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ratriemann

? ratriemann;

Calculate

ratriemann

? ratriemann;

Calculate

ratriemann

Run Example
(%i1)/*  Schwarzschild simplesave as name.macLoad from the file menu - >
 File|Batch file*/kill(all);
(%o0)                                done
(%i1) load(ctensor);
(%o1)          /usr/share/maxima/5.21.1/share/tensor/ctensor.mac
(%i2) /* set some flags */cframe_flag: false;
(%o2)                                false
(%i3) ratchristof: true;
(%o3)                                true
(%i4) ratriemann : true;
(%o4)                                true
(%i5) ratfac : true;
(%o5)                                true
(%i6) ctrgsimp: true;
(%o6)                                true
(%i7) /* define the dimension */dim: 4;
(%o7)                                  4
(%i8) /* list the coordinates */ct_coords: [t,x,y,z];
(%o8)                            [t, x, y, z]
(%i9) /* set up the metric *//* assign to lg a matrix of zeros  ':' means 'assign' */lg:zeromatrix(4,4);
                                [ 0  0  0  0 ]
                                [            ]
                                [ 0  0  0  0 ]
(%o9)                           [            ]
                                [ 0  0  0  0 ]
                                [            ]
                                [ 0  0  0  0 ]
(%i10) /* now add the Schwarzschild coefficients */lg[1,1]:1;
(%o10)                                 1
(%i11) lg[2,2]:-1;
(%o11)                                - 1
(%i12) lg[3,3]:-(1+f(t-x))^2;
                                               2
(%o12)                         - (f(t - x) + 1)
(%i13) lg[4,4]:-(1+g(t-x))^2;
                                               2
(%o13)                         - (g(t - x) + 1)
(%i14) /* make the inverse matrix */ug:invert(lg);
               [ 1   0           0                  0         ]
               [                                              ]
               [ 0  - 1          0                  0         ]
               [                                              ]
               [                  1                           ]
               [ 0   0   - ---------------          0         ]
(%o14)         [                         2                    ]
               [           (f(t - x) + 1)                     ]
               [                                              ]
               [                                     1        ]
               [ 0   0           0          - --------------- ]
               [                                            2 ]
               [                              (g(t - x) + 1)  ]
(%i15) /* get Christoffels */christof(mcs);
                                       d
                                       -- (f(t - x))
                                       dt
(%t15)                    mcs        = -------------
                             1, 3, 3   f(t - x) + 1

                                       d
                                       -- (g(t - x))
                                       dt
(%t16)                    mcs        = -------------
                             1, 4, 4   g(t - x) + 1

                                       d
                                       -- (f(t - x))
                                       dx
(%t17)                    mcs        = -------------
                             2, 3, 3   f(t - x) + 1

                                       d
                                       -- (g(t - x))
                                       dx
(%t18)                    mcs        = -------------
                             2, 4, 4   g(t - x) + 1

                                d
(%t19)            mcs        = (-- (f(t - x))) (f(t - x) + 1)
                     3, 3, 1    dt

                                 d
(%t20)           mcs        = - (-- (f(t - x))) (f(t - x) + 1)
                    3, 3, 2      dx

                                d
(%t21)            mcs        = (-- (g(t - x))) (g(t - x) + 1)
                     4, 4, 1    dt

                                 d
(%t22)           mcs        = - (-- (g(t - x))) (g(t - x) + 1)
                    4, 4, 2      dx

(%o22)                               done
(%i23) /* calculate and display Ricci tensor, ( 'dis' = true ) */ricci(true);
                      2                           2
                     d                           d
(%t23) ric     = - ((--- (f(t - x))) g(t - x) + (--- (g(t - x))) f(t - x)
          1, 1         2                           2
                     dt                          dt
                2                2
               d                d
             + --- (g(t - x)) + --- (f(t - x)))/((f(t - x) + 1) (g(t - x) + 1))
                 2                2
               dt               dt

                       2                             2
                      d                             d
(%t24) ric     = - ((----- (f(t - x))) g(t - x) + (----- (g(t - x))) f(t - x)
          1, 2       dt dx                         dt dx
             2                  2
            d                  d
         + ----- (g(t - x)) + ----- (f(t - x)))/((f(t - x) + 1) (g(t - x) + 1))
           dt dx              dt dx

                      2                           2
                     d                           d
(%t25) ric     = - ((--- (f(t - x))) g(t - x) + (--- (g(t - x))) f(t - x)
          2, 2         2                           2
                     dx                          dx
                2                2
               d                d
             + --- (g(t - x)) + --- (f(t - x)))/((f(t - x) + 1) (g(t - x) + 1))
                 2                2
               dx               dx

                                      2
                                     d
(%t26) ric     = - ((f(t - x) + 1) ((--- (f(t - x))) g(t - x)
          3, 3                         2
                                     dx
     2
    d                           d               d
 - (--- (f(t - x))) g(t - x) + (-- (f(t - x))) (-- (g(t - x)))
      2                         dx              dx
    dt
                                      2                2
    d               d                d                d
 - (-- (f(t - x))) (-- (g(t - x))) + --- (f(t - x)) - --- (f(t - x))))
    dt              dt                 2                2
                                     dx               dt
/(g(t - x) + 1)

                       2                           2
                      d                           d
(%t27) ric     = - (((--- (g(t - x))) f(t - x) - (--- (g(t - x))) f(t - x)
          4, 4          2                           2
                      dx                          dt
    2                                                  2
   d                 d               d                d
 + --- (g(t - x)) + (-- (f(t - x))) (-- (g(t - x))) - --- (g(t - x))
     2               dx              dx                 2
   dx                                                 dt
    d               d
 - (-- (f(t - x))) (-- (g(t - x)))) (g(t - x) + 1))/(f(t - x) + 1)
    dt              dt

(%o27)                               done
(%i28) /* Riemann with all indexes down */lriemann(true);
                                   2
                                  d
(%t28)         lriem           = (--- (f(t - x))) (f(t - x) + 1)
                    3, 3, 1, 1      2
                                  dt

                                   2
                                  d
(%t29)        lriem           = (----- (f(t - x))) (f(t - x) + 1)
                   3, 3, 1, 2    dt dx

                                   2
                                  d
(%t30)        lriem           = (----- (f(t - x))) (f(t - x) + 1)
                   3, 3, 2, 1    dt dx

                                   2
                                  d
(%t31)         lriem           = (--- (f(t - x))) (f(t - x) + 1)
                    3, 3, 2, 2      2
                                  dx

                                   2
                                  d
(%t32)         lriem           = (--- (g(t - x))) (g(t - x) + 1)
                    4, 4, 1, 1      2
                                  dt

                                   2
                                  d
(%t33)        lriem           = (----- (g(t - x))) (g(t - x) + 1)
                   4, 4, 1, 2    dt dx

                                   2
                                  d
(%t34)        lriem           = (----- (g(t - x))) (g(t - x) + 1)
                   4, 4, 2, 1    dt dx

                                   2
                                  d
(%t35)         lriem           = (--- (g(t - x))) (g(t - x) + 1)
                    4, 4, 2, 2      2
                                  dx

                           d               d
(%t36) lriem           = ((-- (f(t - x))) (-- (g(t - x)))
            4, 4, 3, 3     dx              dx
                  d               d
               - (-- (f(t - x))) (-- (g(t - x)))) (f(t - x) + 1) (g(t - x) + 1)
                  dt              dt

(%o36)                               done
(%i37) /* Riemann wi