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Non dimensionalisation is a vary common task, and I was suprised that I couldn't find a maple tool to automate it . Has anyone developed their own package for it?

I want to automatically do it to the system equations for some Dynamical systems to make some of the other processing I do with them easier.

I was hoping to start with somehting in the form of 

Diff(x[1],t)=f[1](p[1]....p[n],x[1]...x[m])

...

Diff(x[m],t)=f[m](p[1]....p[n],x[1]...x[m])

where each f[i] is some kind of quotient of multivariate polynomials in the variables and parameters:
and end up with something like

Diff(y[1],s)=f[1](q[1]....q[p],y[1]...y[m])

...

Diff(y[m],s)=f[m](q[1]....q[p],y[1]...y[m])

where p<n

Hi,

I'm trying to work out whether or not Maple will be able to help me out with some algebra involving lots of indices, but I can't seem to work out how the gauge indices features work in the Physics package. For example I would like to define a gauge field carrying an SU(3) index and a spacetime index. The SU(3) index should run from 1 to 8 and the spacetime index from 1 to 5. I think I have worked out how it works with spacetime indices but I can't seem to find any documentation on the gauge indices.

Is there are a way to set the "dimension" of the gauge index, like we can set the space time dimension?

 

Cheers


The year 2015 has been one with interesting and relevant developments in the MathematicalFunctions  and FunctionAdvisor projects.

• 

Gaps were filled regarding mathematical formulas, with more identities for all of BesselI, BesselK, BesselY, ChebyshevT, ChebyshevU, Chi, Ci, FresnelC, FresnelS, GAMMA(z), HankelH1, HankelH2, InverseJacobiAM, the twelve InverseJacobiPQ for P, Q in [C,D,N,S], KelvinBei, KelvinBer, KelvinKei, KelvinKer, LerchPhi, arcsin, arcsinh, arctan, ln;

• 

Developments happened in the Mathematical function package, to both compute with symbolic sequences and symbolic nth order derivatives of algebraic expressions and functions;

• 

The input FunctionAdvisor(differentiate_rule, mathematical_function) now returns both the first derivative (old behavior) and the nth symbolic derivative (new behavior) of a mathematical function;

• 

A new topic, plot, used as FunctionAdvisor(plot, mathematical_function), now returns 2D and 3D plots for each mathematical function, following the NIST Digital Library of Mathematical Functions;

• 

The previously existing FunctionAdvisor(display, mathematical_function) got redesigned, so that it now displays more information about any mathematical function, and organized into a Section with subsections for each of the different topics, making it simpler to find the information one needs without getting distracted by a myriad of formulas that are not related to what one is looking for.

More mathematics

 

More mathematical knowledge is in place, more identities, differentiation rules of special functions with respect to their parameters, differentiation of functions whose arguments involve symbolic sequences with an indeterminate number of operands, and sum representations for special functions under different conditions on the functions' parameters.

Examples

   

More powerful symbolic differentiation (nth order derivative)

 

Significative developments happened in the computation of the nth order derivative of mathematical functions and algebraic expressions involving them.

Examples

   

Mathematical handling of symbolic sequences

 

Symbolic sequences enter various formulations in mathematics. Their computerized mathematical handling, however, was never implemented - only a representation for them existed in the Maple system. In connection with this, a new subpackage, Sequences , within the MathematicalFunctions package, has been developed.

Examples

   

Visualization of mathematical functions

 

When working with mathematical functions, it is frequently desired to have a rapid glimpse of the shape of the function for some sampled values of their parameters. Following the NIST Digital Library of Mathematical Functions, a new option, plot, has now been implemented.

Examples

   

Section and subsections displaying properties of mathematical functions

 

Until recently, the display of a whole set of mathematical information regarding a function was somehow cumbersome, appearing all together on the screen. That display was and is still available via entering, for instance for the sin function, FunctionAdvisor(sin) . That returns a table of information that can be used programmatically.

With time however, the FunctionAdvisor evolved into a consultation tool, where a better organization of the information being displayed is required, making it simpler to find the information we need without being distracted by a screen full of complicated formulas.

To address this requirement, the FunctionAdvisor now returns the information organized into a Section with subsections, built using the DocumentTools package. This enhances the presentation significantly.

Examples

   

These developments can be installed in Maple 2015 as usual, by downloading the updates (bundled with the Physics and Differential Equations updates) from the Maplesoft R&D webpage for Mathematical Functions and Differential Equations


Download MathematicalFunctionsAndFunctionAdvisor.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

I am tryng to write and solve within Maple the equations of movement for a single body subject to a gravitatinal field, F_=-GMm/r2

But I get an error message when trying to define the Angular Momentum which doesn't make sense for me.

Thank you for any help on this topíc.


restart; with(Physics[Vectors]); conventions, Setup(mathematicalnotation = true);
conventions, [mathematicalnotation = true]
r_ := rho*_rho;
r_ := rho _rho
_rho(t);
_rho(t)
rho(t);
rho(t)
v_ := diff(rho(t)*_rho(t), t);
/ d \ / d \
v_ := |--- rho(t)| _rho(t) + rho(t) |--- phi(t)| _phi(t)
\ dt / \ dt /
a_ := diff(%, t);
/ 2 \
| d | / d \ / d \
a_ := |---- rho(t)| _rho(t) + 2 |--- rho(t)| |--- phi(t)| _phi(t)
| 2 | \ dt / \ dt /
\ dt /

/ 2 \ 2
| d | / d \
+ rho(t) |---- phi(t)| _phi(t) - rho(t) |--- phi(t)| _rho(t)
| 2 | \ dt /
\ dt /

eq[1] := -G*M*_rho(t)/r^2-a_ = 0;
/ / 2 \ 2\
| G M | d | / d \ |
eq[1] := _rho(t) |- --- - |---- rho(t)| + rho(t) |--- phi(t)| |
| 2 | 2 | \ dt / |
\ r \ dt / /

/ / 2 \
| / d \ / d \ | d |
+ _phi(t) |-2 |--- rho(t)| |--- phi(t)| - rho(t) |---- phi(t)|
| \ dt / \ dt / | 2 |
\ \ dt /

\
|
| = 0
|
/
Eq[1, 2] := seq(Component(lhs(eq[1]), n) = 0, n = 1 .. 2);
/ 2 \ 2
G M | d | / d \
Eq[1, 2] := - --- - |---- rho(t)| + rho(t) |--- phi(t)| = 0,
2 | 2 | \ dt /
r \ dt /

/ 2 \
/ d \ / d \ | d |
-2 |--- rho(t)| |--- phi(t)| - rho(t) |---- phi(t)| = 0
\ dt / \ dt / | 2 |
\ dt /
NULL;

