path: root/trumpet.nb

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beta*M*x*(x^2 + y^2 + z^2)*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^3 + 
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3*C*x*Sqrt[1 + C^2/R[r]^4 - (2*M)/R[r]])*R[r]^6 - 
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(2*M)/R[r]])*R[r]^7))/
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z^2)*R[r]^4 + 
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x)*(r + x) + 2*beta*(C*x - beta*M*(x^2 + y^2 + z^2))*R[r]^3 + 
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       beta*x*(x^2 + y^2 + z^2)*(-1 + Sqrt[1 + C^2/R[r]^4 - \
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                                                   2      2   2           3   \
    4
                                         (-1 + beta ) R[r]  (C  - 2 M R[r]  + \
R[r] )
{{-((Sqrt[--------------------------------------------------------------------\
--------------------------------------] 
              2  2                                          2    2    2       \
3       2   2    2    2      4       6
          beta  C  (r - x) (r + x) + 2 beta (C x - beta M (x  + y  + z )) \
R[r]  + beta  (x  + y  + z ) R[r]  - R[r]
 
                                                                            2 \
                                           2
                  2        2    2    2      2    2    2                    C  \
    2 M            2    2    2            C      2 M
        (beta (-(C  x (3 (x  + y  + z ) + (x  + y  + z ) (-2 - 5 Sqrt[1 + \
----- - ----]) + 3 (2 x  + y  + z ) Sqrt[1 + ----- - ----])) - 
                                                                              \
4   R[r]                                     4   R[r]
                                                                          R[r]\
                                         R[r]
 
                                        2                                     \
                            2
                   2  2  4             C      2 M                2    2    2  \
    2    2    2            C      2 M        3
             2 beta  M  r  x Sqrt[1 + ----- - ----] + beta C M (x  + y  + z ) \
(3 x  + y  + z ) Sqrt[1 + ----- - ----]) R[r]  + 
                                          4   R[r]                            \
                              4   R[r]
                                      R[r]                                    \
                          R[r]
 
                                  2                                           \
                       2                                   2
              3    4             C      2 M       4         2    2    2     2 \
   2                  C      2 M       2                  C      2 M
          beta  M r  x Sqrt[1 + ----- - ----] R[r]  + (C (-x  - y  - z  - (y  \
+ z ) (-1 + Sqrt[1 + ----- - ----]) + x  (1 + 2 Sqrt[1 + ----- - ----])) + 
                                    4   R[r]                                  \
                         4   R[r]                            4   R[r]
                                R[r]                                          \
                     R[r]                                R[r]
 
                                                         2                    \
                                        2
                          2    2    2                   C      2 M         6  \
          2    2    2                  C      2 M        7
             2 beta M x (x  + y  + z ) (1 - 2 Sqrt[1 + ----- - ----])) R[r]  \
+ beta x (x  + y  + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / 
                                                           4   R[r]           \
                                          4   R[r]
                                                       R[r]                   \
                                      R[r]
 
                2 3/2  4     3   2           3       4
      ((1 - beta )    r  R[r]  (C  - 2 M R[r]  + R[r] ))), (y (beta C 
 
                                            2                                 \
    2                                         2
              2    2    2                  C      2 M          2    2         \
   C      2 M         2    2    2            C      2 M
         (C (y  + z  + x  (1 + 5 Sqrt[1 + ----- - ----]) + 2 (y  + z ) Sqrt[1 \
+ ----- - ----] - 3 (x  + y  + z ) Sqrt[1 + ----- - ----]) - 
                                              4   R[r]                        \
      4   R[r]                                  4   R[r]
                                          R[r]                                \
  R[r]                                      R[r]
 
                                              2                               \
                               2                               2
                      2    2    2            C      2 M        3             \
2    2    2                    C      2 M                      C      2 M     \
   6
           beta M x (x  + y  + z ) Sqrt[1 + ----- - ----]) R[r]  + (beta M (x \
 + y  + z ) (-2 + 3 Sqrt[1 + ----- - ----]) - 3 C x Sqrt[1 + ----- - ----]) \
R[r]  - 
                                                4   R[r]                      \
                                 4   R[r]                        4   R[r]
                                            R[r]                              \
                             R[r]                            R[r]
 
