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Posted: ecterrab 13431 Product: Maple

April 20 2016

4 0

Formulating and solving the equivalence problem for Schwarzschild metric in a simple case

In connection with the digitizing in Maple 2016 of the database of solutions to Einstein's equations of the book Exact Solutions to Einstein Field Equations. I was recently asked about a statement found in the "What is new in Physics in Maple 2016" page:

In the Maple PDEtools package, you have the mathematical tools - including a complete symmetry approach - to work with the underlying [Einstein’s] partial differential equations. [By combining that functionality with the one in the Physics and Physics:-Tetrads package] you can also formulate and, depending on 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.

This question posed is a reasonable one: "could you please provide one example?" This post provides that example.

First of all the existing science behind: in my opinion, the main reference regarding the equivalence problem is at the paper "A Review of the Geometrical Equivalence of Metrics in General Relativity", General Relativity and Gravitation, Vol. 12, No. 9, 1980, by A. Karlhede (University of Stockholm). This approach got refined later by others and, generally speaking, it is currently know as the Cartan-Karlhede method, summarized in chapter 9.2 of the book Exact Solutions to Einstein Field Equations. whose solutions were all digitized within the Physics and DifferentialGeometry packages for Maple 2016. This method of Chapter 9.2 (see also Tetrads and Weyl scalars in canonical form, Mapleprimes post), however, is not the only approach to the problem, and sometimes simpler methods can handle the problem faster, or just in simpler forms.

The example worked out below is actually the example from Karlhede's paper just mentioned, on pages 704 - 706: "Show that the Schwarzschild metric and its form written in terms of isotropic spherical coordinates are equivalent, and derive the transformation that relates them". Because this problem happens to be simple for nowadays computer algebra, below I also tackle it modified, slightly more difficult variants of it. The approach shown works for more complicated cases as well.

Below we tackle Karlhede's paper-problem using: one PDEtools command, the Physics:-TransformCoordinates, the Physics:-Weyl command to compute the Weyl scalars and the Physics:-Tetrads:-PetrovType to see the Petrov type of the metrics involved. The transformation resolving the equivalence is explicitly derived.

Start loading the Physics and Tetrads package. To reproduce the computations below, as usual, update your Physics library with the one available for download at the Maplesoft R&D Physics webpage

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

(1)

To formulate the problem, set first some symbols to represent the changed metric, changed mass and changed coordinates - no mathematics at this point

(2)

Set now a new coordinates system, call it Y, involving the new coordinates (in the paper they are represented with a tilde on top of the letters)

$`Default differentiation variables for d_, D_ and dAlembertian are: `*{Y = (`&tfr;`, `&rfr;`, `ϑ`, `ϕ`)}$

(3)

According to eq.(7.6) of the paper, the line element of Schwarzschild solution in isotropic spherical coordinates is given by

`#msup(mi("ds"),mn("2"))` := ((1-mt/(2*rt))/(1+mt/(2*rt)))^2*d_(tt)^2-(1+mt/(2*rt))^4*(d_(rt)^2+rt^2*d_(thetat)^2+rt^2*sin(thetat)^2*d_(phit)^2)

(-2*`&rfr;`+`&mfr;`)^2*Physics:-d_(`&tfr;`)^2/(2*`&rfr;`+`&mfr;`)^2-(1/16)*(2*`&rfr;`+`&mfr;`)^4*(Physics:-d_(`&rfr;`)^2+`&rfr;`^2*Physics:-d_(vartheta)^2+`&rfr;`^2*sin(vartheta)^2*Physics:-d_(varphi)^2)/`&rfr;`^4

(4)

Set this to be the metric

Check it out

Physics:-g_[mu, nu] = Matrix(%id = 18446744078306516254)

(5)

In connection with the transformation used further below, compute now the Petrov type and the Weyl scalars for this metric, just to have an idea of what is behind this metric.

(6)

(7)

We see that the Weyl scalars are already in canonical form (see post in Mapleprimes about canonical forms): only and the important thing: it depends on only one coordinate, .

