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einstein-false-invert-kill-load-ratfac-ratriemann-rinvariant-scurvature-sin-true

/* Schwarzschild sim...

load(ctensor);

/* set some flags */c...

Calculate

einstein-false-invert-kill-load-ratfac-ratriemann-rinvariant-scurvature-sin-true

/* Schwarzschild sim...

load(ctensor);

/* set some flags */c...

Calculate

einstein-false-invert-kill-load-ratfac-ratriemann-rinvariant-sin-true

/* Schwarzschild sim...

load(ctensor);

/* set some flags */c...

Calculate

einstein-ident-load-true

load(ctensor);

ct_coords:[t,x,y,z];

lg:ident(4);

Calculate

einstein-false-invert-kill-load-ratfac-ratriemann-rinvariant-scurvature-true

/* Schwarzschild sim...

load(ctensor);

/* set some flags */c...

Calculate

einstein-false-invert-kill-load-ratfac-ratriemann-rinvariant-scurvature-sin-true

/* Schwarzschild sim...

load(ctensor);

/* set some flags */c...

Calculate

einstein-false-ident-load

load(ctensor);

ct_coords:[t,x,y,z];

lg:ident(4);

Calculate

einstein-false-ident-load

load(ctensor);

ct_coords:[t,x,y,z];

lg:ident(4);

Calculate

einstein-false-invert-kill-load-ratfac-ratriemann-rinvariant-scurvature-sin-true

/* Schwarzschild sim...

load(ctensor);

/* set some flags */c...

Calculate

einstein-false-invert-kill-load-ratfac-ratriemann-rinvariant-scurvature-true

/* Schwarzschild sim...

load(ctensor);

/* set some flags */c...

Calculate

einstein

Run Example
(%i1)load(ctensor);
(%o1)          /usr/share/maxima/5.21.1/share/tensor/ctensor.mac
(%i2) ct_coords:[t,r,theta,phi];
(%o2)                         [t, r, theta, phi]
(%i3) lg:matrix([-exp(-2*A/r),0,0,0],[0,1,0,0],[0,0,exp(-2*A/r)*r^2,0],[0,0,0,exp(-2*A/r)*r^2*sin(theta)^2]);
             [       2 A                                        ]
             [     - ---                                        ]
             [        r                                         ]
             [ - %e       0      0                 0            ]
             [                                                  ]
             [     0      1      0                 0            ]
             [                                                  ]
(%o3)        [                      2 A                         ]
             [                    - ---                         ]
             [                2      r                          ]
             [     0      0  r  %e                 0            ]
             [                                                  ]
             [                                              2 A ]
             [                                            - --- ]
             [                            2    2             r  ]
             [     0      0      0       r  sin (theta) %e      ]
(%i4) ug:matrix([-exp(2*A/r),0,0,0],[0,1,0,0],[0,0,exp(2*A/r)/r^2,0],[0,0,0,exp(2*A/r)/(r^2*sin(theta)^2)]);
                     [     2 A                           ]
                     [     ---                           ]
                     [      r                            ]
                     [ - %e     0    0          0        ]
                     [                                   ]
                     [    0     1    0          0        ]
                     [                                   ]
                     [               2 A                 ]
                     [               ---                 ]
                     [                r                  ]
(%o4)                [             %e                    ]
                     [    0     0  -----        0        ]
                     [               2                   ]
                     [              r                    ]
                     [                                   ]
                     [                          2 A      ]
                     [                          ---      ]
                     [                           r       ]
                     [                        %e         ]
                     [    0     0    0    -------------- ]
                     [                     2    2        ]
                     [                    r  sin (theta) ]
(%i5) christof(mcs);
                                               2 A
                                             - ---
                                                r
                                         A %e
(%t5)                       mcs        = ---------
                               1, 1, 2       2
                                            r

                                             A
(%t6)                           mcs        = --
                                   1, 2, 1    2
                                             r

                                           A + r
(%t7)                         mcs        = -----
                                 2, 3, 3     2
                                            r

                                           A + r
(%t8)                         mcs        = -----
                                 2, 4, 4     2
                                            r

                                                   2 A
                                                 - ---
                                                    r
(%t9)                   mcs        = - (A + r) %e
                           3, 3, 2

                                         cos(theta)
(%t10)                      mcs        = ----------
                               3, 4, 4   sin(theta)

                                                               2 A
                                                             - ---
                               2                 2              r
(%t11)      mcs        = - (sin (theta) A + r sin (theta)) %e
               4, 4, 2

