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 2018 has been the consolidation of the functionality introduced in previous releases, together with significant enhancements, mainly in the handling of differential (quantum or not) tensorial operators, new ways to minimize the number of tensor components taking its symmetries into account, automatic handling of collision of indices in tensorial expressions, automatic setting of the EnergyMomentum tensor when loading solutions to Einstein's equations from the database of solutions, automatic setting of the algebras for the Dirac, Pauli and Gell-Mann matrices when Physics is loaded, simplification of Dirac matrices, a new package Physics:-Cactus related to Numerical Relativity and several other improvements.
Taking all together, there are more than 300 enhancements throughout the entire package, increasing robustness, versatility and functionality, extending once more the range of Physics-related algebraic computations that can be done using computer algebra software, and in a natural way.
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 2018.
Automatic handling of collision of tensor indices in products
User defined algebraic differential operators
The Physics:-Cactus package for Numerical Relativity
Automatic setting of the EnergyMomentumTensor for metrics of the database of solutions to Einstein's equations
Minimize the number of tensor components according to its symmetries, relabel, redefine or count the number of independent tensor components
New functionality and display for inert names and inert tensors
Automatic setting of Dirac, Pauli and Gell-Mann algebras
Simplification of products of Dirac matrices
New Library routines to perform matrix operations in expressions involving spinors with omitted indices
Miscellaneous improvements
The design of products of tensorial expressions that have contracted indices got enhanced. The idea: repeated indices in certain subexpressions are actually dummies. So suppose Ta,b and Bb are tensors, then in Ttrace=Taaaa, a is just dummy, therefore Taaaa⁢Ba=Tbbbb⁢Ba is a well defined object. The new design automatically maps input like Taaaa⁢Ba into Tbbbb⁢Ba.
restart;
withPhysics: Setupspacetimeindices=lowercaselatin, quiet
spacetimeindices=lowercaselatin
DefineTa,b, Bb
Defined objects with tensor properties
Bb,γa,σa,Ta,b,∂a,ga,b,δa,b,εa,b,c,d
This shows the automatic handling of collision of indices
Ta,a Ba
Tbbbb⁢Ba
Ta,a2
Taaaa⁢Tbbbb
Consider now the case of three tensors
DefineAa,Ca
Aa,Bb,Ca,γa,σa,Ta,b,∂a,ga,b,δa,b,εa,b,c,d
Aa Ba Ca
Aa⁢Ba⁢Ca
The product above has indeed the index a repeated more than once, therefore none of its occurrences got automatically transformed into contravariant in the output, and Check detects the problem interrupting with an error message
Check
Error, (in Physics:-Check) wrong use of the summation rule for repeated indices: `a repeated 3 times`, in A[a]*B[a]*C[a]
However, it is now also possible to indicate, using parenthesis, that the product of two of these tensors actually form a subexpression, so that the following two tensorial expressions are well defined, where the dummy is automatically replaced making that explicit
Aa ⋅ Ba⋅ Ca
Ab⁢B⁢b⁢b⁢Ca
Aa⋅Ba ⋅ Ca
Aa⁢Bb⁢C⁢b⁢b
This change in design makes concretely simpler the use of indices in that it eliminates the need for manually replacing dummies. For example, consider the tensorial expression for the angular momentum in terms of the coordinates and momentum vectors, in 3 dimensions
Setupcoordinates = cartesian, dimension = 3, metric = euclidean,quiet
coordinatesystems=X,dimension=3,metric=1,1=1,2,2=1,3,3=1
Define Lj,pk respectively representing angular and linear momentum
DefineLj,pk
γa,Lj,σa,Xa,∂a,ga,b,pk,δa,b,εa,b,c
Introduce the tensorial expression for La
La = LeviCivitaa,b,c⋅Xb ⋅pc
La=εa,b,c⁢Xb⁢pc
The left-hand side has one free index, a, while the right-hand side has two dummy indices b and c
Check,all
The repeated indices per term are: ...,...,...; the free indices are: ...
∅,a=b,c,a
If we want to compute L→2=La2 we can now take the square of (11) directly, and the dummy indices on the right-hand side are automatically handled, there is now no need to manually substitute the repeated indices to avoid their collision
2
La2=εa,b,c⁢Xb⁢pc⁢εa,d,e⁢Xd⁢pe
The repeated indices on the right-hand side are now a,b,c,d,e
Check, all
a,∅=a,b,c,d,e,∅
A new keyword in Setup: differentialoperators, allows for defining differential operators (not necessarily linear) with respect to indicated differentiation variables, so that they are treated as noncommutative operands in products, as we do with paper and pencil. These user-defined differential operators can also be vectorial and/or tensorial or inert. When desired, one can use Library:-ApplyProductOfDifferentialOperators to transform the products in the function application of these operators. This new functionality is a generalization of the differential operators ∂μ and ▿μ, and can used beyond Physics.
