Noncoordinate basis

Sections in this chapter:

Introduction

Definition and notation for noncoordinate basis

Computations that depends on the metric

Index notation with noncoordinate basis

Related functions

For other chapters go to Index

Introduction

Until now, we have worked using exclusively coordinate bases (also named natural or holonomic). However, it is often convenient to work in noncoordinate bases. TTC allows this kind of calculations and in this chapter we shall explain how to perform them.

Definition and notation for noncoordinate basis

Associated to a coordinate system xⁱ always exists a natural basis

and its dual dxⁱ and in TTC you can use them without defining them. Nevertheless, if you want to use a noncoordinate basis e_i and eⁱ you have to declare it using InputBasis. The full name of a noncoordinate basis is a couple { coords,basis}, being coords the name of the coordinate system to which basis is related. Accordingly, the input TTC representation of e_i and eⁱ is, assuming that there exist only one nonnatural basis called basis, given by a generalization Natural basis

basis[- i]e_i
basis[ i]eⁱ

If there exist another nonnatural basis called basis but related to another coordinates only the first basis introduced use the name basis in order to write the basis. The second basis introduced needs the full name

ZZ[{coords,basis}][- i]e_i
ZZ[{coords,basis}] [ i]eⁱ

You can short by hand by a not used symbol , say e, assigning e= ZZ[{coords,basis}]. On the other hand, given a noncoordinate basis with a name {cc, bb}, the output of basis elements are represented by

e_ibb[-i]
eⁱbb[i]

Having declared a noncoordinate basis, you can already do all the algebraic operationsdescribed in Tensor operations.

For instance, you can declare an orthonormal basis in polar coordinates and then compute the interior contraction and the symmetric part of the tensor

In[2]:= InputCoordinates[polar, {r, phi}];

In[3]:= InputBasis[{polar, ortho}]

Out[3]= {{polar, ortho}, {1, 2}}

In[4]:= ortho[2]

Out[4]= ortho[2]

In[5]:= InteriorContraction[ortho[-1], ortho[1, 2]]

Out[5]= ortho[2]

In[6]:= t = r ortho[1, 2] + r Sin[phi] ortho[2, 1]

Out[6]= r ortho[1]*.ortho[2] + r Sin[phi] ortho[2]*.ortho[1]

In[7]:= Symmetric[t]

         r   r Sin[phi]
Out[7]= (- + ----------) ortho[1]*.ortho[2] +
         2       2

      r   r Sin[phi]
>    (- + ----------) ortho[2]*.ortho[1]
      2       2

As you can see, the output of InputBasis prints the full name of the basis together with the list {1, 2}. This last element appears because the complete syntax of InputBasis allows an optional argument consisting of a list with the symbols you want TTC labels the basis elements. If you do not input this argument, TTC uses 1,

2, etc. as labels.

In order to use the differential operators LieD and ExteriorD, the relation between the noncoordinate and coordinate basis have to be given. This can be achieved with InputBasisChange

You can input any of the four possibles relations

and TTC inmediately computes the remaining ones. You can see these relations using BasisVectorChange and BasisFormChange.

Next, we give the relation

between the orthonormal and natural basis in polar coordinates using vectors and obtain the same relation in all the alternative manners. Afterwards, we compute the exterior derivative of the 1-form

In[8]:= InputBasisChange[{polar, ortho}, polar,
              {ortho[-1] -> polar[-1], ortho[-2] -> polar[-2]/r}]

In[9]:= BasisFormChange[{polar, ortho}, polar] // ColumnForm

Out[9]= ortho[1] -> dr
         ortho[2] -> r dphi

In[10]:= BasisVectorChange[polar, {polar, ortho}] // ColumnForm

Out[10]= Dr -> ortho[-1]
         Dphi -> r ortho[-2]

In[11]:= BasisFormChange[polar, {polar, ortho}] // ColumnForm

Out[11]= dr -> ortho[1]
                 1
         dphi -> - ortho[2]
                 r

In[12]:= w = (1 + Tan[phi]) ortho[1] + r^2 ortho[2];

In[13]:= ExteriorD[w]

                        2
                Sec[phi]
Out[13]= (3 r - ---------) ortho[1]*^ortho[2]
                    r

Associated to every basis there exists a set of scalars named structure coefficients and defined through the Lie braket

You can obtain one of them with StructureCoefficient and see the whole set with ShowStructureCoefficient.

