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# 28. ctensor

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## 28.1 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

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## 28.2 Functions and Variables for ctensor

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### 28.2.1 Initialization and setup

Function: csetup ()

  A function in the `ctensor` (component tensor) package which initializes the package and allows the user to enter a metric interactively. See `ctensor` for more details.

Function: cmetric (dis)
Function: cmetric ()

  A function in the `ctensor` (component tensor) package that computes the metric inverse and sets up the package for further calculations.

  If `cframe_flag` is `false`, the function computes the inverse metric `ug` from the (user-defined) matrix `lg`. The metric determinant is also computed and stored in the variable `gdet`. Furthermore, the package determines if the metric is diagonal and sets the value of `diagmetric` accordingly. If the optional argument dis is present and not equal to `false`, the user is prompted to see the metric inverse.

  If `cframe_flag` is `true`, the function expects that the values of `fri` (the inverse frame matrix) and `lfg` (the frame metric) are defined. From these, the frame matrix `fr` and the inverse frame metric `ufg` are computed.

Function: ct_coordsys (coordinate_system, extra_arg)
Function: ct_coordsys (coordinate_system)

  Sets up a predefined coordinate system and metric. The argument coordinate_system can be one of the following symbols:

       SYMBOL             Dim Coordinates     Description/comments
       ------------------------------------------------------------------
       cartesian2d           2  [x,y]             Cartesian 2D coordinate
                                                  system
       polar                 2  [r,phi]           Polar coordinate system
       elliptic              2  [u,v]             Elliptic coord. system
       confocalelliptic      2  [u,v]             Confocal elliptic
                                                  coordinates
       bipolar               2  [u,v]             Bipolar coord. system
       parabolic             2  [u,v]             Parabolic coord. system
       cartesian3d           3  [x,y,z]           Cartesian 3D coordinate
                                                  system
       polarcylindrical      3  [r,theta,z]       Polar 2D with
                                                  cylindrical z
       ellipticcylindrical   3  [u,v,z]           Elliptic 2D with
                                                  cylindrical z
       confocalellipsoidal   3  [u,v,w]           Confocal ellipsoidal
       bipolarcylindrical    3  [u,v,z]           Bipolar 2D with
                                                  cylindrical z
       paraboliccylindrical  3  [u,v,z]           Parabolic 2D with
                                                  cylindrical z
       paraboloidal          3  [u,v,phi]         Paraboloidal coords.
       conical               3  [u,v,w]           Conical coordinates
       toroidal              3  [u,v,phi]         Toroidal coordinates
       spherical             3  [r,theta,phi]     Spherical coord. system
       oblatespheroidal      3  [u,v,phi]         Oblate spheroidal
                                                  coordinates
       oblatespheroidalsqrt  3  [u,v,phi]
       prolatespheroidal     3  [u,v,phi]         Prolate spheroidal
                                                  coordinates
       prolatespheroidalsqrt 3  [u,v,phi]
       ellipsoidal           3  [r,theta,phi]     Ellipsoidal coordinates
       cartesian4d           4  [x,y,z,t]         Cartesian 4D coordinate
                                                  system
       spherical4d           4  [r,theta,eta,phi] Spherical 4D coordinate
                                                  system
       exteriorschwarzschild 4  [t,r,theta,phi]   Schwarzschild metric
       interiorschwarzschild 4  [t,z,u,v]         Interior Schwarzschild
                                                  metric
       kerr_newman           4  [t,r,theta,phi]   Charged axially
                                                  symmetric metric

  `coordinate_system` can also be a list of transformation functions, followed by a list containing the coordinate variables. For instance, you can specify a spherical metric as follows:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
            r*sin(theta),[r,theta,phi]]);
      (%o2)                                done
      (%i3) lg:trigsimp(lg);
                                 [ 1  0         0        ]
                                 [                       ]
                                 [     2                 ]
      (%o3)                      [ 0  r         0        ]
                                 [                       ]
                                 [         2    2        ]
                                 [ 0  0   r  cos (theta) ]
      (%i4) ct_coords;
      (%o4)                           [r, theta, phi]
      (%i5) dim;
      (%o5)                                  3

  Transformation functions can also be used when `cframe_flag` is `true`:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) cframe_flag:true;
      (%o2)                                true
      (%i3) ct_coordsys([r*cos(theta)*cos(phi),r*cos(theta)*sin(phi),
            r*sin(theta),[r,theta,phi]]);
      (%o3)                                done
      (%i4) fri;
      (%o4)
       [cos(phi)cos(theta) -cos(phi) r sin(theta) -sin(phi) r cos(theta)]
       [                                                                ]
       [sin(phi)cos(theta) -sin(phi) r sin(theta)  cos(phi) r cos(theta)]
       [                                                                ]
       [    sin(theta)           r cos(theta)                0          ]

      (%i5) cmetric();
      (%o5)                                false
      (%i6) lg:trigsimp(lg);
                                 [ 1  0         0        ]
                                 [                       ]
                                 [     2                 ]
      (%o6)                      [ 0  r         0        ]
                                 [                       ]
                                 [         2    2        ]
                                 [ 0  0   r  cos (theta) ]

  The optional argument extra_arg can be any one of the following:

  `cylindrical` tells `ct_coordsys` to attach an additional cylindrical coordinate.

  `minkowski` tells `ct_coordsys` to attach an additional coordinate with negative metric signature.

  `all` tells `ct_coordsys` to call `cmetric` and `christof(false)` after setting up the metric.