A new routine Library:-GetDifferentiationVariables also acts on a differential operator and tells who are the corresponding differentiation variables
Example:
In Quantum Mechanics, in the coordinates representation, the component of the momentum operator along the x axis is given by the differential operator
p__x=−i ℏ∂∂x
The purpose of the exercises below is thus to derive the commutation rules, in the coordinates representation, between an arbitrary function of the coordinates and the related momentum, departing from the differential representation
pn=−i⁢ℏ⁢∂n
FX,p→−=ⅈ ℏ ∇FX
restart:withPhysics:withPhysicsVectors:interfaceimaginaryunit = i:
Start setting the problem:
all ofx,y,z,p__x,p__y,p__z are Hermitian operators
all of x,y,z commute between each other
tell the system only that the operators x, y, z are the differentiation variables of the corresponding (differential) operators p__x,p__y,p__z but do not tell what is the form of the operators
Setupmathematicalnotation = true,differentialoperators=p_,x,y,z,hermitianoperators=p,x,y,z,algebrarules=%Commutatorx,y=0,%Commutatorx,z=0,%Commutatory,z=0,quiet
algebrarules=x,y−=0,x,z−=0,y,z−=0,differentialoperators=p→,x,y,z,hermitianoperators=p,x,y,z,mathematicalnotation=true
Assuming F⁡X is a smooth function, the idea is to apply the commutator F⁡X,p→− to an arbitrary ket of the Hilbert space ψx,y,z, perform the operation explicitly after setting a differential operator representation for p→, and from there get the commutation rule between F⁡X and p→.
Start introducing the commutator, to proceed with full control of the operations we use the inert form %Commutator
aliasX = x,y,z:
CompactDisplayFX
F⁡X⁢will now be displayed as⁢F
%CommutatorFX,p_⋅Ketψ,X
F,p→−⁢ψx,y,z
For illustration purposes only (not necessary), expand this commutator
=expand
F,p→−⁢ψx,y,z=F⁢p→⁢ψx,y,z−p→⁢F⁢ψx,y,z
Note that p→, F⁡X and the ket ψx,y,z are operands in the products above and that they do not commute: we indicated that the coordinates x, y, z are the differentiation variables of p→. This emulates what we do when computing with these operators with paper and pencil, where we represent the application of a differential operator as a product operation.
This representation can be transformed into the (traditional in computer algebra) application of the differential operator when desired, as follows:
=Library:-ApplyProductsOfDifferentialOperators
F,p→−⁢ψx,y,z=F⁢p→⁡ψx,y,z−p→⁡F⁢ψx,y,z
Note that, in p→⁡F⁡X⁢ψx,y,z, the application of p→ is not expanded: at this point nothing is known about p→ , it is not necessarily a linear operator. In the Quantum Mechanics problem at hands, however, it is. So give now the operator p→ an explicit representation as a linear vectorial differential operator (we use the inert form %Nabla, ∇, to be able to proceed with full control one step at a time)
p_≔f→−ⅈ⋅ℏ⋅%Nablaf
p→≔f↦−ⅈ⁢ℏ⁢∇f
The expression (19) becomes
F,p→−⁢ψx,y,z=−ⅈ⁢ℏ⁢F⁢∇ψx,y,z+ⅈ⁢ℏ⁢∇F⁢ψx,y,z
Activate now the inert operator ∇ and simplify taking into account the algebra rules for the coordinate operators x,y−=0,x,z−=0,y,z−=0
Simplifyvalue
F,p→−⁢ψx,y,z=ⅈ⁢ℏ⁢i∧⁢Fx⁢ψx,y,z+ⅈ⁢ℏ⁢j∧⁢Fy⁢ψx,y,z+ⅈ⁢ℏ⁢k∧⁢Fz⁢ψx,y,z
To make explicit the gradient in disguise on the right-hand side, factor out the arbitrary ket ψx,y,z
Factor
F,p→−⁢ψx,y,z=ⅈ⁢ℏ⁢Fx⁢i∧+Fy⁢j∧+Fz⁢k∧⁢ψx,y,z
Combine now the expanded gradient into its inert (not-expanded) form
subsGradient=%GradientFX,
F,p→−⁢ψx,y,z=ⅈ⁢ℏ⁢∇F⁢ψx,y,z
Since (24) is true for all ψx,y,z, this ket can be removed from both sides of the equation. One can do that either taking coefficients (see Coefficients) or multiplying by the "formal inverse" of this ket, arriving at the (expected) form of the commutation rule between F⁡X and p→
⋅InverseKetψ,x,y,z
F,p→−=ⅈ⁢ℏ⁢∇F
Tensor notation, X__m,Pn−=ⅈ ℏ⁢gm,n
The computation rule for position and momentum, this time in tensor notation, is performed in the same way, just that, additionally, specify that the space indices to be used are lowercase Latin letters, and set the relationship between the differential operators and the coordinates directly using tensor notation. You can also specify that the metric is Euclidean, but that is not necessary: the default metric of the Physics package, a Minkowski spacetime, includes a 3D subspace that is Euclidean, and the default signature, (- - - +), is not a problem regarding this computation.
