ADMEquations - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim

Physics[ThreePlusOne][ADMEquations] - The ADM Equations as the 3+1 decomposition of the Einstein equations

Calling Sequence


ADMEquations(keyword = ...)


keyword = ...


optional, the left-hand side can be inert and the right-hand side can be true (default) or false; or the left-hand side can be output and the right hand side can be 4D (default) or 3D, to return with the free indices of spacetime or space kind.



The ADMEquations command returns the ADM equations in inert form, that is, the 3+1 decomposition of Einstein's equations expressed in terms of the Lapse, Shift, the 3D metric gamma3_, the ExtrinsicCurvature and the EnergyMomentum tensor.


You can set the values of the Lapse and Shift using Setup and its lapseandshift keyword. There are three possible values for lapseandshift: standard, arbitrary, or a list of algebraic expressions representing the Lapse and Shift. The value chosen determines the values of the components of all the ThreePlusOne tensors and so of tensorial expressions involving them (e.g. the ExtrinsicCurvature and ADMEquations). Those components are always computed first in terms of the Lapse and Shift and the space part gi,j of the 4D metric, then in a second step, if lapseandshift = standard, the Lapse and the Shift are replaced by their expressions in terms of the g0,μ part of the 4D metric, according to α2=g0,0-1 and βj=g0,j, or if lapseandshift was set passing a list to Setup then in terms of the values indicated in that list (these values are not used to set the g0,μ part of the 4D metric). When lapseandshift = arbitrary, the second step is not performed and the Lapse and Shift evaluate to themselves, representing an arbitrary value for them. In the three cases, standard, arbitrary, or a list of algebraic expressions, the components of the ThreePlusOne tensors are computed in terms of a metric with line element ds2=α2dt2+γi,jdxi+βiidtdxj+βjjdt, where α and βi have the Lapse and Shift values mentioned, and γi,j=gi,j is the ThreePlusOne:-gamma3_ metric of the 3D hypersurface. This design permits working with any 4D metric set and, without changing its value, experimenting with different values of the Lapse and Shift (different values of g0,μ) for the 3+1 decomposition of Einstein's equations. See also LapseAndShiftConditions.


The output of ADMEquations consists of a column vector with four equations: the Hamiltonian and momentum constraints, and the evolution equations for the ExtrinsicCurvature and for the 3D gamma3_ metric. These equations are expressed using the inert form of the tensors involved in order to allow for different kinds of manipulations. Once the system has the desired form, the inert representations can be transformed into active ones and the computations represented be performed using the value command, possibly followed by convert to g_, with or without the option only = {Lapse, Shift}, to express the Lapse and Shift in terms of the spacetime metric. Before or after that you can also use SumOverRepeatedIndices, TensorArray or Decompose to do the corresponding manipulations on these ADM equations. These different steps can be performed in any preferred order.


Alternatively, by passing the option inert = false, first all occurrences of the Lapse and Shift get rewritten in terms of the spacetime metric g_, and then the inert representations are transformed into active using value.


The free indices in the equations returned are 4D spacetime indices. To request these equations with 3D space indices use the option output = 3D (on the worksheet, when you input 3D it will get converted to the product 3*D, which for the purpose of requesting 3D indices works fine, the same way as the symbol `3D`)


To define the EnergyMomentum tensor that enters the equations returned by ADMEquations, use the Define command.


with(Physics): with(ThreePlusOne):


Setting lowercaselatin_is letters to represent space indices

Defined as 4D spacetime tensors see ?Physics,ThreePlusOne,γμ,ν,μ,Γμ,ν,α,Rμ,ν,Rμ,ν,α,β,βμ,nμ,tμ,Κμ,ν

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

Systems of spacetime coordinates are:X=x,y,z,t



Setup(mathematicalnotation = true);



Set the Schwarzschild metric in spherical coordinates (see g_)



Systems of spacetime coordinates are:X=r,θ,φ,t

Default differentiation variables for d_, D_ and dAlembertian are:X=r,θ,φ,t

The Schwarzschild metric in coordinates r,θ,φ,t

Parameters: m

Signature: + + + -




Check the current (default) value of the EnergyMomentum tensor




This value coincides with the value of this tensor for a Schwarzschild solution in vacuum. Check the definition of the EnergyMomentum tensor, in a curved spacetime, it is the source of the gravitational field




The above are actually Einstein's equations with the Einstein tensor on the right-hand side. Compute a tensor array with the equation components of this tensorial equation




The ADM equations are the 3+1 split of Einstein's equations, presented by ADMEquations as a column vector of four equations: first the two constraints then the two evolution equations




Where  represents the LieDerivative and t and β indexing  respectively represent the TimeVector and the Shift. Note the display of tensors in gray; they are all inert representations - to activate them use value.