  If the global variable `verbose` is set to `true`, `ct_coordsys` displays the values of `dim`, `ct_coords`, and either `lg` or `lfg` and `fri`, depending on the value of `cframe_flag`.

Function: init_ctensor ()

  Initializes the `ctensor` package.

  The `init_ctensor` function reinitializes the `ctensor` package. It removes all arrays and matrices used by `ctensor`, resets all flags, resets `dim` to 4, and resets the frame metric to the Lorentz-frame.

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### 28.2.2 The tensors of curved space

The main purpose of the `ctensor` package is to compute the tensors of curved space(time), most notably the tensors used in general relativity.

When a metric base is used, `ctensor` can compute the following tensors:

     lg  -- ug
       \      \
        lcs -- mcs -- ric -- uric
                  \      \       \
                   \      tracer - ein -- lein
                    \
                     riem -- lriem -- weyl
                         \
                          uriem

`ctensor` can also work using moving frames. When `cframe_flag` is set to `true`, the following tensors can be calculated:

     lfg -- ufg
         \
     fri -- fr -- lcs -- mcs -- lriem -- ric -- uric
          \                       |  \      \       \
           lg -- ug               |   weyl   tracer - ein -- lein
                                  |\
                                  | riem
                                  |
                                  \uriem

Function: christof (dis)

  A function in the `ctensor` (component tensor) package. It computes the Christoffel symbols of both kinds. The argument dis determines which results are to be immediately displayed. The Christoffel symbols of the first and second kinds are stored in the arrays `lcs[i,j,k]` and `mcs[i,j,k]` respectively and defined to be symmetric in the first two indices. If the argument to `christof` is `lcs` or `mcs` then the unique non-zero values of `lcs[i,j,k]` or `mcs[i,j,k]`, respectively, will be displayed. If the argument is `all` then the unique non-zero values of `lcs[i,j,k]` and `mcs[i,j,k]` will be displayed. If the argument is `false` then the display of the elements will not occur. The array elements `mcs[i,j,k]` are defined in such a manner that the final index is contravariant.

Function: ricci (dis)

  A function in the `ctensor` (component tensor) package. `ricci` computes the covariant (symmetric) components `ric[i,j]` of the Ricci tensor. If the argument dis is `true`, then the non-zero components are displayed.

Function: uricci (dis)

  This function first computes the covariant components `ric[i,j]` of the Ricci tensor. Then the mixed Ricci tensor is computed using the contravariant metric tensor. If the value of the argument dis is `true`, then these mixed components, `uric[i,j]` (the index `i` is covariant and the index `j` is contravariant), will be displayed directly. Otherwise, `ricci(false)` will simply compute the entries of the array `uric[i,j]` without displaying the results.

Function: scurvature ()

  Returns the scalar curvature (obtained by contracting the Ricci tensor) of the Riemannian manifold with the given metric.

Function: einstein (dis)

  A function in the `ctensor` (component tensor) package. `einstein` computes the mixed Einstein tensor after the Christoffel symbols and Ricci tensor have been obtained (with the functions `christof` and `ricci`). If the argument dis is `true`, then the non-zero values of the mixed Einstein tensor `ein[i,j]` will be displayed where `j` is the contravariant index. The variable `rateinstein` will cause the rational simplification on these components. If `ratfac` is `true` then the components will also be factored.

Function: leinstein (dis)

  Covariant Einstein-tensor. `leinstein` stores the values of the covariant Einstein tensor in the array `lein`. The covariant Einstein-tensor is computed from the mixed Einstein tensor `ein` by multiplying it with the metric tensor. If the argument dis is `true`, then the non-zero values of the covariant Einstein tensor are displayed.

Function: riemann (dis)

  A function in the `ctensor` (component tensor) package. `riemann` computes the Riemann curvature tensor from the given metric and the corresponding Christoffel symbols. The following index conventions are used:

                      l      _l       _l       _l   _m    _l   _m
       R[i,j,k,l] =  R    = |      - |      + |    |   - |    |
                      ijk     ij,k     ik,j     mk   ij    mj   ik

  This notation is consistent with the notation used by the `itensor` package and its `icurvature` function. If the optional argument dis is `true`, the non-zero components `riem[i,j,k,l]` will be displayed. As with the Einstein tensor, various switches set by the user control the simplification of the components of the Riemann tensor. If `ratriemann` is `true`, then rational simplification will be done. If `ratfac` is `true` then each of the components will also be factored.

  If the variable `cframe_flag` is `false`, the Riemann tensor is computed directly from the Christoffel-symbols. If `cframe_flag` is `true`, the covariant Riemann-tensor is computed first from the frame field coefficients.

Function: lriemann (dis)

  Covariant Riemann-tensor (`lriem[]`).

  Computes the covariant Riemann-tensor as the array `lriem`. If the argument dis is `true`, unique nonzero values are displayed.

  If the variable `cframe_flag` is `true`, the covariant Riemann tensor is computed directly from the frame field coefficients. Otherwise, the (3,1) Riemann tensor is computed first.

  For information on index ordering, see `riemann`.

Function: uriemann (dis)

  Computes the contravariant components of the Riemann curvature tensor as array elements `uriem[i,j,k,l]`. These are displayed if dis is `true`.

Function: rinvariant ()

  Forms the Kretchmann-invariant (`kinvariant`) obtained by contracting the tensors

      lriem[i,j,k,l]*uriem[i,j,k,l].