                                             2
               2    2    2                  C      2 M        7
        beta (x  + y  + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / 
                                               4   R[r]
                                           R[r]
 
                                                           2      2   2       \
    3       4
      4                                          (-1 + beta ) R[r]  (C  - 2 M \
R[r]  + R[r] )
    (r  R[r] \
Sqrt[-------------------------------------------------------------------------\
---------------------------------] 
                      2  2                                          2    2    \
2       3       2   2    2    2      4       6
                  beta  C  (r - x) (r + x) + 2 beta (C x - beta M (x  + y  + \
z )) R[r]  + beta  (x  + y  + z ) R[r]  - R[r]
 
           2  2    2    2                              2    2    2       3    \
   2   2    2    2      4       6
      (beta  C  (-y  - z ) + 2 beta (-(C x) + beta M (x  + y  + z )) R[r]  - \
beta  (x  + y  + z ) R[r]  + R[r] )), 
 
                                                 2                            \
         2                                         2
                   2    2    2                  C      2 M          2    2    \
        C      2 M         2    2    2            C      2 M
   (z (beta C (C (y  + z  + x  (1 + 5 Sqrt[1 + ----- - ----]) + 2 (y  + z ) \
Sqrt[1 + ----- - ----] - 3 (x  + y  + z ) Sqrt[1 + ----- - ----]) - 
                                                   4   R[r]                   \
           4   R[r]                                  4   R[r]
                                               R[r]                           \
       R[r]                                      R[r]
 
                                              2                               \
                               2                               2
                      2    2    2            C      2 M        3             \
2    2    2                    C      2 M                      C      2 M     \
   6
           beta M x (x  + y  + z ) Sqrt[1 + ----- - ----]) R[r]  + (beta M (x \
 + y  + z ) (-2 + 3 Sqrt[1 + ----- - ----]) - 3 C x Sqrt[1 + ----- - ----]) \
R[r]  - 
                                                4   R[r]                      \
                                 4   R[r]                        4   R[r]
                                            R[r]                              \
                             R[r]                            R[r]
 
                                             2
               2    2    2                  C      2 M        7
        beta (x  + y  + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / 
                                               4   R[r]
                                           R[r]
 
                                                           2      2   2       \
    3       4
      4                                          (-1 + beta ) R[r]  (C  - 2 M \
R[r]  + R[r] )
    (r  R[r] \
Sqrt[-------------------------------------------------------------------------\
---------------------------------] 
                      2  2                                          2    2    \
2       3       2   2    2    2      4       6
                  beta  C  (r - x) (r + x) + 2 beta (C x - beta M (x  + y  + \
z )) R[r]  + beta  (x  + y  + z ) R[r]  - R[r]
 
           2  2    2    2                              2    2    2       3    \
   2   2    2    2      4       6
      (beta  C  (-y  - z ) + 2 beta (-(C x) + beta M (x  + y  + z )) R[r]  - \
beta  (x  + y  + z ) R[r]  + R[r] ))}, 
 
                                                 2                            \
         2                                         2
                   2    2    2                  C      2 M          2    2    \
        C      2 M         2    2    2            C      2 M
  {(y (beta C (C (y  + z  + x  (1 + 5 Sqrt[1 + ----- - ----]) + 2 (y  + z ) \
Sqrt[1 + ----- - ----] - 3 (x  + y  + z ) Sqrt[1 + ----- - ----]) - 
                                                   4   R[r]                   \
           4   R[r]                                  4   R[r]
                                               R[r]                           \
       R[r]                                      R[r]
 
