Begleitende Vielbeine (Maxima Manual)

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

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.

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.