  This object is not automatically simplified since it can be very large.

Function: weyl (dis)

  Computes the Weyl conformal tensor. If the argument dis is `true`, the non-zero components `weyl[i,j,k,l]` will be displayed to the user. Otherwise, these components will simply be computed and stored. If the switch `ratweyl` is set to `true`, then the components will be rationally simplified; if `ratfac` is `true` then the results will be factored as well.

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### 28.2.3 Taylor series expansion

The `ctensor` package has the ability to truncate results by assuming that they are Taylor-series approximations. This behavior is controlled by the `ctayswitch` variable; when set to true, `ctensor` makes use internally of the function `ctaylor` when simplifying results.

The `ctaylor` function is invoked by the following `ctensor` functions:

        Function     Comments
        ---------------------------------
        christof()   For mcs only
        ricci()
        uricci()
        einstein()
        riemann()
        weyl()
        checkdiv()

Function: ctaylor ()

  The `ctaylor` function truncates its argument by converting it to a Taylor-series using `taylor`, and then calling `ratdisrep`. This has the combined effect of dropping terms higher order in the expansion variable `ctayvar`. The order of terms that should be dropped is defined by `ctaypov`; the point around which the series expansion is carried out is specified in `ctaypt`.

  As an example, consider a simple metric that is a perturbation of the Minkowski metric. Without further restrictions, even a diagonal metric produces expressions for the Einstein tensor that are far too complex:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) ratfac:true;
      (%o2)                                true
      (%i3) derivabbrev:true;
      (%o3)                                true
      (%i4) ct_coords:[t,r,theta,phi];
      (%o4)                         [t, r, theta, phi]
      (%i5) lg:matrix([-1,0,0,0],[0,1,0,0],[0,0,r^2,0],
                      [0,0,0,r^2*sin(theta)^2]);
                              [ - 1  0  0         0        ]
                              [                            ]
                              [  0   1  0         0        ]
                              [                            ]
      (%o5)                   [          2                 ]
                              [  0   0  r         0        ]
                              [                            ]
                              [              2    2        ]
                              [  0   0  0   r  sin (theta) ]
      (%i6) h:matrix([h11,0,0,0],[0,h22,0,0],[0,0,h33,0],[0,0,0,h44]);
                                  [ h11   0    0    0  ]
                                  [                    ]
                                  [  0   h22   0    0  ]
      (%o6)                       [                    ]
                                  [  0    0   h33   0  ]
                                  [                    ]
                                  [  0    0    0   h44 ]
      (%i7) depends(l,r);
      (%o7)                               [l(r)]
      (%i8) lg:lg+l*h;
            [ h11 l - 1      0          0                 0            ]
            [                                                          ]
            [     0      h22 l + 1      0                 0            ]
            [                                                          ]
      (%o8) [                        2                                 ]
            [     0          0      r  + h33 l            0            ]
            [                                                          ]
            [                                    2    2                ]
            [     0          0          0       r  sin (theta) + h44 l ]
      (%i9) cmetric(false);
      (%o9)                                done
      (%i10) einstein(false);
      (%o10)                               done
      (%i11) ntermst(ein);
      [[1, 1], 62]
      [[1, 2], 0]
      [[1, 3], 0]
      [[1, 4], 0]
      [[2, 1], 0]
      [[2, 2], 24]
      [[2, 3], 0]
      [[2, 4], 0]
      [[3, 1], 0]
      [[3, 2], 0]
      [[3, 3], 46]
      [[3, 4], 0]
      [[4, 1], 0]
      [[4, 2], 0]
      [[4, 3], 0]
      [[4, 4], 46]
      (%o12)                               done

  However, if we recompute this example as an approximation that is linear in the variable `l`, we get much simpler expressions:

      (%i14) ctayswitch:true;
      (%o14)                               true
      (%i15) ctayvar:l;
      (%o15)                                 l
      (%i16) ctaypov:1;
      (%o16)                                 1
      (%i17) ctaypt:0;
      (%o17)                                 0
      (%i18) christof(false);
      (%o18)                               done
      (%i19) ricci(false);
      (%o19)                               done
      (%i20) einstein(false);
      (%o20)                               done
      (%i21) ntermst(ein);
      [[1, 1], 6]
      [[1, 2], 0]
      [[1, 3], 0]
      [[1, 4], 0]
      [[2, 1], 0]
      [[2, 2], 13]
      [[2, 3], 2]
      [[2, 4], 0]
      [[3, 1], 0]
      [[3, 2], 2]
      [[3, 3], 9]
      [[3, 4], 0]
      [[4, 1], 0]
      [[4, 2], 0]
      [[4, 3], 0]
      [[4, 4], 9]
      (%o21)                               done
      (%i22) ratsimp(ein[1,1]);
                               2      2  4               2     2
      (%o22) - (((h11 h22 - h11 ) (l )  r  - 2 h33 l    r ) sin (theta)
                                    r               r r

                                  2               2      4    2
                    - 2 h44 l    r  - h33 h44 (l ) )/(4 r  sin (theta))
                             r r                r

  This capability can be useful, for instance, when working in the weak field limit far from a gravitational source.

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### 28.2.4 Frame fields

When the variable `cframe_flag` is set to true, the `ctensor` package performs its calculations using a moving frame.

Function: frame_bracket (fr, fri, diagframe)

  The frame bracket (`fb[]`).

