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Physics

Maple provides a state-of-the-art environment for algebraic computations in Physics, with emphasis on ensuring that the computational experience is as natural as possible. The theme of the Physics project for Maple 2015 has been Vector Analysis, symbolic Tensor manipulations, Quantum Mechanics, and General Relativity. With more than 400 enhancements throughout the entire package to increase robustness and versatility, two new commands, Assume and SubstituteTensor, a new Tetrads subpackage with 13 commands, as well as 26 new Physics:-Library commands to support further explorations and extensions, and an enlargement of the database of solutions to Einstein's equations with more than 100 new metrics, Maple 2015 extends again the range of Physics-related algebraic computations that can be done, in a natural way, using computer algebra software.


As part of its commitment to providing the best possible environment for algebraic computations in Physics, Maplesoft launched a Maple Physics: Research and Development web site with Maple 18, which enabled users to download research versions, ask questions, and provide feedback. The results from this accelerated exchange with people around the world have been incorporated into the Physics package in Maple 2015. Examples illustrating the use of the new capabilities in the context of more general problems are found in the MaplePrimes post Computer Algebra for Theoretical Physics.

 

Simplification

Tensors

Tetrads in General Relativity

More Metrics in the Database of Solutions to Einstein's Equations

Commutators, AntiCommutators, and Dirac notation in quantum mechanics

New Assume command and new enhanced Mode: automaticsimplification

Vectors Package

The Physics Library

Miscellaneous

Simplification

Simplification is perhaps the most common operation performed in a computer algebra system. In Physics, this typically entails simplifying tensorial expressions, or expressions involving noncommutative operators that satisfy certain commutator/anticommutator rules, or sums and integrals involving quantum operators and Dirac delta functions in the summands and integrands. Relevant enhancements were introduced in Maple 2015 for all these cases, including enhancements in the simplification of:

• 

Products of LeviCivita tensors in curved spacetimes when LeviCivita represents the Galilean pseudo-tensor (related to Setup(levicivita = Galilean)), instead of its generalization to curved spaces (related to Setup(levicivita = nongalilean)).

• 

Tensorial expressions in general that have spacetime, space, and/or tetrad contracted indices, possibly at the same time.

• 

New option tryhard, that resolves zero recognition in an important number of nontrivial situations.

• 

Expressions involving the Dirac function.

• 

Vectorial expressions involving cylindrical or spherical coordinates and related unit vectors.

• 

Expressions simplified with respect to side relations (equations) in the presence of quantum vectorial equations.

• 

Expressions involving products of quantum operators entering parameterized algebra rules.

• 

Expressions involving vectorial quantum operators simplified with respect to other vectorial equations.

• 

Add support for the simplification and integration of spherical harmonics (SphericalY ) relevant in quantum mechanics.

Examples

• 

Enhancements in the simplification of tensorial expressions.

restart;withPhysics: Setupnotation = true, coordinates = X

* Partial match of 'notation' against keyword 'mathematicalnotation'

* Partial match of 'coordinates' against keyword 'coordinatesystems'

Default differentiation variables for d_, D_ and dAlembertian are: X=x1,x2,x3,x4

Systems of spacetime Coordinates are: X=x1,x2,x3,x4

coordinatesystems=X,mathematicalnotation=true

(1)
• 

Define a tensor lμX and define the metric in terms of the components of this tensor so that gμ,ν=ημ,ν+lμlν and ημ,ν is a Minkowski metric.

lμ,ημ,ν=rhsg_Minkowski

The Minkowski metric in cartesian coordinates

Defined objects with tensor properties

γμ,σμ,Xμ,μ,ημ,ν,gμ,ν,lμ,δμ,ν,εα,β,μ,ν

(2)
• 

Avoid redundant display:

PDEtools:-declarelX

lx1,x2,x3,x4will now be displayed asl

(3)
• 

The metric:

g_mu,nu = ημ,ν+lμXlνX

gμ,ν=ημ,ν+lμXlνX

(4)
• 

New: you can define it directly using a tensorial equation like (4), using either Define or Setup.

