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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)

Calling Sequence

Lapse

Shift[ν]

UnitNormalVector[μ]

TimeVector[μ]

Shift[keyword]

UnitNormalVector[keyword]

TimeVector[keyword]

Parameters

μ

-

the index, as a name representing integer numbers between 0 and the spacetime dimension, it can also be one of these numbers

keyword

-

optional, it can be definition, determinant, line_element, matrix, nonzero, if passed together with mu, its covariant or contravariant character is taken into account

Description

• 

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=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 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 [5]). 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

γi,j=gi,j

  

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 Τxi,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, Τxi,t=t. The Lapse is thus defined as

α=μΤ·μΤ12

  

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 Τxi,t as

nμ=αμΤ

  

from where n__μnμ=−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 Τxi,t=constant, is also expressed in terms of g_ and the UnitNormalVector as

γμ,ν=nμnν+gμ,ν

  

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 Τxi,t identifying the 3D hypersurface as

tμμΤ=1

  

from where tμnμ=α. The Shift can be defined as the projection of the TimeVector onto the 3D hypersurface, i.e.

βμ=γμ,νtν

  

from where it follows the relationship between these three vectors

tμ=αnμ+βμ

• 

From a physical point of view, choosing Τxi,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 [1], 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 [1], 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.

ds2=dt2+gi,jdxidxj

  

Under a time rescaling t'=Gt and a general transformation of the space coordinates x'i=Fixj,t, and taking into account γi,j=gi,j, this infinitesimal interval changes into

ds2=α2+βiβidt2+2βidtdxi+γi,jdxidxj

  

where now βμ0 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 [1] 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

g0,0=1α2,g0,i=βiα2,γi,j=gi,j+βiβjα2

• 

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.

Examples

withPhysics:

Setupmathematicalnotation=true,coordinates=cartesian

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

_______________________________________________________

coordinatesystems=X,mathematicalnotation=true

(1)

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 (+ + + -)

withThreePlusOne:

_______________________________________________________

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

_______________________________________________________

(2)

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:

g_

g_μ,ν=1000010000100001

(3)

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

Setupspacetimeindices,spaceindices

spaceindices=lowercaselatin_is,spacetimeindices=greek

(4)

The definition of gamma3_ and of the UnitNormalVector and the TimeVector

gamma3_definition

γμ,ν=nμnν+gμ,ν

(5)

gamma3_i,j,definition

γi,j=gi,j

(6)

UnitNormalVectordefinition

nμ=αμt

(7)

TimeVectordefinition

tμ=αnμ+βμ

(8)

ExtrinsicCurvaturedefinition

Κμ,ν=nγμ,ν2

(9)

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

g_sc

_______________________________________________________

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