computes different forms of a null tetrad for the current spacetime metric - Maple Programming Help

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Physics[Tetrads][NullTetrad] - computes different forms of a null tetrad for the current spacetime metric

Physics[Tetrads][OrthonormalTetrad] - computes different forms of an orthonormal tetrad for the current spacetime metric

Physics[Tetrads][IsTetrad] - returns true or false according to whether a given matrix is a null or orthonormal tetrad and indicates what kind of the tetrad is

Calling Sequence

NullTetrad(method = ..., firstvector = ...)

OrthonormalTetrad(method = ..., firstvector = ...)

IsTetrad(T, orthonormal, null, quiet)

Parameters

method = ..

-

optional, indicates the method to be used, that could be GramSchmidt or Eigenvectors

firstvector = ..

-

optional, a Vector or a list with the vector components, as many as the dimension of spacetime; indicates the first departing vector to be used with the GramSchmidt method

T

-

a Matrix representing a tetrad

orthonormal

-

optional, to indicate to IsTetrad to check only for an orthonormal tetrad

null

-

optional, to indicate to IsTetrad to check only for an null tetrad

quiet

-

optional, to indicate to IsTetrad to avoid displaying the type of tetrad on the screen

Description

  

NullTetrad and OrthonormalTetrad compute tetrads for the spacetime metric set. Recalling, given the metric of a local (tetrad) system of references, there are infinitely many tetrads (transformations, represented in Physics by the Tetrads:-e_ command) relating the components of a tensor in the global (spacetime) and local (tetrad) systems of references. These two commands NullTetrad and OrthonormalTetrad permit computing different tetrads, that may result more convenient in different contexts, for example when computing the Weyl and Ricci scalars or performing a Petrov classification.

• 

IsTetrad is a complementary command: it returns true or false depending on whether the given matrix T is a tetrad, optionally indicating the type of tetrad.

Examples

withPhysics:Setupmathematicalnotation=true;withTetrads

mathematicalnotation=true

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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)

Set the spacetime metric to something non-flat, for instance take the metric [13,7,5] of the Exact solutions book referenced at the end of this page:

g_13,7,5

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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

_______________________________________________________

g_μ,ν=ε0000εAt20000ⅇ2A1xBt22coshx212Bt2ⅇ2A1xcoshxsinhx002Bt2ⅇ2A1xcoshxsinhxⅇ2A1xBt22coshx21

(2)

Compact the display of the functions At,Bt:

PDEtools:-declareAt,Bt

Atwill now be displayed asA

Btwill now be displayed asB

(3)

The default orthonormal tetrad is:

e_[]

e_a,μ=ε0000Atε0000BtⅇA1xcosh2xBtⅇA1xsinh2xcosh2x000BtⅇA1xcosh2x

(4)

It is possible to test whether these tetrads satisfy the tetrad definition

e_definition

𝔢a,μ𝔢bμbμ=ηa,b

(5)

using IsTetrad

IsTetrad

Type of tetrad: orthonormal

true

(6)

To compute with a null tetrad instead of an orthonormal tetrad, set the tetrad or tetradmetric to null using Setup.

Setuptetradmetric=null

tetradmetric=1,2=−1,3,4=1

(7)

Note: this form of the null tetrad metric is consistent with time in position 4 (default) - and not with time in position 1. To see the more frequent form of the tetradmetric

Matrix4,1,2=1,3,4=1,shape=symmetric

0100100000010010

(8)

you can either set the signature of the underlying inertial system of references (Minkowski) indicating time (different sign) in the first place, or set the tetradmetric directly using Setup(tetradmetric = the_matrix_above).

