## Physics in Maple 2017: Special, General and towards...

by: Maple 2017

 Physics

Maple provides a state-of-the-art environment for algebraic and tensorial computations in Physics, with emphasis on ensuring that the computational experience is as natural as possible.

The theme of the Physics project for Maple 2017 has been the consolidation of the functionality introduced in previous releases, together with significant enhancements and new functionality in General Relativity, in connection with classification of solutions to Einstein's equations and tensor representations to work in an embedded 3D curved space - a new ThreePlusOne  package. This package is relevant in numerical relativity and a Hamiltonian formulation of gravity. The developments also include first steps in connection with computational representations for all the objects entering the Standard Model in particle physics.

Classification of solutions to Einstein's equations and the Tetrads package

In Maple 2016, the digitizing of the database of solutions to Einstein's equations  was finished, added to the standard Maple library, with all the metrics from "Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; and Herlt, E., Exact Solutions to Einstein's Field Equations". These metrics can be loaded to work with them, or change them, or searched using g_  (the Physics command representing the spacetime metric that also sets the metric to your choice in one go) or using the command DifferentialGeometry:-Library:-MetricSearch .

In Maple 2017, the Physics:-Tetrads  package has been vastly improved and extended, now including new commands like PetrovType  and SegreType  to classify these metrics, and the TransformTetrad  now has an option canonicalform to automatically derive a transformation and put the tetrad in canonical form (reorientation of the axis of the local system of references), a relevant step in resolving the equivalence between two metrics.

Examples

 Petrov and Segre types, tetrads in canonical form

Equivalence for Schwarzschild metric (spherical and Kruskal coordinates)

 Formulation of the problem (remove mixed coordinates)
 Solving the Equivalence

The ThreePlusOne (3 + 1) new Maple 2017 Physics package

ThreePlusOne , is a package to cast Einstein's equations in a 3+1 form, that is, representing spacetime as a stack of nonintersecting 3-hypersurfaces Σ. This  description is key in the Hamiltonian formulation of gravity as well as in the study of gravitational waves, black holes, neutron stars, and in general to study the evolution of physical system in general relativity by running numerical simulations as traditional initial value (Cauchy) problems. ThreePlusOne includes computational representations for the spatial metric  that is induced by  on the 3-dimensional hypersurfaces, and the related covariant derivative, Christoffel symbols and Ricci and Riemann tensors, the Lapse, Shift, Unit normal and Time vectors and Extrinsic curvature related to the ADM equations.

The following is a list of the available commands:

 ADMEquations Christoffel3 D3_ ExtrinsicCurvature gamma3_ Lapse Ricci3 Riemann3 Shift TimeVector UnitNormalVector

The other four related new Physics  commands:

 • Decompose , to decompose 4D tensorial expressions (free and/or contracted indices) into the space and time parts.
 • gamma_ , representing the three-dimensional metric tensor, with which the element of spatial distance is defined as  .
 • Redefine , to redefine the coordinates and the spacetime metric according to changes in the signature from any of the four possible signatures(− + + +), (+ − − −), (+ + + −) and ((− + + +) to any of the other ones.
 • EnergyMomentum , is a computational representation for the energy-momentum tensor entering Einstein's equations as well as their 3+1 form, the ADMEquations .

Examples

 >
 (2.1.1)
 >
 (2.1.2)

Note the different color for , now a 4D tensor representing the metric of a generic 3-dimensional hypersurface induced by the 4D spacetime metric . All the ThreePlusOne tensors are displayed in black to distinguish them of the corresponding 4D or 3D tensors. The particular hypersurface  operates is parameterized by the Lapse   and the Shift  .

The induced metric is defined in terms of the UnitNormalVector   and the 4D metric  as

 >
 (2.1.3)

where  is defined in terms of the Lapse   and the derivative of a scalar function t that can be interpreted as a global time function

 >
 (2.1.4)

The TimeVector  is defined in terms of the Lapse   and the Shift   and this vector   as

 >
 (2.1.5)

The ExtrinsicCurvature  is defined in terms of the LieDerivative  of

 >
 (2.1.6)

The metric is also a projection tensor in that it projects 4D tensors into the 3D hypersurface Σ. The definition for any 4D tensor that is also a 3D tensor in Σ, can thus be written directly by contracting their indices with . In the case of Christoffel3 , Ricci3  and Riemann3,  these tensors can be defined by replacing the 4D metric  by  and the 4D Christoffel symbols  by the ThreePlusOne  in the definitions of the corresponding 4D tensors. So, for instance

 >
 (2.1.7)
 >
 (2.1.8)
 >
 (2.1.9)

When working with the ADM formalism, the line element of an arbitrary spacetime metric can be expressed in terms of the differentials of the coordinates , the Lapse , the Shift  and the spatial components of the 3D metric gamma3_ . From this line element one can derive the relation between the Lapse , the spatial part of the Shift , the spatial part of the gamma3_  metric and the  components of the 4D spacetime metric.

