Physics[ThreePlusOne][Lapse] - The Lapse in the 3+1 decomposition of Einstein's equations (ADM formalism)
Physics[ThreePlusOne][Shift] - The Shift in the 3+1 decomposition of Einstein's equations (ADM formalism)
Physics[ThreePlusOne][UnitNormalVector] - The UnitNormalVector in the 3+1 decomposition of Einstein's equations (ADM formalism)
Physics[ThreePlusOne][TimeVector] - The TimeVector in the 3+1 decomposition of Einstein's equations (ADM formalism)
the index, as a name representing integer numbers between 0 and the spacetime dimension, it can also be one of these numbers
optional, it can be definition, determinant, line_element, matrix, nonzero, if passed together with mu, its covariant or contravariant character is taken into account
The Lapse, Shift, UnitNormalVector and TimeVector are related quantities that, essentially, represent the 4-coordinate degrees of freedom that exist in the representation of a gravitational field (the Lapse and Shift), and identify a particular choice of 3D spacelike hypersurface in the decomposition of spacetime into 3 + 1, its space and time parts (UnitNormalVector and TimeVector). This decomposition is relevant in the context of numerical relativity, and in a Hamiltonian description of gravity.
Regarding the Lapse, Shift, you can set their values using Setup and its lapseandshift keyword; there are three possible values: standard, arbitrary, or a list of algebraic expressions representing them. The value chosen determines the values of the components of all the ThreePlusOne tensors (e.g. ExtrinsicCurvature) and so of tensorial expressions involving them (e.g. the 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=−α2⁢dt2+γi,j⁡dxi+β⁢i⁢i⁢dt⁡dxj+β⁢j⁢j⁢dt, 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 presentation in the rest of this page assumes the signature (+ + + -), automatically set when ThreePlusOne is loaded and the denominations and definitions are the standard ones of the ADM formalism (ref ). The signature (+ + + -) is physically equivalent to (- + + +), but has the computational advantage of having space indices running from 1 to 3 (instead of from 2 to 4) and the value 0 of a spacetime index pointing to position 4 (instead of position 1). With any of these two signatures, the space components (i, j) of the 4D gamma3_μ,ν are related to the space components (i, j) of the 4D spacetime metric gμ,ν by
where, as is the case of all the tensors of ThreePlusOne, the 3D metric is displayed in black instead of blue.
From a computational point of view, the Lapse is a scalar, and the Shift is a purely spatial 4D vector, that is, its contravariant timelike component is equal to 0. The UnitNormalVector is a timelike vector (all its covariant space components are zero), perpendicular to the hypersurface determined by the Lapse and Shift, and the TimeVector is neither spacelike nor timelike, but in the adapted coordinates used its contravariant space components are zero. All of them can be defined in terms of a global time function Τ⁡xi,t=constant. In a generic system of references, where a 3+1 split formulates spacetime as a 3D spacelike hypersurface parameterized by Τ that evolves in time, the Lapse, assumed to be positive, is the lapse of proper time between two 3D neighboring hypersurfaces, measured by observers moving along the direction normal to the hypersurfaces (normal, or Eulerian observers). In the adapted coordinates used in the implementation in Physics, Τ⁡xi,t=t. The Lapse is thus defined as
where ▿ is the 4D covariant derivative operator D_. The UnitNormalVector is a timelike unit vector, normal to the 3D hypersurface, i.e. it is the 4-velocity of a normal observer whose worldline is normal to the 3D hypersurface. The UnitNormalVector is defined in terms of Τ⁡xi,t as
from where n__μ⁢n⁢μ=−1. The 3D metric gamma3_, that contains information on the space part of the 4D metric g_ and that projects into this 3D spacelike hypersurface determined by Τ⁡xi,t=constant, is also expressed in terms of g_ and the UnitNormalVector as
The TimeVector is tangent to the time lines (lines of constant spatial coordinates), so it can also be expressed in terms of the global function Τ⁡xi,t identifying the 3D hypersurface as
from where tμ⁢nμ=−α. The Shift can be defined as the projection of the TimeVector onto the 3D hypersurface, i.e.
from where it follows the relationship between these three vectors
From a physical point of view, choosing Τ⁡xi,t=t as the time coordinate, the Lapse and Shift arise naturally when defining simultaneity in the general theory of relativity as the possibility of synchronizing clocks located at different points in space (ref , sec. (84)). For that synchronization to be possible, all the components g0,j of the spacetime metric must be equal to zero, and that condition can always be achieved, in any gravitational field, by an appropriate choice of coordinates (ref , sec. (84) and (97)). If, in addition, g0,0=1, the reference system is synchronous. In this referential the time lines (the TimeVector is tangent to them) are geodesics in the 4D spacetime and normal to the hypersurface t=constant (because from g0,j=0 follows that βμ=0 and so tμ and nμ are parallel), and the 4D infinitesimal interval is written as.