restart; withPhysics:interfaceimaginaryunit = i:
Setupmathematicalnotation=true,coordinates = cartesian,spaceindices = lowercaselatin,algebrarules=%Commutatorx,y=0,%Commutatorx,z=0,%Commutatory,z=0,hermitianoperators = X,P,p,differentialoperators=Pm,x,y,z,quiet
algebrarules=x,y−=0,x,z−=0,y,z−=0,coordinatesystems=X,differentialoperators=Pm,x,y,z,hermitianoperators=P,p,t,x,y,z,mathematicalnotation=true,spaceindices=lowercaselatin
Define now the tensor Pm
DefinePm,quiet
γμ,Pm,σμ,Xμ,∂μ,gμ,ν,γa,b,δμ,ν,εα,β,μ,ν
Introduce now the Commutator, this time in active form, to show how to reobtain the non-expanded form at the end by resorting the operands in products
CommutatorXm,Pn⋅Ketψ,x,y,z
Xm,Pn−⁢ψx,y,z
Expand first (not necessary) to see how the operator Pn is going to be applied
Xm,Pn−⁢ψx,y,z=Xm⁢Pn⁢ψx,y,z−Pn⁢Xm⁢ψx,y,z
Now expand and directly apply in one ago the differential operator Pn
Xm,Pn−⁢ψx,y,z=Xm⁢Pn⁡ψx,y,z−Pn⁡Xm⁢ψx,y,z
Introducing the explicit differential operator representation for Pn, here again using the inert ∂n to keep control of the computations step by step
Pn≔f→−ⅈ⋅ℏ⋅%d_nf
Pn≔f↦−ⅈ⁢ℏ⁢∂n⁡f
The expanded and applied commutator (30) becomes
Xm,Pn−⁢ψx,y,z=−ⅈ⁢ℏ⁢Xm⁢∂n⁡ψx,y,z+ⅈ⁢ℏ⁢∂n⁡Xm⁢ψx,y,z
Activate now the inert operators ∂n and simplify taking into account Einstein's rule for repeated indices
Xm,Pn−⁢ψx,y,z=ⅈ⁢ℏ⁢gm,n⁢ψx,y,z
Since the ket ψx,y,z is arbitrary, we can take coefficients (or multiply by the formal Inverse of this ket as done in the previous section). For illustration purposes, we use Coefficients and note how it automatically expands the commutator
Coefficients,Ketψ,x,y,z
Xm⁢Pn−Pn⁢Xm=ⅈ⁢ℏ⁢gm,n
One can undo this (frequently undesired) expansion of the commutator by sorting the products on the left-hand side using the commutator between Xm and Pn
Library:-SortProducts,Pn,Xm,usecommutator
Xm,Pn−=ⅈ⁢ℏ⁢gm,n
And that is the result we wanted to compute.
