Example 1.
We find the Killing vectors for the metric for the Poincare half-plane.
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P >
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| (2.1) |
P >
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| (2.2) |
We check the result using the LieDerivative command.
P >
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| (2.3) |
Alternatively, we can use the keyword argument output = to calculate the general Killing vector in terms of 3 arbitrary constants.
P >
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| (2.4) |
We calculate the structure equations for this Lie algebra of Killing vectors using the LieAlgebraData command, initialize the resulting Lie algebra, and check that it is semi-simple with the LieAlgebra Query command.
P >
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| (2.5) |
P >
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| (2.6) |
Sym >
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| (2.7) |
Example 2.
In this example we consider a metric which depends upon 2 parameters and . We find the Killing vectors of for different values of these parameters.
P >
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| (2.8) |
M >
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| (2.9) |
To invoke case-splitting, we use the keyword argument parameters. With this calling sequence KillingVectors will return a sequence of Killing vectors and, as the last element in the sequence, the special paramenter values used to calculate these Killing vectors.
M >
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| (2.10) |
Four cases are found:
M >
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| (2.11) |
M >
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| (2.12) |
Case 1. 6 Killing vectors
M >
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| (2.13) |
Case 2. { 1 Killing vector
M >
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| (2.14) |
Case 3. 2 Killing vectors
M >
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| (2.15) |
Case 4. 1 Killing vector
M >
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| (2.16) |
The case defines the flat Euclidean metric. We can exclude this case by using the keyword argument auxiliaryequations.
M >
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| (2.17) |
Example 3.
In this example we consider a metric which depends upon an arbitrary function. We find the Killing vectors of for different values of this function.
P >
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| (2.18) |
M >
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| (2.19) |
We exclude the case f(x) = 0.
M >
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| (2.20) |
In the generic case where f(x) is arbitrary there is just 1 Killing vector. When f(x) is a constant there are 2 Killing vectors. If f(x) is a generic quadratic function there are 2 Killing vectors while if f(x) is a perfect square there are 3 Killing vectors.
Let's check this last case by direct calculation.
M >
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| (2.21) |
M >
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| (2.22) |
Example 4.
We use an orthonormal frame to find the Killing vectors for the Godel metric. First we set up a 4-dimensional spacetime M with coordinates . Then we define a coframe, calculate the structure equations for this coframe, and initialize the result as a frame called "Godel".
Sym >
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| (2.23) |
M >
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| (2.24) |
M >
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| (2.25) |
M >
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| (2.26) |
Here is the Godel metric, first in the orthonormal frame and then in the coordinate frame (see Exact Solutions, page 178).
Godel >
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| (2.27) |
Godel >
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| (2.28) |
M >
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| (2.29) |
Here are the 5 Killing vectors for the Godel metric in the adapted frame.
M >
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| (2.30) |
Here are the structure equations for the Lie algebra of Killing vectors.
Godel >
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| (2.31) |
Godel >
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| (2.32) |
We can use the LieAlgebras package to decompose this Lie algebra. The command Decompose returns a basis in which the algebra is decomposed into a direct sum of subalgebras.
Sym >
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| (2.33) |
Sym >
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| (2.34) |
Sym >
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| (2.35) |
L1 >
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| (2.36) |