Define

Defined objects with tensor properties

𝒟μ,γμ,σμ,Rμ,ν,Rμ,ν,α,β,Cμ,ν,α,β,Xμ,μ,ημ,ν,gμ,ν,lμ,Γμ,ν,α,Gμ,ν,δμ,ν,εα,β,μ,ν

(5)
• 

Verify the resulting gμ,ν:

g_

gμ,ν=1+l1X2l1Xl2Xl1Xl3Xl1Xl4Xl1Xl2X1+l2X2l2Xl3Xl2Xl4Xl1Xl3Xl2Xl3X1+l3X2l3Xl4Xl1Xl4Xl2Xl4Xl3Xl4X1+l4X2

(6)
• 

New: you can query about the definition you gave for any tensor directly from the tensor itself.

g_definition

gμ,ν=ημ,ν+lμXlνX

(7)
• 

Enhanced simplification capabilities: show that this expression involving derivatives of lμX is equal to zero.

d_αlλXηα,βα,βd_βlκXlκκXlλλX+ηα,βα,βd_βlκXd_λlαXlκκXlλλX+d_λlκXd_ρlρρXlκκXlλλX+d_βd_λlκXlββXlκκXlλλX12lμμXlννXd_ρlρρXd_μlνX12lμμXlννXlρρXd_ρd_μlνX12lμμXlννXd_ρlρρXd_νlμX12lμμXlννXlρρXd_ρd_νlμX+12lμμXlννXηα,ρα,ρd_ρlμXd_αlνX12lμμXlννXηα,ρα,ρd_μlαXd_ρlνX+12lμμXlννXηα,ρα,ρd_αlμXd_ρlνX12lμμXlννXηα,ρα,ρd_νlαXd_ρlμX12lμμXlννXlααXlρρXd_ρlνXd_αlμX+12lμμXlννXlααXlρρXd_μlαXd_ρlνX12lμμXlννXlααXlρρXd_ρlμXd_αlνX+12lμμXlννXlααXlρρXd_νlαXd_ρlμX

αlλXηα,βα,ββlκXlκκXlλλX+ηα,βα,ββlκXλlαXlκκXlλλX+λlκXρlρρXlκκXlλλX+βλlκXlββXlκκXlλλXlμμXlννXρlρρXμlνX2lμμXlννXlρρXρμlνX2lμμXlννXρlρρXνlμX2lμμXlννXlρρXρνlμX2+lμμXlννXηα,ρα,ρρlμXαlνX2lμμXlννXηα,ρα,ρμlαXρlνX2+lμμXlννXηα,ρα,ραlμXρlνX2lμμXlννXηα,ρα,ρνlαXρlμX2lμμXlννXlααXlρρXρlνXαlμX2+lμμXlννXlααXlρρXμlαXρlνX2lμμXlννXlααXlρρXρlμXαlνX2+lμμXlννXlααXlρρXνlαXρlμX2

(8)

Simplify

0

(9)
• 

A new tryhard option significantly improves zero recognition, which also got improved regardless of the new option.

Set spacetimeindices to be represented with lowercase Latin letters to make the input simpler.

Setupspacetimeindices = lowercaselatin

spacetimeindices=lowercaselatin

(10)

DefineAa, Ba, b, Fa, b, c, Ha, b, c, d, Ja, b, c, d, e

Defined objects with tensor properties

Aa,Ba,b,𝒟μ,γμ,Fa,b,c,Ha,b,c,d,Ja,b,c,d,e,σμ,Rμ,ν,Rμ,ν,α,β,Cμ,ν,α,β,Xμ,μ,ημ,ν,gμ,ν,lμ,Γμ,ν,α,Gμ,ν,δμ,ν,εα,β,μ,ν

(11)

 

Consider the following tensorial expressions.