For this purpose, define a tensor representing the differentials of the coordinates and an alias

 >
 (2.1.10)
 >

The expression for the line element in terms of the Lapse  and Shift   is (see [2], eq.(2.123))

 >
 (2.1.11)

Compare this expression with the 3+1 decomposition of the line element in an arbitrary system. To avoid the automatic evaluation of the metric components, work with the inert form of the metric %g_

 >
 (2.1.12)
 >
 (2.1.13)

The second and third terms on the right-hand side are equal

 >
 (2.1.14)
 >
 (2.1.15)

Taking the difference between this expression and the one in terms of the Lapse  and Shift  we get

 >
 (2.1.16)

Taking coefficients, we get equations for the Shift , the Lapse  and the spatial components of the metric gamma3_

 >
 (2.1.17)
 >
 (2.1.18)
 >
 (2.1.19)

Using these equations, these quantities can all be expressed in terms of the time and space components of the 4D metric  and

 >
 (2.1.20)
 >
 (2.1.21)
 >
 (2.1.22)

References

 [1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.
 [2] Alcubierre, M., Introduction to 3+1 Numerical Relativity, International Series of Monographs on Physics 140, Oxford University Press, 2008.
 [3] Baumgarte, T.W., Shapiro, S.L., Numerical Relativity, Solving Einstein's Equations on a Computer, Cambridge University Press, 2010.
 [4] Gourgoulhon, E., 3+1 Formalism and Bases of Numerical Relativity, Lecture notes, 2007, https://arxiv.org/pdf/gr-qc/0703035v1.pdf.
 [5] Arnowitt, R., Dese, S., Misner, C.W., The Dynamics of General Relativity, Chapter 7 in Gravitation: an introduction to current research (Wiley, 1962), https://arxiv.org/pdf/gr-qc/0405109v1.pdf

Examples: Decompose, gamma_

 >
 >
 (2.2.1)

Define  now an arbitrary tensor

 >
 (2.2.2)

So  is a 4D tensor with only one free index, where the position of the time-like component is the position of the different sign in the signature, that you can query about via

 >
 (2.2.3)

To perform a decomposition into space and time, set - for instance - the lowercase latin letters from i to s to represent spaceindices and

 >
 (2.2.4)

Accordingly, the 3+1 decomposition of  is

 >
 (2.2.5)

The 3+1 decomposition of the inert representation %g_[mu,nu] of the 4D spacetime metric; use the inert representation when you do not want the actual components of the metric appearing in the output

 >
 (2.2.6)

Note the position of the component %g_[0, 0], related to the trailing position of the time-like component in the signature .

Compare the decomposition of the 4D inert with the decomposition of the 4D active spacetime metric

 >
 (2.2.7)
 >
 (2.2.8)

Note that in general the 3D space part of  is not equal to the 3D metric  whose definition includes another term (see [1] Landau & Lifshitz, eq.(84.7)).

 >
 (2.2.9)

The 3D space part of  is actually equal to the 3D metric

 >
 (2.2.10)

To derive the formula  for the covariant components of the 3D metric, Decompose into 3+1 the identity

 >
 (2.2.11)

To the side, for illustration purposes, these are the 3 + 1 decompositions, first excluding the repeated indices, then excluding the free indices

 >
 (2.2.12)
 >
 (2.2.13)

Compare with a full decomposition

 >
 (2.2.14)

is a symmetric matrix of equations involving non-contracted occurrences of ,  and . Isolate, in , , that you input as %g_[~j, ~0], and substitute into

 >
 (2.2.15)
 >
 (2.2.16)

Collect , that you input as %g_[~j, ~i]

 >
 (2.2.17)

Since the right-hand side is the identity matrix and, from , , the expression between parenthesis, multiplied by -1, is the reciprocal of the contravariant 3D metric , that is the covariant 3D metric , in accordance to its definition for the signature

 >
 (2.2.18)
 >

References

 [1] Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.
 Example: Redefine

Tensors in Special and General Relativity

A number of relevant changes happened in the tensor routines of the Physics package, towards making the routines pack more functionality both for special and general relativity, as well as working more efficiently and naturally, based on Maple's Physics users' feedback collected during 2016.