L_ := `&x`(r_, m*v_);
Error, (in Physics:-Vectors:-&x) found the unit vector _rho present also as a function _rho(t); either one form or the other - not both - can be present in an algebraic expression

 

http://www.maplesoft.com/support/help/Maple/view.aspx?path=Physics/.

i see bra and ket expression are so beautiful,

however,

how do real valued eigenvectors involve in calculation of bra and ket style computation?

m1 := <Old_Asso_eigenvector[2][1][1],Old_Asso_eigenvector[2][1][2],Old_Asso_eigenvector[2][1][3]>;
m2 := <Old_Asso_eigenvector[2][2][1],Old_Asso_eigenvector[2][2][2],Old_Asso_eigenvector[2][2][3]>;
m3 := <Old_Asso_eigenvector[2][3][1],Old_Asso_eigenvector[2][3][2],Old_Asso_eigenvector[2][3][3]>;

m1 := <Old_Asso_eigenvector[2][1][1],Old_Asso_eigenvector[2][2][1],Old_Asso_eigenvector[2][3][1]>;
m2 := <Old_Asso_eigenvector[2][1][2],Old_Asso_eigenvector[2][2][2],Old_Asso_eigenvector[2][3][2]>;
m3 := <Old_Asso_eigenvector[2][1][3],Old_Asso_eigenvector[2][2][3],Old_Asso_eigenvector[2][3][3]>;
ord := GramSchmidt([m1, m2, m3]);
ord := Basis([m1, m2, m3]);

ord[1].ord[2];  # expect = 1
ord[1].ord[3];  # expect = 1
ord[2].ord[1];  # expect = 1
ord[2].ord[3];  # expect = 1
ord[3].ord[1];  # expect = 1
ord[3].ord[2];  # expect = 1

is there a function get orthonormal basis ?

On the Physics Research and development page there are only 3 versions of the Physics package available.  What happened to the earlier final updates for Maple 16, 15 etc...

What is the latest one that can work on Maple12?  Was there one for M12?  What is the earliest versions of Maple the Research & Development Physics packages can work on?

Need help for manipulating tensor with the physics package.

I ask some questions about this.  But each time, I am refer to the help pages.  If I ask again some help, it is because I can't not start with the information on the help file.  It is written for people that already know General Relativity (GR).

 

So this time, I have created a document (attach to this post) where I ask specific queations on manipulations.  My goal is to ccrreate a document that I will put on the Applications Center.  I promess that those who will help me on this will be cited in the document.  This way, I hope to create an introduction on how to use tensors for beginner like me.

 

Then, with this help, I am sure I will be able to better understand the help page of the packages.  I am doing this as someone who is starting to learn GR and have to be able to better understand the manipulations of tensor and getting the grasp of teh meaning of all those tensor.  For exemple, the concept of parallel transport on a curve surface.

 

Thank you in advance for all the troubling I give you with this demand.

 

Regards,Parallel_Transport.mw

 

 

 

 

 

 

 

--------------------------------------
Mario Lemelin
Maple 2015 Ubuntu 14.04 - 64 bits
Maple 2015 Win 7 - 64 bits messagerie : mario.lemelin@cgocable.ca téléphone :  (819) 376-0987

I ran into a problem with the physics package that I subsequently solved. But I am wondering whether this would be a candidate for an SCR and/or be considered a bug.

The calculation I am trying is actually (so far) very simple.

I define a Hamiltonian H:

H := sqrt(p_^2*c^2+m^2*c^4); # note the square of vector p_

p_:=p1*_i+p2*_j+p3*_k;

H;

So far so good. Now I want to take the differentials of H against the components of p:

diff(H,p1) assuming c::real;

Hmm... I am not sure why the p2 and p3 still show up; but then, the product between the unit vectors should be 0 for different ones and one for equal unit vectors so maybe this is ok.

But H behaves weird: I can simplify it:

but if I try to do anything with it, it barfs:

dH+0;

Error, (in Physics:-Vectors:-+) invalid operation * between vectors _i and _j

As it turns out the issue is the square of the vector p_. Maple (or rather, Physics) does not recognize that it needs to expand p_^2 as p_.p_ and seems to treat is like p_*p_.

I would like Edgardo---& others more experienced with the Physics package than I am---to look at this. I do not understand the Physics package well enough to judge whether overloading the exponentiation operator to make this work is the right thing.

The example works once I replace p_^2 by p_.p_. But the ^2 notation is fairly standard usage so it feels slightly awkward.

Thanks,

Mac Dude.

Derivation_of_H.mw

Hello, I am working on an elektromagnetism assignment, and have hit a slight problem with the Physics[Vectors] package. The cartesian unit vectors are set to i, j and k. but at my school we have always used x, y and z for both the cartesian unit vectors and the cartesian coordinates. Is there a way to change this so that i am able to use _x, _y and _z istead?

 

P.S. This is my first post so sorry if it is badly done.

in harmonic ocillator version, why it has error after it dsolve?

with(Physics[Vectors]):
with(SumTools):
Setup(mathematicalnotation = true):

Qs is Qss*cos(m*t+phi);

H := 1/(8*Pi*c^2)*Summation(Commutator(diff(Qs*cos(w*t+phi),t)*diff(conjugate(Qs*cos(w*t+phi)),t)+w^2*Qs*cos(w*t+phi)*conjugate(Qs*cos(w*t+phi))),ks=1..3)-1/c*(Summation(conjugate(Qs*cos(w*t+phi)),ks=1..3));

or

H := 1/(8*Pi*c^2)*Summation(Commutator(diff(Qs,t)*diff(conjugate(Qs),t)+w^2*Qs*conjugate(Qs)),ks=1..3);

#Qs = p, Ps := Pss*sin(m*t+phi) Ps = qs

i use

H := 1/(8*Pi*c^2)*Summation(Commutator(diff(Qs*cos(w*t+phi),t)*diff(conjugate(Qs*cos(w*t+phi)),t)+w^2*Qs*cos(w*t+phi)*conjugate(Qs*cos(w*t+phi))),ks=1..3)-1/c*(Summation(conjugate(Qs*cos(w*t+phi)),ks=1..3));

 HJ := subs(Qs= diff(f(q,P,t), q), H);

H:=subs( f(q,P,t) = f1(q) + f2(t), HJ);
sol:=dsolve({rhs(H)=E,lhs(H)=E});
S:=rhs(sol[1][1]+sol[1][2]);
p:=diff(S,q);
Q:=diff(S,E);

H := 1/(2*m)*diff(f(q,P,t), q)^2 = -diff(f(q,P,t),t);
assume f = f1(q) + f2(t);
H := 1/(2*m)*diff(f1(q), q)^2 = -diff(f2(t),t);
dsolve(H);

set each side of equation be a constant E which is anticipated integration constant
how to solve to S = q*sqrt(2*m*E) - E*t

and

p = diff(S,q) = sqrt(2*m*E);
Q = diff(S,E) = q*m/sqrt(2*m*E) - t;


Exact solutions to Einstein’s equations” is one of those books that are difficult even to imagine: the authors reviewed more than 4,000 papers containing solutions to Einstein’s equations in the general relativity literature, collecting, classifying, discarding repetitions in disguise, and organizing the whole material into chapters according to the physical properties of these solutions. The book is already in its second edition and it is a monumental piece of work.