  Computes the frame bracket according to the following definition:

         c          c         c        d     e
      ifb   = ( ifri    - ifri    ) ifr   ifr
         ab         d,e       e,d      a     b

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### 28.2.5 Algebraic classification

A new feature (as of November, 2004) of `ctensor` is its ability to compute the Petrov classification of a 4-dimensional spacetime metric. For a demonstration of this capability, see the file `share/tensor/petrov.dem`.

Function: nptetrad ()

  Computes a Newman-Penrose null tetrad (`np`) and its raised-index counterpart (`npi`). See `petrov` for an example.

  The null tetrad is constructed on the assumption that a four-diemensional orthonormal frame metric with metric signature (-,+,+,+) is being used. The components of the null tetrad are related to the inverse frame matrix as follows:

      np  = (fri  + fri ) / sqrt(2)
        1       1      2

      np  = (fri  - fri ) / sqrt(2)
        2       1      2

      np  = (fri  + %i fri ) / sqrt(2)
        3       3         4

      np  = (fri  - %i fri ) / sqrt(2)
        4       3         4

Function: psi (dis)

  Computes the five Newman-Penrose coefficients `psi[0]`...`psi[4]`. If `psi` is set to `true`, the coefficients are displayed. See `petrov` for an example.

  These coefficients are computed from the Weyl-tensor in a coordinate base. If a frame base is used, the Weyl-tensor is first converted to a coordinate base, which can be a computationally expensive procedure. For this reason, in some cases it may be more advantageous to use a coordinate base in the first place before the Weyl tensor is computed. Note however, that constructing a Newman-Penrose null tetrad requires a frame base. Therefore, a meaningful computation sequence may begin with a frame base, which is then used to compute `lg` (computed automatically by `cmetric` and then `ug`. At this point, you can switch back to a coordinate base by setting `cframe_flag` to false before beginning to compute the Christoffel symbols. Changing to a frame base at a later stage could yield inconsistent results, as you may end up with a mixed bag of tensors, some computed in a frame base, some in a coordinate base, with no means to distinguish between the two.

Function: petrov ()

  Computes the Petrov classification of the metric characterized by `psi[0]`...`psi[4]`.

  For example, the following demonstrates how to obtain the Petrov-classification of the Kerr metric:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) (cframe_flag:true,gcd:spmod,ctrgsimp:true,ratfac:true);
      (%o2)                                true
      (%i3) ct_coordsys(exteriorschwarzschild,all);
      (%o3)                                done
      (%i4) ug:invert(lg)$
      (%i5) weyl(false);
      (%o5)                                done
      (%i6) nptetrad(true);
      (%t6) np =

      [ sqrt(r - 2 m)           sqrt(r)                                 ]
      [---------------   ---------------------    0            0        ]
      [sqrt(2) sqrt(r)   sqrt(2) sqrt(r - 2 m)                          ]
      [                                                                 ]
      [ sqrt(r - 2 m)            sqrt(r)                                ]
      [---------------  - ---------------------   0            0        ]
      [sqrt(2) sqrt(r)    sqrt(2) sqrt(r - 2 m)                         ]
      [                                                                 ]
      [                                          r      %i r sin(theta) ]
      [       0                    0          -------   --------------- ]
      [                                       sqrt(2)       sqrt(2)     ]
      [                                                                 ]
      [                                          r       %i r sin(theta)]
      [       0                    0          -------  - ---------------]
      [                                       sqrt(2)        sqrt(2)    ]

                                   sqrt(r)         sqrt(r - 2 m)
      (%t7) npi = matrix([- ---------------------,---------------, 0, 0],
                            sqrt(2) sqrt(r - 2 m) sqrt(2) sqrt(r)

                sqrt(r)            sqrt(r - 2 m)
      [- ---------------------, - ---------------, 0, 0],
         sqrt(2) sqrt(r - 2 m)    sqrt(2) sqrt(r)

                 1               %i
      [0, 0, ---------, --------------------],
             sqrt(2) r  sqrt(2) r sin(theta)

                 1                 %i
      [0, 0, ---------, - --------------------])
             sqrt(2) r    sqrt(2) r sin(theta)

      (%o7)                                done
      (%i7) psi(true);
      (%t8)                              psi  = 0
                                            0

      (%t9)                              psi  = 0
                                            1

                                                m
      (%t10)                             psi  = --
                                            2    3
                                                r

      (%t11)                             psi  = 0
                                            3

      (%t12)                             psi  = 0
                                            4
      (%o12)                               done
      (%i12) petrov();
      (%o12)                                 D

  The Petrov classification function is based on the algorithm published in "Classifying geometries in general relativity: III Classification in practice" by Pollney, Skea, and d'Inverno, Class. Quant. Grav. 17 2885-2902 (2000). Except for some simple test cases, the implementation is untested as of December 19, 2004, and is likely to contain errors.

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### 28.2.6 Torsion and nonmetricity

`ctensor` has the ability to compute and include torsion and nonmetricity coefficients in the connection coefficients.

The torsion coefficients are calculated from a user-supplied tensor `tr`, which should be a rank (2,1) tensor. From this, the torsion coefficients `kt` are computed according to the following formulae:

                  m          m      m
           - g  tr   - g   tr   - tr   g
              im  kj    jm   ki     ij  km
    kt   = -------------------------------
      ijk                 2

      k     km
    kt   = g   kt
      ij         ijm

Note that only the mixed-index tensor is calculated and stored in the array `kt`.