Additionally, to see this rule in matrix form,
TensorArray−
x,P1−=ⅈ⁢ℏx,P2−=0x,P3−=0y,P1−=0y,P2−=ⅈ⁢ℏy,P3−=0z,P1−=0z,P2−=0z,P3−=ⅈ⁢ℏ
In the above, we use equation (35) multiplied by -1 to avoid a minus sign in all the elements of (36), due to having worked with the default signature (- - - +); this minus sign is not necessary if in the Setup at the beginning one also sets signature=`+ + + -`
For display purposes, to see this matrix expressed in terms of the geometrical components of the momentum p→ , redefine the tensor Pn explicitly indicating its Cartesian components
DefinePm=p__x,p__y,p__z,quiet
x,p__x−=ⅈ⁢ℏx,p__y−=0x,p__z−=0y,p__x−=0y,p__y−=ⅈ⁢ℏy,p__z−=0z,p__x−=0z,p__y−=0z,p__z−=ⅈ⁢ℏ
Finally, in a typical situation, these commutation rules are to be taken into account in further computations, and for that purpose they can be added to the setup via
Setup
algebrarules=x,p__x−=ⅈ⁢ℏ,x,p__y−=0,x,p__z−=0,x,y−=0,x,z−=0,y,p__x−=0,y,p__y−=ⅈ⁢ℏ,y,p__z−=0,y,z−=0,z,p__x−=0,z,p__y−=0,z,p__z−=ⅈ⁢ℏ
For example, from herein computations are performed taking into account that
%Commutator = Commutatorx, p__x
x,p__x−=ⅈ⁢ℏ
There are 991 metrics in the database of solutions to Einstein's equations, based on the book "Exact solutions to Einstein's equations". One can check this number via
nopsDifferentialGeometry:-Library:-RetrieveStephani,1
991
For the majority of these solutions, the book also presents, explicit or implicitly, the form of the Energy-Momentum tensor. New in Maple 2018, we added to the database one more entry indicating the components of the corresponding EnergyMomentum tensor, covering, in Maple 2018.0, 686 out of these 991 solutions.
The design of the EnergyMomentum tensor got slightly adjusted to take these new database entries into account, so that when you load one of these solutions, if the corresponding entry for the EnergyMomentumTensor is already in the database, it is automatically loaded together with the solution.
In addition, it is now possible to define the tensor components using the Define command, or redefine any of its components using the new Library:-RedefineTensorComponent routine (see Physics,Library)
Examples
restart; withPhysics:
Consider the metric of Chapter 12, equation number 16.1
g_12,18,1
Systems of spacetime Coordinates are: X=τ,r,θ,φ
Default differentiation variables for d_, D_ and dAlembertian are: X=τ,r,θ,φ
The Bertotti (1959), Kramer (1978), Levi-Civita (1917), Robinson (1959) metric in coordinates τ,r,θ,φ
Parameters: k,κ0,β
Resetting the signature of spacetime from "- - - +" to `- + + +` in order to match the signature in the database of metrics:
gμ,ν=−k2r20000k2r20000k20000k2⁢sin⁡θ2
New, the covariant components of the EnergyMomentum tensor got automatically loaded, given by
EnergyMomentum
Τμ,ν=1r2⁢κ00000−1r2⁢κ000001κ00000sin⁡θ2κ0
One can verify this checking for the tensor's definition
EnergyMomentumdefinition
Τμ,ν=Gμ,ν8⁢π,Τ⁢μ,ν⁢μ,ν=r2k4⁢κ00000−r2k4⁢κ000001k4⁢κ000001k4⁢sin⁡θ2⁢κ0
Take now the tensor components of the first defining equation of (44)
TensorArray1,simplifier=simplify
1r2⁢κ0=18⁢r2⁢π0=00=00=00=0−1r2⁢κ0=−18⁢r2⁢π0=00=00=00=01κ0=18⁢π0=00=00=00=0sin⁡θ2κ0=sin⁡θ28⁢π
where κ0=8 π is related to Newton's constant.
To see the continuity equations for the components of Τμ,ν, use for instance the inert version of the covariant derivative operator D_ and the TensorArray command
%D_mu=D_muEnergyMomentummu,nu
▿μ⁡Τ⁢μν⁢μν=0
TensorArray
∂1⁡−1k2⁢κ0+∂2⁡0+∂3⁡0+∂4⁡0=0∂1⁡0+∂2⁡−1k2⁢κ0+∂3⁡0+∂4⁡0=0∂1⁡0+∂2⁡0+∂3⁡1k2⁢κ0+∂4⁡0=0∂1⁡0+∂2⁡0+∂3⁡0+∂4⁡1k2⁢κ0=0
value
0=00=00=00=0
The EnergyMomentum tensor can also be (re)defined in any particular way (a correct definition must satisfy ▿μΤ⁢μν⁢μν=0).
