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## SAGE tip: GRmodule. Day 07.

This code is supposed to be (if some one does the work in the future) located in sage.tensor.differential_form_element.

The code presented below is a slight modification of Joris code for differential forms manipulation on SAGE.

## Needed modules

from sage.symbolic.ring import SymbolicRing, SR
from sage.rings.ring_element import RingElement
from sage.algebras.algebra_element import AlgebraElement
from sage.rings.integer import Integer
from sage.combinat.permutation import Permutation

The advantage of using this is that, tensors defined here are an Algebra element, not just a python object as in the previous code.

The sage.combinat.permutation won’t be used (yet), but could be useful if tensor symmetries are defined.

#### TensorFormatter

class TensorFormatter:
r"""
This class contains all the functionality to print a tensor in a
graphically pleasing way.  This class is called by the _latex_ and
_repr_ methods of the Tensor class.
"""
def __init__(self, space):
r"""
Construct a tensor formatter.  See
TensorFormatter for more information.
"""
self._space = space

def repr(self, comp, fun):
r"""
String representation of a primitive tensor, i.e. a function
times a tensor product of d's of the coordinate functions.

INPUT:

- comp -- a subscript of a differential form.

- fun -- the component function of this form.

EXAMPLES::

sage: from sage.tensor.tensor_element import TensorFormatter
sage: x, y, z = var('x, y, z')
sage: U = CoordinatePatch((x, y, z))
sage: D = TensorFormatter(U)
sage: D.repr((0, 1), z^3)
'z^3*dx@dy'

"""

str = "@".join( \
[('d%s' % self._space.coordinate(c).__repr__()) for c in comp])

if fun == 1 and len(comp) > 0:
# We have a non-trivial form whose component function is 1,
# so we just return the formatted form part and ignore the 1.
return str
else:
funstr = fun._repr_()

if not self._is_atomic(funstr):
funstr = '(' + funstr + ')'

if len(str) > 0:
return funstr + "*" + str
else:
return funstr

def latex(self, comp, fun):
r"""
Latex representation of a primitive differential form, i.e. a function
times a tensor product of d's of the coordinate functions.

INPUT:

- comp -- a subscript of a differential form.

- fun -- the component function of this form.

EXAMPLES::

sage: from sage.tensor.tensor_element import TensorFormatter
sage: x, y, z = var('x, y, z')
sage: U = CoordinatePatch((x, y, z))
sage: D = TensorFormatter(U)
sage: D.latex((0, 1), z^3)
'z^{3} d x \otimes d y'

"""

from sage.misc.latex import latex

str = " \otimes ".join( \
[('d %s' % latex(self._space.coordinate(c))) for c in comp])

if fun == 1 and len(comp) > 0:
return str
else:
funstr = latex(fun)

if not self._is_atomic(funstr):
funstr = '(' + funstr + ')'

return funstr + " " + str

def _is_atomic(self, str):
r"""
Helper function to check whether a given string expression
is atomic.

EXAMPLES::

sage: x, y, z = var('x, y, z')
sage: U = CoordinatePatch((x, y, z))
sage: from sage.tensor.tensor_element import TensorFormatter
sage: D = TensorFormatter(U)
sage: D._is_atomic('a + b')
False
sage: D._is_atomic('(a + b)')
True
"""
level = 0
for n, c in enumerate(str):
if c == '(':
level += 1
elif c == ')':
level -= 1

if c == '+' or c == '-':
if level == 0 and n > 0:
return False
return True

The only I’ve changed here is “DifferentialForm” by “Tensor” and “\wedge” by “\otimes”

The above code allows to write the tensor product in a basis, $dx^1\otimes\cdots\otimes dx^n$. The chosen symbol for denoting the tensor product was @.

## Tensor Class

This code is incomplete due to:

• I’ve not defined the TensorsAlgebra, which should be done in parallel.
• There are a lot of attributes not presented in this class.
• class Tensor(AlgebraElement):
r"""
Tensor class.
"""

def __init__(self, parent, degree, fun = None):
r"""
Construct a tensor.

INPUT:

- parent -- Parent algebra of tensors.

- degree -- Degree of the tensor.

- fun (default: None) -- Initialize this differential form with the given function.  If the degree is not zero, this argument is silently ignored.

EXAMPLES::

sage: x, y, z = var('x, y, z')
sage: F = Tensors(); F
Algebra of tensors in the variables x, y, z
sage: f = Tensor(F, 0, sin(z)); f
sin(z)

"""

if not isinstance(parent, Tensors):
raise TypeError, "Parent not an algebra of tensors."

RingElement.__init__(self, parent)

self._degree = degree
self._components = {}

if degree == 0 and fun is not None:
self.__setitem__([], fun)

def __getitem__(self, subscript):
r"""
Return a given component of the tensor.

INPUT:

- subscript: subscript of the component.  Must be an integer
or a list of integers.

EXAMPLES::

sage: x, y, z = var('x, y, z')
sage: F = Tensors(); F
Algebra of tensors in the variables x, y, z
sage: f = Tensor(F, 0, sin(x*y)); f
sin(x*y)
sage: f[()]
sin(x*y)
"""

if isinstance(subscript, (Integer, int)):
subscript = (subscript, )
else:
subscript = tuple(subscript)

dim = self.parent().base_space().dim()
if any([s >= dim for s in subscript]):
raise ValueError, "Index out of bounds."

if len(subscript) != self._degree:
raise TypeError, "%s is not a subscript of degree %s" %\
(subscript, self._degree)

"""sign, subscript = sort_subscript(subscript)"""

if subscript in self._components:
return sign*self._components[subscript]
else:
return 0

def __setitem__(self, subscript, fun):
r"""
Modify a given component of the tensor.