Now: we want to see if this metric (5) is equivalent to Schwarzschild metric in standard spherical coordinates

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

Physics:-g_[mu, nu] = Matrix(%id = 18446744078795590102)

(8)

The equivalence we want to resolve is regarding an arbitrary relationship between the masses used in (5) and (8) and a generic change of variables from X to Y

(9)

Using a differential equation mindset, the formulation of the equivalence between (8) and (5) under the transformation (9) is actually simple: change variables in (8), using (9) and the Physics:-TransformCoordinates command (this is the command that changes variables in tensorial expressions), then equate the result to (5), then try to solve the problem for the unknowns , and .

We note at this point, however, that the Weyl scalars for Schwarzschild metric in this standard form (8) are also in canonical form of Petrov type D and also depend on only one variable, r

(10)

(11)

The fact that the Weyl scalars in both cases ((7) and (11)) are in canonical form (only ) and in both cases this scalar depends on only one coordinate is already an indicator that the transformation involved changes only one variable in terms of the other one. So one could just search for a transformation of the form and resolve the problem instantly. Still, to make the problem slightly more general, consider instead a generic transformation for r in terms of all of

(12)

(13)

Transform the coordinates in the metric (because of having used PDEtools:-declare, derivatives of the unknowns R are displayed indexed, for compact notation)

Matrix(%id = 18446744078873927542)

(14)

Proceed equating (14) to (5) to obtain a set of equations that entirely formulates the problem

${0 = (diff(R(Y), `&rfr;`))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), `&rfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), vartheta))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), vartheta))*(diff(R(Y), `&rfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), vartheta))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), (-2*`&rfr;`+`&mfr;`)^2/(2*`&rfr;`+`&mfr;`)^2 = ((diff(R(Y), `&tfr;`))^2*R(Y)^2-4*(-(1/2)*R(Y)+m)^2)/(R(Y)*(-R(Y)+2*m)), -(1/16)*(2*`&rfr;`+`&mfr;`)^4/`&rfr;`^4 = (diff(R(Y), `&rfr;`))^2*R(Y)/(-R(Y)+2*m), -(1/16)*(2*`&rfr;`+`&mfr;`)^4/`&rfr;`^2 = -(diff(R(Y), vartheta))^2*R(Y)/(R(Y)-2*m)-R(Y)^2, -(1/16)*(2*`&rfr;`+`&mfr;`)^4*sin(vartheta)^2/`&rfr;`^2 = 2*((1/2)*(diff(R(Y), varphi))^2+(cos(vartheta)-1)*R(Y)*(cos(vartheta)+1)*(-(1/2)*R(Y)+m))*R(Y)/(-R(Y)+2*m)}$

(15)

This problem, shown in Karlhede's paper as the example of the approach he summarized, is solvable using the differential equation commands of PDEtools (in this case casesplit) in one go and no time, obtaining the same solution shown in the paper with equation number (7.10), the problem actually admits two solutions

$PDEtools:-casesplit({0 = (diff(R(Y), `&rfr;`))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), `&rfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), vartheta))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), vartheta))*(diff(R(Y), `&rfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), vartheta))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), (-2*`&rfr;`+`&mfr;`)^2/(2*`&rfr;`+`&mfr;`)^2 = ((diff(R(Y), `&tfr;`))^2*R(Y)^2-4*(-(1/2)*R(Y)+m)^2)/(R(Y)*(-R(Y)+2*m)), -(1/16)*(2*`&rfr;`+`&mfr;`)^4/`&rfr;`^4 = (diff(R(Y), `&rfr;`))^2*R(Y)/(-R(Y)+2*m), -(1/16)*(2*`&rfr;`+`&mfr;`)^4/`&rfr;`^2 = -(diff(R(Y), vartheta))^2*R(Y)/(R(Y)-2*m)-R(Y)^2, -(1/16)*(2*`&rfr;`+`&mfr;`)^4*sin(vartheta)^2/`&rfr;`^2 = 2*((1/2)*(diff(R(Y), varphi))^2+(cos(vartheta)-1)*R(Y)*(cos(vartheta)+1)*(-(1/2)*R(Y)+m))*R(Y)/(-R(Y)+2*m)}, [R, mt])$

(16)

By all means this does not mean this differential equation approach is better than the general approach mentioned in the paper (also in section 9.2 of the Exact Solutions book). This presentation above only makes the point of the paragraph mentioned at the beginning of this worksheet "... [in Maple 2016] 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."

