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HELP FOR: Killing Operators and Functions
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SYNOPSIS:
- The operator KillingTest[] can be used to test if a vector is a Killing
or homothetic Killing vector. It takes as its argument the name of a
vector (a single index GRTensorII object). For example:
> grcalc ( KillingTest[v] );
Displaying the result of this calculation (using grdisplay()) will indicate
if the vector was determined to be a Killing vector.
A Killing vector, k , is defined to be a vector which satisfies the
equation:
k + k - 2 Phi g = 0 . (1)
a;b b;a ab
If Phi = 0, the vector is a proper Killing vector (or simply Killing
vector);
If Phi = constant, the vector is called `homothetic'; and
If Phi is a general function of the coordinates then the vector is
a `conformal' Killing vector.
a
( In general Phi = Expansion(k)/ndim = k /ndim )
;a
- NOTE: The determination of whether or not a vector is a Killing vector
requires that Eq (1) holds. The left-hand side of this equation is
stored in the GRTensorII object ktest[v](dn,dn). In order to ensure
that the algorithm work properly, it is necessary to ensure that
this object is simplified as much as possible using gralter().
The conformal factor is stored in the GRTensorII object conf[v], and
if this can not be evaluated to zero the vector is labelled as a
conformal/homothetic Killing vector rather than a Killing vector.
Thus it is important that this object also be simplified as much as
possible before using KillingTest.
If upon examining ktest[v](dn,dn) and conf[v] it is found that these
objects have not been completely simplified, then use
> grclear ( KillingTest[v] ):
to clear the results of the KillingTest[] calculation, then apply
the necessary simplifications to conf[v] and ktest[v](dn,dn) using
gralter(), then calculate KillingTest[v] again using grcalc().
- The function KillingCoords() exists to create coordinate vectors and then
check which (if any) are Killing vectors. It does not take any arguments.
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Examples:
> qload ( schw ):
> grdef ( `v{^a} := kdelta( x{^a}, t)` ):
Created definition for v(up)
> grcalc ( KillingTest[v] ):
CPU Time = .850
> grdisplay ( KillingTest[v] ):
Killing Test Result
For the schw metric.
Killing Test = v is a Killing vector.
> KillingCoords():
Killing Coordinate Test Results
Coordinate vector = [r, theta, phi, t]
coord1(up) = [1, 0, 0, 0], coord1 is NOT a Killing vector.
coord2(up) = [0, 1, 0, 0], coord2 is NOT a Killing vector.
coord3(up) = [0, 0, 1, 0], coord3 is a Killing vector.
coord4(up) = [0, 0, 0, 1], coord4 is a Killing vector.
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See also: grt_operators.
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