New functionality

 • Implement conversions to most of the tensors of general relativity (relevant in connection with functional differentiation)
 • New setting in the Physics Setup  allows for specifying the cosmologicalconstant and a default tensorsimplifier

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

## GRTensorIII: Component Tensor Calculations for...

by: Maple 2017

The MapleCloud now provides the GRTensorIII package for component tensor calculations in general relativity. This package is an update of the established GRTensorII package, last updated in 1999.  In early June, I had the opportunity to present this update at the recent Atlantic General Relativity meeting. The GR community was delighted to have an updated version that works well with more recent versions of Maple. Several talks specifically called out the key role GRTensorIII had in establishing the results presented.

GRTensorIII supports the calculation of the standard curvature objects in relativity. It also provides the command which allows the definition of new tensor objects via a simple definition string (without programming). This command is the key reason for GRTensor’s continued use in the GR community. GRTensorIII also provides a new, more direct API to define a spacetime via the command. This command allows for the metric or line element definition within a worksheet – removing the need for storing the metric in a file and allowing example worksheets to be self-contained. GRTensorIII has significant new functionality to support the definition of hyper-surfaces and the calculation of junction conditions and thin-shell stress energy. An extensive series of example worksheets based on Eric Poisson’s “A Relativist’s Toolkit” are included.

GRTensorIII is available from the MapleCloud window within Maple in the list of available packages or via a download from Maple. It can be loaded directly via the command line as:

It is also available on github. The source code is available on github in a separate repository: grtensor3src .

GRTensorIII was developed in collaboration with Kayll Lake and Denis Pollney. I would like to thank Maple for access to the beta program and Eric Poisson for testing and feedback.

## HOW TO SOLVE EINSTIEN EQUATION ON MAPLE...

hi,

i'am beginers in  the maple programmation, i want to solve the einstien equation in the spherical coordinate,

## How can I see a list of the metric accepted in the...

I would like to see the list of metrics recognize by Maple with their acornym.  For example,:

>Setup(coordinates = spherical, metric = kerr)

## How to define a metric tensor in 3-d (r, theta, p...

Helle everybohy,

I need to setup a metric tensor in 3-d but with the varalble r, and .  So I try this:

>with(Physics); Setup(mathematicalnotation = true, dimension = 3)

>Setup(coordinates = spherical[r, theta, varphi], metric = M)

where M is the metric that I need to use.  But the last command does not work.  Il I don't write [r, theta, varphi], it work but it's r, theta and t.

Mario Lemelin

mario.lemelin@cgocable.ca

## How to do symbolic calculation in GR...

Hello everybody,

In the linearised theory of gravity, I want to do some symbolic calculations.  First, I need to set that:

Then I want to see how the Christoffel symbols will change by putting the above in this:

Any hint someone?  I really appreciate the help for learning the Physics package.  Thank you in advance.

Mario

## d'Alembertian in General Relativity don't give wah...

I have to prove the following:

So I do not need the explicit derivative of the function (r,t) . The metric is:

ds^2=(1-rg/r)*dt^2-(1-rg/r)^(-1)*dr^2

I am in the case of a collapsing star that emit radiation during the collapsing.  And I do not need to have a rotating black hole so that the reason I dont have dt*dr term in the metric, and I fix and .  So if you look in the Maple file attach to this post, I don't manage to obtain what I need to prove the equality between the two aspect of the same calculation.

Plese, take into account that I am sort of novice with the Physcis package and that the question is not part of an exam.

Mario Lemelin

dAlembertian.mw

## Computer Algebra in Theoretical Physics: the IOP...

by: Maple

Below is the worksheet with the whole material presented yesterday in the webinar, “Applying the power of computer algebra to theoretical physics”, broadcasted by the “Institute of Physics” (IOP, England). The material was very well received, rated 4.5 out of 5 (around 30 voters among the more than 300 attendants), and generated a lot of feedback. The webinar was recorded so that it is possible to watch it (for free, of course, click the link above, it will ask you for registration, though, that’s how IOP works).