 

As good as it is, however, the project resulted only in printed material, a textbook constituted of paper and ink. In 2006, when the DifferentialGeometry package was rewritten to enter the Maple library, one of the first things that passed through our minds was to bring the whole of “Exact solutions to Einstein’s equations” into Maple.

 

It took some time to start but in 2010, for Maple 14, we featured the first 26 solutions from this book. In Maple 15 this number jumped to 61. For Maple 17 we decided to emphasize the general relativity functionality of the DifferentialGeometry package, and Maple 18 added 50 more, featuring in total 225 of these solutions - great! but still far from the whole thing …

 

And this is when we decided to “step on the gas” - go for it, the whole book. One year later, working in collaboration with Denitsa Staicova from Bulgarian Academy of Sciences, Maple 2015 appeared with 330 solutions to Einstein’s equations. Today we have already implemented 492 solutions, and for the first time we can see the end of the tunnel: we are targeting finishing the whole book by the end of this year.

 

Wow2! This is a terrific result. First, because these solutions are key in the area of general relativity, and at this point what we have in Maple is already the most thorough digitized database of solutions to Einstein’s equations in the world. Second, and not any less important, because within Maple this knowledge comes alive. The solutions are fully searcheable and are set by a simple call to the Physics:-g_  spacetime metric command, and that automatically sets the related coordinates, Christoffel symbols , Ricci  and Riemann  tensors, orthonormal and null tetrads , etc. All of this happens on the fly, and all the mathematics within the Maple library are ready to work with these solutions. Having everything come alive completely changes the game. The ability to search the database according to the physical properties of the solutions, their classification, or just by parts of keywords also makes the whole book concretely more useful.

 

And, not only are these solutions to Einstein’s equations brought to life in a full-featured way through the Physics  package: they can also be reached through the DifferentialGeometry:-Library:-MetricSearch  applet. Almost all of the mathematical operations one can perform on them are also implemented as commands in DifferentialGeometry .

 

Finally, in the Maple PDEtools package , we already have all the mathematical tools to start resolving the equivalence problem around these solutions. That is: to answer whether a new solution is or not new, or whether it can be obtained from an existing solution by transformations of coordinates of different kinds. And we are going for it.

 

What follows is a basic illustration of what has already been implemented. As usual, in order to reproduce these results, you need to update your Physics library from the Maplesoft R&D Physics webpage.

 

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

with(Physics)

g_[sc]

`Systems of spacetime Coordinates are: `*{X = (r, theta, phi, t)}

 

`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (r, theta, phi, t)}

 

`The Schwarzschild metric in coordinates `[r, theta, phi, t]

 

`Parameters: `[m]

 

g[mu, nu] = (Matrix(4, 4, {(1, 1) = r/(-r+2*m), (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -r^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -r^2*sin(theta)^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = (r-2*m)/r}))

(1)

And that is all we do :) Although the strength in Physics  is to compute with tensors using indicial notation, all of the tensor components and related properties of this metric are also derived on the fly (and no, they are not in any database). For instance these are the definition in terms of Christoffel symbols , and the covariant components of the Ricci tensor

Ricci[definition]

Physics:-Ricci[mu, nu] = Physics:-d_[alpha](Physics:-Christoffel[`~alpha`, mu, nu], [X])-Physics:-d_[nu](Physics:-Christoffel[`~alpha`, mu, alpha], [X])+Physics:-Christoffel[`~beta`, mu, nu]*Physics:-Christoffel[`~alpha`, beta, alpha]-Physics:-Christoffel[`~beta`, mu, alpha]*Physics:-Christoffel[`~alpha`, nu, beta]

(2)

Ricci[]

Physics:-Ricci[mu, nu] = Matrix(%id = 18446744078179871670)

(3)

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

Riemann[invariants]

r[0] = 0, r[1] = 0, r[2] = 0, r[3] = 0, w[1] = 6*m^2/r^6, w[2] = 6*m^3/r^9, m[1] = 0, m[2] = 0, m[3] = 0, m[4] = 0, m[5] = 0

(4)

The related Weyl scalars  in the context of the Newman-Penrose formalism

Weyl[scalars]

psi__0 = 0, psi__1 = 0, psi__2 = -m/r^3, psi__3 = 0, psi__4 = 0

(5)

 

These are the 2x2 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

"Christoffel[~1,alpha,beta,matrix]"

Physics:-Christoffel[`~1`, alpha, beta] = Matrix(%id = 18446744078160684686)

(6)

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

Christoffel[1, alpha, beta, matrix]

Physics:-Christoffel[1, alpha, beta] = Matrix(%id = 18446744078160680590)

(7)

One could query the database, directly from the spacetime metrics, 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"]]

(8)

These solutions can be set in one go from the metrics command, just by indicating the number with which it appears in "Exact Solutions to Einstein's Equations"

g_[[12, 16, 1]]

`Systems of spacetime Coordinates are: `*{X = (t, x, theta, phi)}

 

`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (t, x, theta, phi)}

 

`The Bertotti (1959), Kramer (1978), Levi-Civita (1917), Robinson (1959) metric in coordinates `[t, x, theta, phi]

 

`Parameters: `[k, kappa0, beta]

 

g[mu, nu] = (Matrix(4, 4, {(1, 1) = -k^2*sinh(x)^2, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = k^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = k^2, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = k^2*sin(theta)^2}))

(9)

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

"Ricci[~]"

Physics:-Ricci[`~mu`, `~nu`] = Matrix(%id = 18446744078179869750)

(10)

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

with(Tetrads)

`Setting lowercaselatin letters to represent tetrad indices `

 

0, "%1 is not a command in the %2 package", Tetrads, Physics

 

0, "%1 is not a command in the %2 package", Tetrads, Physics

 

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

(11)

This is the tetrad related to the book's metric with number 12.16.1

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078160685286)

(12)