(%t12)               mcs        = - cos(theta) sin(theta)
                        4, 4, 3

(%o12)                               done
(%i13) riemann(true);
                                            2 A          2 A
                                          - ---        - ---
                                             r     2      r
                                    2 A %e        A  %e
(%t13)             riem           = ----------- - ----------
                       1, 2, 1, 2        3             4
                                        r             r

                                                      2 A
                                                    - ---
                                                       r
                                        A (A + r) %e
(%t14)               riem           = - -----------------
                         1, 3, 1, 3             4
                                               r

                                                      2 A
                                                    - ---
                                                       r
                                        A (A + r) %e
(%t15)               riem           = - -----------------
                         1, 4, 1, 4             4
                                               r

                                             2
                                            A  - 2 r A
(%t16)                   riem           = - ----------
                             2, 2, 1, 1          4
                                                r

                                          2
                                   (A + r)    2 (A + r)   1
(%t17)            riem           = -------- - --------- + --
                      2, 3, 2, 3       4          3        2
                                      r          r        r

                                          2
                                   (A + r)    2 (A + r)   1
(%t18)            riem           = -------- - --------- + --
                      2, 4, 2, 4       4          3        2
                                      r          r        r

                                                       2 A
                                                     - ---
                                          2             r
                                        (A  + r A) %e
(%t19)               riem           = - ------------------
                         3, 3, 1, 1              2
                                                r

                                                   2 A
                                                 - ---
                                             2      r
                                            A  %e
(%t20)                   riem           = - ----------
                             3, 3, 2, 2          2
                                                r

                                                   2 A
                                                 - ---
                                             2      r
                                      (A + r)  %e
(%t21)               riem           = ---------------- - 1
                         3, 4, 3, 4           2
                                             r

                                                                   2 A
                                                                 - ---
                                2         2        2                r
                            (sin (theta) A  + r sin (theta) A) %e
(%t22)   riem           = - ------------------------------------------
             4, 4, 1, 1                          2
                                                r

                                                         2 A
                                                       - ---
                                         2         2      r
                                      sin (theta) A  %e
(%t23)             riem           = - ----------------------
                       4, 4, 2, 2                2
                                                r

                             2 A                   2 A
                           - ---                   ---
                              r    2    2           r       2         2
(%t24) riem           = (%e      (r  sin (theta) %e    - sin (theta) A
           4, 4, 3, 3
                                               2             2    2           2
                                      - 2 r sin (theta) A - r  sin (theta)))/r

(%o24)                               done
(%i25) ricci(true);
                            2 A                   2 A           2 A
                          - ---                 - ---         - ---
                      2      r                     r             r
                     A  %e        2 A (A + r) %e        2 A %e
(%t25)     ric     = ---------- + ------------------- - -----------
              1, 1        4                4                 3
                         r                r                 r

                                   2    2
                          2 (A + r)    A    4 (A + r)   2 A   2
(%t26)        ric     = - ---------- - -- + --------- + --- - --
                 2, 2          4        4       3        3     2
                              r        r       r        r     r

                                             2 A
                                           - ---       2 A
                                              r      - ---
                             3 A (A + r) %e             r
(%t27)           ric     = - ------------------- - %e      + 1
                    3, 3              2
                                     r

                                                           2 A
                                                         - ---
                           2                 2              r
                   3 A (sin (theta) A + r sin (theta)) %e
(%t28) ric     = - -------------------------------------------
          4, 4                          2
                                       r
                                                              2 A
                                                            - ---
                                                 2             r       2
                                            - sin (theta) %e      + sin (theta)

(%o28)                               done
(%i29) R:scurvature();
                 2 A                   2 A           2 A
               - ---                 - ---         - ---    2 A
           2      r                     r             r     ---
          A  %e        2 A (A + r) %e        2 A %e          r
(%o29) - (---------- + ------------------- - -----------) %e
               4                4                 3
              r                r                 r
                                               2 A
                                             - ---                   2 A
               2                 2              r                  - ---
       3 A (sin (theta) A + r sin (theta)) %e           2             r
 + ((- ------------------------------------------- - sin (theta) %e
                            2
                           r
                  2 A
                  ---
      2            r     2    2
 + sin (theta)) %e   )/(r  sin (theta))
                      2 A
                    - ---       2 A        2 A
                       r      - ---        ---
      3 A (A + r) %e             r          r
   (- ------------------- - %e      + 1) %e
               2                                          2    2
              r                                  2 (A + r)    A    4 (A + r)
 + ------------------------------------------- - ---------- - -- + ---------
                        2                             4        4       3
                       r                             r        r       r
   2 A   2
 + --- - --
    3     2
   r     r
(%i30) einstein(true);
                                    2 A
                                    ---
                                2    r       2            2
                               r  %e    - 3 A  - 2 r A - r
(%t30)             ein     = - ----------------------------
                      1, 1                   4
                                            r