Anyway, you can reproduce the presentation with the worksheet below (mw file linked at the end, or the corresponding pdf also linked with all the input lines executed). As usual, to reproduce the input/output you need to have installed the latest version of Physics, available in the Maplesoft R&D Physics webpage.

 Why computer algebra? ... and why computer algebra? We can concentrate more on the ideas instead of on the algebraic manipulations   We can extend results with ease   We can explore the mathematics surrounding a problem   We can share results in a reproducible way
 Representation issues that were preventing the use of computer algebra in Physics Notation and related mathematical methods that were missing: coordinate free representations for vectors and vectorial differential operators, covariant tensors distinguished from contravariant tensors, functional differentiation, relativity differential operators and sum rule for tensor contracted (repeated) indices Bras, Kets, projectors and all related to Dirac's notation in Quantum Mechanics   Inert representations of operations, mathematical functions, and related typesetting were missing:   inert versus active representations for mathematical operations ability to move from inert to active representations of computations and viceversa as necessary hand-like style for entering computations and textbook-like notation for displaying results   Key elements of the computational domain of theoretical physics were missing:   ability to handle products and derivatives involving commutative, anticommutative and noncommutative variables and functions ability to perform computations taking into account custom-defined algebra rules of different kinds (commutator, anticommutator and bracket rules, etc.)

Examples

 The Maple computer algebra environment

Classical Mechanics

 Inertia tensor for a triatomic molecule

Classical Field Theory

 *The field equations for the  model
 *Maxwell equations departing from the 4-dimensional Action for Electrodynamics
 *The Gross-Pitaevskii field equations for a quantum system of identical particles

Quantum mechanics

 *The quantum operator components of   satisfy
 Quantization of the energy of a particle in a magnetic field

Unitary Operators in Quantum Mechanics

 *Eigenvalues of an unitary operator and exponential of Hermitian operators

Properties of unitary operators

Consider two set of kets  and , each of them constituting a complete orthonormal basis of the same space.

One can always build an unitary operator  that maps one basis to the other, i.e.:

 *Verify that  implies on
 *Show that is unitary
 *Show that the matrix elements of  in the  and   basis are equal
 Show that  and have the same spectrum

Schrödinger equation and unitary transform

Consider a ket  that solves the time-dependant Schrödinger equation:

and consider

,

where  is a unitary operator.

Does  evolves according a Schrödinger equation

and if yes, which is the expression of ?

 Solution

Translation operators using Dirac notation

In this section, we focus on the operator

 Settings
 The Action (translation) of the operator  on a ket
 Action of  on an operator

General Relativity

 *Exact Solutions to Einstein's Equations

*"Physical Review D" 87, 044053 (2013)

Given the spacetime metric,

a) Compute the Ricci and Weyl scalars

b) Compute the trace of

where  is some function of the radial coordinate,  is the Ricci tensor,  is the covariant derivative operator and  is the stress-energy tensor

c) Compute the components of the traceless part of   of item b)

d) Compute an exact solution to the nonlinear system of differential equations conformed by the components of   obtained in c)

Background: paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by P. Fiziev.

 a) The Ricci and Weyl scalars
 b) The trace of
 b) The components of the traceless part of
 c) An exact solution for the nonlinear system of differential equations conformed by the components of

*The Equivalence problem between two metrics

From 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.
 Example from: A. Karlhede, "A Review of the Geometrical Equivalence of Metrics in General Relativity", General Relativity and Gravitation, Vol. 12, No. 9, 1980
 *Equivalence for Schwarzschild metric (spherical and Krustal coordinates)

Tetrads and Weyl scalars in canonical form

Generally speaking a canonical form is obtained using transformations that leave invariant the tetrad metric in a tetrad system of references, so that theWeyl scalars are fixed as much as possible (conventionally, either equal to 0 or to 1).

Bringing a tetrad in canonical form is a relevant step in the tackling of the equivalence problem between two spacetime metrics.

The implementation is as in "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, page 388:

 Residual invariance Petrov type I 0 1 0 none Petrov type II 0 0 1 0 none Petrov type III 0 0 0 1 0 none Petrov type D 0 0 0 0 remains invariant under rotations of Class III Petrov type N 0 0 0 0 1 remains invariant under rotations of Class II

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.