So now the metric of the local (tetrad) system of references is:

eta_[]

eta_a,b=0100100000010010

(9)

and the null tetrad is:

e_[]

e_a,μ=122ε122Atε00122ε122Atε0000122BtⅇA1xcosh2x122BtⅇA1xsinh2xIcosh2x00122BtⅇA1xcosh2x122BtⅇA1xsinh2x+Icosh2x

(10)

IsTetrad

Type of tetrad: null

true

(11)

You can compute different forms of an orthonormal tetrad or a null tetrad using the OrthonormalTetrad and NullTetrad commands, requesting the use of different methods, or passing a starting vector. For instance, this is the default form of an orthonormal tetrad for the current spacetime metric:

OrthonormalTetrad

ε0000Atε0000BtⅇA1xcosh2xBtⅇA1xsinh2xcosh2x000BtⅇA1xcosh2x

(12)

There are two methods available for computing these tetrads: the GramSchmidt (generalization to curved spaces) and Eigenvectors (based on computing the Eigenvectors of a related Matrix). By default, the Physics package routines decide on what method is more convenient to use for the the spacetime metric set. In this case the routines use the GramSchmidt method; you can check the other method via:

OrthonormalTetradmethod=Eigenvectors

ε0000Atε0000122BtⅇA1x2coshx2+2coshxsinhx1122BtⅇA1x2coshx2+2coshxsinhx100122BtⅇA1x2coshx22coshxsinhx1122BtⅇA1x2coshx22coshxsinhx1

(13)

If this is an orthonormal tetrad you prefer, you can set it via Setup(e_ = ...), where ... is the matrix you prefer (see the previous output). To see what the null tetrads corresponding to these two orthonormal tetrads would be:

NullTetrad

122ε122Atε00122ε122Atε0000122BtⅇA1xcosh2x122BtⅇA1xsinh2xIcosh2x00122BtⅇA1xcosh2x122BtⅇA1xsinh2x+Icosh2x

(14)

NullTetradmethod=Eigenvectors

122ε122Atε00122ε122Atε000012BtⅇA1xI2coshx2+2coshxsinhx1+2coshx22coshxsinhx12coshx2+2coshxsinhx12coshx22coshxsinhx112BtⅇA1xI2coshx2+2coshxsinhx1+2coshx22coshxsinhx12coshx2+2coshxsinhx12coshx22coshxsinhx10012BtⅇA1xI2coshx2+2coshxsinhx1+2coshx22coshxsinhx12coshx2+2coshxsinhx12coshx22coshxsinhx112BtⅇA1xI2coshx2+2coshxsinhx1+2coshx22coshxsinhx12coshx2+2coshxsinhx12coshx22coshxsinhx1

(15)

When using the GramSchmidt method you can also specify the first vector. Recalling, the method works iterating the computation of the next vector (line of the tetrad) starting from a first vector. For example, instead of starting from εAt,0,0,0, start from 0,0,0,1 compare with default result of OrthonormalTetrad lines above.

OrthonormalTetradfirstvector=0,0,0,1

000IBtⅇA1xcosh2x0Atε00Iε00000BtⅇA1xsinhx2cosh2x1+cosh2x2coshxsinhx2cosh2xBt2ⅇA1x1+cosh2x2coshx21

(16)

IsTetrad

Type of tetrad: orthonormal

true

(17)

NullTetradfirstvector=0,0,0,1

0122Atε012I2BtⅇA1xcosh2x0122Atε012I2BtⅇA1xcosh2x12I2ε0IBtⅇA1xsinhxcosh2x1+cosh2x2Icoshxsinhx2cosh2xBtⅇA1x1+cosh2x2coshx2112I2ε0IBtⅇA1xsinhxcosh2x1+cosh2x2Icoshxsinhx2cosh2xBtⅇA1x1+cosh2x2coshx21

(18)

IsTetrad

Type of tetrad: null

true

(19)

See Also

d_, D_, e_, eta_, g_, gamma_, l_, lambda_, m_, mb_, n_, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Ricci, Setup, Tetrads,, , TransformTetrad, Weyl

Compatibility

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The Physics[Tetrads][NullTetrad], Physics[Tetrads][OrthonormalTetrad] and Physics[Tetrads][IsTetrad] commands were introduced in Maple 2015.

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