In any case this problem above is rather easy for the computer. Consider a slightly more difficult problem, where . For example:

Physics:-g_[mu, nu] = Matrix(%id = 18446744078854733566)

(17)

Tackle now the same problem

${0 = (diff(R(Y), `&rfr;`))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), `&rfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), vartheta))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), vartheta))*(diff(R(Y), `&rfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), vartheta))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), (-2*`&rfr;`+1/`&mfr;`^2)^2/(2*`&rfr;`+1/`&mfr;`^2)^2 = ((diff(R(Y), `&tfr;`))^2*R(Y)^2-4*(-(1/2)*R(Y)+m)^2)/(R(Y)*(-R(Y)+2*m)), -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4/`&rfr;`^4 = (diff(R(Y), `&rfr;`))^2*R(Y)/(-R(Y)+2*m), -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4/`&rfr;`^2 = -(diff(R(Y), vartheta))^2*R(Y)/(R(Y)-2*m)-R(Y)^2, -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4*sin(vartheta)^2/`&rfr;`^2 = 2*((1/2)*(diff(R(Y), varphi))^2+(cos(vartheta)-1)*R(Y)*(cos(vartheta)+1)*(-(1/2)*R(Y)+m))*R(Y)/(-R(Y)+2*m)}$

(18)

The solutions to the equivalence between (17) and (5) are then given by

$PDEtools:-casesplit({0 = (diff(R(Y), `&rfr;`))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), `&rfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), varphi))*(diff(R(Y), vartheta))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), vartheta))*(diff(R(Y), `&rfr;`))*R(Y)/(-R(Y)+2*m), 0 = (diff(R(Y), vartheta))*(diff(R(Y), `&tfr;`))*R(Y)/(-R(Y)+2*m), (-2*`&rfr;`+1/`&mfr;`^2)^2/(2*`&rfr;`+1/`&mfr;`^2)^2 = ((diff(R(Y), `&tfr;`))^2*R(Y)^2-4*(-(1/2)*R(Y)+m)^2)/(R(Y)*(-R(Y)+2*m)), -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4/`&rfr;`^4 = (diff(R(Y), `&rfr;`))^2*R(Y)/(-R(Y)+2*m), -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4/`&rfr;`^2 = -(diff(R(Y), vartheta))^2*R(Y)/(R(Y)-2*m)-R(Y)^2, -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4*sin(vartheta)^2/`&rfr;`^2 = 2*((1/2)*(diff(R(Y), varphi))^2+(cos(vartheta)-1)*R(Y)*(cos(vartheta)+1)*(-(1/2)*R(Y)+m))*R(Y)/(-R(Y)+2*m)}, [R, mt])$

`casesplit/ans`([R(Y) = -(1/4)*(m-2*`&rfr;`)^2/`&rfr;`, `&mfr;`^2 = -1/m], [`&mfr;` <> 0]), `casesplit/ans`([R(Y) = (1/4)*(2*`&rfr;`+m)^2/`&rfr;`, `&mfr;`^2 = 1/m], [`&mfr;` <> 0])

(19)

Moreover, despite that the Weyl scalars suggest that a transformation of only one variable is sufficient to solve the problem, one could also consider a more general transformation, of more variables. Provided we exclude (because there is around and that would take us to solve differential equations for , that involve things like ), and also to speed up matters let's remove the change in , consider an arbitrary change in r and t

(20)

(21)

So our transformation now involve two arbitrary variables, each one depending on all the four coordinates, and a more complicated function . Change variables (because of having used PDEtools:-declare, derivatives of the unknowns R and are displayed indexed, for compact notation)

Matrix(%id = 18446744078309268046)

(22)