One can check these directly; for instance this is the definition of the tetrad, where the right-hand side is the tetrad metric

e_[definition]

Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b]

(13)

This shows that, for the components given by (12), the definition holds

TensorArray(Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b])

Matrix(%id = 18446744078195401422)

(14)

One frequently works with a different signature and null tetrads; set that, and everything gets automatically recomputed for the metric 12.16.1 accordingly

Setup(signature = "+---", tetradmetric = null)

[signature = `+ - - -`, tetradmetric = {(1, 2) = 1, (3, 4) = -1}]

(15)

eta_[]

eta[a, b] = (Matrix(4, 4, {(1, 1) = 0, (1, 2) = 1, (1, 3) = 0, (1, 4) = 0, (2, 1) = 1, (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0, (3, 4) = -1, (4, 1) = 0, (4, 2) = 0, (4, 3) = -1, (4, 4) = 0}))

(16)

e_[]

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078191417574)

(17)

TensorArray(Physics:-Tetrads:-e_[a, mu]*Physics:-Tetrads:-e_[b, `~mu`] = Physics:-Tetrads:-eta_[a, b])

Matrix(%id = 18446744078191319390)

(18)

The related 16 Riemann invariant

Riemann[invariants]

r[0] = 0, r[1] = 1/k^4, r[2] = 0, r[3] = (1/4)/k^8, w[1] = 0, w[2] = 0, m[1] = 0, m[2] = 0, m[3] = 0, m[4] = 0, m[5] = 0

(19)

The ability to query rapidly, set things in one go, change everything again etc. are at this point fantastic. For instance, these are the metrics by Kaigorodov; next are those published in 1962

g_[Kaigorodov]

____________________________________________________________

 

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

 

____________________________________________________________

 

[12, 35, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous", "SimpleTransitive"]]

(20)

g_[`1962`]

____________________________________________________________

 

[12, 13, 1] = ["Authors" = ["Ozsvath, Schucking (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["Homogeneous", "PlaneWave"], "Comments" = ["geodesically complete, no curvature singularities"]]

 

____________________________________________________________

 

[12, 14, 1] = ["Authors" = ["Petrov (1962)"], "PrimaryDescription" = "Vacuum", "SecondaryDescription" = ["Homogeneous", "SimpleTransitive"]]

 

____________________________________________________________

 

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

 

____________________________________________________________

 

[12, 35, 1] = ["Authors" = ["Kaigorodov (1962)", "Cahen (1964)", "Siklos (1981)", "Ozsvath (1987)"], "PrimaryDescription" = "Einstein", "SecondaryDescription" = ["Homogeneous", "SimpleTransitive"]]

 

____________________________________________________________

 

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

 

____________________________________________________________

 

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

 

____________________________________________________________

 

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

 

____________________________________________________________

 

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

 

____________________________________________________________

 

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

(21)

 

The search can be done visually, by properties; this is the only solution in the database that is a Pure Ratiation 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, 74, 1]]

`Systems of spacetime Coordinates are: `*{X = (u, eta, r, y)}

 

`Default differentiation variables for d_, D_ and dAlembertian are: `*{X = (u, eta, r, y)}

 

`The Frolov and Khlebnikov (1975) metric in coordinates `[u, eta, r, y]

 

`Parameters: `[kappa0, m(u), b, d]

 

"`Comments: `With _m(u) = constant, the metric is Ricci flat and becomes 28.24 in Stephani."

 

g[mu, nu] = (Matrix(4, 4, {(1, 1) = (2*m(u)^3-6*m(u)^2*eta*r-r^2*(-6*eta^2+b)*m(u)+r^3*(-2*eta^3+b*eta+d))/(r*m(u)^2), (1, 2) = -r^2/m(u), (1, 3) = -1, (1, 4) = 0, (2, 1) = -r^2/m(u), (2, 2) = r^2/(-2*eta^3+b*eta+d), (2, 3) = 0, (2, 4) = 0, (3, 1) = -1, (3, 2) = 0, (3, 3) = 0, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = r^2*(-2*eta^3+b*eta+d)}))

(22)

 

The related Riemann invariants:

Riemann[invariants]

r[0] = 0, r[1] = 0, r[2] = 0, r[3] = 0, w[1] = 6*m(u)^2/r^6, w[2] = -6*m(u)^3/r^9, m[1] = 0, m[2] = 0, m[3] = 0, m[4] = 0, m[5] = 0

(23)

To conclude, how many solutions from the book have we already implemented?

DifferentialGeometry:-Library:-Retrieve("Stephani", 1)