Define the EnergyMomentum tensor indicating the functionality in the definition in terms of W to be constant energy (i.e. no functionality) and the flux density Sμ and stress σμ,ν tensors depending on X. For this purpose, use the new option minimizetensorcomponents to make explicit the symmetry of the stress tensor σμ,ν
DefineSmu, sigmamu,nu,symmetric, minimizetensorcomponents
▿μ,γμ,σμ,Rμ,ν,Rμ,ν,α,β,Sμ,Cμ,ν,α,β,Xμ,∂μ,gμ,ν,σμ,ν,Γμ,ν,α,Gμ,ν,Τμ,ν,δμ,ν,εα,β,μ,ν
The symmetry of σμ,ν is now explicit in that its matrix form is symmetric
sigma
σμ,ν=σ1,1σ1,2σ1,3σ1,4σ1,2σ2,2σ2,3σ2,4σ1,3σ2,3σ3,3σ3,4σ1,4σ2,4σ3,4σ4,4
The new routines for testing tensor symmetries
Library:-IsTensorialSymmetricsigmamu,nu
true
The symmetry is regarding interchanging positions of the 1st and 2nd indices
Library:-GetTensorSymmetryPropertiessigma
1,2,∅
So this is the form of the EnergyMomentum with all its components - but for the total energy - depending on the coordinates
EnergyMomentumμ,ν=Matrix⁡4,μ,ν→ifμ=4thenifν=4thenWelseSν⁡Xfielifν=4thenSμ⁡Xelseσμ,ν⁡Xfi
Τμ,ν=σ1,1⁡Xσ1,2⁡Xσ1,3⁡XS1⁡Xσ2,1⁡Xσ2,2⁡Xσ2,3⁡XS2⁡Xσ3,1⁡Xσ3,2⁡Xσ3,3⁡XS3⁡XS1⁡XS2⁡XS3⁡XW
CompactDisplay⁡
S⁡X⁢will now be displayed as⁢S
sigma⁡X⁢will now be displayed as⁢σ
Define now Τμ,ν with these components
Define⁡
Τμ,ν=σ1,1σ1,2σ1,3S1σ1,2σ2,2σ2,3S2σ1,3σ2,3σ3,3S3S1S2S3W
To see the continuity equations for the components of Τμ,ν, use again the inert version of the covariant derivative operator D_ and the TensorArray command
%D_μ=D_μ⁡EnergyMomentumμ,ν
For a more convenient reading, present the result as a vector column
Vectorcolumn⁡TensorArray⁡
∂1⁡−r2⁢σ1,1k2+∂2⁡r2⁢σ1,2k2+∂3⁡σ1,3k2+∂4⁡S1k2⁢sin⁡θ2−2⁢r⁢σ1,2k2+cos⁡θ⁢σ1,3sin⁡θ⁢k2=0∂1⁡−r2⁢σ1,2k2+∂2⁡r2⁢σ2,2k2+∂3⁡σ2,3k2+∂4⁡S2k2⁢sin⁡θ2−r⁢σ1,1k2−r⁢σ2,2k2+cos⁡θ⁢σ2,3sin⁡θ⁢k2=0∂1⁡−r2⁢σ1,3k2+∂2⁡r2⁢σ2,3k2+∂3⁡σ3,3k2+∂4⁡S3k2⁢sin⁡θ2−cos⁡θ⁢Wsin⁡θ3⁢k2−2⁢r⁢σ2,3k2+cos⁡θ⁢σ3,3sin⁡θ⁢k2=0∂1⁡−r2⁢S1k2+∂2⁡r2⁢S2k2+∂3⁡S3k2+∂4⁡Wk2⁢sin⁡θ2+cos⁡θ⁢S3sin⁡θ⁢k2−2⁢r⁢S2k2=0
Comparing the specific form (43) for the EnergyMomentum loaded from the database of solutions to Einstein's equations with the general form (56), one can ask the formal question of whether there are other forms for the EnergyMomentum satisfying the continuity equations (58).