e__1AcHd,h,c,bFh,d,fHa,e,e,fAhHa,c,c,fFe,d,fHd,e,h,b

e__1Ha,eefa,eefHd,h,c,bFh,d,fh,d,fAccHa,ccfa,ccfHd,e,h,bFe,d,fe,d,fAhh

(12)

e__2Jf,a,b,g,cFe,f,dFe,d,gJe,a,b,f,cFd,e,gFd,g,f

e__2Je,a,b,f,cFdegdegFd,g,fd,g,f+Jf,a,b,g,cFe,dge,dgFe,f,de,f,d

(13)

e__3Jg,c,k,j,eHi,h,a,hHe,b,i,gFd,k,jHh,b,g,jFd,k,eJj,c,k,e,hHg,i,a,i

e__3Jj,c,k,e,hHg,i,aig,i,aiHhbg,jhbg,jFdk,edk,e+Jg,c,k,j,eHeb,igeb,igHih,ahih,ahFdk,jdk,j

(14)

There are two, three and four free indices, and five, four and six repeated indices respectively in each of these three expressions, e__1,e__2,e__3 (see Check).

mapu  Checku,all,quiet, e__1,e__2,e__3

c,d,e,f,h,c,d,e,f,h,a,b,d,e,f,g,d,e,f,g,a,b,c,e,g,h,i,j,k,e,g,h,i,j,k,a,b,c,d

(15)

Simplifye__1,tryhard

0

(16)

Simplifye__2,tryhard

0

(17)

Simplifye__3

0

(18)
• 

Improve the simplification with respect to side relations (equations) in the presence of quantum vectorial equations.

withVectors:

Setupredo, hermitianoperators = L,r_, p_, x,y,z, p, quantumoperators = A, B

hermitianoperators=L,p,p,r,x,y,z,quantumoperators=A,B,L,p,p,r,x,y,z

(19)
• 

The identification of the vectorial character of expressions got improved. For example, this commutator is actually a non-projected vector.

Commutatorp_·p_,r_

Normp2,r

(20)

 

Identify

5

(21)
• 

Consider an expression involving vectorial quantum operators.

interfaceimaginaryunit = i

I

(22)

_ixpzzpx+_jypzzpy

ixpzzpx+jypzzpy

(23)
• 

Simplify (23) taking into account the definition of angular momentum.

ypzzpy=Lx,zpxxpz=Ly

ypzzpy=Lx,zpxxpz=Ly

(24)

simplify,

iLy+jLx

(25)
• 

Enhancements in the simplification and integration of spherical harmonics (SphericalY) relevant in quantum mechanics.

186486407293ⅇ6Itheta1cosphi2320270252cosphi21351352π+SphericalY8,6,phi,theta

7293ⅇ6θ1cosφ232027025cosφ2213513528648640π+Y86φ,θ

(26)

simplify

0

(27)

conjugateSphericalY8,6,phi,theta*1+1/5*sin6*theta*sin5*phi*sinphi

Y86φ,θ&conjugate0;1+sin6θsin5φ5sinφ

(28)

IntInt,theta=0..2 Pi, phi=0..Pi

0π02πY86φ,θ&conjugate0;1+sin6θsin5φ5sinφⅆθⅆφ

(29)

value

3781920π327293

(30)

 

Tensors

A number of relevant changes happened in the tensor routines of the Physics package, towards making the routines pack more functionality, the simplification more powerful, and the handling of symmetries, substitutions, and other operations more flexible and natural.

• 

Physics now works with four kinds of Minkowski spaces (different signatures) to accommodate the typical situations seen in textbooks; to these, correspond the signatures +---, ---+, -+++ and +++-.

• 

Allow setting the metric by specifying the signature directly, as in g_[`-`] or g_[`+---`], or Setupmetric=`---+` or Setupgμ,ν=`---+`.

• 

The signature keyword of the Physics Setup is now in use, to set the metric and to indicate the form of the orthonormal tetrad, in turn used to derive the form of a null tetrad.

• 

Automatic detection of the position of t as the time variable when you set the coordinates automatically sets the signature of the default Minkowski spacetime metric accordingly to ---+ or +---.