Construct the set of Partial Differential Equations to be tackled

${0 = (-4*(diff(Tau(Y), `&rfr;`))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&tfr;`))+(diff(R(Y), `&rfr;`))*(diff(R(Y), `&tfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), varphi))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&rfr;`))+(diff(R(Y), varphi))*(diff(R(Y), `&rfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), varphi))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&tfr;`))+(diff(R(Y), varphi))*(diff(R(Y), `&tfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), varphi))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), vartheta))+(diff(R(Y), varphi))*(diff(R(Y), vartheta))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), vartheta))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&rfr;`))+(diff(R(Y), vartheta))*(diff(R(Y), `&rfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), vartheta))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&tfr;`))+(diff(R(Y), vartheta))*(diff(R(Y), `&tfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), (-2*`&rfr;`+1/`&mfr;`^2)^2/(2*`&rfr;`+1/`&mfr;`^2)^2 = (-4*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&tfr;`))^2+(diff(R(Y), `&tfr;`))^2*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4/`&rfr;`^4 = (-4*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&rfr;`))^2+(diff(R(Y), `&rfr;`))^2*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4/`&rfr;`^2 = (diff(Tau(Y), vartheta))^2*(R(Y)-2*m)/R(Y)-(diff(R(Y), vartheta))^2*R(Y)/(R(Y)-2*m)-R(Y)^2, -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4*sin(vartheta)^2/`&rfr;`^2 = (-4*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), varphi))^2+2*((1/2)*(diff(R(Y), varphi))^2+(cos(vartheta)-1)*R(Y)*(cos(vartheta)+1)*(-(1/2)*R(Y)+m))*R(Y)^2)/(R(Y)*(-R(Y)+2*m))}$

(23)

Solve the problem running a differential elimination (actually without solving any differential equations): there are more than two solutions

$sol := PDEtools:-casesplit({0 = (-4*(diff(Tau(Y), `&rfr;`))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&tfr;`))+(diff(R(Y), `&rfr;`))*(diff(R(Y), `&tfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), varphi))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&rfr;`))+(diff(R(Y), varphi))*(diff(R(Y), `&rfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), varphi))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&tfr;`))+(diff(R(Y), varphi))*(diff(R(Y), `&tfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), varphi))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), vartheta))+(diff(R(Y), varphi))*(diff(R(Y), vartheta))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), vartheta))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&rfr;`))+(diff(R(Y), vartheta))*(diff(R(Y), `&rfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), 0 = (-4*(diff(Tau(Y), vartheta))*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&tfr;`))+(diff(R(Y), vartheta))*(diff(R(Y), `&tfr;`))*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), (-2*`&rfr;`+1/`&mfr;`^2)^2/(2*`&rfr;`+1/`&mfr;`^2)^2 = (-4*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&tfr;`))^2+(diff(R(Y), `&tfr;`))^2*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4/`&rfr;`^4 = (-4*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), `&rfr;`))^2+(diff(R(Y), `&rfr;`))^2*R(Y)^2)/(R(Y)*(-R(Y)+2*m)), -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4/`&rfr;`^2 = (diff(Tau(Y), vartheta))^2*(R(Y)-2*m)/R(Y)-(diff(R(Y), vartheta))^2*R(Y)/(R(Y)-2*m)-R(Y)^2, -(1/16)*(2*`&rfr;`+1/`&mfr;`^2)^4*sin(vartheta)^2/`&rfr;`^2 = (-4*(-(1/2)*R(Y)+m)^2*(diff(Tau(Y), varphi))^2+2*((1/2)*(diff(R(Y), varphi))^2+(cos(vartheta)-1)*R(Y)*(cos(vartheta)+1)*(-(1/2)*R(Y)+m))*R(Y)^2)/(R(Y)*(-R(Y)+2*m))}, [R, mt])$

(24)

Consider for instance the first one

(25)

Compute the actual solution behind this case :

>	$pdsolve(`casesplit/ans`([R(Y) = (1/4)(2`&rfr;`+m)^2/`&rfr;`, `&mfr;`^2 = 1/m, diff(Tau(Y), `&tfr;`) = -1, diff(Tau(Y), `&rfr;`) = 0, diff(Tau(Y), vartheta) = 0, diff(Tau(Y), varphi) = 0], [`&mfr;` <> 0]), {R, Tau, mt})$

${`&mfr;` = -1/m^(1/2), R(Y) = (1/4)*(2*`&rfr;`+m)^2/`&rfr;`, Tau(Y) = -`&tfr;`+_C1}, {`&mfr;` = 1/m^(1/2), R(Y) = (1/4)*(2*`&rfr;`+m)^2/`&rfr;`, Tau(Y) = -`&tfr;`+_C1}$

(26)

The fact that the time t appears defined in terms of the transformed time involving an arbitrary constant is expected: the time does not enter the metric, it only enters through derivatives of entering the Jacobian of the transformation used to change variables in tensorial expressions (the metric) in (22).

