(maxima.info)Introduction to ctensor

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Enter node , (file) or (file)node

26.1 Introduction to ctensor
============================

'ctensor' is a component tensor manipulation package.  To use the
'ctensor' package, type 'load("ctensor")'.  To begin an interactive
session with 'ctensor', type 'csetup()'.  You are first asked to specify
the dimension of the manifold.  If the dimension is 2, 3 or 4 then the
list of coordinates defaults to '[x,y]', '[x,y,z]' or '[x,y,z,t]'
respectively.  These names may be changed by assigning a new list of
coordinates to the variable 'ct_coords' (described below) and the user
is queried about this.  Care must be taken to avoid the coordinate names
conflicting with other object definitions.

   Next, the user enters the metric either directly or from a file by
specifying its ordinal position.  The metric is stored in the matrix
'lg'.  Finally, the metric inverse is computed and stored in the matrix
'ug'.  One has the option of carrying out all calculations in a power
series.

   A sample protocol is begun below for the static, spherically
symmetric metric (standard coordinates) which will be applied to the
problem of deriving Einstein's vacuum equations (which lead to the
Schwarzschild solution) as an example.  Many of the functions in
'ctensor' will be displayed for the standard metric as examples.

     (%i1) load("ctensor");
     (%o1)      /share/tensor/ctensor.mac
     (%i2) csetup();
     Enter the dimension of the coordinate system:
     4;
     Do you wish to change the coordinate names?
     n;
     Do you want to
     1. Enter a new metric?

     2. Enter a metric from a file?

     3. Approximate a metric with a Taylor series?
     1;

     Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General
     Answer 1, 2, 3 or 4
     1;
     Row 1 Column 1:
     a;
     Row 2 Column 2:
     x^2;
     Row 3 Column 3:
     x^2*sin(y)^2;
     Row 4 Column 4:
     -d;

     Matrix entered.
     Enter functional dependencies with the DEPENDS function or 'N' if none
     depends([a,d],x);
     Do you wish to see the metric?
     y;
                               [ a  0       0        0  ]
                               [                        ]
                               [     2                  ]
                               [ 0  x       0        0  ]
                               [                        ]
                               [         2    2         ]
                               [ 0  0   x  sin (y)   0  ]
                               [                        ]
                               [ 0  0       0       - d ]
     (%o2)                                done
     (%i3) christof(mcs);
                                                 a
                                                  x
     (%t3)                          mcs        = ---
                                       1, 1, 1   2 a

                                                  1
     (%t4)                           mcs        = -
                                        1, 2, 2   x

                                                  1
     (%t5)                           mcs        = -
                                        1, 3, 3   x

                                                 d
                                                  x
     (%t6)                          mcs        = ---
                                       1, 4, 4   2 d

                                                   x
     (%t7)                          mcs        = - -
                                       2, 2, 1     a

                                                cos(y)
     (%t8)                         mcs        = ------
                                      2, 3, 3   sin(y)

                                                    2
                                               x sin (y)
     (%t9)                      mcs        = - ---------
                                   3, 3, 1         a

     (%t10)                   mcs        = - cos(y) sin(y)
                                 3, 3, 2

                                                 d
                                                  x
     (%t11)                         mcs        = ---
                                       4, 4, 1   2 a
     (%o11)                               done