• 

New keywords with special meaning when indexing the Physics (also the user defined) tensors:
· `~`; for example g_[`~`] returns the all-contravariant matrix form of the metric.
· definition; for example Ricci[definition] returns the definition of the Ricci tensor; works also with user-defined tensors.
· scalars; for example Weyl[scalars] and Ricci[scalars] return the five Weyl and seven Ricci scalars used to perform a Petrov classification and in the Newman-Penrose formalism.
· scalarsdefinition, and invariantsdefinition; for example Weyl[scalarsdefinition] or Riemann[invariantsdefinition] return the corresponding definitions for the scalars and invariants.
· nullvectors; for example, when the new Tetrads subpackage is loaded, e_[nullvectors] returns a sequence of null vectors with their products normalized according to the Newman-Penrose formalism.
· matrix; this keyword was introduced in previous releases, and in Maple 2015 it can appear after a space index (not spacetime), in which case a matrix with only the space components is returned.

• 

Tensorial expressions can now have spacetime indices (related to a global system of references) and tetrad indices (related to a local system of references) at the same time, or they be rewritten in one (spacetime) or the other (tetrad) frames.

• 

The matrix keyword can be used with spacetime, space, or tetrad indices, resulting in the corresponding matrix

• 

Implement automatic determination of symmetry under permutation of tensor indices when the tensor is defined as a matrix.

• 

New conversions from the Weyl to the Ricci tensors, and from Weyl to the Christoffel symbols.

• 

New option evaluatetrace = true or false within convert/Ricci, to avoid automatically evaluating the Ricci trace when performing conversions that involve this trace.

• 

New option 'evaluate' to convert/g_, convert/Christoffel and convert/Ricci. With this option set to false, it is possible to see the algebraic form of the result (that is, of the tensors involved) before evaluating it.

• 

The Maple 18 Library:-SubstituteTensor command, got enhanced and transformed into one of the main Physics commands, that substitutes tensorial equation(s) Eqs into an expression, taking care of the free and repeated indices, such that: 1) equations in Eqs are interpreted as mappings having the free indices as parameters, 2) repeated indices in Eqs do not clash with repeated indices in the expression and 3) spacetime, space, and tetrad indices are handled independently, so they can all be present in Eqs and in the expression at the same time. This new command can also substitute algebraic sub-expressions of type product or sum within the expression, generalizing and unifying the functionality of the subs and algsubs commands for algebraic tensor expressions.

Examples

• 

Minkowski spacetime: four different conventions.

restart; withPhysics:CoordinatesX

Default differentiation variables for d_, D_ and dAlembertian are: X=x1,x2,x3,x4

Systems of spacetime Coordinates are: X=x1,x2,x3,x4

X

(31)

By default, the metric has a signature - - - +, with time in the last position.

g_

x0

x4

(32)

You can now set the signature of a Minkowski spacetime directly from the metric, or using the Setup keyword signature to use any of the four conventions frequently found in textbooks (- - - +), (+ - - -), (+++ -), and (- +++), and to indicate an Euclidean spacetime you can indicate (++++) or just +.

g_`+`

Changing the signature of the tensor spacetime to: + + + +

The Euclidean metric in coordinates x1,x2,x3,x4

g_`+++-`

The Minkowski metric, with signature + + + -, in coordinates x1,x2,x3,x4

Using Setup's signature keyword:

Setupsignature = `- +++`

signature=- + + +

(33)

g_

• 

The position of the different sign in a Minkowski spacetime always, either the first or the last symbol in the signature, refers to the time variable. So if you indicate t as one of the coordinates, say in the first position, and the signature is not in agreement with time in the first position, it is now automatically corrected and the metric set. For example, in this moment, due to (33), time is implicitly assumed to be in the first position; set coordinates with t in the last (fourth) position.

CoordinatesX=x,y,z,t

Detected `t`, the time variable, in position 4. Changing the signature of the spacetime metric accordingly, to: + + + -

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

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

X

(34)

g_

• 

New shortcut for the matrix form of the all contravariant components of a tensor.

