElems, d_, D_, e_, eta_, g_, IsTetrad, l_, m_, mb_, n_, NullTetrad, OrthonormalTetrad, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, TensorArray, Tetrads,, TransformTetrad

References

  

Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.

Compatibility

• 

The Physics[Tetrads][gamma_] and Physics[Tetrads][lambda_] commands were introduced in Maple 2015.

• 

For more information on Maple 2015 changes, see Updates in Maple 2015.