The nonmetricity coefficients are calculated from the user-supplied nonmetricity vector `nm`. From this, the nonmetricity coefficients `nmc` are computed as follows:

                 k    k        km
           -nm  D  - D  nm  + g   nm  g
       k      i  j    i   j         m  ij
    nmc  = ------------------------------
       ij                2

where D stands for the Kronecker-delta.

When `ctorsion_flag` is set to `true`, the values of `kt` are substracted from the mixed-indexed connection coefficients computed by `christof` and stored in `mcs`. Similarly, if `cnonmet_flag` is set to `true`, the values of `nmc` are substracted from the mixed-indexed connection coefficients.

If necessary, `christof` calls the functions `contortion` and `nonmetricity` in order to compute `kt` and `nm`.

Function: contortion (tr)

  Computes the (2,1) contortion coefficients from the torsion tensor tr.

Function: nonmetricity (nm)

  Computes the (2,1) nonmetricity coefficients from the nonmetricity vector nm.

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### 28.2.7 Miscellaneous features

Function: ctransform (M)

  A function in the `ctensor` (component tensor) package which will perform a coordinate transformation upon an arbitrary square symmetric matrix M. The user must input the functions which define the transformation. (Formerly called `transform`.)

Function: findde (A, n)

  returns a list of the unique differential equations (expressions) corresponding to the elements of the n dimensional square array A. Presently, n may be 2 or 3. `deindex` is a global list containing the indices of A corresponding to these unique differential equations. For the Einstein tensor (`ein`), which is a two dimensional array, if computed for the metric in the example below, `findde` gives the following independent differential equations:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) derivabbrev:true;
      (%o2)                                true
      (%i3) dim:4;
      (%o3)                                  4
      (%i4) lg:matrix([a, 0, 0, 0], [ 0, x^2, 0, 0],
                                    [0, 0, x^2*sin(y)^2, 0], [0,0,0,-d]);
                                [ a  0       0        0  ]
                                [                        ]
                                [     2                  ]
                                [ 0  x       0        0  ]
      (%o4)                     [                        ]
                                [         2    2         ]
                                [ 0  0   x  sin (y)   0  ]
                                [                        ]
                                [ 0  0       0       - d ]
      (%i5) depends([a,d],x);
      (%o5)                            [a(x), d(x)]
      (%i6) ct_coords:[x,y,z,t];
      (%o6)                            [x, y, z, t]
      (%i7) cmetric();
      (%o7)                                done
      (%i8) einstein(false);
      (%o8)                                done
      (%i9) findde(ein,2);
                                                  2
      (%o9) [d  x - a d + d, 2 a d d    x - a (d )  x - a  d d  x
              x                     x x         x        x    x

                                                    2          2
                                + 2 a d d   - 2 a  d , a  x + a  - a]
                                         x       x      x
      (%i10) deindex;
      (%o10)                     [[1, 1], [2, 2], [4, 4]]

Function: cograd ()

  Computes the covariant gradient of a scalar function allowing the user to choose the corresponding vector name as the example under `contragrad` illustrates.

Function: contragrad ()

  Computes the contravariant gradient of a scalar function allowing the user to choose the corresponding vector name as the example below for the Schwarzschild metric illustrates:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) derivabbrev:true;
      (%o2)                                true
      (%i3) ct_coordsys(exteriorschwarzschild,all);
      (%o3)                                done
      (%i4) depends(f,r);
      (%o4)                               [f(r)]
      (%i5) cograd(f,g1);
      (%o5)                                done
      (%i6) listarray(g1);
      (%o6)                            [0, f , 0, 0]
                                            r
      (%i7) contragrad(f,g2);
      (%o7)                                done
      (%i8) listarray(g2);
                                     f  r - 2 f  m
                                      r        r
      (%o8)                      [0, -------------, 0, 0]
                                           r

Function: dscalar ()

  computes the tensor d'Alembertian of the scalar function once dependencies have been declared upon the function. For example:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) derivabbrev:true;
      (%o2)                                true
      (%i3) ct_coordsys(exteriorschwarzschild,all);
      (%o3)                                done
      (%i4) depends(p,r);
      (%o4)                               [p(r)]
      (%i5) factor(dscalar(p));
                                2
                          p    r  - 2 m p    r + 2 p  r - 2 m p
                           r r           r r        r          r
      (%o5)               --------------------------------------
                                             2
                                            r

Function: checkdiv ()

  computes the covariant divergence of the mixed second rank tensor (whose first index must be covariant) by printing the corresponding n components of the vector field (the divergence) where n = `dim`. If the argument to the function is `g` then the divergence of the Einstein tensor will be formed and must be zero. In addition, the divergence (vector) is given the array name `div`.

Function: cgeodesic (dis)

  A function in the `ctensor` (component tensor) package. `cgeodesic` computes the geodesic equations of motion for a given metric. They are stored in the array `geod[i]`. If the argument dis is `true` then these equations are displayed.

Function: bdvac (f)

  generates the covariant components of the vacuum field equations of the Brans- Dicke gravitational theory. The scalar field is specified by the argument f, which should be a (quoted) function name with functional dependencies, e.g., `'p(x)`.

  The components of the second rank covariant field tensor are represented by the array `bd`.

Function: invariant1 ()

  generates the mixed Euler- Lagrange tensor (field equations) for the invariant density of R^2. The field equations are the components of an array named `inv1`.

