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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 2016 has been the consolidation of the new functionality introduced during the last three releases, through more than 300 enhancements across the board, together with signification enhancements and new functionality in General Relativity. Most notably, for the 100th anniversary of the presentation of the theory of relativity, in Maple 2016 we are bringing to completion the digitalization of the solutions to Einstein's equations collected, from more than 4000 papers, in the classic book "Exact Solutions to Einstein's Equations" by H. Stephani - et al.

In addition, Maple 2016 implements new general functionality that is relevant within and beyond Physics, including a new Factor command with the ability to perform factorization in expressions involving products of noncommutative operators and the ability to compute with differential operators algebraically, that is, operating with them using multiplication to express application. Combining these two developments, it is now possible to factorize algebraic expressions involving differential operators and so, for example, solve partial differential equations through factorization.

Taking all together, the more than 350 enhancements throughout the entire package increased robustness, versatility and functionality, extending again 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 2016.

 

 

Completion of the Database of Solutions to Einstein's Equations

Operatorial Algebraic Expressions Involving the Differential Operators μ, 𝒟μ and  (nabla)

Factorization of Expressions Involving Noncommutative Operators

Tensors in Special and General Relativity

Vectors Package

The Physics Library

Redesigned Functionality and Miscellaneous

Completion of the Database of Solutions to Einstein's Equations

A database of solutions to Einstein's equations was added to the Maple library in Maple 15 with a selection of metrics from "Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; and Herlt, E.,  Exact Solutions to Einstein's Field Equations". More metrics from this book were added for Maple 16, Maple 17, and Maple 18, up to 225 metrics. In Maple 2016, for the 100th anniversary of the presentation of the theory of Relativity by A. Einstein, we brought to completion the digitalization of all the 971 metric solutions collected from more than 4000 papers presented in the General Relativity literature. All these metrics can be loaded or searched using g_ (the Physics command representing the spacetime metric that also sets the metric to your choice in one go) or using the command DifferentialGeometry:-Library:-MetricSearch. When a metric is loaded, all the General Relativity tensors (Christoffel, Ricci, Einstein, and Riemann) are automatically computed on background and available for further computations. These metrics can also be changed in various ways and, automatically, all the General Relativity tensors are recomputed on background. You can work with these metrics, the ones in the database or any other one you set, using the Tetrads formalism too, with the commands of the Physics:-Tetrads package.

In the Maple PDEtools package, you have the mathematical tools - including a complete symmetry approach - to work with the underlying partial differential equations. By combining the functionality of the Physics:-Tetrads package, the Physics:-TransformCoordinates command, and the ability to compute Riemann and Weyl invariants, you can also formulate and, depending on the the metrics also resolve, the equivalence problem; that is: to answer whether or not, given two metrics, they can be obtained from each other by a transformation of coordinates, as well as compute the transformation.

Examples

Load Physics, set the metric to Schwarzschild (and everything else automatically) in one go

restart; withPhysics:

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

That is all you need to do: although the strength in Physics is to compute with tensors using indicial notation, by setting the metric as in  all of the tensor components of the General Relativity tensors are also derived on the fly and ready for use. For instance these are the definition in terms of Christoffel symbols, and the covariant components of the Ricci tensor for Schwarzschild solution

Riccidefinition

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

(1)

Ricci

Rμ,ν=0000000000000000

(2)

These are the 16 Riemann invariants for Schwarzschild solution, using the formulas by Carminati and McLenaghan

Riemanninvariants

r0=0,r1=0,r2=0,r3=0,w1=6m2r6,w2=6m3r9,m1=0,m2=0,m3=0,m4=0,m5=0

(3)

The related Weyl scalars in the context of the Newman-Penrose formalism; the definition is in terms of the Weyl tensor and the tetrad of tensors lμ,nμ,mμ,m&conjugate0;μ of the Newman-Penrose formalism

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

(4)

Weylscalars

ψ__0=0,ψ__1=0,ψ__2=mr3,ψ__3=0,ψ__4=0

(5)

 

These are the matrix components of the Christoffel symbols of the second kind (that describe, in coordinates, the effects of parallel transport in curved surfaces), when the first of its three indices is equal to 1; contravariant indices are prefixed by ~

Christoffel~1,alpha,beta,matrix

Γ1α,β1α,β=mrr+2m0000r+2m0000r+2msinθ200002m2+mrr3

(6)