[[8, 33, 1], [8, 34, 1], [12, 6, 1], [12, 7, 1], [12, 8, 1], [12, 8, 2], [12, 8, 3], [12, 8, 4], [12, 8, 5], [12, 8, 6], [12, 8, 7], [12, 8, 8], [12, 9, 1], [12, 9, 2], [12, 9, 3], [12, 12, 1], [12, 12, 2], [12, 12, 3], [12, 12, 4], [12, 13, 1], [12, 14, 1], [12, 16, 1], [12, 18, 1], [12, 19, 1], [12, 21, 1], [12, 23, 1], [12, 23, 2], [12, 23, 3], [12, 24.1, 1], [12, 24.2, 1], [12, 24.3, 1], [12, 26, 1], [12, 27, 1], [12, 28, 1], [12, 29, 1], [12, 30, 1], [12, 31, 1], [12, 32, 1], [12, 34, 1], [12, 35, 1], [12, 36, 1], [12, 37, 1], [12, 37, 2], [12, 37, 3], [12, 37, 4], [12, 37, 5], [12, 37, 6], [12, 37, 7], [12, 37, 8], [12, 37, 9], [12, 38, 1], [12, 38, 2], [12, 38, 3], [12, 38, 4], [12, 38, 5], [13, 2, 1], [13, 2, 2], [13, 2, 3], [13, 7, 1], [13, 7, 2], [13, 7, 3], [13, 7, 4], [13, 7, 5], [13, 7, 6], [13, 7, 7], [13, 7, 8], [13, 14, 1], [13, 14, 2], [13, 14, 3], [13, 19, 1], [13, 31, 1], [13, 32, 1], [13, 46, 1], [13, 48, 1], [13, 49, 1], [13, 49, 2], [13, 51, 1], [13, 53, 1], [13, 59, 1], [13, 59, 2], [13, 60, 1], [13, 60, 2], [13, 60, 3], [13, 60, 4], [13, 60, 5], [13, 60, 6], [13, 60, 7], [13, 60, 8], [13, 61, 1], [13, 61, 2], [13, 62, 1], [13, 62, 2], [13, 62, 4], [13, 62, 6], [13, 63, 1], [13, 63, 2], [13, 63, 3], [13, 63, 4], [13, 64, 1], [13, 64, 2], [13, 64, 3], [13, 64, 4], [13, 65, 1], [13, 69, 1], [13, 71, 1], [13, 72, 1], [13, 73, 1], [13, 74, 1], [13, 74, 2], [13, 74, 3], [13, 76, 1], [13, 77, 1], [13, 77, 2], [13, 79, 1], [13, 79, 2], [13, 80, 1], [13, 81, 1], [13, 83, 1], [13, 84, 1], [13, 84, 2], [13, 84, 3], [13, 85, 1], [13, 85, 2], [13, 86, 1], [13, 87, 1], [14, 6.1, 1], [14, 6.2, 1], [14, 6.3, 1], [14, 7, 1], [14, 8.1, 1], [14, 8.2, 1], [14, 8.3, 1], [14, 9.1, 1], [14, 9.2, 1], [14, 10, 1], [14, 10, 2], [14, 12, 1], [14, 12, 2], [14, 12, 3], [14, 14, 1], [14, 14, 2], [14, 15, 1], [14, 15.1, 2], [14, 15.2, 2], [14, 15.3, 2], [14, 16, 1], [14, 16, 2], [14, 17, 1], [14, 18, 1], [14, 18, 2], [14, 19, 1], [14, 20, 1], [14, 21, 1], [14, 21, 2], [14, 21, 3], [14, 22, 1], [14, 23, 1], [14, 24, 1], [14, 25, 1], [14, 26, 1], [14, 26, 2], [14, 26, 3], [14, 26, 4], [14, 27, 1], [14, 28, 1], [14, 28, 2], [14, 28, 3], [14, 29, 1], [14, 30, 1], [14, 31, 1], [14, 32, 1], [14, 33, 1], [14, 35, 1], [14, 37, 1], [14, 38, 1], [14, 38, 2], [14, 38, 3], [14, 39, 1], [14, 39, 2], [14, 39, 3], [14, 39, 4], [14, 39, 5], [14, 39, 6], [14, 40, 1], [14, 41, 1], [14, 42, 1], [14, 46, 1], [15, 3, 1], [15, 3, 2], [15, 4, 1], [15, 4, 2], [15, 4, 3], [15, 9, 1], [15, 10, 1], [15, 12, 1], [15, 12, 2], [15, 12, 3], [15, 12, 4], [15, 12, 5], [15, 12, 6], [15, 17, 1], [15, 17, 2], [15, 17, 3], [15, 17, 4], [15, 18, 1], [15, 19, 1], [15, 19, 2], [15, 20, 1], [15, 21, 1], [15, 21, 2], [15, 22, 1], [15, 23, 1], [15, 23, 2], [15, 24, 1], [15, 24, 2], [15, 25, 1], [15, 25, 2], [15, 26, 1], [15, 26, 2], [15, 27, 1], [15, 27, 2], [15, 27, 3], [15, 27, 4], [15, 27, 5], [15, 27, 6], [15, 27, 7], [15, 27, 8], [15, 28, 1], [15, 29, 1], [15, 30, 1], [15, 31, 1], [15, 32, 1], [15, 34, 1], [15, 34, 2], [15, 34, 3], [15, 43, 1], [15, 43, 2], [15, 43, 3], [15, 50, 1], [15, 50, 2], [15, 50, 3], [15, 50, 4], [15, 50, 5], [15, 50, 6], [15, 62, 1], [15, 62, 2], [15, 62, 3], [15, 63, 1], [15, 63, 2], [15, 63, 3], [15, 65, 1], [15, 65, 2], [15, 66, 1], [15, 66, 2], [15, 66, 3], [15, 75, 1], [15, 75, 2], [15, 75, 3], [15, 77, 1], [15, 77, 2], [15, 77, 3], [15, 78, 1], [15, 79, 1], [15, 81, 1], [15, 81, 2], [15, 81, 3], [15, 82, 1], [15, 82, 2], [15, 82, 3], [15, 83, 1.1], [15, 83, 1.2], [15, 83, 2], [15, 83, 3.1], [15, 83, 3.2], [15, 83, 4], [15, 84, 1], [15, 85, 1], [15, 85, 2], [15, 85, 3], [15, 86, 1], [15, 86, 2], [15, 86, 3], [15, 87, 1], [15, 87, 2], [15, 87, 3], [15, 87, 4], [15, 87, 5], [15, 88, 1], [15, 89, 1], [15, 90, 1], [16, 1, 1], [16, 1, 2], [16, 1, 3], [16, 1, 4], [16, 1, 5], [16, 1, 6], [16, 1, 7], [16, 1, 8], [16, 1, 9], [16, 1, 10], [16, 1, 11], [16, 1, 12], [16, 1, 13], [16, 1, 14], [16, 1, 15], [16, 1, 16], [16, 1, 17], [16, 1, 18], [16, 1, 19], [16, 1, 20], [16, 1, 21], [16, 1, 22], [16, 1, 23], [16, 1, 24], [16, 1, 25], [16, 1, 26], [16, 1, 27], [16, 14, 1], [16, 14, 2], [16, 14, 3], [16, 14, 4], [16, 14, 5], [16, 14, 6], [16, 14, 7], [16, 14, 8], [16, 14, 9], [16, 14, 10], [16, 14, 11], [16, 14, 12], [16, 14, 13], [16, 14, 14], [16, 14, 15], [16, 14, 16], [16, 14, 17], [16, 14, 18], [16, 14, 19], [16, 14, 20], [16, 18, 1], [16, 19, 1], [16, 20, 1], [16, 22, 1], [16, 24, 1], [16, 24, 2], [16, 43, 1], [16, 45, 1], [16, 45, 2], [16, 46, 1], [16, 47, 1], [16, 50, 1], [16, 51, 1], [16, 54, 1], [16, 61, 1], [16, 63, 1], [16, 66, 1], [16, 66, 2], [16, 66, 3], [16, 67, 1], [16, 71, 1], [16, 72, 1], [16, 73, 1], [16, 74, 1], [16, 75, 1], [16, 76, 1], [16, 77, 1], [16, 77, 2], [16, 77, 3], [16, 78, 1], [17, 4, 1], [17, 4, 2], [17, 5, 1], [17, 9, 1], [17, 14, 1], [17, 15, 1], [17, 15, 2], [17, 16, 1], [17, 20, 1], [17, 22, 1], [17, 23, 1], [17, 24, 1], [17, 24, 2], [17, 26, 1], [17, 27, 1], [17, 27, 2], [17, 28, 1], [17, 28, 2], [17, 29, 1], [17, 29, 2], [17, 30, 1], [17, 31, 1], [18, 2, 1], [18, 2, 2], [18, 2, 3], [18, 2, 4], [18, 2, 5], [18, 2, 6], [18, 2, 7], [18, 2, 8], [18, 48, 1], [18, 48, 2], [18, 49, 1], [18, 50, 1], [18, 64, 1], [18, 64, 2], [18, 64, 3], [18, 65, 1], [18, 66, 1], [18, 67, 1], [18, 71, 1], [18, 75, 1], [19, 17, 1], [19, 17, 2], [19, 21, 1], [20, 3, 1], [20, 4, 1], [20, 5, 1], [20, 7, 1], [20, 8, 1], [20, 9, 1], [20, 10, 1], [20, 11, 1], [20, 12, 1], [20, 13, 1], [20, 15, 1], [20, 16, 1], [20, 17, 1], [20, 20, 1], [20, 21, 1], [20, 23, 1], [20, 27, 1], [20, 28, 1], [20, 29, 1], [20, 32, 1], [20, 34, 1], [20, 36, 1], [20, 38, 1], [20, 38, 2], [20, 38, 3], [20, 44, 1], [20, 46, 1], [20, 54, 1], [20, 57, 1], [20, 57, 2], [28, 16, 1], [28, 17, 1], [28, 21, 1], [28, 21, 2], [28, 21, 3], [28, 21, 4], [28, 21, 5], [28, 21, 6], [28, 21, 7], [28, 24, 1], [28, 25, 1], [28, 26, 1], [28, 26, 2], [28, 26, 3], [28, 41, 1], [28, 43, 1], [28, 44, 1], [28, 44, 2], [28, 44, 3], [28, 44, 4], [28, 44, 5], [28, 44, 6], [28, 45, 1], [28, 45, 2], [28, 46, 1], [28, 46, 2], [28, 53, 1], [28, 53, 2], [28, 55, 1], [28, 55, 2], [28, 56.1, 1], [28, 56.2, 2], [28, 56.2, 3], [28, 56.3, 1], [28, 56.4, 1], [28, 56.5, 1], [28, 56.6, 1], [28, 58.2, 1], [28, 58.3, 1], [28, 58.3, 2], [28, 58.4, 1], [28, 60, 1], [28, 61, 1], [28, 64, 1], [28, 66, 1], [28, 67, 1], [28, 68, 1], [28, 72, 1], [28, 73, 1], [28, 74, 1]]

