Also available:

## Trouble with Dagger?!...

To my amazement, the following code hangs

restart:
with(Physics):
Physics:-Version();
Setup(anticommutativeprefix = psi):
Dagger(psi);

at the last line, or produces the output "Error, (in sprintf) too many levels of recursion" if I wait long enough.

What is going on? Am I just being stupid? The version line verifies to me that I am using the Physics package as shipped with Maple 2017, version date 17th of May 2017. If I outcomment the Setup line, then Dagger does not hang. Do others experience the same behaviour?

## Could someone help me with grtensor?...

I need a help from someone who knows the GRTensor commands.

If you create a tensor of rank 2 using the grdef ("F {(a) (b)}"). Until then, okay!

However, I would like to define each of the 16 tensor components and I do not know how to do.

Could someone help me?

Thank you very much!

## 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.

## can we start the value of indices in the christo...

right_file.mw

i want to start i from 0 ,1 ,2 , 3 but it does not work I rhink the defaul value is from 1 to 4. can we begin it with zero.

## indices mistake Chiristoffel symbol...

A lot of efforts I performed to obtained the right results, somehow I remain sucessful but the results are not exactly same as expected. I request you to kindly see at the christoffel symbol. I also sent the right results.

some_mistakes_in_christoffel.mw

you can see the right results in the given below attachment

right_christoffel_symbols.pdf

as you can comprare the results of both file. the obtain crystoffel are not same as the requied.

## Computational physics with DE equations ...

I need help with these two questions. anything is helpful

## Declare a multilinear operator as real valued...

I'm trying to define some multilinear forms to study differential geometry. What I need is only symbolic. My intention is symplify computations involving multilinear forms.

For example, to create an symbolic inner product "g" I used the command "define" like in this post:

http://www.mapleprimes.com/questions/203480-Define-And-Use-Abstract-Linear-Operator

So I tipped:

define(g, orderless, multilinear);

My doubt is: how can I declare that g(x,y) is always scalar?

With it I would simplify things like g(g(z,w)*x,y) = g(z,w)*g(x,y)

In my case, specifically, I type:

v:=(X,Y,Z)->g(Y,Z)*X-g(X,Z)*Y;

r:=(X,Y,Z,W)->g(v(X,Y,Z),W)-g(Y,T)*g(v(X,T,Z),W)+g(X,T)*g(v(Y,T,Z),W);

expand(r(X,Y,Z,W));

and the result is:

g(W,g(Y,Z)*X)-g(W,g(X,Z)*Y)-g(T,Y)*g(W,g(T,Z)*X)+g(T,Y)*g(W,g(X,Z)*T)+g(T,X)*g(W,g(T,Z)*Y)-g(T,X)*g(W,g(Y,Z)*T)

But I would enjoy that it were:

g(Y,Z)*g(W,X)-g(X,Z)*g(W,Y)-g(T,Z)*g(T,Y)*g(W,X)+g(X,Z)*g(T,Y)*g(W,T)+g(T,Z)*g(T,X)*g(W,Y)-g(Y,Z)*g(T,X)*g(W,T)

Is there a way to declare that g(x,y) is always scalar?

Thanks.

## 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,

## expand expression have free and dummy index?...

Hi

How Maple knows free and dummy index?

## Is it possible to plot the magnetic field of maxwe...

How to plot magnetic field of maxwell equations like vector field

## complex tensor expression with Pauli matrices...

I attempt to understand is it possible whether to calculate the next expression by means of mathematica:

where

{,}- anti-communicator;

a=0,1,2,3;

\tau^{0}-unity matrix; \tau^{i} - Pauli matrix;

and:

finally - Levi-Civita symbol; -Hermitian conjugation.

Thank you for your kind replies .

## problem with multiplication and Pauli matrices...

I need hepl.  I work with the physics paсkage and I set:

with(Physics);
with(StandardModel);
Setup(mathematicalnotation = true)

Coordinates(X)

Define(p)

Setup(su2)

CompactDisplay(p(X))

U := 1+i*(1/f)*p[a](X)*Psigma[a]

H := v*U

DD[mu] := (d_[mu]+2*i*nu_5*KroneckerDelta[mu, 0])*Psigma[0]

And next:

Trace(DD[mu]*H*DD[mu]*H) or simplify(DD[mu]*H*DD[mu]*H)

Maple speaks:

Error, (in Physics:-Trace) invalid input: union received Physics:-d_[mu] = F1, which is not valid for its 1st argument

What I am doing wrong? (f, v, nu_5 is constans; a=1,2,3 and mu=0,1,2,3, Psigma[0] is unit matrix 2x2)

And if I write:

simplify(Trace(Psigma[a]*Psigma[a]))

Maple doesn't understand that it equals six.

## problem with multiplication ...

I need yours hepl.  I work with the physics paсkage and I set:

with(Physics)

Setup(mathematicalnotation = true)

Coordinates(X)

Setup(Dgammarepresentation = standard)

Setup(spaceindices = uppercaselatin)

Define(m, m5, y, p, mm, pp)

I try to square the next value:

W := Dgamma[mu]*d_[mu]+M+Psigma[A]*aa[A]-mm*Dgamma[0]-m5*Dgamma[0]*Dgamma[5]+I*Dgamma[5]*Psigma[B]*pp[B]+I*Dgamma[5]*y

("*" is multiplication)

W*W

And after that I want to simplify it:

Simplify(W*W)

I guess that matter is owing to d_[~mu]. If I remove this term:

E:=Psigma[A]*aa[A]-mm*Dgamma[0]-m5*Dgamma[0]*Dgamma[5]+I*Dgamma[5]*Psigma[B]*pp[B]+I*Dgamma[5]*y

And if i do:

E*E

Then next error emerges:

What is it?

## 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.