The determination of appropriate transformation parameters to be used in these rotations, as well as the sequence of transformations happens all automatically by using the optional argument, canonicalform of TransformTetrad .

 >
 (7.4.1)
 Petrov type I
 Petrov type II
 Petrov type III
 Petrov type N
 Petrov type D

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

## Tetrads and Weyl scalars in canonical form

by: Maple

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:

 Residual invariance Petrov type I 0 1 0 none Petrov type II 0 0 1 0 none Petrov type III 0 0 0 1 0 none Petrov type D 0 0 0 0 remains invariant under rotations of Class III Petrov type N 0 0 0 0 1 remains invariant under rotations of Class II

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.

 >
 (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)

 >
 (3)

The Weyl scalars

 >
 (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)

 >
 (8)

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

 >
 (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

 >
 (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)

 >
 (12)

Check the Petrov type

 >
 (13)

 >
 (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)

 >
 (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

 >
 (19)
 >
 (20)

The Petrov type and the original tetrad

 >
 (21)
 >
 (22)

This tetrad results in the following scalars

 >
 (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

 >
 (24)

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

 >
 >
 (25)

Great!

An Example of Petrov type N

 >
 (26)
 >
 (27)
 >
 (28)

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

 >
 (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

 >
 (31)
 >
 >
 (32)

All as expected.

An Example of Petrov type D

 >
 (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

 >
 (36)
 >
 (37)

Transform the  tetrad, set it and recompute the Weyl scalars

 >
 (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"

 >
 (40)
 >
 (41)

The Weyl tensor and its scalars all vanish:

 >
 (42)
 >
 (43)
 >
 >

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

## Using the Physics package for special relativity...

Hello everybody,

I want to use the physics package in the context of special relativity.  I am mostly interrested with the Lorenzts transformations, not having to unprotect gamma all the time, the 4-vectors for space, time and momentum in (ct,x,y).  For exemple, to be able to calculate the invariant (ct)^2-(r^2).  A small document (or worksheet) would be very nice of you.

I understande that someone could say that I want to use a gun to kill a fly.  But this is a process that will lead me in using it in general relativity and using tensors in my calculations.  It would be very interresting to have a Student,Physics package.  Don't you think so?

```--------------------------------------
Mario Lemelin```
```Maple 18 Ubuntu 13.10 - 64 bitsMaple 18 Win 7 -  64 bits
messagerie : mario.lemelin@cgocable.ca
téléphone :  (819) 376-0987```

## General Relativity using Computer Algebra

by: Maple

I was recently asked about performing some General Relativity computations from a paper by Plamen Fiziev, posted in the arXiv in 2013. It crossed my mind that this question is also instrumental to illustrate how these General Relativity algebraic computations can be performed using the Physics package. The pdf and mw links at the end show the same contents but with the Sections expanded.

General Relativity using Computer Algebra

Problem: for the spacetime metric,

a) Compute the trace of

where  is some function of the radial coordinate,  is the Ricci tensor,  is the covariant derivative operator and  is the stress-energy tensor

b) Compute the components of the traceless part of   of item a)

c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of   obtained in b)

Background: The equations of items a) and b) appear in a paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by Plamen Fiziev, a Maple user.  These equations model a problem in the context of a Branse-Dicke theory with vanishing parameter  The Brans–Dicke theory is in many respects similar to Einstein's theory, but the gravitational "constant" is not actually presumed to be constant - it can vary from place to place and with time - and the gravitational interaction is mediated by a scalar field. Both Brans–Dicke's and Einstein's theory of general relativity are generally held to be in agreement with observation.

The computations below aim at illustrating how this type of computation can be performed using computer algebra, and so they focus only on the algebraic aspects, not the physical interpretation of the results.

 a) The trace of
 b) The components of the traceless part of
 c) An exact solution for the nonlinear system of differential equations conformed by the components of

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

## The Physics project at Maplesoft

by:

(Presentation in Spain a month ago with a full description of the project and its current status)

A computational environment for Physicists

"Algebraic manipulations in Physics and related numerical exploration and visualization come together within computer algebra systems"

 Project background
 Three reasons for the underuse of Computer Algebra Systems in Physics
 The Physics project goals
 Status of things in Maple 17
 Examples

Edgardo S. Cheb-Terrab
Physics, Maplesoft