To answer that question, rewrite this system of equations for the flux density Sμ and stress σμ,ν tensors as a set, and solve it for them
convertvalue,setofequations
−r2⁢S1τk2+2⁢r⁢S2+r2⁢S2rk2+S3θk2+cos⁡θ⁢S3sin⁡θ⁢k2−2⁢r⁢S2k2=0,−r2⁢σ1,1τk2+2⁢r⁢σ1,2+r2⁢σ1,2rk2+σ1,3θk2+S1φk2⁢sin⁡θ2−2⁢r⁢σ1,2k2+cos⁡θ⁢σ1,3sin⁡θ⁢k2=0,−r2⁢σ1,2τk2+2⁢r⁢σ2,2+r2⁢σ2,2rk2+σ2,3θk2+S2φk2⁢sin⁡θ2−r⁢σ1,1k2−r⁢σ2,2k2+cos⁡θ⁢σ2,3sin⁡θ⁢k2=0,−r2⁢σ1,3τk2+2⁢r⁢σ2,3+r2⁢σ2,3rk2+σ3,3θk2+S3φk2⁢sin⁡θ2−cos⁡θ⁢Wsin⁡θ3⁢k2−2⁢r⁢σ2,3k2+cos⁡θ⁢σ3,3sin⁡θ⁢k2=0
This system in fact admits much more general solutions than (43):
pdsolve
S1=S1,S2=S2,S3=∫r2⁢S1τ⁢sin⁡θ−sin⁡θ⁢S2r⁢r2ⅆθ+_F1⁡τ,r,φsin⁡θ,σ1,1=σ1,1,σ1,2=∫−σ1,1τ⁢cos⁡θ2⁢r2−r2⁢σ1,1τ+cos⁡θ⁢σ1,3⁢sin⁡θ−σ1,3θ⁢cos⁡θ2+σ1,3θ+S1φr2⁢sin⁡θ2ⅆr+_F2⁡τ,θ,φ,σ1,3=σ1,3,σ2,2=∫r2⁢_F2τ⁢cos⁡4⁢θ+4⁢∫r2⁢σ1,1τ,τ⁢cos⁡2⁢θ−r2⁢σ1,1τ,τ−σ1,3τ,θ⁢cos⁡2⁢θ+σ1,3τ⁢sin⁡2⁢θ+σ1,3τ,θ+2⁢S1φ,τ2⁢r2ⅆr⁢r2⁢cos⁡2⁢θ−4⁢r2⁢_F2τ⁢cos⁡2⁢θ+r⁢σ1,1⁢cos⁡4⁢θ−4⁢r⁢σ1,1⁢cos⁡2⁢θ−4⁢∫r2⁢σ1,1τ,τ⁢cos⁡2⁢θ−r2⁢σ1,1τ,τ−σ1,3τ,θ⁢cos⁡2⁢θ+σ1,3τ⁢sin⁡2⁢θ+σ1,3τ,θ+2⁢S1φ,τ2⁢r2ⅆr⁢r2+3⁢r2⁢_F2τ+4⁢S2φ⁢cos⁡2⁢θ−_F3θ⁢cos⁡4⁢θ+_F3⁡τ,θ,φ⁢sin⁡4⁢θ−8⁢cos⁡θ⁢∫3⁢r2⁢σ1,3τ⁢sin⁡θ−σ1,3τ⁢r2⁢sin⁡3⁢θ+4⁢r2⁢∫sin⁡θ⁢S2φ,r−sin⁡θ⁢S1φ,τⅆθ−3⁢σ3,3θ⁢sin⁡θ−cos⁡θ⁢σ3,3+σ3,3⁢cos⁡3⁢θ+σ3,3θ⁢sin⁡3⁢θ+4⁢cos⁡θ⁢W−4⁢_F1φ4⁢r2ⅆr+4⁢_F3θ⁢cos⁡2⁢θ−2⁢_F3⁡τ,θ,φ⁢sin⁡2⁢θ+3⁢r⁢σ1,1−4⁢S2φ−8⁢∫σ1,3τ,θ⁢r2⁢cos⁡4⁢θ−4⁢σ1,3τ,θ⁢r2⁢cos⁡2⁢θ−4⁢S2φ,r⁢r2⁢cos⁡2⁢θ+4⁢S1φ,τ⁢r2⁢cos⁡2⁢θ−24⁢r2⁢∫sin⁡θ⁢S2φ,r−sin⁡θ⁢S1φ,τⅆθ⁢cos⁡θ+3⁢σ1,3τ,θ⁢r2+4⁢S2φ,r⁢r2−4⁢S1φ,τ⁢r2−σ3,3θ,θ⁢cos⁡4⁢θ+4⁢σ3,3θ,θ⁢cos⁡2⁢θ+σ3,3θ⁢sin⁡4⁢θ−4⁢σ3,3⁢cos⁡2⁢θ−2⁢σ3,3θ⁢sin⁡2⁢θ+24⁢_F1φ⁢cos⁡θ−8⁢W⁢cos⁡2⁢θ−3⁢σ3,3θ,θ+4⁢σ3,3−16⁢W8⁢r2ⅆr−3⁢_F3θr⁢−4⁢cos⁡2⁢θ+3+cos⁡4⁢θⅆr+_F4⁡τ,θ,φr,σ2,3=∫−σ1,3τ⁢sin⁡θ⁢cos⁡θ2⁢r2+r2⁢σ1,3τ⁢sin⁡θ+σ3,3θ⁢sin⁡θ⁢cos⁡θ2+σ3,3⁢cos⁡θ3+r2⁢∫sin⁡θ⁢S2φ,r−S1φ,τⅆθ−σ3,3θ⁢sin⁡θ−cos⁡θ⁢σ3,3+cos⁡θ⁢W−_F1φr2⁢sin⁡θ3ⅆr+_F3⁡τ,θ,φ,σ3,3=σ3,3
This solution can be verified in different ways, for instance using pdetest showing it cancels the PDE system (59) for Sμ and stress σμ,ν
pdetest,
0
Minimize the number of tensor components according to its symmetries, and relabel, redefine or count the number of independent tensor components