In[16]:= ShowStructureCoefficient[{polar, ortho}]

   Non null structure coefficients:

       2     1
   c     = -(-)
    1 2      r

Computations that depends on the metric

All we have said in chapter Riemannian manifolds remains unchanged. Just a few comments and precissions about notations
In a noncoordinate basis the definition of the connection coefficients have to be generalized to

where is the directional derivative ofin the direction ofand are the structure coefficients just defined with the last index lowered. The TTC function used to compute this connection is Christoffel although in the literature this name is reserved to the case of natural basis. ShowChristoffel also works with noncoordinate basis. On the other hand, the expression for the covariant derivative

becomes

but AbsoluteD and CovariantD are used as in section:Covariant derivative Finally, the components of the Riemann tensor are computed from

and Riemann, Ricci and Curvature are computed in the same way.

In the following examples the metric in the orthonormal basis of polar coordinates is defined and the inverse metric computed. Then, we get the dual and the covariant derivative of the form previously defined. Finally, we check that the Riemann tensor is null. (The examples in this section are a continuation of those of section: Definition and notation for noncoordinate basis

In[14]:= g = ortho[1, 1] + ortho[2, 2]

Out[14]= ortho[1]*.ortho[1] + ortho[2]*.ortho[2]

In[15]:= InputMetric[eucl, {polar, ortho}, g];

In[16]:= InverseMetric[eucl, {polar, ortho}]

Out[16]= ortho[-2]*.ortho[-2] + ortho[-1]*.ortho[-1]

In[17]:= InputMetricSign[eucl, {polar, ortho}, 1]

Out[17]= 1

In[18]:= Hodge[eucl][w]

           2
Out[18]= -r  ortho[1] + (1 + Tan[phi]) ortho[2]

In[19]:= ShowChristoffel[eucl, {polar, ortho}]

   Non null Christoffel symbols:

    1        1
   K     = -(-)
     2 2     r


    2      1
   K     = -
     1 2   r

In[20]:= AbsoluteD[eucl][w]

                       2
               Sec[phi]
Out[20]= (-r + ---------) ortho[1]*.ortho[2] + 2 r ortho[2]*.ortho[1] +
                   r

     1 + Tan[phi]
>    ------------ ortho[2]*.ortho[2]
          r

In[24]:= Riemann[eucl, {polar, ortho}]

Out[24]= 0

Index notation with noncoordinate basis

The index notation showed in chapter Index notation can be perfectly used even if the tensors that take part are expressed in a noncoordinate basis. No modifications are needed, neither in the inputs nor in the way of proceeding. The user only has to follow faithfully the instructions given in that chapter.

Related functions

There are some other TTC functions related to the basis problem that are not described in the above section. These functions are:

Basis           InputScalarBasis                ScalarBasisQ
BasisNames      BasisSymbols                    Dimension
ClearBasis

Basis is used to get the basis in which a tensor has been expanded. When Basis acts upon a scalar it cannot find the basis and asks the user to input it. Since Basis is used internally, it can happen that TTC stops a computation and demands for the basis of a scalar. To avoid this nuisance and when one is going to use the same basis during the current session it can be established once and forever the basis of all the scalars that will appear with InputScalarBasis. To tell TTC whether to stop and ask or not, the variable ScalarBasisQ has to be set to True (to ask) or to False (to avoid the asking).

BasisNames, BasisSymbols and Dimension are merely informative commands. BasisNames is the analogous of CoordinateNames for basis. It shows the noncoordinate basis declared in the session along with the coordinates to which they are related and the symbols used to represent the basis elements. Likewise BasisSymbols returns a list of these symbols in the same way as Coordinates gives a list of the coordinates of a system. On the other hand, Dimension does what is expected: it gives the dimension of the specified basis.

ClearBasis serves to erase a declared basis but, like ClearCoordinates, it has other effects: it erases all objects builded from the erased basis. A more detailed explanation of how it works can be found in section How to clear

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