a Maplet for searching the DifferentialGeometry libraries of metrics - Maple Programming Help

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Library[MetricSearch] - a Maplet for searching the DifferentialGeometry libraries of metrics

Calling Sequences

MetricSearch()

MetricSearch(PropList)

Parameters

PropList    - a list of  spacetime metric properties to search for.

Description

 • The DifferentialGeometry Library package contains an extensive database of spacetime metrics and matter fields which give solutions to the Einstein equations of general relativity. The command MetricSearch allows the Maple user to search the database for metrics with specified properties.  The first calling sequence launches an easy to use Maplet. The second calling sequence provides the same search capabilities in a command line format.
 • The command MetricSearch() initializes a maplet which searches the DifferentialGeometry Library for metrics with user-specified properties. The current search criteria are summarized in the following table.

 Physical Properties Primary Description.  Search for solutions to the Einstein equations with a prescribed class of energy-momentum tensors. Secondary Description. Search for metrics with a prescribed class of mathematical or physical properties. Keyword. Keyword descriptions include metric names or authors. Algebraic Properties Petrov Type. Algebraic classification of the Weyl tensor. Plebanski-Petrov Type. Algebraic classification of the Ricci tensor. This is the Petrov type of the Plebanski tensor. Segre Type.  Algebraic classification of the Ricci tensor. The Segre type specifies the normal form of a linear transformation which is self-adjoint with respect to a 4-dimensional Lorentz signature metric. Isometry Properties Isometry Dimension.  The number of Killing vectors. KillingVectors Orbit Dimension.  The number of pointwise independent Killing vectors; or the dimension of the orbit of the group of isometries.   Orbit Type.  The signature of the induced metric on the orbits.  SubspaceType Isotropy Type.   For any Lorentz signature 4-dimensional spacetime, the infinitesimal isotropy representation defines a subalgebra of $\mathrm{so}\left(3,1\right)$. These subalgebras have been classified and are labeled .

 • Once properties are selected by checking appropriate boxes, pressing the Search button will return all metrics in the DifferentialGeometry library which possess all of the indicated properties. The format of the result is a string representing the source reference and a sequence of lists indicating the equation numbers in the reference where the metrics appear. All the information in the DifferentialGeometry library for each metric can be obtained with the Retrieve command.
 • Information regarding the metric found by a search of the DifferentialGeometry library can be retrieved in one of two ways. One method is to enter the reference name (string), equation number, and the name of an initialized manifold (created in the calling worksheet using DGsetup) into the text boxes in the Retrieve section of the MetricSearch Maplet. Then pressing the Retrieve button will return a list of the spacetime fields defining the solution (e.g., metric, electromagnetic field, etc.) to the calling worksheet.  All the information in the DifferentialGeometry library for each metric can be obtained with the Retrieve command.
 • The Clear button resets all check boxes and clears all text boxes. The Close button will close the Maplet.
 • The second calling sequence accepts a list of desired properties, each specified by an equation . The possibilities are:

 metricproperty propertyvalue Related Commands "PrimaryDescription" "Dust", "Einstein", "EinsteinMaxwell", "PerfectFluid", "PureRadiation", "Vacuum" "SecondaryDescription" "Homogeneous", "PlaneWave", "PPWave", "PureRadiation", "RobertsonWalker", "SimplyTransitive", "Static" "KeywordDescription" a list of strings: author names, metric names, etc. "PetrovType" "I", "II", "III", "D", "N", "O" "PlebanskiPetrovType" "I", "II", "III", "D", "N", "O" "SegreType" For example, "[1,(111)]" "IsometryDimension" 0, 1, 2, 3, 4, 5, 6, 7, 10 "OrbitDimension" 0, 1, 2, 3, 4 "OrbitType" "Null", "Riemannian", "PseudoRiemannian" "IsotropyType "F1", "F2", "F3", "F4", "F5", "F6", "F7", "F8", "F9", "F10", "F11", "F12", "F13", "F14", "F15"

 • Currently the DifferentialGeometry library contains selected metrics from:

 1 Stephani, H.; Kramer, D.; MacCallum, M.; Hoenselaers, C.; and Herlt, E. Exact Solutions to Einstein's Field Equations. 2nd ed. Cambridge Monographs on Mathematical Physics, 2003.

2. Hawking, Stephen; and Ellis, G. F. R. (1973). The Large Scale Structure of Space-Time. Cambridge: Cambridge University Press.

Examples

 > with(DifferentialGeometry): with(Tensor): with(LieAlgebras) :with(Library):

Example 1

We find examples of metrics which are homogeneous Einstein metrics of Petrov type III. First initialize a manifold with coordinates, e.g.,

 > DGsetup([x1, x2, x3, x4], M);
 ${\mathrm{frame name: M}}$ (2.1)

Start the MetricSearch Maplet.

 M > F := MetricSearch():

Check the Einstein box in the PhysicalProperties-Primary Description section, check type III in the Algebraic Properties-Petrov Type section, check 4 in the Orbit Dimension section.