Since previous releases, you can always request the matrix (or array in the case of more than 2 indices) form of a tensor by adding the keyword matrix as a last index. For example, g_μ,ν,matrix . A shortcut notation for the all covariant components is to omit the indices, as in . A new shortcut notation for the all contravariant components is to pass only they tilde `~`. To see the difference, consider a non-Minkowski spacetime, for instance set it in one go via:

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

The all contravariant components of the metric are:

g_`~`

• 

It is now possible to also use the keyword matrix passing only space (not spacetime) indices, resulting in a matrix with only the space components.

Setupspaceindices=lowercase_is

spaceindices=lowercaselatin_is

(35)

g_~i,~j, matrix

gi,ji,j=r+2mr0001r20001r2sinθ2

(36)

 

• 

All the tensor predefined in the Physics package, or defined using defining equations and the Define command, accept a new keyword, definition, that returns their definition. For example:

Riccidefinition

Rμ,ν=αΓαμ,ναμ,ννΓαμ,ααμ,α+Γβμ,νβμ,νΓαβ,ααβ,αΓβμ,αβμ,αΓαν,βαν,β

(37)

Einsteindefinition

Gμ,ν=Rμ,νgμ,νRαααα2

(38)

Define a tensor using a tensorial equation:

PDEtools:-declareAX

Ar,θ,φ,twill now be displayed asA

(39)

Fmu,nu = d_muAnuX  d_nuAmuX

Fμ,ν=μAνXνAμX

(40)

Note that for consistency of free indices, if you now define the left-hand side Fμ,ν as a tensor, in the right-hand side Aμ is also a tensor and so is automatically defined as well.

Define

Defined objects with tensor properties

Aμ,𝒟μ,γμ,Fμ,ν,σμ,Rμ,ν,Rμ,ν,α,β,Cμ,ν,α,β,Xμ,μ,gμ,ν,Γμ,ν,α,Gμ,ν,δμ,ν,εα,β,μ,ν

(41)

Check the definition of Fμ,ν:

Fdefinition

Fμ,ν=μAνXνAμX

(42)
• 

A new keyword scalars for the Weyl and Ricci tensors, generates the Weyl and Ricci scalars of the Newman-Penrose formalism. Correspondingly, another new keyword scalarsdefinition return the definition of these scalars.

Weylscalars

ψ__0=0,ψ__1=0,ψ__2=mr3,ψ__3=0,ψ__4=0

(43)

Weylscalarsdefinition

ψ__0=Cμ,ν,α,βμ,ν,α,βlμmνlαmβ,ψ__1=Cμ,ν,α,βμ,ν,α,βlμnνlαmβ,ψ__2=Cμ,ν,α,βμ,ν,α,βlμmνm&conjugate0;αnβ,ψ__3=Cμ,ν,α,βμ,ν,α,βlμnνm&conjugate0;αnβ,ψ__4=Cμ,ν,α,βμ,ν,α,βnμm&conjugate0;νnαm&conjugate0;β

(44)

In the definition earlier, the tensors lμ,nμ,mμ,m&conjugate0;μ are the null tensors of the Newman-Penrose formalism, implemented in Maple 2015 within the new Tetrads package.

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,lμ,nμ,mμ,m&conjugate0;μ

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

(45)

In this new Tetrads package, 𝔢a,μ is the tetrad (vierbein), by default an orthonormal tetrad that can be set to be a null tetrad using Setuptetrad=null, γa,b,c,λa,b,c are respectively the Ricci rotation coefficients and the lambda tensor defined 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))

For instance, the form of the orthonormal tetrad for the Schwarzschild metric set in  is:

e_

𝔢a,μ=rr+2m0000−Ir0000−Irsinθ0000r+2mr

(46)

A new keyword for the tetrad 𝔢a,μ is nullvectors:

e_nullvectors

lμ=2r2r+2m002r+2m2r,nμ=2r2r+2m002r+2m2r,mμ=0I22r2rsinθ20,m&conjugate0;μ=0I22r2rsinθ20

(47)

The matrix keyword can be used with different kinds of indices, representing different objects. For example, the spacetime components of the electromagnetic tensor defined in equation (40) are:

