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.



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 3+1 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 gamma[i, j] that is induced by g[mu, nu] 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:















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  `#mrow(msup(mi("dl"),mrow(mo("⁢"),mn("2"))),mo("="),msub(mi("γ",fontstyle = "normal"),mrow(mi("i"),mo(","),mi("j"))),mo("⁢"),msup(mi("dx"),mi("i")),mo("⁢"),msup(mi("dx"),mi("j")))`.


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 .




restart; with(Physics); Setup(coordinatesystems = cartesian)

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


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


[coordinatesystems = {X}]



`Setting lowercaselatin_is letters to represent space indices `


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


`Changing the signature of spacetime from `(`- - - +`)*` to `(`+ + + -`)*` in order to match the signature customarily used in the ADM formalism`


[ADMEquations, Christoffel3, D3_, ExtrinsicCurvature, Lapse, Ricci3, Riemann3, Shift, TimeVector, UnitNormalVector, gamma3_]


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

The induced metric gamma[mu, nu]is defined in terms of the UnitNormalVector  n[mu] and the 4D metric g[mu, nu] as


Physics:-ThreePlusOne:-gamma3_[mu, nu] = Physics:-ThreePlusOne:-UnitNormalVector[mu]*Physics:-ThreePlusOne:-UnitNormalVector[nu]+Physics:-g_[mu, nu]


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


Physics:-ThreePlusOne:-UnitNormalVector[mu] = -Physics:-ThreePlusOne:-Lapse*Physics:-D_[mu](t)


The TimeVector  is defined in terms of the Lapse  alpha and the Shift  beta[mu] and this vector  n[mu] as


Physics:-ThreePlusOne:-TimeVector[mu] = Physics:-ThreePlusOne:-Lapse*Physics:-ThreePlusOne:-UnitNormalVector[mu]+Physics:-ThreePlusOne:-Shift[mu]


The ExtrinsicCurvature  is defined in terms of the LieDerivative  of  gamma[mu, nu]


Physics:-ThreePlusOne:-ExtrinsicCurvature[mu, nu] = -(1/2)*Physics:-LieDerivative[Physics:-ThreePlusOne:-UnitNormalVector](Physics:-ThreePlusOne:-gamma3_[mu, nu])


The metric gamma[mu, nu]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 gamma[mu, nu]. In the case of Christoffel3 , Ricci3  and Riemann3,  these tensors can be defined by replacing the 4D metric g[mu, nu] by gamma[mu, nu] and the 4D Christoffel symbols GAMMA[mu, nu, alpha] by the ThreePlusOne GAMMA[mu, nu, alpha] in the definitions of the corresponding 4D tensors. So, for instance


Physics:-ThreePlusOne:-Christoffel3[mu, nu, alpha] = (1/2)*Physics:-ThreePlusOne:-gamma3_[mu, `~beta`]*(Physics:-d_[alpha](Physics:-ThreePlusOne:-gamma3_[beta, nu], [X])+Physics:-d_[nu](Physics:-ThreePlusOne:-gamma3_[beta, alpha], [X])-Physics:-d_[beta](Physics:-ThreePlusOne:-gamma3_[nu, alpha], [X]))



Physics:-ThreePlusOne:-Ricci3[mu, nu] = Physics:-d_[alpha](Physics:-ThreePlusOne:-Christoffel3[`~alpha`, mu, nu], [X])-Physics:-d_[nu](Physics:-ThreePlusOne:-Christoffel3[`~alpha`, mu, alpha], [X])+Physics:-ThreePlusOne:-Christoffel3[`~beta`, mu, nu]*Physics:-ThreePlusOne:-Christoffel3[`~alpha`, beta, alpha]-Physics:-ThreePlusOne:-Christoffel3[`~beta`, mu, alpha]*Physics:-ThreePlusOne:-Christoffel3[`~alpha`, nu, beta]



Physics:-ThreePlusOne:-Riemann3[mu, nu, alpha, beta] = Physics:-g_[mu, lambda]*(Physics:-d_[alpha](Physics:-ThreePlusOne:-Christoffel3[`~lambda`, nu, beta], [X])-Physics:-d_[beta](Physics:-ThreePlusOne:-Christoffel3[`~lambda`, nu, alpha], [X])+Physics:-ThreePlusOne:-Christoffel3[`~lambda`, upsilon, alpha]*Physics:-ThreePlusOne:-Christoffel3[`~upsilon`, nu, beta]-Physics:-ThreePlusOne:-Christoffel3[`~lambda`, upsilon, beta]*Physics:-ThreePlusOne:-Christoffel3[`~upsilon`, nu, alpha])


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 dx^mu, 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 g[0, j] components of the 4D spacetime metric.