                                              2                               \
                               2                               2
                      2    2    2            C      2 M        3             \
2    2    2                    C      2 M                      C      2 M     \
   6
           beta M x (x  + y  + z ) Sqrt[1 + ----- - ----]) R[r]  + (beta M (x \
 + y  + z ) (-2 + 3 Sqrt[1 + ----- - ----]) - 3 C x Sqrt[1 + ----- - ----]) \
R[r]  - 
                                                4   R[r]                      \
                                 4   R[r]                        4   R[r]
                                            R[r]                              \
                             R[r]                            R[r]
 
                                             2
               2    2    2                  C      2 M        7
        beta (x  + y  + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / 
                                               4   R[r]
                                           R[r]
 
                                                           2      2   2       \
    3       4
      4                                          (-1 + beta ) R[r]  (C  - 2 M \
R[r]  + R[r] )
    (r  R[r] \
Sqrt[-------------------------------------------------------------------------\
---------------------------------] 
                      2  2                                          2    2    \
2       3       2   2    2    2      4       6
                  beta  C  (r - x) (r + x) + 2 beta (C x - beta M (x  + y  + \
z )) R[r]  + beta  (x  + y  + z ) R[r]  - R[r]
 
           2  2    2    2                              2    2    2       3    \
   2   2    2    2      4       6
      (beta  C  (-y  - z ) + 2 beta (-(C x) + beta M (x  + y  + z )) R[r]  - \
beta  (x  + y  + z ) R[r]  + R[r] )), 
 
                                                  2      2   2           3    \
   4
                                        (-1 + beta ) R[r]  (C  - 2 M R[r]  + \
R[r] )
   (Sqrt[---------------------------------------------------------------------\
-------------------------------------] 
             2  2                                          2    2    2       \
3       2   2    2    2      4       6
         beta  C  (r - x) (r + x) + 2 beta (C x - beta M (x  + y  + z )) R[r] \
 + beta  (x  + y  + z ) R[r]  - R[r]
 
                                                                 2            \
                              2
             2     2    2    2     2    2    2                  C      2 M    \
    2      2    2            C      2 M
      (beta C  x (x  + y  + z  + (x  + y  + z ) (-2 + Sqrt[1 + ----- - ----]) \
- (x  - 2 y  + z ) Sqrt[1 + ----- - ----]) + 
                                                                   4   R[r]   \
                                4   R[r]
                                                               R[r]           \
                            R[r]
 
                                                           2                  \
                        2                                      2
            3                2    2    2                  C      2 M          \
2    2                 C      2 M           2    2            C      2 M
        R[r]  (-2 beta M x (x  + y  + z ) (-1 + Sqrt[1 + ----- - ----]) + C \
(x  + x  (-1 + Sqrt[1 + ----- - ----]) + (-2 y  + z ) Sqrt[1 + ----- - ----]) \
+ 
                                                             4   R[r]         \
                          4   R[r]                               4   R[r]
                                                         R[r]                 \
                      R[r]                                   R[r]
 
                                                  2
                    2    2    2                  C      2 M                   \
         2   4   2           3       4
           beta x (x  + y  + z ) (-1 + Sqrt[1 + ----- - ----]) R[r]))) / \
(Sqrt[1 - beta ] r  (C  - 2 M R[r]  + R[r] )), 
                                                    4   R[r]
                                                R[r]
 
                       2
                      C      2 M                      3
   -3 C y z Sqrt[1 + ----- - ----] (-(beta C x) + R[r] ) Sqrt[
                         4   R[r]
                     R[r]
    
                                                2      2   2           3      \
 4
                                      (-1 + beta ) R[r]  (C  - 2 M R[r]  + \
R[r] )
       -----------------------------------------------------------------------\
-----------------------------------]
           2  2                                          2    2    2       3  \
     2   2    2    2      4       6
       beta  C  (r - x) (r + x) + 2 beta (C x - beta M (x  + y  + z )) R[r]  \
+ beta  (x  + y  + z ) R[r]  - R[r]
   ---------------------------------------------------------------------------\
------------------------------------------------------------------------------\
-------------
                                                                             2\
   4   2           3       4
                                                                Sqrt[1 - beta \
] r  (C  - 2 M R[r]  + R[r] )
 
