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HELP FOR: nprotate
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CALLING SEQUENCE:  nprotate ( class, parm1, parm2 )

      PARAMETERS:  class - rotation type (see below).

                   parm1 - first rotation parameter:
                             Re(a) for class I rotations,
                             Re(b) for class II rotations,
                             A for class III rotations.

                   parm2 - second rotation parameter:
                             Im(a) for class I rotations,
                             Im(b) for class II rotations,
                             theta for class III rotations.

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SYNOPSIS:

- Performs a rotation of a null tetrad.

- Rotation classes for null tetrads {l,n,m,mbar} are defined as follows:

    I: leaves the vector l unchanged, rotation given by a complex valued
       parameter, a;
   II: leaves the vector n unchanged, rotation given by a complex valued
       parameter, b; and
  III: leave the directions of l, n, unchanged and rotate m,mbar in the
       (m,mbar) plane, rotation given by A*exp(i*theta) where A and theta
       are real.

- Explicit formulas for the effects of a rotation class on the tetrad are:

  Class I:     l -> l,
               n -> n + (a*)m + (a)mbar + (a)(a*)l,
               m -> m + (a)l,
               mbar -> mbar + (a*)l.

  Class II:    l -> l + (b*)m + (b)mbar + (b)(b*)n,
               n -> n,
               m -> m + (b)n,
               mbar -> mbar + (b*)n.

  Class III:   l -> (1/A)l,
               n -> (A)n,
               m -> exp(i (theta))m,
               mbar -> exp(-i (theta))mbar.

  In the above equations, (a), (b) are complex, (A), (theta) are real.
  The complex conjugate of (a) is represented by (a*).

- Note, in the current versions of GRTensorII, only rotations of the basis
  can be performed. The action of basis rotations on other curvature tensors
  has not yet been implemented. Thus curvature tensors must be recalculated
  from the rotated basis.

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EXAMPLES:

> qload ( schw ):

                              Default metric = schw

                                   Coordinates

                              For the schw metric.

                      1        2            3          4
                    x   = r, x   = theta, x   = phi, x   = t

                                  Line element

                              For the schw metric.

    2
  ds  =

           2
        d r       2         2     2           2       2                      2
     --------- + r   d theta   + r  sin(theta)   d phi   + (- 1 + 2 m/r)  d t
     1 - 2 m/r

> nptetrad ( [ 0, 0, 0, 1 ] ):

Calculated detg for schw. (0.034000 sec.) 
Calculated g(up,up) for schw. (0.116000 sec.) 

> grdisplay ( e(bdn,up) ):

                            For the schw spacetime:
 
                                 Basis vectors
 
       b
e [(a)]  =
 
 [             1/2                                                  1/2       ]
 [            2                                                    2          ]
 [  1/2 ----------------       0              0          1/2 ---------------- ]
 [      /      r    \1/2                                     /  - r + 2 m\1/2 ]
 [      |- ---------|                                        |- ---------|    ]
 [      \  - r + 2 m/                                        \      r    /    ]
 [                                                                            ]
 [              1/2                                                 1/2       ]
 [             2                                                   2          ]
 [ - 1/2 ----------------      0              0          1/2 ---------------- ]
 [       /      r    \1/2                                    /  - r + 2 m\1/2 ]
 [       |- ---------|                                       |- ---------|    ]
 [       \  - r + 2 m/                                       \      r    /    ]
 [                                                                            ]
 [                              1/2             1/2                           ]
 [                             2             I 2                              ]
 [            0            1/2 ----   1/2 ------------             0          ]
 [                               r        r sin(theta)                        ]
 [                                                                            ]
 [                              1/2              1/2                          ]
 [                             2              I 2                             ]
 [            0            1/2 ----  - 1/2 ------------            0          ]
 [                               r         r sin(theta)                       ]
 

> nprotate ( III, sqrt(2)*sqrt(1-2*m/r), 0 ):

Rotated tetrad stored as e(bdn,up).

> gralter ( e(bdn,up), radsimp ):

Component Alteration of a grtensor object:
Applying routine simplify[sqrt] to object e(bdn,up)

> grdisplay ( e(bdn,up) ):

                            For the schw spacetime:
 
                                 Basis vectors
 
              [                                                      r     ]
              [     -1/2         0              0          - 1/2 --------- ]
              [                                                  - r + 2 m ]
              [                                                            ]
              [   - r + 2 m                                                ]
              [ - ---------      0              0                 1        ]
              [       r                                                    ]
          b   [                                                            ]
   e [(a)]  = [                   1/2             1/2                      ]
              [                  2             I 2                         ]
              [      0       1/2 ----   1/2 ------------          0        ]
              [                    r        r sin(theta)                   ]
              [                                                            ]
              [                   1/2              1/2                     ]
              [                  2              I 2                        ]
              [      0       1/2 ----  - 1/2 ------------         0        ]
              [                    r         r sin(theta)                  ]
 

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SEE ALSO: grt_basis, nptetrad, makeg, grsaveg.
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