For this purpose, define a tensor representing the differentials of the coordinates and an alias  dt = `#msup(mi("dx"),mn("0"))`


`Defined objects with tensor properties`


{Physics:-ThreePlusOne:-D3_[mu], Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-ThreePlusOne:-Ricci3[mu, nu], Physics:-ThreePlusOne:-Shift[mu], Physics:-d_[mu], dx[mu], Physics:-g_[mu, nu], Physics:-ThreePlusOne:-gamma3_[mu, nu], Physics:-gamma_[i, j], Physics:-ThreePlusOne:-Christoffel3[mu, nu, alpha], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu], Physics:-ThreePlusOne:-Riemann3[mu, nu, alpha, beta], Physics:-ThreePlusOne:-TimeVector[mu], Physics:-ThreePlusOne:-ExtrinsicCurvature[mu, nu], Physics:-ThreePlusOne:-UnitNormalVector[mu], Physics:-SpaceTimeVector[mu](X)}


"alias(dt = dx[~0]):"

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

ds^2 = (-Lapse^2+Shift[i]^2)*dt^2+2*Shift[i]*dt*dx[`~i`]+gamma_[i, j]*dx[`~i`]*dx[`~j`]

ds^2 = (-Physics:-ThreePlusOne:-Lapse^2+Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`])*dt^2+2*Physics:-ThreePlusOne:-Shift[i]*dt*dx[`~i`]+Physics:-gamma_[i, j]*dx[`~i`]*dx[`~j`]


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_

ds^2 = %g_[mu, nu]*dx[`~mu`]*dx[`~nu`]

ds^2 = %g_[mu, nu]*dx[`~mu`]*dx[`~nu`]


Decompose(ds^2 = %g_[mu, nu]*dx[`~mu`]*dx[`~nu`])

ds^2 = %g_[0, 0]*dt^2+%g_[0, j]*dt*dx[`~j`]+%g_[i, 0]*dt*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`]


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

op(2, rhs(ds^2 = dt^2*%g_[0, 0]+dt*%g_[0, j]*dx[`~j`]+dt*%g_[i, 0]*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`])) = op(3, rhs(ds^2 = dt^2*%g_[0, 0]+dt*%g_[0, j]*dx[`~j`]+dt*%g_[i, 0]*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`]))

%g_[0, j]*dt*dx[`~j`] = %g_[i, 0]*dt*dx[`~i`]


subs(%g_[0, j]*dt*dx[`~j`] = %g_[i, 0]*dt*dx[`~i`], ds^2 = dt^2*%g_[0, 0]+dt*%g_[0, j]*dx[`~j`]+dt*%g_[i, 0]*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`])

ds^2 = %g_[0, 0]*dt^2+2*%g_[i, 0]*dt*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`]


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

simplify((ds^2 = dt^2*%g_[0, 0]+2*dt*%g_[i, 0]*dx[`~i`]+%g_[i, j]*dx[`~i`]*dx[`~j`])-(ds^2 = (-Physics:-ThreePlusOne:-Lapse^2+Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`])*dt^2+2*Physics:-ThreePlusOne:-Shift[i]*dt*dx[`~i`]+Physics:-gamma_[i, j]*dx[`~i`]*dx[`~j`]))

0 = (Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0])*dt^2+2*dx[`~i`]*(%g_[i, 0]-Physics:-ThreePlusOne:-Shift[i])*dt-dx[`~i`]*dx[`~j`]*(Physics:-gamma_[i, j]-%g_[i, j])


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

eq[1] := coeff(coeff(rhs(0 = (Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0])*dt^2+2*dx[`~i`]*(%g_[i, 0]-Physics:-ThreePlusOne:-Shift[i])*dt-dx[`~i`]*dx[`~j`]*(Physics:-gamma_[i, j]-%g_[i, j])), dt), dx[`~i`]) = 0

2*%g_[i, 0]-2*Physics:-ThreePlusOne:-Shift[i] = 0


eq[2] := coeff(rhs(0 = (Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0])*dt^2+2*dx[`~i`]*(%g_[i, 0]-Physics:-ThreePlusOne:-Shift[i])*dt-dx[`~i`]*dx[`~j`]*(Physics:-gamma_[i, j]-%g_[i, j])), dt^2)

Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0]


eq[3] := coeff(coeff(rhs(0 = (Physics:-ThreePlusOne:-Lapse^2-Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]+%g_[0, 0])*dt^2+2*dx[`~i`]*(%g_[i, 0]-Physics:-ThreePlusOne:-Shift[i])*dt-dx[`~i`]*dx[`~j`]*(Physics:-gamma_[i, j]-%g_[i, j])), dx[`~i`]), dx[`~j`]) = 0

-Physics:-gamma_[i, j]+%g_[i, j] = 0


Using these equations, these quantities can all be expressed in terms of the time and space components of the 4D metric g[0, 0] and g[i, j]

isolate(eq[1], Shift[i])

Physics:-ThreePlusOne:-Shift[i] = %g_[i, 0]


isolate(eq[2], Lapse^2)

Physics:-ThreePlusOne:-Lapse^2 = Physics:-ThreePlusOne:-Shift[i]*Physics:-ThreePlusOne:-Shift[`~i`]-%g_[0, 0]


isolate(eq[3], gamma_[i, j])

