LieD doesn't seem to work with WeightedWithBasis

[Leo Stein]

Hello again, and sorry to be filling your inboxes,
When I started using xAct, I had copied the commands for defining a
manifold and a metric from one of the examples; this example had
passed to DefMetric the option WeightedWithBasis->AIndex, and I just
kept it. I'm not sure what this option is for. Anyway, I now find that
I can not expand Lie derivatives in terms of the covariant derivative
associated with the aforementioned metric. Mathematica complains:
"$RecursionLimit::reclim : Recursion depth of 256 exceeded."

I have found that removing the option WeightedWithBasis->AIndex from
the call to DefMetric makes LieD[v,CD][T[...]] work as expected.

First, I don't know what WeightedWithBasis is for, and the
documentation does not seem to explain it very well. Could someone
knowledgeable in this group explain what this option does?

Second, this issue with LieD[v,CD] is likely a bug -- maybe an
underspecified definition which leads to circular rule applications.

Thanks,
Leo Stein

[JMM]

Hi Leo,

On Mar 17, 7:16 pm, Leo Stein <leo.st...@gmail.com> wrote:
> Hello again, and sorry to be filling your inboxes,
> When I started using xAct, I had copied the commands for defining a
> manifold and a metric from one of the examples; this example had
> passed to DefMetric the option WeightedWithBasis->AIndex, and I just
> kept it. I'm not sure what this option is for.

Let me explain briefly. In order to compute the determinant of a
metric you need to use a particular basis. If you change from one
basis to another basis then the determinant of a metric changes as the
(Jacobian) determinant squared of the matrix of the change of basis.
Therefore when you say that the Levi-Civita connection CD of the
metric g obeys CD[-a][ Detg[] ] = 0, this is a property which is
obeyed only for a given basis, and not for others. The option
WeightedWithBasis allows you to specify which basis is that. Of
course, there is only a real choice in xCoba, because xTensor only
handles the generic coordinated basis AIndex. With xCoba you can
handle any number of bases, both coordinated and noncoordinated. In
other words, there are Jacobians in xCoba, but not in xTensor.

Once you have defined that, you can declare tensors which are
densities in a given basis, for instance

DefTensor[ T[a,b], M, WeightOfTensor -> -2 AIndex ]

The power of this notation is that you can have tensors which are
densities in various bases at the same time, and this is represented
in xAct by weights which are linear combinations of basis names. For
instance the Jacobian from basis B1 to basis B2 is naturally a scalar
density with weight B2 - B1.

> Anyway, I now find that
> I can not expand Lie derivatives in terms of the covariant derivative
> associated with the aforementioned metric. Mathematica complains:
> "$RecursionLimit::reclim : Recursion depth of 256 exceeded."

I have not been able to reproduce this problem. Please, send me a
small notebook with the error.

> I have found that removing the option WeightedWithBasis->AIndex from
> the call to DefMetric makes LieD[v,CD][T[...]] work as expected.

I guess it will work as expected for 0-weight tensors, but not for
densities.

> First, I don't know what WeightedWithBasis is for, and the
> documentation does not seem to explain it very well.

Densities are documented in section 4 of xCobaDoc.nb, but you are
right that the interaction between derivatives and densities is not
documented at all.

> Could someone knowledgeable in this group explain what this option does?
>
> Second, this issue with LieD[v,CD] is likely a bug -- maybe an
> underspecified definition which leads to circular rule applications.

Could be. Send me an example and I'll have a look.

Cheers,
Jose.

>
> Thanks,
> Leo Stein

[Leo Stein]

Thanks for the help. I've uploaded LieDtest.nb to the group. There I define two metrics on the same manifold (though this is just to illustrate that one works and the other does not; it doesn't have anything to do with having two metrics, as far as my investigation goes). When LieD is called with the derivative operators from the different metrics, one works and one does not.

Cheers

Leo

[Leo Stein]

Hi xActers,

I think I figured out what is going on.

Expanding a LieD calls lieDcovDdiff, which has in it the following code:

... +

If[WeightedCovDQ[covd],

 With[{i1 = DummyIn[vb], i2 = DummyIn[vb], 

   metric = MetricOfCovD[covd]},

  WeightOf[expr, WeightedWithBasis[covd]]/2 expr

   metric[i1, i2] LieD[vector, covd][metric[-i1, -i2]]],

 0]

This recursively expresses a Lie derivative of some tensor density in terms of the usual terms you calculate plus a contribution for have a different density. The recursion is LieD[][metric[-i1,-i2]] and so has a weight of 0. This means that when _this_ lie derivative is expanded, there should be no additional term (since WeightOf[metric[...], ...] should be zero).

The issue seems to be that Mathematica tries to expand LieD[] and recurse before evaluating WeightOf and finding that the whole term should be 0.

I have the following workaround: replace the first argument to If:

WeightedCovDQ[covd]

should be replaced with

WeightedCovDQ[covd] && WeightOf[expr, WeightedWithBasis[covd]] != 0

JMM, since you were not able to reproduce this bug, does this mean that my Mathematica evaluation order is different from yours, or I have a different version of the code?

Cheers

Leo

[JMM]

Hi Leo,

Sorry for the delay in replying. Yes, I had forgotten that I had fixed
this a few days after the last release. I used essentially the same
solution you suggest. This is the diff between the last release and my
current development version :

7174c7288,7290
<
With[{i1=DummyIn[vb],i2=DummyIn[vb],metric=MetricOfCovD[covd]},
---
>
> With[{i1=DummyIn[vb],i2=DummyIn[vb],metric=MetricOfCovD[covd],
> weight=WeightOf[expr,WeightedWithBasis[covd]]},
7176,7177c7292,7294
< WeightOf[expr,WeightedWithBasis[covd]]/2 expr

metric[i1,
< i2]LieD[vector,covd][metric[-i1,-i2]]],

---
> If[weight\[NotEqual]0,
> weight/2 expr metric[i1,i2]LieD[vector,covd][metric[-i1,-i2]],
> 0]],

Thanks for reporting the problem and the solution.

Cheers,
Jose.

> On Thu, Mar 18, 2010 at 10:21 AM, Leo Stein <leo.st...@gmail.com> wrote:
> > Thanks for the help. I've uploaded LieDtest.nb to the group. There I define
> > two metrics on the same manifold (though this is just to illustrate that one
> > works and the other does not; it doesn't have anything to do with having two
> > metrics, as far as my investigation goes). When LieD is called with the
> > derivative operators from the different metrics, one works and one does not.
>
> > Cheers
> > Leo
>