Summary: the approach shown above, based on formulating the problem for the transformation functions of the equivalence and solving for them the differential equations using the commands in PDEtools, after restricting the generality of the transformation functions by looking at the form of the Weyl scalars, works well for other cases too, specially now that, in Maple 2016, the Weyl scalars can be expressed also in canonical form in one go (see previous Mapleprimes post on "Tetrads and Weyl scalars in canonical form"). Also important: in Maple 2016 it is present the functionality necessary to implement the approach of section 9.2 of the Exact solutions book as well.

Download Equivalence_-_Schwarzschild.mw

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

Posted: ecterrab 13431 Product: Maple

April 20 2016

4 0

Tetrads and Weyl scalars in canonical form

The material below is about a new development that didn't arrive in time for the launch of Maple 2016 (March) and that complements in a relevant way the ones introduced in Physics in Maple 2016. It is at topic in general relativity, the computation of a canonical form of a tetrad, so that, generally speaking (skipping a technical description) the Weyl scalars are fixed as much as possible (either equal to 0 or to 1) regarding transformations that leave invariant the tetrad metric in a tetrad system of references. Bringing a tetrad in canonical form is a relevant step in the tackling of the equivalence problem between two spacetime metrics (Mapleprimes post), and it is relevant in connection with the digitizing in Maple 2016 of the database of solutions to Einstein's equations of the book Exact Solutions to Einstein Field Equations.

The reference for this development is the book "General Relativity, an Einstein century survey", edited by S.W. Hawking (Cambridge) and W. Israel (U. Alberta, Canada), specifically Chapter 7 written by S. Chandrasekhar, and more specifically exploring what is said in page 388 about the Petrov classification.

A canonical form for the tetrad and Weyl scalars admits alternate forms; the implementation is as implicit in page 388:

The transformations (rotations of the tetrad system of references) used are of Class I, II and III as defined in Chandrasekar's chapter - equations (7.79) in page 384, (7.83) and (7.84) in page 385. Transformations of Class I can be performed with the command Physics:-Tetrads:-TransformTetrad using the optional argument nullrotationwithfixedl_, of Class II using nullrotationwithfixedn_ and of Class III by calling TransformTetrad(spatialrotationsm_mb_plan, boostsn_l_plane), so with the two optional arguments simultaneously.

In this development, a new optional argument, canonicalform got implemented to TransformTetrad so that the whole sequence of three transformations of Classes I, II and III is performed automatically, in one go. Regarding the canonical form of the tetrad, the main idea is that from the change in the Weyl scalars one can derive the parameters entering tetrad transformations that result in a canonical form of the tetrad.

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

(1)

(Note the Tetrads:-PetrovType command, unfinished in the first release of Maple 2016.) To run the following computations you need to update your Physics library to the latest version from the Maplesoft R&D Physics webpage, so with this datestamp or newer:

(2)

An Example of Petrov type I

There are six Petrov types: I, II, III, D, N and O. Start with a spacetime metric of Petrov type "I" (the numbers always refer to the equation number in the "Exact solutions to Einstein's field equations" textbook)

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

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

$g[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 2*y, (2, 1) = 0, (2, 2) = a^2/x^2, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = a^2/x^2, (3, 4) = 0, (4, 1) = 2*y, (4, 2) = 0, (4, 3) = 0, (4, 4) = x^2-4*y^2}))$

(3)

The Weyl scalars

psi__0 = (1/4)*((4*I)*x^3*abs(x)^3-abs(x)^6+abs(x)^4*x^2+abs(x)^2*x^4-x^6)/(a^2*abs(x)^4*x^2), psi__1 = 0, psi__2 = -(1/4)*(x^2+abs(x)^2)*(x^4+abs(x)^4)/(a^2*abs(x)^4*x^2), psi__3 = 0, psi__4 = (1/4)*((4*I)*x^3*abs(x)^3-abs(x)^6+abs(x)^4*x^2+abs(x)^2*x^4-x^6)/(a^2*abs(x)^4*x^2)

(4)

... there is abs around. Let's assume everything is positive to simplify formulas, use Capital Physics:-Assume (the lower case assume command redefines the assumed variables, so it is not compatible with Physics, DifferentialGeometry and VectorCalculus among others).