                                                      2                       \
              2                                         2
                        2    2    2                  C      2 M          2    \
2            C      2 M         2    2    2            C      2 M
    }, {(z (beta C (C (y  + z  + x  (1 + 5 Sqrt[1 + ----- - ----]) + 2 (y  + \
z ) Sqrt[1 + ----- - ----] - 3 (x  + y  + z ) Sqrt[1 + ----- - ----]) - 
                                                        4   R[r]              \
                4   R[r]                                  4   R[r]
                                                    R[r]                      \
            R[r]                                      R[r]
 
                                              2                               \
                               2                               2
                      2    2    2            C      2 M        3             \
2    2    2                    C      2 M                      C      2 M     \
   6
           beta M x (x  + y  + z ) Sqrt[1 + ----- - ----]) R[r]  + (beta M (x \
 + y  + z ) (-2 + 3 Sqrt[1 + ----- - ----]) - 3 C x Sqrt[1 + ----- - ----]) \
R[r]  - 
                                                4   R[r]                      \
                                 4   R[r]                        4   R[r]
                                            R[r]                              \
                             R[r]                            R[r]
 
                                             2
               2    2    2                  C      2 M        7
        beta (x  + y  + z ) (-1 + Sqrt[1 + ----- - ----]) R[r] )) / 
                                               4   R[r]
                                           R[r]
 
                                                           2      2   2       \
    3       4
      4                                          (-1 + beta ) R[r]  (C  - 2 M \
R[r]  + R[r] )
    (r  R[r] \
Sqrt[-------------------------------------------------------------------------\
---------------------------------] 
                      2  2                                          2    2    \
2       3       2   2    2    2      4       6
                  beta  C  (r - x) (r + x) + 2 beta (C x - beta M (x  + y  + \
z )) R[r]  + beta  (x  + y  + z ) R[r]  - R[r]
 
           2  2    2    2                              2    2    2       3    \
   2   2    2    2      4       6
      (beta  C  (-y  - z ) + 2 beta (-(C x) + beta M (x  + y  + z )) R[r]  - \
beta  (x  + y  + z ) R[r]  + R[r] )), 
 
                       2
                      C      2 M                      3
   -3 C y z Sqrt[1 + ----- - ----] (-(beta C x) + R[r] ) Sqrt[
                         4   R[r]
                     R[r]
    
                                                2      2   2           3      \
 4
                                      (-1 + beta ) R[r]  (C  - 2 M R[r]  + \
R[r] )
       -----------------------------------------------------------------------\
-----------------------------------]
           2  2                                          2    2    2       3  \
     2   2    2    2      4       6
       beta  C  (r - x) (r + x) + 2 beta (C x - beta M (x  + y  + z )) R[r]  \
+ beta  (x  + y  + z ) R[r]  - R[r]
   ---------------------------------------------------------------------------\
------------------------------------------------------------------------------\
-------------
                                                                             2\
   4   2           3       4
                                                                Sqrt[1 - beta \
] r  (C  - 2 M R[r]  + R[r] )
 
                                                     2      2   2           3 \
      4
                                           (-1 + beta ) R[r]  (C  - 2 M R[r]  \
+ R[r] )
    , (Sqrt[------------------------------------------------------------------\
----------------------------------------] 
                2  2                                          2    2    2     \
  3       2   2    2    2      4       6
            beta  C  (r - x) (r + x) + 2 beta (C x - beta M (x  + y  + z )) \
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                                                                 2            \
                              2
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                                                                   4   R[r]   \
                                4   R[r]
                                                               R[r]           \
                            R[r]
 
                                                           2                  \
                        2                                     2
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2    2                 C      2 M        2      2            C      2 M
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                                                             4   R[r]         \
                          4   R[r]                              4   R[r]
                                                         R[r]                 \
                      R[r]                                  R[r]
 
                                                  2
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         2   4   2           3       4
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