Function: invariant2 ()

  *** NOT YET IMPLEMENTED ***

  generates the mixed Euler- Lagrange tensor (field equations) for the invariant density of `ric[i,j]*uriem[i,j]`. The field equations are the components of an array named `inv2`.

Function: bimetric ()

  *** NOT YET IMPLEMENTED ***

  generates the field equations of Rosen's bimetric theory. The field equations are the components of an array named `rosen`.

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### 28.2.8 Utility functions

Function: diagmatrixp (M)

  Returns `true` if M is a diagonal matrix or (2D) array.

Function: symmetricp (M)

  Returns `true` if M is a symmetric matrix or (2D) array.

Function: ntermst (f)

  gives the user a quick picture of the "size" of the doubly subscripted tensor (array) f. It prints two element lists where the second element corresponds to NTERMS of the components specified by the first elements. In this way, it is possible to quickly find the non-zero expressions and attempt simplification.

Function: cdisplay (ten)

  displays all the elements of the tensor ten, as represented by a multidimensional array. Tensors of rank 0 and 1, as well as other types of variables, are displayed as with `ldisplay`. Tensors of rank 2 are displayed as 2-dimensional matrices, while tensors of higher rank are displayed as a list of 2-dimensional matrices. For instance, the Riemann-tensor of the Schwarzschild metric can be viewed as:

      (%i1) load(ctensor);
      (%o1)       /share/tensor/ctensor.mac
      (%i2) ratfac:true;
      (%o2)                                true
      (%i3) ct_coordsys(exteriorschwarzschild,all);
      (%o3)                                done
      (%i4) riemann(false);
      (%o4)                                done
      (%i5) cdisplay(riem);
                [ 0               0                   0           0     ]
                [                                                       ]
                [                              2                        ]
                [      3 m (r - 2 m)   m    2 m                         ]
                [ 0  - ------------- + -- - ----      0           0     ]
                [            4          3     4                         ]
                [           r          r     r                          ]
                [                                                       ]
      riem    = [                                m (r - 2 m)            ]
          1, 1  [ 0               0              -----------      0     ]
                [                                     4                 ]
                [                                    r                  ]
                [                                                       ]
                [                                           m (r - 2 m) ]
                [ 0               0                   0     ----------- ]
                [                                                4      ]
                [                                               r       ]

                                      [    2 m (r - 2 m)       ]
                                      [ 0  -------------  0  0 ]
                                      [          4             ]
                                      [         r              ]
                           riem     = [                        ]
                               1, 2   [ 0        0        0  0 ]
                                      [                        ]
                                      [ 0        0        0  0 ]
                                      [                        ]
                                      [ 0        0        0  0 ]

                                      [         m (r - 2 m)    ]
                                      [ 0  0  - -----------  0 ]
                                      [              4         ]
                                      [             r          ]
                           riem     = [                        ]
                               1, 3   [ 0  0        0        0 ]
                                      [                        ]
                                      [ 0  0        0        0 ]
                                      [                        ]
                                      [ 0  0        0        0 ]

                                      [            m (r - 2 m) ]
                                      [ 0  0  0  - ----------- ]
                                      [                 4      ]
                                      [                r       ]
                           riem     = [                        ]
                               1, 4   [ 0  0  0        0       ]
                                      [                        ]
                                      [ 0  0  0        0       ]
                                      [                        ]
                                      [ 0  0  0        0       ]

                                     [       0         0  0  0 ]
                                     [                         ]
                                     [       2 m               ]
                                     [ - ------------  0  0  0 ]
                          riem     = [    2                    ]
                              2, 1   [   r  (r - 2 m)          ]
                                     [                         ]
                                     [       0         0  0  0 ]
                                     [                         ]
                                     [       0         0  0  0 ]

                   [     2 m                                         ]
                   [ ------------  0        0               0        ]
                   [  2                                              ]
                   [ r  (r - 2 m)                                    ]
                   [                                                 ]
                   [      0        0        0               0        ]
                   [                                                 ]
        riem     = [                         m                       ]
            2, 2   [      0        0  - ------------        0        ]
                   [                     2                           ]
                   [                    r  (r - 2 m)                 ]
                   [                                                 ]
                   [                                         m       ]
                   [      0        0        0         - ------------ ]
                   [                                     2           ]
                   [                                    r  (r - 2 m) ]

                                      [ 0  0       0        0 ]
                                      [                       ]
                                      [            m          ]
                                      [ 0  0  ------------  0 ]
                           riem     = [        2              ]
                               2, 3   [       r  (r - 2 m)    ]
                                      [                       ]
                                      [ 0  0       0        0 ]
                                      [                       ]
                                      [ 0  0       0        0 ]

                                      [ 0  0  0       0       ]
                                      [                       ]
                                      [               m       ]
                                      [ 0  0  0  ------------ ]
                           riem     = [           2           ]
                               2, 4   [          r  (r - 2 m) ]
                                      [                       ]
                                      [ 0  0  0       0       ]
                                      [                       ]
                                      [ 0  0  0       0       ]

                                            [ 0  0  0  0 ]
                                            [            ]
                                            [ 0  0  0  0 ]
                                            [            ]
                                 riem     = [ m          ]
                                     3, 1   [ -  0  0  0 ]
                                            [ r          ]
                                            [            ]
                                            [ 0  0  0  0 ]

                                            [ 0  0  0  0 ]
                                            [            ]
                                            [ 0  0  0  0 ]
                                            [            ]
                                 riem     = [    m       ]
                                     3, 2   [ 0  -  0  0 ]
                                            [    r       ]
                                            [            ]
                                            [ 0  0  0  0 ]

