#19823: Addcode for computation of the Schouten, Cotton, and Cotton-York
tensors of
a riemannian metric
-------------------------------------+-------------------------------------
Reporter: pang | Owner: pang
Type: enhancement | Status: needs_work
Priority: trivial | Milestone:
Component: geometry | Resolution:
Keywords: differentiable | Merged in:
manifold, Riemannian metric, | Reviewers: Eric Gourgoulhon
Cotton tensor, Cotton-York | Work issues:
tensor, Schouten tensor | Commit:
Authors: pang | 99edbe55fa1793f1db80c11c129716d96794bd08
Report Upstream: N/A | Stopgaps:
Branch: |
public/manifolds/Schouten_Cotton_York|
Dependencies: #19209 |
-------------------------------------+-------------------------------------
Changes (by egourgoulhon):
* status: needs_review => needs_work
Comment:
The introduction of a class attribute for the derived objects is a good
idea! However, I would suggest that its name is changed from
`DERIVED_OBJECTS` to `_derived_objects`. Indeed, it seems that in Sage,
capital letters are not used for attributes. Moreover, the heading
underscore indicates a "private" attribute, not exposed to a tab search.
Otherwise, a user searching for some derivative via `g.der` + TAB, might
find this attribute.
Here are some remarks about each of the three tensors:
- '''Schouten tensor:'''
The LaTeX formula defining the Schouten tensor in the documentation
should
use the same notations than in the rest of the documentation of class
`PseudoRiemannianMetric`:
Ricci should be denoted by 'Ric' (cf. method `ricci`) and the Ricci
scalar should be denoted by
'r', not by 's' (cf. method `ricci_scalar`)
- '''Cotton tensor:'''
The LaTeX formula defining the Cotton tensor in the documentation is not
correct: it should be
{{{
C_{ijk} = \nabla_k S_{ij} - \nabla_j S_{ik}
}}}
(the Cotton tensor is antisymmetric in its last two indices).
Besides, in the code, the line
{{{
cot = nabla(s).antisymmetrize(0,2)
}}}
must be replaced by
{{{
cot = 2 * nabla(s).antisymmetrize(1,2)
}}}
Indeed, the convention for the index positions of a covariant derivative
is
`(nabla(s))_{ijk} = \nabla_k s_{ij}`, i.e. the derivation index is the
last one, cf. the note
at
http://sagemanifolds.obspm.fr/doc/19209/reference/manifolds/sage/manifolds/differentiable/affine_connection.html
. Note also the factor 2 to compensate for the 1/2 factor introduced by
the
antisymmetrization.
Besides, I have the impression that in the literature, the Cotton tensor
is generaly defined by
{{{
C_{ijk} = (n-2) ( \nabla_k S_{ij} - \nabla_j S_{ik} )
}}}
i.e. it differs from your definition by a factor n-2, cf. for instance
https://en.wikipedia.org/wiki/Cotton_tensor.
Of course in dimension 3, both definitions coincide, but could you
please
check the literature for n > 3 ?
- '''Cotton-York tensor:'''
Since in the rest of the documentation, $\epsilon$ stands for the Levi-
Civita
tensor (cf. the method `volume_form`), there should be no $\sqrt{\det
g}$ in the LaTeX formula
defining the Cotton-York tensor. Moreover, the indices seem ill-placed;
the
correct formula is (cf. Eq. (4) in York's paper, Phys. Rev. Lett.
'''26''', 1656 (1971)
or https://en.wikipedia.org/wiki/Cotton_tensor)
{{{
CY_{ij} = \frac{1}{2} \epsilon^{kl}_{\quad i} C_{jlk}
= \epsilon^{kl}_{\quad i} \nabla_k S_{lj}
}}}
Accordingly, in the code, the ligne
{{{
cy = -cot.contract(0,2,eps,0,1)
}}}
must be replaced by
{{{
cy = (1/2) * eps.contract(0, 1, cot, 2, 1)
}}}
Besides, the test
{{{
if n < 3:
}}}
must be replaced by
{{{
if n != 3:
}}}
--
Ticket URL: <http://trac.sagemath.org/ticket/19823#comment:12>
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