In Physics, the Christoffel symbols of the first kind are represented by the same object (one command, Christoffel, not two) just by taking the first index covariant, as we do when computing with paper and pencil

Christoffel1,alpha,beta,matrix

Γ1,α,β=mr+2m20000r0000rsinθ20000mr2

(7)

One could query the database, directly from the spacetime metrics, for example about the solutions (metrics) to Einstein's equations related to Levi-Civita, the Italian mathematician

g_civi

____________________________________________________________

12,16,1=Authors=Bertotti (1959),Kramer (1978),Levi-Civita (1917),Robinson (1959),PrimaryDescription=EinsteinMaxwell,SecondaryDescription=Homogeneous

____________________________________________________________

12,18,1=Authors=Bertotti (1959),Kramer (1978),Levi-Civita (1917),Robinson (1959),PrimaryDescription=EinsteinMaxwell,SecondaryDescription=Homogeneous

____________________________________________________________

12,19,1=Authors=Bertotti (1959),Kramer (1978),Levi-Civita (1917),Robinson (1959),PrimaryDescription=EinsteinMaxwell,SecondaryDescription=Homogeneous,Comments=_lambda=_zeta

____________________________________________________________

22,7,1=Authors=Levi-Civita (1917), Frehland (1971),PrimaryDescription=Vacuum,SecondaryDescription=Cylindrically-Symmetric,Comments=Locally static, Weyl class_m=0,1 - flat, _m=1/2, 2, -1 - PetrovType D

(8)

Each triad of numbers indicates the chapter and equation number with which each of these metrics appear in "Exact Solutions to Einstein's Field Equations". For example, [12, 16, 1] is the metric found in Chapter 12 with equation number 16.1. These solutions can be set in one go from the metrics command, just by indicating the triad of numbers as follows

g_12,16,1

Systems of spacetime Coordinates are: X=t,x,θ,φ

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

The Bertotti (1959), Kramer (1978), Levi-Civita (1917), Robinson (1959) metric in coordinates t,x,θ,φ

Parameters: k,κ0,β

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

Automatically, everything gets set accordingly; these are the contravariant components of the related Ricci tensor

Ricci~

Rμ,νμ,ν=1k4sinhx200001k400001k400001k4sinθ2

(9)

One works with the Newman-Penrose formalism frequently using tetrads (local system of references); the Physics subpackage for this is Tetrads

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

(10)

To query about the definition of any of these tensors, of the Tetrads or Physics packages, index them with the keyword definition

l_definition

lμlμμ=0,lμnμμ=−1,lμmμμ=0,lμm&conjugate0;μμ=0,gμ,ν=lμnνlνnμ+mμm&conjugate0;ν+mνm&conjugate0;μ

(11)

gamma_definition

γa,b,c=𝒟ν𝔢a,μ𝔢bμbμ𝔢cνcν

(12)

This is the orthonormal tetrad 𝔢a,μ related to the metric set in  set by the package (you can change these tetrads in different ways using the TransformTetrad command)

e_

𝔢a,μ=sinhxk0000k0000k0000sinθk

(13)

You can check these components directly using the definition. In this case, the right-hand side is the (orthonormal) tetrad metric

e_definition

𝔢a,μ𝔢bμbμ=ηa,b

(14)

eta_

This shows that, for the tetrad components given by (13), the definition (14) holds

TensorArray

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

(15)

One frequently works with a different signature and null tetrads; set that, and everything related automatically

Setuptetradmetric = null

tetradmetric=1,2=−1,3,4=1

(16)

eta_

e_

𝔢a,μ=2sinhxk22k2002sinhxk22k200002k2I22sinθk002k2I22sinθk

(17)

Verifying (17) using the definition (14)

TensorArray

0=0−1=−10=00=0−1=−10=00=00=00=00=00=01=10=00=01=10=0

(18)

You can also change the signature and everything gets automatically recomputed as well, from the components of the tensors to the definition of Weyl scalars. You can query the value of the signature using Setup

Setupsignature

signature=- + + +

(19)

Set the signature to be + - - - , compare the components of the null tetrad metric with the components of  and verify the tetrad 𝔢a,μ

Setupsignature = `+---`

signature=+ - - -

(20)

eta_

e_

𝔢a,μ=I22sinhxkI22k00I22sinhxkI22k0000I22k2sinθk200I22k2sinθk2

(21)

TensorArray

0=01=10=00=01=10=00=00=00=00=00=0−1=−10=00=0−1=−10=0

(22)

The related 16 Riemann invariants

Riemanninvariants

r0=0,r1=1k4,r2=0,r3=14k8,w1=0,w2=0,m1=0,m2=0,m3=0,m4=0,m5=0

(23)

The ability to query rapidly, set things in one go, change everything and have all the quantities automatically adjusted are at the realm of the design of the General Relativity functionality of the Physics package, resulting in a rather flexible computational environment.