                                    2 A
                                    ---
                                2    r       2            2
                               r  %e    - 3 A  - 4 r A - r
(%t31)             ein     = - ----------------------------
                      2, 2                   4
                                            r

                                          2
                                       3 A  - r A
(%t32)                       ein     = ----------
                                3, 3        4
                                           r

                                          2
                                       3 A  - r A
(%t33)                       ein     = ----------
                                4, 4        4
                                           r

(%o33)                               done
(%i34) 
Run Example
/*  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 with all indexes up */uriemann(false);
(%o37)                               done
(%i38) /* calculate Einstein tensor */einstein(false);
(%o38)                               done
(%i39) /* show Einstein tensor as a matrix */cdisplay(ein);
                   2                           2
                  d                           d
ein = matrix([- ((--- (f(t - x))) g(t - x) + (--- (g(t - x))) f(t - x)
                    2                           2
                  dx                          dx
    2
   d                 d               d
 + --- (g(t - x)) + (-- (f(t - x))) (-- (g(t - x)))
     2               dx              dx
   dx
                                      2
    d               d                d
 - (-- (f(t - x))) (-- (g(t - x))) + --- (f(t - x)))
    dt              dt                 2
                                     dx
                                      2
                                     d
/((f(t - x) + 1) (g(t - x) + 1)), ((----- (f(t - x))) g(t - x)
                                    dt dx
      2                            2                  2
     d                            d                  d
 + (----- (g(t - x))) f(t - x) + ----- (g(t - x)) + ----- (f(t - x)))
    dt dx                        dt dx              dt dx
/((f(t - x) + 1) (g(t - x) + 1)), 0, 0], 
       2                             2
      d                             d
[- ((----- (f(t - x))) g(t - x) + (----- (g(t - x))) f(t - x)
     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
((--- (f(t - x))) g(t - x) + (--- (g(t - x))) f(t - x)
    2                           2
  dt                          dt
                                      2
    d               d                d
 - (-- (f(t - x))) (-- (g(t - x))) + --- (g(t - x))
    dx              dx                 2
                                     dt
                                      2
    d               d                d
 + (-- (f(t - x))) (-- (g(t - x))) + --- (f(t - x)))
    dt              dt                 2
                                     dt
/((f(t - x) + 1) (g(t - x) + 1)), 0, 0], 
          2                2
         d                d
         --- (g(t - x)) - --- (g(t - x))
           2                2
         dx               dt
[0, 0, - -------------------------------, 0], 
                  g(t - x) + 1
             2                2
            d                d
            --- (f(t - x)) - --- (f(t - x))
              2                2
            dx               dt
[0, 0, 0, - -------------------------------])
                     f(t - x) + 1

(%o39)                               done
(%i40) /* ricci curvature */scurvature();
             2                           2
            d                           d
(%o40) (2 ((--- (f(t - x))) g(t - x) - (--- (f(t - x))) g(t - x)
              2                           2
            dx                          dt
     2                           2                          2
    d                           d                          d
 + (--- (g(t - x))) f(t - x) - (--- (g(t - x))) f(t - x) + --- (g(t - x))
      2                           2                          2
    dx                          dt                         dx
                                      2
    d               d                d
 + (-- (f(t - x))) (-- (g(t - x))) - --- (g(t - x))
    dx              dx                 2
                                     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
/((f(t - x) + 1) (g(t - x) + 1))
(%i41) /* Kretschmann scalar */rinvariant();
           d               d                 d               d              2
       4 ((-- (f(t - x))) (-- (g(t - x))) - (-- (f(t - x))) (-- (g(t - x))))
           dx              dx                dt              dt
(%o41) ----------------------------------------------------------------------
                                        2               2
                          (f(t - x) + 1)  (g(t - x) + 1)
       2                     2
      d              2      d              2        2
   4 (--- (g(t - x)))    4 (--- (g(t - x)))        d               2
        2                     2                8 (----- (g(t - x)))
      dx                    dt                    dt dx
 + ------------------- + ------------------- - ---------------------
                   2                     2                      2
     (g(t - x) + 1)        (g(t - x) + 1)         (g(t - x) + 1)
       2                     2
      d              2      d              2        2
   4 (--- (f(t - x)))    4 (--- (f(t - x)))        d               2
        2                     2                8 (----- (f(t - x)))
      dx                    dt                    dt dx
 + ------------------- + ------------------- - ---------------------
                   2                     2                      2
     (f(t - x) + 1)        (f(t - x) + 1)         (f(t - x) + 1)
(%i42) /* Geodesic equations */cgeodesic(true);
                d                        dz 2    d               dz 2
(%t42) geod  = (-- (g(t - x))) g(t - x) (--)  + (-- (g(t - x))) (--)
           1    dt                       ds      dt              ds
                                                                             2
                    d                        dy 2    d               dy 2   d t
                 + (-- (f(t - x))) f(t - x) (--)  + (-- (f(t - x))) (--)  + ---
                    dt                       ds      dt              ds       2
                                                                            ds