                                     [   m                      ]
                                     [ - -   0   0       0      ]
                                     [   r                      ]
                                     [                          ]
                                     [        m                 ]
                                     [  0   - -  0       0      ]
                          riem     = [        r                 ]
                              3, 3   [                          ]
                                     [  0    0   0       0      ]
                                     [                          ]
                                     [              2 m - r     ]
                                     [  0    0   0  ------- + 1 ]
                                     [                 r        ]

                                          [ 0  0  0    0   ]
                                          [                ]
                                          [ 0  0  0    0   ]
                                          [                ]
                               riem     = [            2 m ]
                                   3, 4   [ 0  0  0  - --- ]
                                          [             r  ]
                                          [                ]
                                          [ 0  0  0    0   ]

                                      [       0        0  0  0 ]
                                      [                        ]
                                      [       0        0  0  0 ]
                                      [                        ]
                           riem     = [       0        0  0  0 ]
                               4, 1   [                        ]
                                      [      2                 ]
                                      [ m sin (theta)          ]
                                      [ -------------  0  0  0 ]
                                      [       r                ]

                                      [ 0        0        0  0 ]
                                      [                        ]
                                      [ 0        0        0  0 ]
                                      [                        ]
                           riem     = [ 0        0        0  0 ]
                               4, 2   [                        ]
                                      [         2              ]
                                      [    m sin (theta)       ]
                                      [ 0  -------------  0  0 ]
                                      [          r             ]

                                    [ 0  0          0          0 ]
                                    [                            ]
                                    [ 0  0          0          0 ]
                                    [                            ]
                         riem     = [ 0  0          0          0 ]
                             4, 3   [                            ]
                                    [                2           ]
                                    [         2 m sin (theta)    ]
                                    [ 0  0  - ---------------  0 ]
                                    [                r           ]

                 [        2                                             ]
                 [   m sin (theta)                                      ]
                 [ - -------------         0                0         0 ]
                 [         r                                            ]
                 [                                                      ]
                 [                         2                            ]
                 [                    m sin (theta)                     ]
      riem     = [        0         - -------------         0         0 ]
          4, 4   [                          r                           ]
                 [                                                      ]
                 [                                          2           ]
                 [                                   2 m sin (theta)    ]
                 [        0                0         ---------------  0 ]
                 [                                          r           ]
                 [                                                      ]
                 [        0                0                0         0 ]

      (%o5)                                done

Function: deleten (L, n)

  Returns a new list consisting of L with the n'th element deleted.

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### 28.2.9 Variables used by `ctensor`

Option variable: dim

  Default value: 4

  An option in the `ctensor` (component tensor) package. `dim` is the dimension of the manifold with the default 4. The command `dim: n` will reset the dimension to any other value `n`.

Option variable: diagmetric

  Default value: `false`

  An option in the `ctensor` (component tensor) package. If `diagmetric` is `true` special routines compute all geometrical objects (which contain the metric tensor explicitly) by taking into consideration the diagonality of the metric. Reduced run times will, of course, result. Note: this option is set automatically by `csetup` if a diagonal metric is specified.

Option variable: ctrgsimp

  Causes trigonometric simplifications to be used when tensors are computed. Presently, `ctrgsimp` affects only computations involving a moving frame.

Option variable: cframe_flag

  Causes computations to be performed relative to a moving frame as opposed to a holonomic metric. The frame is defined by the inverse frame array `fri` and the frame metric `lfg`. For computations using a Cartesian frame, `lfg` should be the unit matrix of the appropriate dimension; for computations in a Lorentz frame, `lfg` should have the appropriate signature.

Option variable: ctorsion_flag

  Causes the contortion tensor to be included in the computation of the connection coefficients. The contortion tensor itself is computed by `contortion` from the user-supplied tensor `tr`.

Option variable: cnonmet_flag

  Causes the nonmetricity coefficients to be included in the computation of the connection coefficients. The nonmetricity coefficients are computed from the user-supplied nonmetricity vector `nm` by the function `nonmetricity`.

Option variable: ctayswitch

  If set to `true`, causes some `ctensor` computations to be carried out using Taylor-series expansions. Presently, `christof`, `ricci`, `uricci`, `einstein`, and `weyl` take into account this setting.

Option variable: ctayvar

  Variable used for Taylor-series expansion if `ctayswitch` is set to `true`.

Option variable: ctaypov

  Maximum power used in Taylor-series expansion when `ctayswitch` is set to `true`.

Option variable: ctaypt

  Point around which Taylor-series expansion is carried out when `ctayswitch` is set to `true`.

System variable: gdet

  The determinant of the metric tensor `lg`. Computed by `cmetric` when `cframe_flag` is set to `false`.

Option variable: ratchristof

  Causes rational simplification to be applied by `christof`.

Option variable: rateinstein

  Default value: `true`

  If `true` rational simplification will be performed on the non-zero components of Einstein tensors; if `ratfac` is `true` then the components will also be factored.

Option variable: ratriemann

  Default value: `true`

  One of the switches which controls simplification of Riemann tensors; if `true`, then rational simplification will be done; if `ratfac` is `true` then each of the components will also be factored.

Option variable: ratweyl

  Default value: `true`

  If `true`, this switch causes the `weyl` function to apply rational simplification to the values of the Weyl tensor. If `ratfac` is `true`, then the components will also be factored.