INPUT:

- subscript: subscript of the component.  Must be an integer or a list of integers.

EXAMPLES::

sage: F = Tensors(); F
Algebra of tensors in the variables x, y, z
sage: f = Tensor(F, 2)
sage: f[1, 2] = x; f
x*dy@dz
"""

if isinstance(subscript, (Integer, int)):
subscript = (subscript, )
else:
subscript = tuple(subscript)

dim = self.parent().base_space().dim()
if any([s >= dim for s in subscript]):
raise ValueError, "Index out of bounds."

if len(subscript) != self._degree:
raise TypeError, "%s is not a subscript of degree %s" %\
(subscript, self._degree)

"""sign, subscript = sort_subscript(subscript)"""
self._components[subscript] = SR(fun)

Ok, so again I’ve changed “DifferentialForm(s)” by “Tensor(s)”, drop the permutation of indices (’cause tensors do not need to be neither symmetric nor anti-symmetric.

I’ll keep working with this code… It’s all by now. Oh! I’ll post next week the rest of the code based in Sergey’s GRPy.

Enjoy.

Dox

## SAGE tip: GRmodule. Day 06.

I’ve just updated the SAGE worksheet which uses the definitions described in the previous posts.

• There are some explanations in text format
• The code has been hidden… because is long.
• Moreover… I’ve discover something really amazing! Joris Vankerschaver‘s code of the differential form package in SAGE. Thus, I could use some ideas from Joris’ code to improve GRmodule. Nice, Isn’t it?

Let’s hope I could so something nice this weekend!

Don’t forget check the worksheet, and post some comment for feedback! ðŸ˜‰

Enjoy!

Dox

## SAGE tip: GRmodule. Day 05

Hi everyone!

This time the Christoffel connection will be defined.

# The code

As usual, here is the code:

class Christoffel(Tensor):
'''The class to represent Christoffel Symbols of the second kind. Please
note that while it inherits from Tensor, Christoffel symbols are
NOT tensors'''

def __init__(self,metr,symbol='C',rank=(1,2),sh=(1,-1,-1)):

# The metric
self.g_down = metr

# Since we have a metric we do indeed have a coordinate system
self.rep  = self.g_down.rep

self.g_up = metr.inverse

# Please note that this call will trigger a call to allocate in
# the Tensor class, but the allocate will actually be the allocate
# defined below
super(Christoffel,self).__init__(symbol,rank,sh,coords=metr.coord)

def allocate(self,rank):
Tensor.allocate(self,rank)
# Now that we have allocated things, time to actually compute things
for i in xrange(self.dim):
for k in xrange(self.dim):
for l in xrange(self.dim):
sum = 0
for m in xrange(self.dim):
term1 = diff(self.g_down[m,k],self.g_down.coords[l])
term2 = diff(self.g_down[m,l],self.g_down.coords[k])
term3 = diff(self.g_down[k,l],self.g_down.coords[m])

tr = self.g_up[i,m] * (term1+term2-term3)

sum += tr
res = sum/2
self.components[i,k,l] = res
self.getNonZero()

This code is almost a copy of Sergey’s one, except for the use of xrange instead of np.arange, and the fact that I’ve dropped the minus signs denoting the shape of the tensors.

## Sage implementation

This time I won’t present a Python file, but a SAGE file, GRmodule.

Enjoy!

Dox

## SAGE tip: GRmodule. Day 04.

Hello everyone. I’ll do it quick, ’cause I’m really tired, and by the way, today… Feb. 8th, I’m turning 30 years old ðŸ™‚

First of all, I create a page in my google site for the GRmodule files. Note that the name is GRmodule and not GR-module, because the dash is not admissible in a module name (beginner mistake!)

#### In the last post…

The Metric class was defined, and the first steps of the implementation were walked.

## Inverting the metric

Once the components of the metric are given, one can call the invert method for assigning the inverse of the metric. Don’t know why, but I’d like to keep track of what’s going on! so, I change a little bit the code from Sergei, and my invert method returns the inverse metric tensor,

  def invert(self):
'''Find the inverse of the metric and store the result in a
Metric object self.inverse'''

'''Create a unit matrix of dimension dim'''
temp = sympy.eye(self.dim)

'''Assign the values of the metric to temp'''
for key in self.components.keys():
id = tuple(np.abs(key))
temp[id] = self.components[key]

'''invert the matrix with inv() from sympy'''
inv = temp.inv()
'''convert the matrix in a dictionary'''
inverse = self._dictkeycopy(self.components)
for i in range(self.dim):
for j in range(self.dim):
inverse[i,j] = inv[i,j]
self.inverse = Metric(self.coord,rank=(2,0),sh=(1,1),symbol='g_inv')
self.inverse.components = inverse
return self.inverse

First a temporal matrix is created, and just for assuring it’s invertible, one creates a unit matrix of dimension dim. That’s what the sympy.eye does!

Secondly, the values of the metric are assigned to temp.

The temp matrix is inverted. The sympy command to inver a matrix is inv.

Next, the inverse matrix is converted in a dictionary (just like in previous cases).

Finally, the characteristic of tensor is given to the inverse matrix object… and it’s returned!, i.e., it can be assigned.

In the implementation file the components of the metric are assigned, the metric is invert… and the result of the inversion is printed.

### How to run it?!

• Download the latest files and store them in the same folder, e.g., GR.
• Open a terminal and move to the GR folder  cd path/to/GR
• In the terminal type  $python Proof-GR-module.py > Result.txt • Open the Result.txt file, i.e., $ emacs Result.txt
• Enjoy life!

DOX

Happy B-day to me!!! ðŸ™‚