 

These are the metrics by Kaigorodov; next are those published in 1962

g_Kaigorodov

____________________________________________________________

12&comma;34&comma;1=Authors=Kaigorodov (1962)&comma;Cahen (1964)&comma;Siklos (1981)&comma;Ozsvath (1987)&comma;PrimaryDescription=Einstein&comma;SecondaryDescription=Homogeneous&comma;Comments=All metrics with _epsilon <> 0 are equivalent to the cases _epsilon = +1, -1, _epsilon = 0 is anti-deSitter space

____________________________________________________________

12&comma;35&comma;1=Authors=Kaigorodov (1962)&comma;Cahen (1964)&comma;Siklos (1981)&comma;Ozsvath (1987)&comma;PrimaryDescription=Einstein&comma;SecondaryDescription=Homogeneous&comma;SimpleTransitive

____________________________________________________________

38&comma;2&comma;1=Authors=Kaigorodov (1962)&comma;PrimaryDescription=Hypersurface-Homogeneous&comma;SecondaryDescription=AlgebraicallySpecial

____________________________________________________________

38&comma;3&comma;1=Authors=Kaigorodov (1962)&comma;PrimaryDescription=Hypersurface-Homogeneous&comma;SecondaryDescription=AlgebraicallySpecial&comma;Comments=Note, there is a typo (the x^4 term) in this metric in Stephani which we were not able to verify from the original. This metric may not be correct!

(24)

g_`1962`

____________________________________________________________

12&comma;13&comma;1=Authors=Ozsvath, Schucking (1962)&comma;PrimaryDescription=Vacuum&comma;SecondaryDescription=Homogeneous&comma;PlaneWave&comma;Comments=geodesically complete, no curvature singularities

____________________________________________________________

12&comma;14&comma;1=Authors=Petrov (1962)&comma;PrimaryDescription=Vacuum&comma;SecondaryDescription=Homogeneous&comma;SimpleTransitive

____________________________________________________________

12&comma;34&comma;1=Authors=Kaigorodov (1962)&comma;Cahen (1964)&comma;Siklos (1981)&comma;Ozsvath (1987)&comma;PrimaryDescription=Einstein&comma;SecondaryDescription=Homogeneous&comma;Comments=All metrics with _epsilon <> 0 are equivalent to the cases _epsilon = +1, -1, _epsilon = 0 is anti-deSitter space

____________________________________________________________

12&comma;35&comma;1=Authors=Kaigorodov (1962)&comma;Cahen (1964)&comma;Siklos (1981)&comma;Ozsvath (1987)&comma;PrimaryDescription=Einstein&comma;SecondaryDescription=Homogeneous&comma;SimpleTransitive

____________________________________________________________

18&comma;2&comma;1=Authors=Ehlers and Kundt (1962)&comma;PrimaryDescription=vacuum solutions &comma;SecondaryDescription=degenerate static vacuum solution&comma;Comments=Table 18.2, case AI, _Psi2=-_b/(2*r^3)

____________________________________________________________

18&comma;2&comma;2=Authors=Ehlers and Kundt (1962)&comma;PrimaryDescription=vacuum solutions &comma;SecondaryDescription=degenerate static vacuum solution&comma;Comments=Table 18.2, case AII, _Psi2=_b*(1/(2*z^3))

____________________________________________________________

18&comma;2&comma;3=Authors=Ehlers and Kundt (1962)&comma;PrimaryDescription=vacuum solutions &comma;SecondaryDescription=degenerate static vacuum solution&comma;Comments=Table 18.2, case AIII, _Psi2=1/(2*z^3)

____________________________________________________________

18&comma;2&comma;4=Authors=Ehlers and Kundt (1962)&comma;PrimaryDescription=vacuum solutions &comma;SecondaryDescription=degenerate static vacuum solution&comma;Comments=Table 18.2, case BI, _Psi2=-_b*(1/(2*r^3))