The result is "Stephani": [12, 35, 1].

Enter "Stephani" into the Reference textbox, enter [12, 35, 1] into the equation number textbox, and enter M into the manifold textbox. Press the Retrieve button. The metric is assigned to $F.$ (If matter fields were present, they would also be assigned to $F$.)

 M > F;

Example 2

We calculate some properties of a given metric and identify the metrics with the same properties in the library database.

 M > DGsetup([t, x, y, phi], M);
 ${\mathrm{frame name: M}}$ (2.2)
 M > g := evalDG(1/x^2*(dx &t dx + dy &t dy) + x^2*dphi &t dphi - (dt -2*y*dphi) &t (dt -2*y*dphi));
 ${g}{:=}{-}{\mathrm{dt}}{}{\mathrm{dt}}{+}{2}{}{y}{}{\mathrm{dt}}{}{\mathrm{dphi}}{+}\frac{{\mathrm{dx}}{}{\mathrm{dx}}}{{{x}}^{{2}}}{+}\frac{{\mathrm{dy}}{}{\mathrm{dy}}}{{{x}}^{{2}}}{+}{2}{}{y}{}{\mathrm{dphi}}{}{\mathrm{dt}}{+}\left({-}{4}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dphi}}{}{\mathrm{dphi}}$ (2.3)

We use the command RainichConditions to see if the space-time is a solution of the Einstein-Maxwell equations (electrovac spacetime).

 M > RainichConditions(g);
 ${\mathrm{true}}$ (2.4)

We use the command KillingVectors to determine the dimension of the group of isometries.

 M > KV := KillingVectors(g);
 ${\mathrm{KV}}{:=}\left[\frac{{1}}{{4}}{}{x}{}{\mathrm{D_x}}{+}\frac{{1}}{{4}}{}{y}{}{\mathrm{D_y}}{-}\frac{{1}}{{4}}{}{\mathrm{φ}}{}{\mathrm{D_phi}}{,}\frac{{1}}{{2}}{}{\mathrm{φ}}{}{\mathrm{D_t}}{+}\frac{{1}}{{4}}{}{\mathrm{D_y}}{,}{-}\frac{{1}}{{4}}{}{\mathrm{D_phi}}{,}\frac{{1}}{{2}}{}{\mathrm{D_t}}\right]$ (2.5)
 M > nops(KV);
 ${4}$ (2.6)

Next we find the Petrov type of the metric.

 M > PetrovType(g);
 ${"I"}$ (2.7)

Search for this metric in the data-base using the command line version of MetricSearch. We find that this is the metric in Stephani, Kramer et al. equation [12, 21, 1].

 M > MetricSearch(["PrimaryDescription" = "EinsteinMaxwell", "PetrovType" = "I", "IsometryDimension" = 4]);
 $\left[\left[{"Stephani"}{,}{1}{,}\left[{12}{,}{21}{,}{1}\right]\right]\right]$ (2.8)
 M > Retrieve("Stephani", 1, [12, 21, 1], manifoldname=M, output=["Metric"]);
 $\left[{-}{\mathrm{dt}}{}{\mathrm{dt}}{+}{2}{}{y}{}{\mathrm{dt}}{}{\mathrm{dphi}}{+}\frac{{{\mathrm{_a}}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}}{{{x}}^{{2}}}{+}\frac{{{\mathrm{_a}}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}}{{{x}}^{{2}}}{+}{2}{}{y}{}{\mathrm{dphi}}{}{\mathrm{dt}}{+}\left({-}{4}{}{{y}}^{{2}}{+}{{x}}^{{2}}\right){}{\mathrm{dphi}}{}{\mathrm{dphi}}\right]$ (2.9)

Example 3

Find the Goedel metric in the data-base and then retrieve it:

 M > S := MetricSearch(["Keywords" = ["Goedel"]]);
 ${S}{:=}\left[\left[{"Stephani"}{,}{1}{,}\left[{12}{,}{26}{,}{1}\right]\right]\right]$ (2.10)
 M > g := Retrieve(op(S[1]), manifoldname = M, output = ["Metric"])[1];
 ${g}{:=}{-}{{\mathrm{_a}}}^{{2}}{}{\mathrm{dt}}{}{\mathrm{dt}}{-}{{\mathrm{_a}}}^{{2}}{}{{ⅇ}}^{{x}}{}{\mathrm{dt}}{}{\mathrm{dphi}}{+}{{\mathrm{_a}}}^{{2}}{}{\mathrm{dx}}{}{\mathrm{dx}}{+}{{\mathrm{_a}}}^{{2}}{}{\mathrm{dy}}{}{\mathrm{dy}}{-}{{\mathrm{_a}}}^{{2}}{}{{ⅇ}}^{{x}}{}{\mathrm{dphi}}{}{\mathrm{dt}}{-}\frac{{1}}{{2}}{}{{\mathrm{_a}}}^{{2}}{}{{ⅇ}}^{{2}{}{x}}{}{\mathrm{dphi}}{}{\mathrm{dphi}}$ (2.11)