Maple provides a stateoftheart 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.


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 3hypersurfaces 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 3dimensional 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 threedimensional 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 energymomentum 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 3dimensional 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 righthand 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.




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 timelike 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 timelike 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 noncontracted 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 righthand 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) 
> 