Variable: lfg

  The covariant frame metric. By default, it is initialized to the 4-dimensional Lorentz frame with signature (+,+,+,-). Used when `cframe_flag` is `true`.

Variable: ufg

  The inverse frame metric. Computed from `lfg` when `cmetric` is called while `cframe_flag` is set to `true`.

Variable: riem

  The (3,1) Riemann tensor. Computed when the function `riemann` is invoked. For information about index ordering, see the description of `riemann`.

  if `cframe_flag` is `true`, `riem` is computed from the covariant Riemann-tensor `lriem`.

Variable: lriem

  The covariant Riemann tensor. Computed by `lriemann`.

Variable: uriem

  The contravariant Riemann tensor. Computed by `uriemann`.

Variable: ric

  The mixed Ricci-tensor. Computed by `ricci`.

Variable: uric

  The contravariant Ricci-tensor. Computed by `uricci`.

Variable: lg

  The metric tensor. This tensor must be specified (as a `dim` by `dim` matrix) before other computations can be performed.

Variable: ug

  The inverse of the metric tensor. Computed by `cmetric`.

Variable: weyl

  The Weyl tensor. Computed by `weyl`.

Variable: fb

  Frame bracket coefficients, as computed by `frame_bracket`.

Variable: kinvariant

  The Kretchmann invariant. Computed by `rinvariant`.

Variable: np

  A Newman-Penrose null tetrad. Computed by `nptetrad`.

Variable: npi

  The raised-index Newman-Penrose null tetrad. Computed by `nptetrad`. Defined as `ug.np`. The product `np.transpose(npi)` is constant:

      (%i39) trigsimp(np.transpose(npi));
                                    [  0   - 1  0  0 ]
                                    [                ]
                                    [ - 1   0   0  0 ]
      (%o39)                        [                ]
                                    [  0    0   0  1 ]
                                    [                ]
                                    [  0    0   1  0 ]

Variable: tr

  User-supplied rank-3 tensor representing torsion. Used by `contortion`.

Variable: kt

  The contortion tensor, computed from `tr` by `contortion`.

Variable: nm

  User-supplied nonmetricity vector. Used by `nonmetricity`.

Variable: nmc

  The nonmetricity coefficients, computed from `nm` by `nonmetricity`.

System variable: tensorkill

  Variable indicating if the tensor package has been initialized. Set and used by `csetup`, reset by `init_ctensor`.

Option variable: ct_coords

  Default value: `[]`

  An option in the `ctensor` (component tensor) package. `ct_coords` contains a list of coordinates. While normally defined when the function `csetup` is called, one may redefine the coordinates with the assignment `ct_coords: [j1, j2, ..., jn]` where the j's are the new coordinate names. See also `csetup`.

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### 28.2.10 Reserved names

The following names are used internally by the `ctensor` package and should not be redefined:

      Name         Description
      ---------------------------------------------------------------------
      _lg()        Evaluates to lfg if frame metric used, lg otherwise
      _ug()        Evaluates to ufg if frame metric used, ug otherwise
      cleanup()    Removes items drom the deindex list
      contract4()  Used by psi()
      filemet()    Used by csetup() when reading the metric from a file
      findde1()    Used by findde()
      findde2()    Used by findde()
      findde3()    Used by findde()
      kdelt()      Kronecker-delta (not generalized)
      newmet()     Used by csetup() for setting up a metric interactively
      setflags()   Used by init_ctensor()
      readvalue()
      resimp()
      sermet()     Used by csetup() for entering a metric as Taylor-series
      txyzsum()
      tmetric()    Frame metric, used by cmetric() when cframe_flag:true
      triemann()   Riemann-tensor in frame base, used when cframe_flag:true
      tricci()     Ricci-tensor in frame base, used when cframe_flag:true
      trrc()       Ricci rotation coefficients, used by christof()
      yesp()

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### 28.2.11 Changes

In November, 2004, the `ctensor` package was extensively rewritten. Many functions and variables have been renamed in order to make the package compatible with the commercial version of Macsyma.

      New Name     Old Name        Description
      ---------------------------------------------------------------------
      ctaylor()    DLGTAYLOR()     Taylor-series expansion of an expression
      lgeod[]      EM              Geodesic equations
      ein[]        G[]             Mixed Einstein-tensor
      ric[]        LR[]            Mixed Ricci-tensor
      ricci()      LRICCICOM()     Compute the mixed Ricci-tensor
      ctaypov      MINP            Maximum power in Taylor-series expansion
      cgeodesic()  MOTION          Compute geodesic equations
      ct_coords    OMEGA           Metric coordinates
      ctayvar      PARAM           Taylor-series expansion variable
      lriem[]      R[]             Covariant Riemann-tensor
      uriemann()   RAISERIEMANN()  Compute the contravariant Riemann-tensor
      ratriemann   RATRIEMAN       Rational simplif. of the Riemann-tensor
      uric[]       RICCI[]         Contravariant Ricci-tensor
      uricci()     RICCICOM()      Compute the contravariant Ricci-tensor
      cmetric()    SETMETRIC()     Set up the metric
      ctaypt       TAYPT           Point for Taylor-series expansion
      ctayswitch   TAYSWITCH       Taylor-series setting switch
      csetup()     TSETUP()        Start interactive setup session
      ctransform() TTRANSFORM()    Interactive coordinate transformation
      uriem[]      UR[]            Contravariant Riemann-tensor
      weyl[]       W[]             (3,1) Weyl-tensor

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