A new keyword in Define and Setup: minimizetensorcomponents, allows for automatically minimizing the number of tensor components taking into account the tensor symmetries. For example, if a 2-tensor in a 4D spacetime is defined a antisymmetric, the number of different tensor components is 6, and the elements of the diagonal are automatically set equal to 0. After setting this keyword to true with Setup, all subsequent definitions of tensors automatically minimize the number of components while using this keyword with Define makes this minimization only happen with the tensors being defined in the call to Define.
Related to this new functionality, 4 new Library routines were added: MinimizeTensorComponents, NumberOfIndependentTensorComponents, RelabelTensorComponents and RedefineTensorComponents
Define an antisymmetric tensor with two indices
DefineFmu,nu,antisymmetric
γμ,Fμ,ν,σμ,∂μ,gμ,ν,δμ,ν,εα,β,μ,ν
Although the system knows that Fμ,ν is antisymmetric,
Fmu,nu + Fnu,mu
Fμ,ν+Fν,μ
Simplify
by default the components of Fμ,ν do not automatically reflect that, it is necessary to use the simplifier of the Physics package, Simplify
F1,2 + F2,1
F1,2+F2,1
Likewise, computing the array form of Fμ,ν we do not see the elements of the diagonal equal to 0, nor the lower-left triangle equal to the upper-right triangle but with a different sign:
TensorArrayFmu,nu
F1,1F1,2F1,3F1,4F2,1F2,2F2,3F2,4F3,1F3,2F3,3F3,4F4,1F4,2F4,3F4,4
This new functionality, here called minimizetensorcomponents, makes the symmetries of the tensor explicitly reflected in its components. There are three ways to use it. First, one can minimize the number of tensor components of a tensor previously defined. For example
Library:-MinimizeTensorComponentsF
0F1,2F1,3F1,4−F1,20F2,3F2,4−F1,3−F2,30F3,4−F1,4−F2,4−F3,40
After this, both (63) and (64) are automatically equal to 0 without having to use Simplify
And the output of TensorArray in (67) becomes equal to (68).
NOTE: after using minimizetensorcomponents in the definition of a tensor, say F, all the keywords implemented for Physics tensors are available for the F:
F
Fμ,ν=0F1,2F1,3F1,4−F1,20F2,3F2,4−F1,3−F2,30F3,4−F1,4−F2,4−F3,40
Ftrace
Fnonzero
Fμ,ν=1,2=F1,2,1,3=F1,3,1,4=F1,4,2,1=−F1,2,2,3=F2,3,2,4=F2,4,3,1=−F1,3,3,2=−F2,3,3,4=F3,4,4,1=−F1,4,4,2=−F2,4,4,3=−F3,4
F~1,mu,matrix
F⁢1μ⁢1μ=0−F1,2−F1,3−F1,4
Alternatively, one can define a tensor, specifying that the symmetries should be taken into account to minimize the number of its components passing the keyword minimizetensorcomponents to Define.