(5)

The scalars are now simpler, although still not in "canonical form" because and .

(6)

The Petrov type

(7)

The call to Tetrads:-TransformTetrad two lines below transforms the current tetrad ,

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078512745638)

(8)

into another tetrad such that the Weyl scalars are in canonical form, which for Petrov "I" type happens when and .

Matrix(%id = 18446744078500192254)

(9)

Despite the fact that the result is a much more complicated tetrad, this is an amazing result in that the resulting Weyl scalars are all fixed (see below). Let's first verify that this is indeed a tetrad, and that now the Weyl scalars are in canonical form

(10)

Set (9) to be the tetrad in use and recompute the Weyl scalars

Inded we now have and

[psi__0 = 0, psi__1 = (-1/2-(3/2)*I)/a^4, psi__2 = (-1+I)/a^2, psi__3 = 1, psi__4 = 0]

(11)

So Weyl scalars computed after setting the canonical tetrad (9) to be the tetrad in use are in canonical form. Great! NOTE: computing the canonicalWeyl scalars is not really the difficult part, and within the code, these scalars (11) are computed before arriving at the tetrad (9). What is really difficult (from the point of view of computational complexity and simplifications) is to compute the actual canonical form of the tetrad (9).

An Example of Petrov type II

Consider this other solution to Einstein's equation (again, the numbers in g_[[24,37,7]] always refer to the equation number in the "Exact solutions to Einstein's field equations" textbook)

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

"`Comments: `Case 6 from Table 24.1:_Psi1(u,x,y): diff(_Psi1(u,x,y),x,x)+diff(_Psi1(u,x,y),y,y)=0, diff(x*diff(_M(u,x,y),x),x)+x*diff(_M(u,x,y),y,y)=_kappa0*(diff(_Psi(u,x,y),x)^2+diff(_Psi(u,x,y),y)^2)"

$g[mu, nu] = (Matrix(4, 4, {(1, 1) = -2*x*(f(x)+y*a), (1, 2) = -x, (1, 3) = 0, (1, 4) = 0, (2, 2) = 0, (2, 3) = 0, (2, 4) = 0, (3, 3) = 1/x^(1/2), (3, 4) = 0, (4, 4) = 1/x^(1/2)}, storage = triangular[upper], shape = [symmetric]))$

(12)

Check the Petrov type

(13)

The starting tetrad

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078835577550)

(14)

results in Weyl scalars not in canonical form:

(15)

For Petrov type "II", the canonical form is as for type "I" but in addition Again let's assume positive, not necessary, but to get simpler formulas around

(16)

Compute now a canonical form for the tetrad, to be used instead of (14)

Matrix(%id = 18446744078835949430)

(17)

Set this tetrad and check the Weyl scalars again

(18)

This result (18) is fantastic. Compare these Weyl scalars with the ones (15) before transforming the tetrad.

An Example of Petrov type III

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

g[mu, nu] = (Matrix(4, 4, {(1, 1) = 0, (1, 2) = exp(-2*z), (1, 3) = 0, (1, 4) = 0, (2, 2) = exp(4*z), (2, 3) = 2*exp(z), (2, 4) = 0, (3, 3) = 2*exp(-2*z), (3, 4) = 0, (4, 4) = 3/abs(Lambda)}, storage = triangular[upper], shape = [symmetric]))

(19)

(20)

The Petrov type and the original tetrad

(21)

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078349449926)

(22)

This tetrad results in the following scalars

psi__0 = -2*Lambda*2^(1/2)+(11/4)*Lambda, psi__1 = -(1/2)*Lambda*2^(1/2)+(3/4)*Lambda, psi__2 = (1/4)*Lambda, psi__3 = -(1/2)*Lambda*2^(1/2)-(3/4)*Lambda, psi__4 = 2*Lambda*2^(1/2)+(11/4)*Lambda

(23)

that are not in canonical form, which for Petrov type III is as in Petrov type II but in addition we should have .