F

Fμ,ν=0rA2XθA1XrA3XφA1XrA4XtA1XθA1XrA2X0θA3XφA2XθA4XtA2XφA1XrA3XφA2XθA3X0φA4XtA3XtA1XrA4XtA2XθA4XtA3XφA4X0

(48)

The components of this tensor in the local inertial (tetrad) system of references are:

Fa,b,matrix

Fa,b=0Ir+2mrA2XθA1Xr32Ir+2mrA3XφA1Xr32sinθtA1XrA4X−Ir+2mrA2XθA1Xr320φA2XθA3Xr2sinθItA2XθA4Xr+2mr−Ir+2mrA3XφA1Xr32sinθθA3XφA2Xr2sinθ0ItA3XφA4Xr+2mrsinθrA4XtA1XIθA4XtA2Xrr+2mIφA4XtA3Xrsinθr+2m0

(49)

Likewise, the nonzero components of the Riemann tensor in the global (spacetime, Greek indices) and local (tetrad, lowercase roman indices from a to h) system of references are:

Riemannnonzero

Rμ,ν,α,β=1,2,1,2=mr+2m,1,2,2,1=mr+2m,1,3,1,3=msinθ2r+2m,1,3,3,1=msinθ2r+2m,1,4,1,4=2mr3,1,4,4,1=2mr3,2,1,1,2=mr+2m,2,1,2,1=mr+2m,2,3,2,3=2rmsinθ2,2,3,3,2=2rmsinθ2,2,4,2,4=r+2mmr2,2,4,4,2=2m2+mrr2,3,1,1,3=msinθ2r+2m,3,1,3,1=msinθ2r+2m,3,2,2,3=2rmsinθ2,3,2,3,2=2rmsinθ2,3,4,3,4=sinθ2r+2mmr2,3,4,4,3=sinθ2r+2mmr2,4,1,1,4=2mr3,4,1,4,1=2mr3,4,2,2,4=2m2+mrr2,4,2,4,2=r+2mmr2,4,3,3,4=sinθ2r+2mmr2,4,3,4,3=sinθ2r+2mmr2

(50)

Riemanna,b,c,d,nonzero

Ra,b,c,d=1,2,1,2=mr3,1,2,2,1=mr3,1,3,1,3=mr3,1,3,3,1=mr3,1,4,1,4=2mr3,1,4,4,1=2mr3,2,1,1,2=mr3,2,1,2,1=mr3,2,3,2,3=2mr3,2,3,3,2=2mr3,2,4,2,4=mr3,2,4,4,2=mr3,3,1,1,3=mr3,3,1,3,1=mr3,3,2,2,3=2mr3,3,2,3,2=2mr3,3,4,3,4=mr3,3,4,4,3=mr3,4,1,1,4=2mr3,4,1,4,1=2mr3,4,2,2,4=mr3,4,2,4,2=mr3,4,3,3,4=mr3,4,3,4,3=mr3

(51)
• 

Implement automatic determination of symmetry under permutation of tensor indices when the tensor is defined as a matrix.

Define a tensor using a symmetric matrix on the right-hand-side:

Mmu, nu = Matrix4, symbol=m, shape=symmetric

Defined objects with tensor properties

Aμ,𝒟μ,γμ,Fμ,ν,Mμ,ν,σμ,Rμ,ν,Rμ,ν,α,β,Cμ,ν,α,β,Xμ,μ,𝔢a,μ,ηa,b,gμ,ν,γa,b,c,lμ,λa,b,c,mμ,m&conjugate0;μ,nμ,Γμ,ν,α,Gμ,ν,δμ,ν,εα,β,μ,ν

(52)

Check whether the system detected that ηα,β is symmetric under permutation of α,β.