(24)

nops([[8, 33, 1], [8, 34, 1], [12, 6, 1], [12, 7, 1], [12, 8, 1], [12, 8, 2], [12, 8, 3], [12, 8, 4], [12, 8, 5], [12, 8, 6], [12, 8, 7], [12, 8, 8], [12, 9, 1], [12, 9, 2], [12, 9, 3], [12, 12, 1], [12, 12, 2], [12, 12, 3], [12, 12, 4], [12, 13, 1], [12, 14, 1], [12, 16, 1], [12, 18, 1], [12, 19, 1], [12, 21, 1], [12, 23, 1], [12, 23, 2], [12, 23, 3], [12, 24.1, 1], [12, 24.2, 1], [12, 24.3, 1], [12, 26, 1], [12, 27, 1], [12, 28, 1], [12, 29, 1], [12, 30, 1], [12, 31, 1], [12, 32, 1], [12, 34, 1], [12, 35, 1], [12, 36, 1], [12, 37, 1], [12, 37, 2], [12, 37, 3], [12, 37, 4], [12, 37, 5], [12, 37, 6], [12, 37, 7], [12, 37, 8], [12, 37, 9], [12, 38, 1], [12, 38, 2], [12, 38, 3], [12, 38, 4], [12, 38, 5], [13, 2, 1], [13, 2, 2], [13, 2, 3], [13, 7, 1], [13, 7, 2], [13, 7, 3], [13, 7, 4], [13, 7, 5], [13, 7, 6], [13, 7, 7], [13, 7, 8], [13, 14, 1], [13, 14, 2], [13, 14, 3], [13, 19, 1], [13, 31, 1], [13, 32, 1], [13, 46, 1], [13, 48, 1], [13, 49, 1], [13, 49, 2], [13, 51, 1], [13, 53, 1], [13, 59, 1], [13, 59, 2], [13, 60, 1], [13, 60, 2], [13, 60, 3], [13, 60, 4], [13, 60, 5], [13, 60, 6], [13, 60, 7], [13, 60, 8], [13, 61, 1], [13, 61, 2], [13, 62, 1], [13, 62, 2], [13, 62, 4], [13, 62, 6], [13, 63, 1], [13, 63, 2], [13, 63, 3], [13, 63, 4], [13, 64, 1], [13, 64, 2], [13, 64, 3], [13, 64, 4], [13, 65, 1], [13, 69, 1], [13, 71, 1], [13, 72, 1], [13, 73, 1], [13, 74, 1], [13, 74, 2], [13, 74, 3], [13, 76, 1], [13, 77, 1], [13, 77, 2], [13, 79, 1], [13, 79, 2], [13, 80, 1], [13, 81, 1], [13, 83, 1], [13, 84, 1], [13, 84, 2], [13, 84, 3], [13, 85, 1], [13, 85, 2], [13, 86, 1], [13, 87, 1], [14, 6.1, 1], [14, 6.2, 1], [14, 6.3, 1], [14, 7, 1], [14, 8.1, 1], [14, 8.2, 1], [14, 8.3, 1], [14, 9.1, 1], [14, 9.2, 1], [14, 10, 1], [14, 10, 2], [14, 12, 1], [14, 12, 2], [14, 12, 3], [14, 14, 1], [14, 14, 2], [14, 15, 1], [14, 15.1, 2], [14, 15.2, 2], [14, 15.3, 2], [14, 16, 1], [14, 16, 2], [14, 17, 1], [14, 18, 1], [14, 18, 2], [14, 19, 1], [14, 20, 1], [14, 21, 1], [14, 21, 2], [14, 21, 3], [14, 22, 1], [14, 23, 1], [14, 24, 1], [14, 25, 1], [14, 26, 1], [14, 26, 2], [14, 26, 3], [14, 26, 4], [14, 27, 1], [14, 28, 1], [14, 28, 2], [14, 28, 3], [14, 29, 1], [14, 30, 1], [14, 31, 1], [14, 32, 1], [14, 33, 1], [14, 35, 1], [14, 37, 1], [14, 38, 1], [14, 38, 2], [14, 38, 3], [14, 39, 1], [14, 39, 2], [14, 39, 3], [14, 39, 4], [14, 39, 5], [14, 39, 6], [14, 40, 1], [14, 41, 1], [14, 42, 1], [14, 46, 1], [15, 3, 1], [15, 3, 2], [15, 4, 1], [15, 4, 2], [15, 4, 3], [15, 9, 1], [15, 10, 1], [15, 12, 1], [15, 12, 2], [15, 12, 3], [15, 12, 4], [15, 12, 5], [15, 12, 6], [15, 17, 1], [15, 17, 2], [15, 17, 3], [15, 17, 4], [15, 18, 1], [15, 19, 1], [15, 19, 2], [15, 20, 1], [15, 21, 1], [15, 21, 2], [15, 22, 1], [15, 23, 1], [15, 23, 2], [15, 24, 1], [15, 24, 2], [15, 25, 1], [15, 25, 2], [15, 26, 1], [15, 26, 2], [15, 27, 1], [15, 27, 2], [15, 27, 3], [15, 27, 4], [15, 27, 5], [15, 27, 6], [15, 27, 7], [15, 27, 8], [15, 28, 1], [15, 29, 1], [15, 30, 1], [15, 31, 1], [15, 32, 1], [15, 34, 1], [15, 34, 2], [15, 34, 3], [15, 43, 1], [15, 43, 2], [15, 43, 3], [15, 50, 1], [15, 50, 2], [15, 50, 3], [15, 50, 4], [15, 50, 5], [15, 50, 6], [15, 62, 1], [15, 62, 2], [15, 62, 3], [15, 63, 1], [15, 63, 2], [15, 63, 3], [15, 65, 1], [15, 65, 2], [15, 66, 1], [15, 66, 2], [15, 66, 3], [15, 75, 1], [15, 75, 2], [15, 75, 3], [15, 77, 1], [15, 77, 2], [15, 77, 3], [15, 78, 1], [15, 79, 1], [15, 81, 