Compute now a canonical form for the tetrad

Matrix(%id = 18446744078500057566)

(24)

Set this one to be the tetrad in use and recompute the Weyl scalars

(25)

Great!

An Example of Petrov type N

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

$g[mu, nu] = (Matrix(4, 4, {(1, 1) = 0, (1, 2) = -(3/2)/(y^2*Lambda), (1, 3) = 0, (1, 4) = 0, (2, 2) = -3/(y^4*Lambda), (2, 3) = 0, (2, 4) = 0, (3, 3) = 3/(y^2*Lambda), (3, 4) = 0, (4, 4) = 3/(y^2*Lambda)}, storage = triangular[upper], shape = [symmetric]))$

(26)

(27)

(28)

The original tetrad and related Weyl scalars are not in canonical form:

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078404437406)

(29)

(30)

For Petrov type "N", the canonical form has and all the other .

Compute a canonical form, set it to be the tetrad in use and recompute the Weyl scalars

Matrix(%id = 18446744078518486190)

(31)

(32)

All as expected.

An Example of Petrov type D

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

$g[mu, nu] = (Matrix(4, 4, {(1, 1) = -B^2*sin(z)^2, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 2) = A^2, (2, 3) = 0, (2, 4) = 0, (3, 3) = A^2*x^2, (3, 4) = 0, (4, 4) = B^2}, storage = triangular[upper], shape = [symmetric]))$

(33)

(34)

(35)

The default tetrad and related Weyl scalars are not in canonical form, which for Petrov type "D" is with and all the other

Physics:-Tetrads:-e_[a, mu] = Matrix(%id = 18446744078503920694)

(36)

(37)

Transform the tetrad, set it and recompute the Weyl scalars

Matrix(%id = 18446744078814996830)

(38)

(39)

Again the expected canonical form of the Weyl scalars, and remains invariant under transformations of Class III.

An Example of Petrov type O

Finally an example of type "O". This corresponds to a conformally flat spacetime, for which the Weyl tensor (and with it all the Weyl scalars) vanishes. So the code just interrupts with "not implemented for conformally flat spactimes of Petrov type O"

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

g[mu, nu] = z

(40)

(41)

The Weyl tensor and its scalars all vanish:

(42)

(43)

Error, (in Tetrads:-CanonicalForm) canonical form is not implemented for flat or conformally flat spacetimes of Petrov type "O"

Download TetradsAndWeylScalarsInCanonicalForm.mw

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

Posted: ecterrab 13431 Product: Maple

December 30 2015

14 7

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 n^th order derivatives of algebraic expressions and functions;

•	The input now returns both the first derivative (old behavior) and the n^th symbolic derivative (new behavior) of a mathematical function;

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

•

The previously existing 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 (n^th 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, . 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

Posted: ecterrab 13431

October 14 2015

9 19

“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 (and everything else automatically) in one go

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

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

(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

(2)

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

(3)

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

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

(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

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

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

[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"

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

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

(9)

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

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

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

(11)

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

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

(13)

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

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

(15)

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)

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

(17)

Matrix(%id = 18446744078191319390)

(18)

The related 16 Riemann invariant

(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

[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)

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

[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, 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:

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

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

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

(22)

The related Riemann invariants:

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?

(24)

(25)

Residual invariance

Petrov type I

Petrov type II

Petrov type III

Petrov type D

remains invariant under rotations of Class III

Petrov type N

remains invariant under rotations of Class II

Posted: ecterrab 13431

September 25 2015

8 0

One of the interesting things about the Physics package is that it was designed from scratch to extend the domain of operations of the Maple system from commutative variables to one that includes commutative, anticommutative and nonocommutative variables, as well as abstract vectors and related (nabla) differential operators. In this line we have, among others, the following Physics commands working with this extended domain: `*` , `.` , `^` , diff , Expand , Normal , Simplify , Gtaylor , and Coefficients .