Under a time rescaling t⁢'=G⁡t and a general transformation of the space coordinates x⁢'i=F⁢i⁡xj,t, and taking into account γi,j=gi,j, this infinitesimal interval changes into
where now βμ≠0 and so the new time lines t=constant (and hence tμ) are no longer orthogonal to the 3D hypersurface whose geometry is described by γ__i,j. In this decomposition of the infinitesimal interval, the 4-coordinate degrees of freedom are represented by the freedom in choosing the Lapse and the Shift (corresponding to different choices of the system of references; conversely, one can depart from this general form of the infinitesimal interval and construct a transformation to a synchronous reference system, where g0,0=1 and g0,j=0, by solving a related Hamilton-Jacobi equation see  sec.(97)). In this system of references (so-called adapted coordinates system of references), the Lapse and Shift are expressed in terms of the contravariant spatial components of the 4D spacetime metric g_ as
Besides being indexed with a spacetime or a space index, the Shift, TimeVector and UnitNormalVector accept the following keywords as an index:
definition: when passed alone, gamma3_ returns its 4D definition in terms of the UnitNormalVector and the spacetime metric g_. When passed with space indices, it returns its expression in terms of the spatial components of g_.
matrix: (synonym: Matrix, array, Array, or no indices whatsoever, as in Shift) returns an Array that when indexed with numerical values from 1 to the dimension of spacetime it returns the value of each of the components of Shift[mu]. If this keyword is passed together with indices, that can be covariant or contravariant, for example Shift[~mu, matrix], the resulting array takes into account the character of the indices.
nonzero: returns a set of equations, with the left-hand side as a sequence of two positive numbers identifying the element of γ__i,j and the corresponding value on the right-hand side. Note that this set is actually the output of the ArrayElems command when passing to it the Matrix obtained with the keyword matrix.
You can change the value of the 3D metric gamma3_ by changing the value of the 4D metric g_, using Setup or g_. Regarding the geometry in this 3D hypersurface, a 3D covariant derivative, related Christoffel symbols, and the Ricci and Riemann tensors are represented respectively by the D3_, Christoffel3, Ricci3 and Riemann3 commands.
The %Lapse, %Shift, %TimeVector and %UnitNormalVector command are the inert forms of the corresponding active commands, that is represent the same object but entering them does not result in performing any computation. To perform the related computations, e.g. as if %Lapse were Lapse, use value.
⁢Systems of spacetime coordinates are:⁢X=x,y,z,t
When Physics is initialized, the default spacetime metric is of Minkowski type. You can see the metric querying Setup, as in Setup(metric), or directly entering the metric as g_, with no indices. The default signature is (- - - +) so that the timelike component in position 4, and when indexing 4D tensors the values 0 and 4 represent the same object.
When loading ThreePlusOne, a kind of letter to represent space indices is automatically set and, in order to match the literature related to the ADM formalism and numerical relativity, the signature is also changed to (+ + + -)
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
so the value 0 of a spacetime index still points to the tensor component in position 4. Note the display of the tensors of ThreePlusOne: it is the same as the analogous 4D spacetime tensors but in black instead of blue. That automatic redefinition of the signature includes a call to the Redefine command to redefine the spacetime metric accordingly:
You can always change the signature as desired using the Setup command and/or redefine the metric accordingly using Redefine. (Note: with the signature (- + + +) the value 0 of an index points to position 1, not position 4.)