1], [15, 81, 2], [15, 81, 3], [15, 82, 1], [15, 82, 2], [15, 82, 3], [15, 83, 1.1], [15, 83, 1.2], [15, 83, 2], [15, 83, 3.1], [15, 83, 3.2], [15, 83, 4], [15, 84, 1], [15, 85, 1], [15, 85, 2], [15, 85, 3], [15, 86, 1], [15, 86, 2], [15, 86, 3], [15, 87, 1], [15, 87, 2], [15, 87, 3], [15, 87, 4], [15, 87, 5], [15, 88, 1], [15, 89, 1], [15, 90, 1], [16, 1, 1], [16, 1, 2], [16, 1, 3], [16, 1, 4], [16, 1, 5], [16, 1, 6], [16, 1, 7], [16, 1, 8], [16, 1, 9], [16, 1, 10], [16, 1, 11], [16, 1, 12], [16, 1, 13], [16, 1, 14], [16, 1, 15], [16, 1, 16], [16, 1, 17], [16, 1, 18], [16, 1, 19], [16, 1, 20], [16, 1, 21], [16, 1, 22], [16, 1, 23], [16, 1, 24], [16, 1, 25], [16, 1, 26], [16, 1, 27], [16, 14, 1], [16, 14, 2], [16, 14, 3], [16, 14, 4], [16, 14, 5], [16, 14, 6], [16, 14, 7], [16, 14, 8], [16, 14, 9], [16, 14, 10], [16, 14, 11], [16, 14, 12], [16, 14, 13], [16, 14, 14], [16, 14, 15], [16, 14, 16], [16, 14, 17], [16, 14, 18], [16, 14, 19], [16, 14, 20], [16, 18, 1], [16, 19, 1], [16, 20, 1], [16, 22, 1], [16, 24, 1], [16, 24, 2], [16, 43, 1], [16, 45, 1], [16, 45, 2], [16, 46, 1], [16, 47, 1], [16, 50, 1], [16, 51, 1], [16, 54, 1], [16, 61, 1], [16, 63, 1], [16, 66, 1], [16, 66, 2], [16, 66, 3], [16, 67, 1], [16, 71, 1], [16, 72, 1], [16, 73, 1], [16, 74, 1], [16, 75, 1], [16, 76, 1], [16, 77, 1], [16, 77, 2], [16, 77, 3], [16, 78, 1], [17, 4, 1], [17, 4, 2], [17, 5, 1], [17, 9, 1], [17, 14, 1], [17, 15, 1], [17, 15, 2], [17, 16, 1], [17, 20, 1], [17, 22, 1], [17, 23, 1], [17, 24, 1], [17, 24, 2], [17, 26, 1], [17, 27, 1], [17, 27, 2], [17, 28, 1], [17, 28, 2], [17, 29, 1], [17, 29, 2], [17, 30, 1], [17, 31, 1], [18, 2, 1], [18, 2, 2], [18, 2, 3], [18, 2, 4], [18, 2, 5], [18, 2, 6], [18, 2, 7], [18, 2, 8], [18, 48, 1], [18, 48, 2], [18, 49, 1], [18, 50, 1], [18, 64, 1], [18, 64, 2], [18, 64, 3], [18, 65, 1], [18, 66, 1], [18, 67, 1], [18, 71, 1], [18, 75, 1], [19, 17, 1], [19, 17, 2], [19, 21, 1], [20, 3, 1], [20, 4, 1], [20, 5, 1], [20, 7, 1], [20, 8, 1], [20, 9, 1], [20, 10, 1], [20, 11, 1], [20, 12, 1], [20, 13, 1], [20, 15, 1], [20, 16, 1], [20, 17, 1], [20, 20, 1], [20, 21, 1], [20, 23, 1], [20, 27, 1], [20, 28, 1], [20, 29, 1], [20, 32, 1], [20, 34, 1], [20, 36, 1], [20, 38, 1], [20, 38, 2], [20, 38, 3], [20, 44, 1], [20, 46, 1], [20, 54, 1], [20, 57, 1], [20, 57, 2], [28, 16, 1], [28, 17, 1], [28, 21, 1], [28, 21, 2], [28, 21, 3], [28, 21, 4], [28, 21, 5], [28, 21, 6], [28, 21, 7], [28, 24, 1], [28, 25, 1], [28, 26, 1], [28, 26, 2], [28, 26, 3], [28, 41, 1], [28, 43, 1], [28, 44, 1], [28, 44, 2], [28, 44, 3], [28, 44, 4], [28, 44, 5], [28, 44, 6], [28, 45, 1], [28, 45, 2], [28, 46, 1], [28, 46, 2], [28, 53, 1], [28, 53, 2], [28, 55, 1], [28, 55, 2], [28, 56.1, 1], [28, 56.2, 2], [28, 56.2, 3], [28, 56.3, 1], [28, 56.4, 1], [28, 56.5, 1], [28, 56.6, 1], [28, 58.2, 1], [28, 58.3, 1], [28, 58.3, 2], [28, 58.4, 1], [28, 60, 1], [28, 61, 1], [28, 64, 1], [28, 66, 1], [28, 67, 1], [28, 68, 1], [28, 72, 1], [28, 73, 1], [28, 74, 1]])

492

(25)

NULL

:)



Download Exact_Solutions_to_Einstein_Equations.mw

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

 

I am trying to use the Define() funtion for the tetrad form of the Ricci Tensor. I use the Setup() function to define the tetradmetric and the tetrad labels. However, the function continues do some strange things with the d_ . It should be just a simple sum over the repeated indices, but I am getting imaginary terms.