More recently, Pascal Szriftgiser (from Laboratoire PhLAM, Université Lille 1, France), suggested a similar approach to factorize expressions involving noncommutative variables. This is a pretty complicated problem though. Pascal's suggestion, however, spinned around an idea beautiful for its simplicity, similar to what is done in the experimental Physics command, PerformOnAnticommutativeSystem , that is, to transform the problem into one that can be treated with the command that works only with commutative variables and from there extract the result for noncommutative ones.The approach has limitations but it is surprising how far one can go using imaginative algebraic manipulations to extend these commands that otherwise only work with commutative variables.

In brief, we now have a new command, Physics:-Factor, with already powerful performance for factorizing algebraic expressions that involve commutative, noncommutative and anticommutative variables, making Maple's mathematical capabilities more advanced in very interesting directions. This command is in fact useful not just in advanced theoretical physics, but for instance also when working with noncommutative symbols representing abstract matrices (that can have dependency, and so they can be differentiated before saying anything about their components, multiplied, and be present int expressions that in turn can be expanded, simplified and now also factorized), and also useful with expressions that include differential operators, now that within Physics you can compute with the the covariant and noncovariant derivatives D_ and d_ algebraically. For instance, how about solving differential equations using Physics:-Factor (reducing their order by means of factoring the involved differential operators) ? :)

What follows are some basic algebraic examples illustrating the novelty, and as usual to reproduce the results in this worksheet you need to update your Physics library with the one available in the Maplesoft R&D Physics webpage.

(1)

(2)

First example, because of using mathematical notation, noncommutative variables are displayed in different color (olive)

alpha^2*Physics:-`^`(a, 2)+alpha*2^(1/2)*Physics:-`*`(a, b)+4*2^(1/2)*lambda*Physics:-`*`(Physics:-`^`(b, 2), c)+4*lambda*alpha*Physics:-`*`(b, c, a)+4*2^(1/2)*lambda*Physics:-`*`(b, c, b)+16*lambda^2*Physics:-`^`(Physics:-`*`(b, c), 2)+4*lambda*alpha*Physics:-`*`(a, b, c)+alpha*2^(1/2)*Physics:-`*`(b, a)+2*Physics:-`^`(b, 2)

(3)

(4)

A more involved example from a physics problem, illustrating that the factorization is also happening within function's arguments, as well as that we can also correctly expand mathematical expressions involving noncommutative variables

(5)

Int(Int(Physics:-`^`(4*lambda*Physics:-`*`(Physics:-Dagger(b(x, y)), c(x, y))+alpha*f(t)*Physics:-`*`(a(x, y), Physics:-Dagger(a(x, y)))+2^(1/2)*g(x, y)*b(x, y), 2), x = -infinity .. infinity), y = -infinity .. infinity)

(6)

So first expand ...

(7)

Now retrieve the original expression by recursing over the arguments and so factoring the integrand

(8)

This following one looks simpler but it is actually more complicated:

(9)

(10)

The complication consists of the fact that the standard factor command, that assumes products are commutative, can never deal with factors like because if products were commutative these factors are equal to 0. Of course we not just us factor but include a number of algebraic manipulations before using it, so that the approach handles these cases nicely anyway

(11)

This other one is more complicated:

(12)

When you expand,

(13)

you see that there are various terms involving the same noncommutative operands, just multiplied in different order. Generally speaking the limitation (n this moment) of the approach is: "there cannot be more than 2 terms in the expanded form containing the same operands" . For instance in (13) the 1st and 4th terms have the same operands, that are actually also present in the 5th term but there you also have and for that reason (involving some additional manipulations) it can be handled:

(14)

Recalling, in all these examples, the task is actually accomplished by the standard factor command, and the manipulations consist of ingeniously rewriting the given problem as one that involves only commutative variables, and from extract the correct result for non commutative variables.

To conclude, here is an example where the approach implemented does not work (yet) because of the limitation mentioned in the previous paragraph:

(15)

(16)

In this expression, the 1st, 2nd, 4th and 5th terms have the same operands and then there are four terms containing the operands . We do have an idea of how this could be done too ... :) To be there in one of the next Physics updates.