Physics[ThreePlusOne][gamma3_] - The gamma3_ 3D metric tensor of a hypersurface defined by Tau(x1, x2, x3, t) = constant
the indices, as names representing integer numbers between 0 and the spacetime dimension, they can also be the numbers themselves
optional, returns the matrix form for the given (covariant or contravariant) indices; if passed without indices it returns the covariant metric
optional, it can be definition, determinant, line_element, matrix, nonzero
The gamma3_[mu, nu], displayed as γμ,ν, is a computational representation for a 4D tensor whose space components are the space 3D metric of a hypersurface defined by some global function Τ⁡x1,x2,x3,t=constant. It is assumed that the hypersurface is spacelike, i.e. the corresponding 3D metric is definite positive with signature (+ + +). Note: to work with gamma3_[mu, nu], a kind of letter to represent a space index must be set first using Setup, for instance entering Setup(spaceindices = lowercaselatin_is).
gamma3_[mu, nu] is a purely spatial tensor that projects 4D tensors into purely spatial tensors, all of whose components are equal to zero when its indices are contravariant and one of them has a timelike value (i.e. contravariant 0). gamma3_[mu, nu] is defined in terms of the spacetime metric g_ and the UnitNormalVector nμ as
Note that, because the hypersurface is spacelike, the UnitNormalVector is timelike.
With one index contravariant and the other covariant, γ⁢⁢νμ projects an arbitrary 4D contravariant vector into the 3D hypersurface. To project higher rank tensors into the spatial hypersurface, each free index has to be contracted in this way. Thus, a projector normal to the hypersurface, that you can define to compute with it using Define, is given by
This projector projects an arbitrary 4D covariant vector into a corresponding timelike vector, i.e. one whose covariant space components are equal to 0.
In the 3+1 formulation, it is relevant the values of the Lapse and Shift, that you can set using Setup and its lapseandshift keyword - for that purpose see setting the Lapse and Shift using Setup.
The presentation in this page also assumes the signature (+ + + -), automatically set when ThreePlusOne is loaded. This 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 of the 4D gamma3_[mu, nu] are related to the space components of the 4D spacetime metric gμ,ν by
When the spacetime indices of gamma3_ assume integer values, it is expected they are between 0 and 3, and the corresponding value of gamma3_ is returned. The value 0 always points to the position of the timelike component, so with the signatures (- - - +) and (+ + + -) indexing with 0 is the same as indexing with 4, while with the signatures (+ - - -) and (- + + +) it is the same as indexing with 1.
An important distinction is made between the 3D spatial metric represented by Physics:-gamma_, related to the infinitesimal distance measured by one observer at a fixed point in space, expressed in terms of the contravariant spatial components of the metric g_ as γi,j=gi,j (see , sec.(84)), and the 3D spatial metric of a hypersurface defined by a fixed constant value of t, the time, represented by Physics:-ThreePlusOne:-gamma3_, related to the infinitesimal spatial distance on this hypersurface, the distance measured by a normal (Eulerian) observer (whose velocity is perpendicular to the hypersurface), and so can be obtained from the covariant 4D one by restricting it to constant time surfaces, namely γi,j=gi,j
The definition of Physics:-ThreePlusOne:-gamma3_ arises 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)). As is well known, 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 are geodesics in the 4D spacetime and normal to the hypersurface t=constant, 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 α is the Lapse and β__i is the Shift, and the new time lines t=constant are no longer orthogonal to the 3D hypersurface whose geometry is described by γ__i,j. Instead, the UnitNormalVector is. In this decomposition of the infinitesimal interval into its time and space parts, the 4-coordinate degrees of freedom are represented by the freedom in choosing the Lapse and the Shift. (Conversely, one can depart from this general form of the infinitesimal interval and construct a synchronous reference system, where g0,0=1 and g0,j=0, by solving a related Hamilton-Jacobi equation see  sec.(97)).
Regarding the geometry in this 3D hypersurface described by γ__i,j, 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 definition of these commands is obtained by replacing g_ by gamma3_ in the definitions of the 4D spacetime D_, Christoffel, Ricci and Riemann tensors. All of D3_, Christoffel3, Ricci3 and Riemann3 are also tensors in 3D when replacing spacetime indices by space indices.