____________________________________________________________

18&comma;2&comma;5=Authors=Ehlers and Kundt (1962)&comma;PrimaryDescription=vacuum solutions &comma;SecondaryDescription=degenerate static vacuum solution&comma;Comments=Table 18.2, case BII, _Psi2=_b*(1/(2*z^3))

____________________________________________________________

18&comma;2&comma;6=Authors=Ehlers and Kundt (1962)&comma;PrimaryDescription=vacuum solutions &comma;SecondaryDescription=degenerate static vacuum solution&comma;Comments=Table 18.2, case BIII, _Psi2=1/(2*z^3)

____________________________________________________________

18&comma;2&comma;7=Authors=Ehlers and Kundt (1962)&comma;PrimaryDescription=vacuum solutions &comma;SecondaryDescription=degenerate static vacuum solution&comma;Comments=Table 18.2, case C, _Psi2=(1/2)*(x+y)^3, + case

____________________________________________________________

18&comma;2&comma;8=Authors=Ehlers and Kundt (1962)&comma;PrimaryDescription=vacuum solutions &comma;SecondaryDescription=degenerate static vacuum solution&comma;Comments=Table 18.2, case C, _Psi2=-(1/2)*(x+y)^3, - case

____________________________________________________________

22&comma;59&comma;1=Authors=Misra and and Radhakrishna (1962)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=Cylindrically-Symmetric&comma;Comments=Null EM field, _W=Int((diff(_Theta(u),u)^2+diff(_eta(u),u)^2),u), u=(t-rho)/sqrt(2), the non-null case is with _Theta(t,rho)=_Psi(t,rho)*cos(_alpha), _Theta(t,rho)=_Psi(t,rho)*sin(_alpha)

____________________________________________________________

26&comma;21&comma;1=Authors=Newman and Tamburino (1962)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=Non-Aligned, Non-Null&comma;Comments=Spherical class

____________________________________________________________

26&comma;22&comma;1=Authors=Newman and Tamburino (1962)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=Non-Aligned, Non-Null&comma;Comments=Cyllindrical class

____________________________________________________________

26&comma;23&comma;1=Authors=Newman and Tamburino (1962)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=Non-Aligned, Non-Null&comma;Comments=Cyllindrical class

____________________________________________________________

27&comma;27&comma;1=Authors=Robinson and Trautman 1962, Debney et al. 1969, Talbot 1969, Robinson et al. 1969a, Lind 1974 &comma;PrimaryDescription=Generic&comma;SecondaryDescription=geodesic, shearfree and diverging null congruence&comma;Comments=rho:=1/(-(r+r0+I*Sigma)): Sigma:=-2*I*P^2*((diff(L,zb)-L*diff(L,u))-(diff(Lb,z)-L*diff(Lb,u))):

____________________________________________________________

27&comma;37&comma;1=Authors=Robinson and Trautman (1962)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=Non-Aligned, non-null&comma;Comments=admits geodesic, shearfree, twistfree null congruence, _rho=-1/r=_rho_b

____________________________________________________________

28&comma;8&comma;1=Authors=Robinson-Trautman (1962)&comma;PrimaryDescription=Vacuum&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=diff(2*_P(u,z,zb)^2*diff(2*_P(u,z,zb)^2*ln(_P(u,z,zb)),z,zb),z,zb)+12*_m(u)*diff(ln(_P(u,z,zb)),u)-4*diff(_m(u),u)=0

____________________________________________________________

28&comma;12&comma;1=Authors=Robinson-Trautman (1962)&comma;PrimaryDescription=Vacuum&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=2*P(u, z, zb)^2*diff(ln(P(u, z, zb)),z,zb)=_K(u), _K(u)=0,+1,-1

____________________________________________________________

28&comma;16&comma;1=Authors=Robinson-Trautman (1962)&comma;PrimaryDescription=Vacuum&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=The coordinate zeta is changed to xi&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;26&comma;1=Authors=Robinson, Trautman (1962)&comma;PrimaryDescription=Vacuum&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=One can use the diffeo r -> -r and u -> -u to make the assumption r > 0&comma;The case _m = 0 is Stephani, [28, 16,1]&comma;The metric is type D at points where r = 3*_m/(xi1+xi2) and type II on either side of this hypersurface. For convenience, it is assumed that 3*_m - r*(xi1 + xi2) > 0&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;26&comma;2=Authors=Robinson, Trautman (1962)&comma;PrimaryDescription=Vacuum&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=One can use the diffeo r -> -r and u -> -u to make the assumption r > 0&comma;The case _m = 0 is Stephani, [28, 16,1].&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;26&comma;3=Authors=Robinson, Trautman (1962)&comma;PrimaryDescription=Vacuum&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=One can use the diffeo r -> -r and u -> -u to make the assumption r > 0&comma;The case _m = 0 is Stephani, [28, 16,1].&comma;AlternativeOrthonormalTetrad1 and AlternativeNullTetrad1 are adapted to the shear-free null geodesic congruence (Robinson-Trautman tetrads)

