Maxima Manual: Begleitende Vielbeine

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#### 20.2.2.5 Begleitende Vielbeine

Maxima now has the ability to perform calculations using moving frames. These can be orthonormal frames (tetrads, vielbeins) or an arbitrary frame.

To use frames, you must first set `iframe_flag` to `true`. This causes the Christoffel-symbols, `ichr1` and `ichr2`, to be replaced by the more general frame connection coefficients `icc1` and `icc2` in calculations. Speficially, the behavior of `covdiff` and `icurvature` is changed.

The frame is defined by two tensors: the inverse frame field (`ifri`, the dual basis tetrad), and the frame metric `ifg`. The frame metric is the identity matrix for orthonormal frames, or the Lorentz metric for orthonormal frames in Minkowski spacetime. The inverse frame field defines the frame base (unit vectors). Contraction properties are defined for the frame field and the frame metric.

When `iframe_flag` is true, many `itensor` expressions use the frame metric `ifg` instead of the metric defined by `imetric` for raising and lowerind indices.

IMPORTANT: Setting the variable `iframe_flag` to `true` does NOT undefine the contraction properties of a metric defined by a call to `defcon` or `imetric`. If a frame field is used, it is best to define the metric by assigning its name to the variable `imetric` and NOT invoke the `imetric` function.

Maxima uses these two tensors to define the frame coefficients (`ifc1` and `ifc2`) which form part of the connection coefficients (`icc1` and `icc2`), as the following example demonstrates:

    (%i1) load("itensor");
    (%o1)      /share/tensor/itensor.lisp
    (%i2) iframe_flag:true;
    (%o2)                                true
    (%i3) ishow(covdiff(v([],[i]),j))$
                                   i        i     %1
    (%t3)                         v   + icc2     v
                                   ,j       %1 j
    (%i4) ishow(ev(%,icc2))$
                                   %1     i       i
    (%t4)                         v   ifc2     + v
                                          %1 j    ,j
    (%i5) ishow(ev(%,ifc2))$
                              %1    i %2                i
    (%t5)                    v   ifg     ifc1        + v
                                             %1 j %2    ,j
    (%i6) ishow(ev(%,ifc1))$
                %1    i %2
               v   ifg     (ifb        - ifb        + ifb       )
                               j %2 %1      %2 %1 j      %1 j %2     i
    (%t6)      -------------------------------------------------- + v
                                       2                             ,j
    (%i7) ishow(ifb([a,b,c]))$
                                                       %3    %4
    (%t7)               (ifri        - ifri       ) ifr   ifr
                             a %3,%4       a %4,%3     b     c

An alternate method is used to compute the frame bracket (`ifb`) if the `iframe_bracket_form` flag is set to `false`:

    (%i8) block([iframe_bracket_form:false],ishow(ifb([a,b,c])))$
                                    %6    %5        %5      %6
    (%t8)              ifri     (ifr   ifr     - ifr     ifr  )
                           a %5     b     c,%6      b,%6    c

Optionsvariable: iframe_flag

  Standardwert: `false`

  To use frames, you must first set `iframe_flag` to `true`. This causes the Christoffel-symbols, `ichr1` and `ichr2`, to be replaced by the more general frame connection coefficients `icc1` and `icc2` in calculations. Speficially, the behavior of `covdiff` and `icurvature` is changed.

  The frame is defined by two tensors: the inverse frame field (`ifri`, the dual basis tetrad), and the frame metric `ifg`. The frame metric is the identity matrix for orthonormal frames, or the Lorentz metric for orthonormal frames in Minkowski spacetime. The inverse frame field defines the frame base (unit vectors). Contraction properties are defined for the frame field and the frame metric.

  When `iframe_flag` is true, many `itensor` expressions use the frame metric `ifg` instead of the metric defined by `imetric` for raising and lowerind indices.

  IMPORTANT: Setting the variable `iframe_flag` to `true` does NOT undefine the contraction properties of a metric defined by a call to `defcon` or `imetric`. If a frame field is used, it is best to define the metric by assigning its name to the variable `imetric` and NOT invoke the `imetric` function.

Function: iframes ()

  Since in this version of Maxima, contraction identities for `ifr` and `ifri` are always defined, as is the frame bracket (`ifb`), this function does nothing.

Variable: ifb

  The frame bracket. The contribution of the frame metric to the connection coefficients is expressed using the frame bracket:

                - ifb      + ifb      + ifb
                     c a b      b c a      a b c
      ifc1    = --------------------------------
          abc                  2

  The frame bracket itself is defined in terms of the frame field and frame metric. Two alternate methods of computation are used depending on the value of `frame_bracket_form`. If true (the default) or if the `itorsion_flag` is `true`:

                d      e                                      f
      ifb =  ifr    ifr   (ifri      - ifri      - ifri    itr   )
         abc    b      c       a d,e       a e,d       a f    d e

  Otherwise:

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

Variable: icc1

  Connection coefficients of the first kind. In `itensor`, defined as

      icc1    = ichr1    - ikt1    - inmc1
          abc        abc       abc        abc

  In this expression, if `iframe_flag` is true, the Christoffel-symbol `ichr1` is replaced with the frame connection coefficient `ifc1`. If `itorsion_flag` is `false`, `ikt1` will be omitted. It is also omitted if a frame base is used, as the torsion is already calculated as part of the frame bracket. Lastly, of `inonmet_flag` is `false`, `inmc1` will not be present.

Variable: icc2

  Connection coefficients of the second kind. In `itensor`, defined as

          c         c        c         c
      icc2   = ichr2   - ikt2   - inmc2
          ab        ab       ab        ab

  In this expression, if `iframe_flag` is true, the Christoffel-symbol `ichr2` is replaced with the frame connection coefficient `ifc2`. If `itorsion_flag` is `false`, `ikt2` will be omitted. It is also omitted if a frame base is used, as the torsion is already calculated as part of the frame bracket. Lastly, of `inonmet_flag` is `false`, `inmc2` will not be present.

Variable: ifc1

  Frame coefficient of the first kind (also known as Ricci-rotation coefficients.) This tensor represents the contribution of the frame metric to the connection coefficient of the first kind. Defined as:

                - ifb      + ifb      + ifb
                     c a b      b c a      a b c
      ifc1    = --------------------------------
          abc                   2

Variable: ifc2

  Frame coefficient of the first kind. This tensor represents the contribution of the frame metric to the connection coefficient of the first kind. Defined as a permutation of the frame bracket (`ifb`) with the appropriate indices raised and lowered as necessary:

          c       cd
      ifc2   = ifg   ifc1
          ab             abd

Variable: ifr

  The frame field. Contracts with the inverse frame field (`ifri`) to form the frame metric (`ifg`).

Variable: ifri

  The inverse frame field. Specifies the frame base (dual basis vectors). Along with the frame metric, it forms the basis of all calculations based on frames.

Variable: ifg

  The frame metric. Defaults to `kdelta`, but can be changed using `components`.

Variable: ifgi

  The inverse frame metric. Contracts with the frame metric (`ifg`) to `kdelta`.

Option variable: iframe_bracket_form

  Default value: `true`

  Specifies how the frame bracket (`ifb`) is computed.

Next: Torsion und Nichtmetrizität, Previous: Tensoren in gekrümmten Räumen, Nach oben: Funktionen und Variablen für itensor [Inhalt][Index]