Define a tensor with the symmetries of the Riemann tensor, that is, a tensor of 4 indices that is symmetric with respect to interchanging the positions of the 1st and 2nd pair of indices and antisymmetric with respect to interchanging the position of its 1st and 2nd indices, or 3rd and 4th indices, and minimizing the number of tensor components
DefineRalpha,beta,mu,nu, symmetric=1,2,3,4, antisymmetric = 1,2,3,4,minimizetensorcomponents
γμ,Fμ,ν,σμ,Rμ,ν,α,β,∂μ,gμ,ν,δμ,ν,εα,β,μ,ν
R1,2,3,4 + R2,1,3,4
Ralpha,beta,mu,nu − Rmu,nu,alpha,beta
One can always retrieve the symmetry properties in the abstract notation used by the Define command using the new Library:-GetTensorSymmetryProperties, its output is ordered, first the symmetric then the antisymmetric properties
Library:-GetTensorSymmetryPropertiesR
1,2,3,4,1,2,3,4
After making the symmetries explicit (and also before that), it frequently s useful to know the number of independent components of a given tensor. For this purpose use the new Library:-NumberOfIndependentTensorComponents
Library:-NumberOfIndependentTensorComponentsR
21
and besides the symmetries, in the case of the Riemann tensor after taking into account the first Bianchi identity, this number of components is further reduced to 20.
A third way of using the new minimizetensorcomponents functionality is using Setup, so that every subsequent definition of tensors with symmetries is automatically performed minimizing the number of components.
Setupminimizetensorcomponents = true
minimizetensorcomponents=true
You can now define without having to include the keyword minimizetensorcomponents in the definition of tensors with symmetries
DefineCalpha,beta,antisymmetric
Cμ,ν,γμ,Fμ,ν,σμ,Rμ,ν,α,β,∂μ,gμ,ν,δμ,ν,εα,β,μ,ν
C
Cμ,ν=0C1,2C1,3C1,4−C1,20C2,3C2,4−C1,3−C2,30C3,4−C1,4−C2,4−C3,40
Two new related functionalities are provided via Library:-RelabelTensorComponents and Library:-RedefineTensorComponent, the first one to have the number of tensor components directly reflected in the names of the components, the second one to redefine only one of these components
Library:-RelabelTensorComponentsC
0C__1C__2C__3−C__10C__4C__5−C__2−C__40C__6−C__3−C__5−C__60
Suppose now we want to make one of these components equal to 1, say C__2
Library:-RedefineTensorComponentC1 ,2 = 1
Cμ,ν=01C__2C__3−10C__4C__5−C__2−C__40C__6−C__3−C__5−C__60
New: as part of the developments of Physics bug regardless of loading the Physics package, inert names are now typeset in gray, the standard for inert functions, with copy & paste working
%x
x
2⋅%x
2⁢x
Note that this was already in place in previous releases regarding inert functions but not regarding inert names. Regarding inert functions, since Maple 2016 their typesetting is also in grey with copy & paste working
%fx
f⁡x
Regarding Physics, having the right typeset also for symbols and tensor names is particularly relevant now that one can compute with differential operators as operands of a product.
withPhysics: withVectors:
Setupop=p_,x,y,z,differentialoperators = p_, x,y,z
* Partial match of 'op' against keyword 'quantumoperators'
differentialoperators=p→,x,y,z,quantumoperators=p→,x,y,z
The active and inert representations of the same differential-vectorial operator are
p_ = %p_
p→=p→
Hence, you can:
a) assign a mapping to p_ while represent it using %p_ when you do not want the mapping to be applied.
b) using %p_ has mathematical and clear typesetting (in gray always means inert) making its use more pleasant / easy to read.
interfaceimaginaryunit=i:
Assign a procedure to the differential-vectorial operator p→
Apply both the active and the inert operators to some function of the coordinates
%p_ = p_⋅ fx,y,z
p→⁢f⁡x,y,z=p→⁢f⁡x,y,z
Apply now the differential operators in products: the left-hand side, inert, remains a product, while the right-hand side, becomes a function application, and so (90) gets applied
Library:-ApplyProductsOfDifferentialOperators
p→⁢f⁡x,y,z=−ⅈ⁢ℏ⁢∇f⁡x,y,z
NOTE: the implementation is such that if p is noncommutative, then so is %p and the same holds regarding their possibly differential operator and tensorial character: the inert versions inherit the properties of their active counterparts, and the same regarding their tensorial character: if p is a tensor, so is %p. In addition, inert tensors are also now displayed the same way as their active versions but in gray, improving the readability of tensorial expressions
Load a curved spacetime, for instance Schwarzschild's metric
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
gμ,ν=r−r+2⁢m0000−r20000−r2⁢sin⁡θ20000r−2⁢mr
Define a tensor and compute its covariant derivative equating the inert with the active form of it
DefineAmu
Aμ,▿μ,γμ,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
CompactDisplayAX
A⁡X⁢will now be displayed as⁢A
Since