                   d                        dz 2    d               dz 2
(%t43) geod  = - ((-- (g(t - x))) g(t - x) (--)  + (-- (g(t - x))) (--)
           2       dx                       ds      dx              ds
                                                                            2
                   d                        dy 2    d               dy 2   d x
                + (-- (f(t - x))) f(t - x) (--)  + (-- (f(t - x))) (--)  - ---)
                   dx                       ds      dx              ds       2
                                                                           ds

                   d                        dz 2    d               dz 2
(%t44) geod  = - ((-- (g(t - x))) g(t - x) (--)  + (-- (g(t - x))) (--)
           3       dy                       ds      dy              ds
                                                                           2
      d                       dy dz      d              dy dz    2        d y
 - 2 (-- (f(t - x))) f(t - x) -- -- - 2 (-- (f(t - x))) -- -- - f (t - x) ---
      dz                      ds ds      dz             ds ds               2
                                                                          ds
               2     2
              d y   d y    d                        dy 2
 - 2 f(t - x) --- - --- - (-- (f(t - x))) f(t - x) (--)
                2     2    dy                       ds
              ds    ds
    d               dy 2      d                       dx dy
 - (-- (f(t - x))) (--)  - 2 (-- (f(t - x))) f(t - x) -- --
    dy              ds        dx                      ds ds
      d              dx dy     dt  d                       dy
 - 2 (-- (f(t - x))) -- -- - 2 -- (-- (f(t - x))) f(t - x) --
      dx             ds ds     ds  dt                      ds
     dt  d              dy                2
 - 2 -- (-- (f(t - x))) --)/(f(t - x) + 1)
     ds  dt             ds

                           2                2     2
                 2        d z              d z   d z
(%t45) geod  = (g (t - x) --- + 2 g(t - x) --- + ---
           4                2                2     2
                          ds               ds    ds
    d                        dz 2    d               dz 2
 + (-- (g(t - x))) g(t - x) (--)  + (-- (g(t - x))) (--)
    dz                       ds      dz              ds
      d                       dy dz      d              dy dz
 + 2 (-- (g(t - x))) g(t - x) -- -- + 2 (-- (g(t - x))) -- --
      dy                      ds ds      dy             ds ds
      d                       dx dz      d              dx dz
 + 2 (-- (g(t - x))) g(t - x) -- -- + 2 (-- (g(t - x))) -- --
      dx                      ds ds      dx             ds ds
     dt  d                       dz     dt  d              dz
 + 2 -- (-- (g(t - x))) g(t - x) -- + 2 -- (-- (g(t - x))) --
     ds  dt                      ds     ds  dt             ds
    d                        dy 2    d               dy 2                2
 - (-- (f(t - x))) f(t - x) (--)  - (-- (f(t - x))) (--) )/(g(t - x) + 1)
    dz                       ds      dz              ds