Library:−IsTensorialSymmetricMα,β

true

(53)

Hence, the indices of this tensor are automatically normalized taking this symmetry into account, so that:

Mα,β

Mα,β

(54)

Mβ,α

Mα,β

(55)

0

(56)

 

Tetrads in General Relativity

The formalism of tetrads in general relativity got implemented within Physics as a new package, Physics:-Tetrads, with 13 commands, mainly the null vectors of the Newman-Penrose formalism, the tetrad tensors 𝔢a,μ,ηa,b,γa,b,c,λa,b,c, respectively: the tetrad, the tetrad metric, the Ricci rotation coefficients, and the lambda tensor, plus five algebraic manipulation commands: IsTetrad, NullTetrad, OrthonormalTetrad, SimplifyTetrad, and TransformTetrad to construct orthonormal and null tetrads of different forms and using different methods.

Examples

The new Tetrads package contains 13 commands for computing in a local (tetrad) frame. Tensor components in the local frame are represented with tetrad indices using a type of letter different than the one representing global spacetime indices. You can set the type of letter using Setup, or just load the new package and the type of letter will be set automatically.

restart;withPhysics:withTetrads;

Setting lowercaselatin 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,lμ,nμ,mμ,m&conjugate0;μ

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

(57)

The most relevant commands are 𝔢a,μ and ηa,b, respectively representing the tetrad (also vierbein; by default, this is an orthonormal tetrad) and the tetrad metric (that is, the metric of the local - by default inertial - frame)

e_definition

𝔢a,μ𝔢bμbμ=ηa,b

(58)

Then γa,b,c are the Ricci rotation coefficients and λa,b,c is a linear combination 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)).

gamma_definition

γa,b,c=𝒟ν𝔢a,μ𝔢bμbμ𝔢cνcν

(59)

lambda_definition

λa,b,c=𝒟ν𝔢a,μ𝒟μ𝔢a,ν𝔢bμbμ𝔢cνcν

(60)

The lμ,nμ,mμ,m&conjugate0;μ tensors are the null tensors of the Newman-Penrose formalism and the commands  IsTetrad,NullTetrad,OrthonormalTetrad,SimplifyTetrad,TransformTetrad are for manipulating and exploring different forms of tetrads.

 

In a flat space, the spacetime and tetrad metrics are the same, so the orthonormal tetrad is just the identity.

g_

eta_

e_

𝔢a,μ=1000010000100001

(61)

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

g_13,7,5

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

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

The metric in coordinates x,y,z,t

Parameters: ε,At,Bt

Comments: _ⅇpsⅈlon=1 or _ⅇpsⅈlon=-1

PDEtools:-declareAt, Bt

Atwill now be displayed asA

Btwill now be displayed asB

(62)

The default orthonormal tetrad is:

e_

𝔢a,μ=−IεAt0000−Icosh2xBtⅇAxIsinh2xBtⅇAxcosh2x000−IBtⅇAxcosh2x0000Iε

(63)

The following null vectors correspond to this tetrad:

e_nullvectors

lμ=I22εAt00I22ε,nμ=I22εAt00I22ε,mμ=0I22cosh2xBtⅇAx2BtⅇAxIsinh2x+12cosh2x0,m&conjugate0;μ=0I22cosh2xBtⅇAx2BtⅇAxIsinh2x12cosh2x0

(64)

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

l_mu2,l_mu n_mu,l_mu m_mu,l_mu mb_mu

lμlμμ,lμnμμ,lμmμμ,m&conjugate0;μlμμ

(65)

mapu  u = SumOverRepeatedIndicesu, 

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

(66)

You can query about their definition 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=lμnν+lνnμmμm&conjugate0;νmνm&conjugate0;μ

(67)

Now that this definition depends on the signature:

Setupsignature

signature=- - - +

(68)

Change the signature, both in sign and placing time in position 1:

Setupsignature=`- +++`

signature=- + + +

(69)

So, now mμm&conjugate0;μμ=1 instead of 1 and there is a change in the sign of the definition in terms of the metric gμ,ν if compared with (67).

m_definition

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

(70)

You can verify these tensorial identities using TensorArray; for example, for the last equation:

1

gμ,ν=lμnνlνnμ+mμm&conjugate0;ν+mνm&conjugate0;μ

(71)

TensorArray,simplifier=simplify@expand

εAt2=εAt20=00=00&e