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Library[MetricSearch] - a Maplet for searching the DifferentialGeometry libraries of metrics
Calling Sequences
MetricSearch()
Parameters
None
Description
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. The possible Petrov types of the Weyl tensor for a space-time metric are:
PetrovType
Plebanksi-Petrov Type. This is the Petrov type of the Plebanksi tensor.
PlebanskiTensor
Segre Type. The Segre type specifies the normal form of a linear transformation which is self-adjoint with respect to a 4-dimensional Lorentz signature metric.
SegreType
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 respesentation defines a subalgebra of . These subalgebras have been classified and are labeled .
IsotropySubalgebra, IsotropyType
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.
Currently the DifferentialGeometry library contains selected metrics from the books:
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
Example 1
We find examples of metrics which are Einstein metrics of Petrov type III. First initialize a manifold with coordinates
Start the MetricSearch Maplet.
Check the Einstein box in the PhysicalProperties-Primary Description section and type III in the Algebraic Properties-Petrov Type 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. Hit Retrieve. The fields for this library entry are assigned to
Example 2
We calculate some properties of a given metric and identify the metrics with the same properties in the library database.
We use the command RainichConditions to see if the space-time is an Einstein-Maxwell space-time.
We use the command KillingVectors to determine the dimension of the group of isometries.
Next we find the Petrov type of the metric.
Start the MetricSearch Maplet, check the Einstein-Maxwell box, the Petrov Type I box, and the Isometry Dimension 4 box. We find that this is the metric in Stephani, Kramer et al. equation [12, 21, 1].
See Also
DifferentialGeometry, Tensor, IsotropySubalgebra, IsotropyType, KillingVectors, PetrovType, PlebanskiTensor, SegreTensor, SubspaceType
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