You can always query about the letters used to represent spacetime and space indices via
The definition of gamma3_ and of the UnitNormalVector and the TimeVector
To set the scenario as a Schwarzschild spacetime in spherical coordinates; you can enter Setup(metric = Schwarzschild) or simpler: take advantage of abbreviations and directly using the spacetime metric g_ command
⁢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
Signature: + + + -
In this case, all the g0,j=0 so the time lines are orthogonal to the hypersurface whose metric is given by γ__μ,ν, so the Shift is equal to 0 and the UnitNormalVector and the TimeVector are parallel
The 4D gamma3_ metric is also a projector onto the 3D hypersurface; you can define a complementary projector, that projects onto the UnitNormalVector orthogonal to the 3D hypersurface (in this case parallel to the time lines due to g0,j=0)
Defined objects with tensor properties
The covariant and contravariant components of this projector
Every 4D tensor can now be decomposed into its part within the 3D hypersurface and its part perpendicular to it. For example, decompose the 4D metric itself:
Compute a tensor array for this tensorial equation: you see that each component on the left hand side is equal to the corresponding one on the right hand side
The space components of 4D tensors are not necessarily equal to the space components of the related 3D tensors. For example, for the Schwarzschild metric set, the components of the 4D Ricci tensor are all equal to zero but those of the 3D Ricci3 tensor are not:
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 dxμ, the Lapse, the Shift and the spatial components of the metric gamma3_. From this line element one can derive the relation between the Lapse, the spatial part of the Shift and the spatial part of the gamma3_ metric, on the one hand, and the g0,j components of the 4D spacetime metric. For this purpose, define a tensor representing the differentials of the coordinates and an alias dt=dx0
The expression for the line element in terms of the Lapse and Shift is (see , eq.(2.123))
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_
The second and third terms on the right-hand side are equal
Taking the difference between this expression and the one in terms of the Lapse and Shift we get
Taking coefficients, we get equations for the Shift, the Lapse and the spatial components of the 3D metric gamma3_
eq1 ≔ coeff⁡coeff⁡rhs⁡,dt,dx`~i`=0
eq2 ≔ coeff⁡rhs⁡,dt2=0
eq3 ≔ coeff⁡coeff⁡rhs⁡,dx`~i`,dx`~j`=0
Using these equations, these quantities can all be expressed in terms of the time and space components of the 4D metric g0,0 and gi,j
When computing with the components of the ThreePlusOne tensors, the default (standard) setup is to express their components in terms of the space components gi,j of the 4D metric, and the Lapse and Shift, these in turn in terms of g0,0 and the spatial part g0,i of the spacetime metric. In summary, all in terms of the components of gμ,ν. This setting is governed by the keyword lapseandshift of the Setup command; to query about its value, enter
With this setting, for instance the components of the ExtrinsicCurvature for the Schwarzschild spacetime are given by
By changing this setting for lapseandshift to arbitrary, you can see the components of the ThreePlusOne tensors in terms of gi,j and arbitrary Lapse and Shift (skipping rewriting them in terms of g0,0 and gi,0);
Where the superscripts of the Shift (β) are not powers but identify the corresponding contravariant components, and because, as we have seen, for a Schwarzschild spacetime in spherical coordinates the components of the Shift are all equal to 0, in view of the expression above, so are the components of the ExtrinsicCurvature.
As it is the case of all the tensors of the Physics package, to compute with a representation for them without actually performing the operation, use the inert form, that is the same tensor name but preceded by the percentage %. To afterwards perform the operation use value. For example,
where the value is expressed in terms of the Lapse instead of g0,0 because of having set lapseandshift = arbitrary using Setup lines above.
Using Setup you can also set the Lapse and the three space contravariant components of the Shift vector βμ to any valid algebraic expression (equivalent to choosing the system of coordinates in a particular way without changing the underlying physics of the gravitational field) by indicating the desired values as a list as follows - for example for Gauss coordinates
With this setting, and for the Schwarzschild metric set,
the components of the UnitNormalVector are given by
In order to perform computations with a 3+1 formulation of a gravitational filed, in addition to the Lapse, Shift and UnitNormalVector it is also necessary to have 3D Christoffel symbols, a covariant derivative, and the Ricci and Riemann tensors on the spacelike hypersurface whose metric is gamma3_. These mathematical objects/tensors are respectively represented on the worksheet by the commands Christoffel3, D3_, Ricci3 and Riemann3. The definitions of Christoffel3, Ricci3 and Riemann3 are the same as the definitions of the corresponding 4D tensors but for replacing g_ by gamma3_ and Christoffel by Christoffel3, followed by projecting the result over the hypersurface, using gamma3_ as a projector. For example, these are the 4D and 3D definitions
In the case of Ricci3 and Riemann3 tensors, for computational convenience, when using the keyword definition the formulas are expressed directly in terms of the 4D Riemann tensor and the ExtrinsicCurvature
ArrayElems, Christoffel, Christoffel3, convert, Coordinates, D_, d_, ExtrinsicCurvature, gamma3_, gamma_, LapseAndShiftConditions, Matrix, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Ricci, Ricci3, Riemann, Riemann3, Setup, ThreePlusOne, value
 Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.
 Alcubierre, M., Introduction to 3+1 Numerical Relativity, International Series of Monographs on Physics 140, Oxford University Press, 2008.
 Baumgarte, T.W., Shapiro, S.L., Numerical Relativity, Solving Einstein's Equations on a Computer, Cambridge University Press, 2010.
 Gourgoulhon, E., 3+1 Formalism and Bases of Numerical Relativity, Lecture notes, 2007, https://arxiv.org/pdf/gr-qc/0703035v1.pdf.
 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
The Physics[ThreePlusOne][Lapse], Physics[ThreePlusOne][Shift], Physics[ThreePlusOne][UnitNormalVector] and Physics[ThreePlusOne][TimeVector] commands were introduced in Maple 2017.
For more information on Maple 2017 changes, see Updates in Maple 2017.
The Physics[ThreePlusOne][Lapse] command was introduced in Maple 2020.
For more information on Maple 2020 changes, see Updates in Maple 2020.
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