____________________________________________________________

28&comma;37&comma;1=Authors=Robinson-Trautman (1962)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=RobinsonTrautman&comma;Comments=diff(2*P^2*diff(2*P(u,z,zb)^2*ln(P(u,z,zb)),z,zb),z,zb)+12*_m(u)*diff(ln(P(u,z,zb)),u)-4*diff(_m(u),u)=4*_kappa0*P^2*_h(u,z,zb)*_hb(u,z,zb), _Q(u,z,zb)*diff(_Qb(u,z,zb),u)-_Qb(u,z,zb)*diff(_Q(u,z,zb),u)=2*_P(u,z,zb)^2*(_hb(u,z,zb)*diff(_Qb(u,z,zb),zb)-_h(u,z,zb)*diff(_Q(u,z,zb),b)), diff(_h(u,z,zb),z)=diff(_Qb(u,z,zb)/2/_P(u,z,zb)^2,u), diff(_m(u,z,zb),z)=_kappa0*_hb(u,z,zb)*_Qb(u,z,zb)

____________________________________________________________

28&comma;43&comma;1=Authors=Robinson, Trautman (1962)&comma;PrimaryDescription=EinsteinMaxwell&comma;SecondaryDescription=PureRadiation&comma;RobinsonTrautman&comma;Comments=h1(u) is the conjugate of h(u)

____________________________________________________________

31&comma;40&comma;1=Authors= Petrov (1962)&comma;PrimaryDescription=Vacuum&comma;SecondaryDescription=Non-diverging

____________________________________________________________

35&comma;19&comma;1=Authors=Ehlers and Kundt 1962&comma;PrimaryDescription=Generic&comma;SecondaryDescription=PPWave&comma;Comments=constant null bivector, _k[mu]=-D_[mu](u)

____________________________________________________________

38&comma;2&comma;1=Authors=Kaigorodov (1962)&comma;PrimaryDescription=Hypersurface-Homogeneous&comma;SecondaryDescription=AlgebraicallySpecial

____________________________________________________________

38&comma;3&comma;1=Authors=Kaigorodov (1962)&comma;PrimaryDescription=Hypersurface-Homogeneous&comma;SecondaryDescription=AlgebraicallySpecial&comma;Comments=Note, there is a typo (the x^4 term) in this metric in Stephani which we were not able to verify from the original. This metric may not be correct!

(25)

 

The search can also be done visually, by properties. The following is the only solution in the database that is a Pure Radiation solution, of Petrov Type "D", Plebanski-Petrov Type "O" and that has Isometry Dimension equal to 1:

DifferentialGeometry:-Library:-MetricSearch

 

Set the solution, and everything related to work with it, in one go

g_28&comma;74&comma;1

Systems of spacetime Coordinates are: X=u&comma;η&comma;r&comma;y

Default differentiation variables for d_, D_ and dAlembertian are: X=u&comma;η&comma;r&comma;y

The Frolov and Khlebnikov (1975) metric in coordinates u&comma;η&comma;r&comma;y

Parameters: &kappa;0&comma;mu&comma;b&comma;d

Comments: Wⅈth m(u) = constant, th&ExponentialE; m&ExponentialE;trⅈc ⅈs Rⅈccⅈ flat an&DifferentialD; b&ExponentialE;com&ExponentialE;s 28.24 ⅈn St&ExponentialE;phanⅈ.

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

 

The related Riemann invariants:

Riemanninvariants

r0=0,r1=0,r2=0,r3=0,w1=6mu2r6,w2=6mu3r9,m1=0,m2=0,m3=0,m4=0,m5=0

(26)

To list all the triads of numbers associated to the solutions digitized in the database (each triad indicates the Chapter and equation number with which each of these metrics appear in the book), enter

DifferentialGeometry:-Library:-RetrieveStephani&comma;1

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