Having set the Schwarzschild metric, if we now compute the components of each of these equations we should obtain 0=0 for all of them. For that purpose, first compute these equations in active form (not inert as in the above), then compute a tensor array for them




TensorArray((8), simplifier=simplify);



To obtain the 3D version of these equations, pass the optional argument 3D that you can input as the product 3*D or as the symbol `3D`

ADMEquations(inert = false, output = 3*D);



Check the repeated and free indices of each of these four equations

map(Check, (10), all);

The repeated indices per term are: ...,...,..., the free indices are: ...

The products in the given expression check ok.

The repeated indices per term are: ...,...,..., the free indices are: ...

The products in the given expression check ok.

The repeated indices per term are: ...,...,..., the free indices are: ...

The repeated indices per term are: ...,...,..., the free indices are: ...



In this result we see the free indices i,j are 3D space indices, and all the repeated ones are 4D spacetime indices. For this 3D version of the ADM equations we also expect all of them identically satisfied and equal to 0

TensorArray((10), simplifier=simplify);



Consider now the Lemaitre-Tolman-Bondi metric, that in the Maple database of solutions to Einstein's equations can be retrieved directly using a portion of the word Tolman as an index to the metric g_



The Tolman metric in coordinates r,θ,φ,t

Parameters: Rt,r,Er

Signature: + + + -





Erwill now be displayed asE

Rt,rwill now be displayed asR


The EnergyMomentum tensor for this metric is given by

EnergyMomentum[~mu, nu] = -rho__M(t, r)*g_[`~mu`, 0]*g_[nu, `~0`]-rho__Lambda*g_[`~mu`, nu];



where ρMt,r is the matter density, uμ=δ0μ is the 4-velocity of the matter that is comoving and we keep the vacuum energy ρΛ=constant for illustration purposes only. Define this tensor and use a CompactDisplay for ρMt,r


Defined objects with tensor properties




ρ__Mt,rwill now be displayed asρ__M


Take now the 4D form of Einstein's equations and derive an expression for ρMt,r as a function of R,E

isolate((5), Einstein[mu, nu]);



EQ4 := TensorArray((5));



The relationship we are looking for is in EQ44,4 and can be simplified further if we introduce Mr, the gravitational mass within the comoving sphere at radius r

M(r) = -1/2*(-(diff(R(t, r), t))^2+2*E(r))*R(t, r);



So simplify the expression obtained for ρM+ρΛ introducing Mr and eliminating E (see simplify, siderelations)

simplify(EQ4[4,4], {(20)}, {E(r)});



Consider now the 3 + 1 ADM equations equivalent to the 4D Einstein equations EQ4, and show that it results in the same expression for ρM+ρΛ and that both systems of equations are one and the same. Start from the active form of the ADM equations

eq := ADMEquations(inert = false);



The expression for ρM+ρΛ in terms of Mr is obtained now from eq1







simplify(isolate((24), rho__M(t, r) + rho__Lambda));



simplify((25), {(20)}, {E(r)});



The second equation, eq2, is identically satisfied




TensorArray(eq[2], simplifier = simplify);



The fourth equation, eq4, is also identically satisfied (basically, this is the definition of the ExtrinsicCurvature)




TensorArray(eq[2], simplifier = simplify);



So it is in eq3 where the evolution of the system is encoded, in terms of the functions ρM,E,R and their derivatives



EQ3 := TensorArray(eq[3], simplifier = simplify);



To demonstrate that this system of equations EQ3 together with the constraint eq1 (23), that is the ADMEquations decomposition, is equivalent to the 4D system of equations EQ4 obtained directly from Einstein's equations in (19), it suffices to show that each of these two systems entirely reduces the other one. For this purpose, convert these arrays of equations to sets of equations

EQ4 := convert(EQ4, setofequations);



EQ3 := convert(EQ3, setofequations) union {(25)};



The differential reductions can now be performed using PDEtools:-ReducedForm

simplify(PDEtools:-ReducedForm(EQ4, EQ3));



The reduction the other way around

simplify(PDEtools:-ReducedForm(EQ3, EQ4));



See Also

Check, Decompose, Einstein, EnergyMomentum, ExtrinsicCurvature, Lapse, LapseAndShiftConditions, LieDerivative, PDEtools:-ReducedForm, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Ricci3, Riemann3, Setup, Shift, SumOverRepeatedIndices, TensorArray, ThreePlusOne, TimeVector, UnitNormalVector, value



[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,