 

  restart; with(Physics); with(Tetrads)

0, "%1 is not a command in the %2 package", Tetrads, Physics

(1)

Physics:-Setup(coordinatesystems = {X}, spacetimeindices = greek, tetradindices = lowercaselatin, tetradmetric = Matrix([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]], shape = symmetric), mathematicalnotation = true)

[coordinatesystems = {X}, mathematicalnotation = true, spacetimeindices = greek, tetradindices = lowercaselatin, tetradmetric = {(1, 2) = 1, (3, 4) = 1}]

(2)

Physics:-Define(GammaT[a, b, c], RicciT[b, c] = -%d_[c](GammaT[`~a`, b, a])+%d_[a](GammaT[`~a`, b, c])-Physics:-`*`(GammaT[`~m`, b, a], GammaT[`~a`, m, c])+Physics:-`*`(GammaT[`~m`, b, c], GammaT[`~a`, m, a])-Physics:-`*`(GammaT[`~a`, b, m], GammaT[`~m`, c, a]-GammaT[`~m`, a, c]))

{Physics:-Dgamma[mu], GammaT[a, b, c], Physics:-Psigma[mu], RicciT[b, c], Physics:-d_[mu], Physics:-Tetrads:-eta_[a, b], Physics:-g_[mu, nu], Physics:-Tetrads:-l_[mu], Physics:-Tetrads:-m_[mu], Physics:-Tetrads:-mb_[mu], Physics:-Tetrads:-n_[mu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-SpaceTimeVector[mu](X)}

(3)

RicciT[1, 2]

-%d_[2](GammaT[`~1`, 1, 1]+GammaT[`~2`, 1, 2]+GammaT[`~3`, 1, 3]+GammaT[`~4`, 1, 4])-GammaT[`~1`, 1, 3]*GammaT[`~3`, 2, 1]-GammaT[`~3`, 1, 3]*GammaT[`~3`, 2, 3]-GammaT[`~4`, 1, 3]*GammaT[`~3`, 2, 4]+GammaT[`~4`, 1, 2]*GammaT[`~1`, 4, 1]+GammaT[`~4`, 1, 2]*GammaT[`~3`, 4, 3]+GammaT[`~4`, 1, 2]*GammaT[`~4`, 4, 4]+GammaT[`~1`, 1, 2]*GammaT[`~3`, 1, 3]+GammaT[`~1`, 1, 2]*GammaT[`~4`, 1, 4]+GammaT[`~2`, 1, 2]*GammaT[`~1`, 2, 1]+GammaT[`~2`, 1, 2]*GammaT[`~3`, 2, 3]+GammaT[`~2`, 1, 2]*GammaT[`~4`, 2, 4]+GammaT[`~3`, 1, 2]*GammaT[`~1`, 3, 1]+GammaT[`~3`, 1, 2]*GammaT[`~3`, 3, 3]+GammaT[`~3`, 1, 2]*GammaT[`~4`, 3, 4]-GammaT[`~1`, 1, 4]*GammaT[`~4`, 2, 1]-GammaT[`~3`, 1, 4]*GammaT[`~4`, 2, 3]-GammaT[`~4`, 1, 4]*GammaT[`~4`, 2, 4]-GammaT[`~1`, 1, 1]*GammaT[`~1`, 2, 1]-GammaT[`~3`, 1, 1]*GammaT[`~1`, 2, 3]-GammaT[`~4`, 1, 1]*GammaT[`~1`, 2, 4]-GammaT[`~1`, 1, 2]*GammaT[`~2`, 2, 1]-GammaT[`~3`, 1, 2]*GammaT[`~2`, 2, 3]-GammaT[`~4`, 1, 2]*GammaT[`~2`, 2, 4]-(1/2)*2^(1/2)*%d_[3](GammaT[`~2`, 1, 2])+(1/2)*2^(1/2)*%d_[3](GammaT[`~1`, 1, 2])+GammaT[`~2`, 3, 2]*GammaT[`~3`, 1, 2]+GammaT[`~2`, 4, 2]*GammaT[`~4`, 1, 2]+GammaT[`~1`, 1, 1]*GammaT[`~1`, 1, 2]+GammaT[`~1`, 1, 2]*GammaT[`~2`, 1, 2]-GammaT[`~1`, 2, 2]*GammaT[`~2`, 1, 1]-GammaT[`~2`, 1, 3]*GammaT[`~3`, 2, 2]-GammaT[`~2`, 1, 4]*GammaT[`~4`, 2, 2]+((1/2)*I)*2^(1/2)*%d_[1](GammaT[`~4`, 1, 2])-((1/2)*I)*2^(1/2)*%d_[4](GammaT[`~4`, 1, 2])+((1/2)*I)*2^(1/2)*%d_[1](GammaT[`~3`, 1, 2])+((1/2)*I)*2^(1/2)*%d_[4](GammaT[`~3`, 1, 2])+((1/2)*I)*2^(1/2)*%d_[2](GammaT[`~2`, 1, 2])+((1/2)*I)*2^(1/2)*%d_[2](GammaT[`~1`, 1, 2])

(4)

 

``

 

Download Problem_with_Defined_Tetrad_Function.mw

In preparation for learning GR, I need somme trick on how to use the DifferentialGEometry package or else. So I want to show you what I want to do (see the link to the document below) and I would like some help on how to do it with tensor.  That would help me a lot.  Then, I will extansion it on a 3-D problem and next in 4-D.

 

Thank you in advance dor your help

 

here is the file : Finding_covariant_and_contravariant.mw

 

--------------------------------------
Mario Lemelin
Maple 2015 Ubuntu 14.04 - 64 bits
Maple 2015 Win 10 - 64 bits messagerie : mario.lemelin@cgocable.ca téléphone :  (819) 376-0987
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