You can change the value of the 3D metric gamma3_ in different ways by changing the value of the 4D metric g_, using Setup or g_ itself as explained in its help page. For example, among others, the simplest way to set the Schwarzschild metric and related coordinates is to enter g_[sc]
Besides being indexed with two indices, gamma3_ accepts 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 them spatial components of g_.
determinant: when passed alone, gamma3_ returns the determinant of the all-covariant γ__i,j . If this keyword is passed together with indices, that can be covariant or contravariant, the resulting determinant takes into account the character of the indices.
line_element: (synonym: lineelement) when passed alone, gamma3_ returns the 3D line element for the 4D current metric expressing the differentials of the coordinates using d_.
matrix: (synonym: Matrix, array, Array, or no indices whatsoever, as in gamma3_) returns a Matrix that when indexed with numerical values from 1 to the dimension of spacetime it returns the value of each of the components of gamma3_. If this keyword is passed together with indices, that can be covariant or contravariant, the resulting Matrix 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.
In turn, some predefined values for the 4D spacetime metric can be set by indexing the metric with a name or a portion of it - see g_ - and in that way you can indirectly set the values of the 3D gamma3_ metric.
The %gamma3_ command is the inert form of gamma3_, so it represents the same tensor but entering it does not result in performing any computation. To perform the related computations as if %gamma3_ were gamma3_, 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 into (+ + + -)
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.)
The definition of gamma3_ and of UnitNormalVector
The 4D and 3D line elements with the Minkowski metric
The symmetry property of gamma3_ is automatically taken into account when the indices have symbolic values
You can always query about the letters used to represent spacetime and space indices via
To set the scenario as a curved spacetime set the metric using Setup, for instance indicating the square of the spacetime interval. This can be done directly from g_. Choose for example the keyword Tolman and this also, automatically, implies on setting spherical coordinates as the differentiation variables for d_
⁢⁢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 Tolman metric in coordinates ⁢⁢⁢r,θ,φ,t
Signature: ⁢+ + + -
Use a compact display for these functions to avoid redundant repeated display of their functionality plus having derivatives displayed indexed by the differentiation variables
E⁡r⁢will now be displayed as⁢E
R⁡t,r⁢will now be displayed as⁢R
The corresponding 3D line element
The matrix form: all of gamma3_, gamma3_[matrix] and gamma3_[mu, nu, matrix] return the all-covariant matrix
Or, for the matrix components of the all contravariant γ&i;,&j;, which as a matrix is equal to the inverse of the all-covariant γμ,ν, you can use gamma3_[~mu, ~nu, matrix], with contravariant indices (prefixed by ~), or directly
The nonzero components
The nonzero components of the inverse of the metric, that is, the all-contravariant γ&i;,&j;
By replacing g_ by gamma3_ in the definitions of Christoffel you obtain the corresponding 3D tensor, i.e., the 3D Christoffel symbols, that allow you to express covariant derivatives in a 3D curved space (not spacetime)
This actually the definition of Christoffel3
The same relationship / definition, also for Ricci3 and Riemann3, is obtained by indexing these tensors with the keyword definition, or using convert with g_ as second argument
The 3D space components of the 4D tensors Christoffel, Ricci and Riemann, that are not tensors in 3D, are not necessarily the same as the 3D space components of Christoffel3, Ricci3 and Riemann3 that are tensors in 3D space. For example, set the scenario as a Schwarzschild spacetime in spherical coordinates; you can do this entering Setup(metric = Schwarzschild) or in the simpler form taking advantage of abbreviations and directly using the spacetime metric g_ command
⁢⁢The Schwarzschild metric in coordinates ⁢⁢⁢r,θ,φ,t
For this spacetime, the components of the Ricci tensor are all equal to 0
while the diagonal space components of the 4D tensor Ricci3 in a Schwarzschild spacetime are different from zero
To work with only the 3D space components that constitute a 3D tensor, use space indices. To see the matrix form restricted to 3D, for example:
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,
ArrayElems, Christoffel, Christoffel3, convert, Coordinates, D_, d_, ExtrinsicCurvature, 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, TimeVector, UnitNormalVector, 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.
The Physics[ThreePlusOne][gamma3_] command was introduced in Maple 2017.
For more information on Maple 2017 changes, see Updates in Maple 2017.
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