(%o45)                               done
(%i46) 
Run Example
load(ctensor);
(%o1)          /usr/share/maxima/5.21.1/share/tensor/ctensor.mac
(%i2) ct_coords:[t,r,theta,phi];
(%o2)                         [t, r, theta, phi]
(%i3) lg:matrix([-e^(-2*c/r),0,0,0],[0,e^(2*c/r),0,0],[0,0,a(r),0],[0,0,0,a(r)*sin(theta)^2]);
                   [    1                                 ]
                   [ - ----   0     0           0         ]
                   [    2 c                               ]
                   [    ---                               ]
                   [     r                                ]
                   [   e                                  ]
                   [                                      ]
(%o3)              [          2 c                         ]
                   [          ---                         ]
                   [           r                          ]
                   [   0     e      0           0         ]
                   [                                      ]
                   [   0      0    a(r)         0         ]
                   [                                      ]
                   [                             2        ]
                   [   0      0     0    a(r) sin (theta) ]
(%i4) ug:matrix([-e^(2*c/r),0,0,0],[0,1/(e^(2*c/r)),0,0],[0,0,1/(a(r)),0],[0,0,0,1/(a(r)*sin(theta)^2)]);
                   [    2 c                               ]
                   [    ---                               ]
                   [     r                                ]
                   [ - e      0     0           0         ]
                   [                                      ]
                   [          1                           ]
                   [   0     ----   0           0         ]
                   [          2 c                         ]
                   [          ---                         ]
(%o4)              [           r                          ]
                   [         e                            ]
                   [                                      ]
                   [                1                     ]
                   [   0      0    ----         0         ]
                   [               a(r)                   ]
                   [                                      ]
                   [                            1         ]
                   [   0      0     0    ---------------- ]
                   [                             2        ]
                   [                     a(r) sin (theta) ]
(%i5) christof(mcs);
                                          c log(e)
(%t5)                        mcs        = --------
                                1, 1, 2    4 c
                                           ---
                                            r   2
                                          e    r

                                          c log(e)
(%t6)                        mcs        = --------
                                1, 2, 1       2
                                             r

                                           c log(e)
(%t7)                       mcs        = - --------
                               2, 2, 2         2
                                              r

                                         d
                                         -- (a(r))
                                         dr
(%t8)                       mcs        = ---------
                               2, 3, 3    2 a(r)

                                         d
                                         -- (a(r))
                                         dr
(%t9)                       mcs        = ---------
                               2, 4, 4    2 a(r)

                                          d
                                          -- (a(r))
                                          dr
(%t10)                     mcs        = - ---------
                              3, 3, 2         2 c
                                              ---
                                               r
                                           2 e

                                         cos(theta)
(%t11)                      mcs        = ----------
                               3, 4, 4   sin(theta)

                                    d             2
                                   (-- (a(r))) sin (theta)
                                    dr
(%t12)              mcs        = - -----------------------
                       4, 4, 2                2 c
                                              ---
                                               r
                                           2 e

(%t13)               mcs        = - cos(theta) sin(theta)
                        4, 4, 3

(%o13)                               done
(%i14) riemann(true);
                                                   2    2
                                   2 c log(e)   2 c  log (e)
(%t14)            riem           = ---------- - ------------
                      1, 2, 1, 2     4 c           4 c
                                     ---           ---
                                      r   3         r   4
                                    e    r        e    r

                                                 d
                                       c log(e) (-- (a(r)))
                                                 dr
(%t15)              riem           = - --------------------
                        1, 3, 1, 3           4 c
                                             ---
                                              r   2
                                          2 e    r  a(r)

                                                 d
                                       c log(e) (-- (a(r)))
                                                 dr
(%t16)              riem           = - --------------------
                        1, 4, 1, 4           4 c
                                             ---
                                              r   2
                                          2 e    r  a(r)

                                                    2    2
                                  2 c log(e) r - 2 c  log (e)
(%t17)           riem           = ---------------------------
                     2, 2, 1, 1                4
                                              r

                         2
                        d
                        --- (a(r))    d         2             d
                          2          (-- (a(r)))    c log(e) (-- (a(r)))
                        dr            dr                      dr
(%t18) riem           = ---------- - ------------ + --------------------
           2, 3, 2, 3     2 a(r)          2                 2
                                       4 a (r)           2 r  a(r)

                         2
                        d
                        --- (a(r))    d         2             d
                          2          (-- (a(r)))    c log(e) (-- (a(r)))
                        dr            dr                      dr
(%t19) riem           = ---------- - ------------ + --------------------
           2, 4, 2, 4     2 a(r)          2                 2
                                       4 a (r)           2 r  a(r)

                                                 d
                                       c log(e) (-- (a(r)))
                                                 dr
(%t20)              riem           = - --------------------
                        3, 3, 1, 1             2 c
                                               ---
                                                r   2
                                            2 e    r