Physics:-gamma_[i, j] = %g_[i, j]




[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,


[5] Arnowitt, R., Dese, S., Misner, C.W., The Dynamics of General Relativity, Chapter 7 in Gravitation: an introduction to current research (Wiley, 1962),



Examples: Decompose, gamma_



Setup(mathematicalnotation = true)

[mathematicalnotation = true]


Define  now an arbitrary tensor A


`Defined objects with tensor properties`


{A, Physics:-Dgamma[mu], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}


So A^mu 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


[signature = `- - - +`]


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

Setup(spaceindices = lowercase_is)

[spaceindices = lowercaselatin_is]


Accordingly, the 3+1 decomposition of A^mu is


Array(%id = 18446744078724512334)


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

Decompose(%g_[mu, nu])

Matrix(%id = 18446744078724507998)


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


g[mu, nu] = (Matrix(4, 4, {(1, 1) = -1, (1, 2) = 0, (1, 3) = 0, (1, 4) = 0, (2, 1) = 0, (2, 2) = -1, (2, 3) = 0, (2, 4) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = -1, (3, 4) = 0, (4, 1) = 0, (4, 2) = 0, (4, 3) = 0, (4, 4) = 1}))


Decompose(g_[mu, nu])

Matrix(%id = 18446744078724494270)


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


Physics:-gamma_[i, j] = -Physics:-g_[i, j]+Physics:-g_[0, i]*Physics:-g_[0, j]/Physics:-g_[0, 0]


The 3D space part of -g[`~mu`, `~nu`] is actually equal to the 3D metric "gamma[]^(i,j)"


Physics:-gamma_[`~i`, `~j`] = -Physics:-g_[`~i`, `~j`]


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

%g_[`~alpha`, `~mu`]*%g_[mu, beta] = KroneckerDelta[`~alpha`, beta]

%g_[`~alpha`, `~mu`]*%g_[mu, beta] = Physics:-KroneckerDelta[beta, `~alpha`]


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

Eq := Decompose(%g_[`~alpha`, `~mu`]*%g_[mu, beta] = Physics[KroneckerDelta][beta, `~alpha`], repeatedindices = false)

Matrix(%id = 18446744078132963318)


Eq := Decompose(%g_[`~alpha`, `~mu`]*%g_[mu, beta] = Physics[KroneckerDelta][beta, `~alpha`], freeindices = false)

%g_[0, beta]*%g_[`~alpha`, `~0`]+%g_[i, beta]*%g_[`~alpha`, `~i`] = Physics:-KroneckerDelta[beta, `~alpha`]


Compare with a full decomposition

Eq := Decompose(%g_[`~alpha`, `~mu`]*%g_[mu, beta] = Physics[KroneckerDelta][beta, `~alpha`])

Matrix(%id = 18446744078724489454)


Eq is a symmetric matrix of equations involving non-contracted occurrences of `#msup(mi("g"),mrow(mn("0"),mo(","),mn("0")))`, `#msup(mi("g"),mrow(mi("j"),mo(","),mo("0")))` and `#msup(mi("g"),mrow(mi("j"),mo(","),mi("i")))`. Isolate, in Eq[1, 2], `#msup(mi("g"),mrow(mi("j"),mo(","),mo("0")))`, that you input as %g_[~j, ~0], and substitute into Eq[1, 1]

"isolate(Eq[1, 2], `%g_`[~j, ~0]);"

%g_[`~j`, `~0`] = -%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0]


subs(%g_[`~j`, `~0`] = -%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0], Eq[1, 1])

-%g_[0, k]*%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0]+%g_[i, k]*%g_[`~j`, `~i`] = Physics:-KroneckerDelta[k, `~j`]


Collect `#msup(mi("g"),mrow(mi("j"),mo(","),mi("i")))`, that you input as %g_[~j, ~i]

collect(-%g_[0, k]*%g_[i, 0]*%g_[`~j`, `~i`]/%g_[0, 0]+%g_[i, k]*%g_[`~j`, `~i`] = Physics[KroneckerDelta][k, `~j`], %g_[`~j`, `~i`])

(-%g_[0, k]*%g_[i, 0]/%g_[0, 0]+%g_[i, k])*%g_[`~j`, `~i`] = Physics:-KroneckerDelta[k, `~j`]


Since the right-hand side is the identity matrix and, from , `#msup(mi("g"),mrow(mi("i"),mo(","),mi("j")))` = -`#msup(mi("γ",fontstyle = "normal"),mrow(mi("i"),mo(","),mi("j")))`, the expression between parenthesis, multiplied by -1, is the reciprocal of the contravariant 3D metric `#msup(mi("γ",fontstyle = "normal"),mrow(mi("i"),mo(","),mi("j")))`, that is the covariant 3D metric gamma[i, j], in accordance to its definition for the signature `- - - +`


Physics:-gamma_[i, j] = -Physics:-g_[i, j]+Physics:-g_[0, i]*Physics:-g_[0, j]/Physics:-g_[0, 0]





[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

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