(%t21) riem           = 
           3, 3, 2, 2
                     2
            2       d              2  d         2                    d
         2 r  a(r) (--- (a(r))) - r  (-- (a(r)))  + 2 c log(e) a(r) (-- (a(r)))
                      2               dr                             dr
                    dr
       - ----------------------------------------------------------------------
                                        2 c
                                        ---
                                         r   2
                                     4 e    r  a(r)

                                         d         2
                                        (-- (a(r)))
                                         dr
(%t22)                 riem           = ------------ - 1
                           3, 4, 3, 4      2 c
                                           ---
                                            r
                                        4 e    a(r)

                                           d             2
                                 c log(e) (-- (a(r))) sin (theta)
                                           dr
(%t23)        riem           = - --------------------------------
                  4, 4, 1, 1                   2 c
                                               ---
                                                r   2
                                            2 e    r

                                        2
                               2       d              2  d         2
(%t24) riem           = - ((2 r  a(r) (--- (a(r))) - r  (-- (a(r)))
           4, 4, 2, 2                    2               dr
                                       dr
                                                                   2 c
                                                                   ---
                                      d              2              r   2
                   + 2 c log(e) a(r) (-- (a(r)))) sin (theta))/(4 e    r  a(r))
                                      dr

                                                2 c
                                                ---
                               d         2       r           2
                             ((-- (a(r)))  - 4 e    a(r)) sin (theta)
                               dr
(%t25)    riem           = - ----------------------------------------
              4, 4, 3, 3                      2 c
                                              ---
                                               r
                                           4 e    a(r)

(%o25)                               done
(%i26) ricci(true);
                              d
                    c log(e) (-- (a(r)))                   2    2
                              dr           2 c log(e)   2 c  log (e)
(%t26)    ric     = -------------------- - ---------- + ------------
             1, 1        4 c                 4 c           4 c
                         ---                 ---           ---
                          r   2               r   3         r   4
                        e    r  a(r)        e    r        e    r

                    2
                   d
                   --- (a(r))    d         2             d
                     2          (-- (a(r)))    c log(e) (-- (a(r)))
                   dr            dr                      dr
(%t27) ric     = - ---------- + ------------ - --------------------
          2, 2        a(r)           2                2
                                  2 a (r)            r  a(r)
                                                                      2    2
                                                      2 c log(e)   2 c  log (e)
                                                    + ---------- - ------------
                                                           3             4
                                                          r             r

                            2
                           d
                           --- (a(r))             d
                             2          c log(e) (-- (a(r)))
                           dr                     dr
(%t28)         ric     = - ---------- - -------------------- + 1
                  3, 3          2 c            2 c
                                ---            ---
                                 r              r   2
                             2 e              e    r

                     2
                    d              2
                   (--- (a(r))) sin (theta)             d             2
                      2                       c log(e) (-- (a(r))) sin (theta)
                    dr                                  dr
(%t29) ric     = - ------------------------ - --------------------------------
          4, 4                 2 c                         2 c
                               ---                         ---
                                r                           r   2
                            2 e                           e    r
                                                                       2
                                                                  + sin (theta)

(%o29)                               done
(%i30) R:scurvature();
(%o30) 
    2
   d              2
  (--- (a(r))) sin (theta)             d             2
     2                       c log(e) (-- (a(r))) sin (theta)
   dr                                  dr                          2
- ------------------------ - -------------------------------- + sin (theta)
              2 c                         2 c
              ---                         ---
               r                           r   2
           2 e                           e    r
---------------------------------------------------------------------------
                                     2
                             a(r) sin (theta)
       2
      d
      --- (a(r))    d         2             d
        2          (-- (a(r)))    c log(e) (-- (a(r)))
      dr            dr                      dr           2 c log(e)
 + (- ---------- + ------------ - -------------------- + ----------
         a(r)           2                2                    3
                     2 a (r)            r  a(r)              r
                           2
                          d
                          --- (a(r))             d
                            2          c log(e) (-- (a(r)))
                          dr                     dr
                        - ---------- - -------------------- + 1
                               2 c            2 c
                  2 c          ---            ---
      2    2      ---           r              r   2
   2 c  log (e)    r        2 e              e    r
 - ------------)/e    + ---------------------------------------
         4                               a(r)
        r
    2 c            d
    ---  c log(e) (-- (a(r)))                   2    2
     r             dr           2 c log(e)   2 c  log (e)
 - e    (-------------------- - ---------- + ------------)
              4 c                 4 c           4 c
              ---                 ---           ---
               r   2               r   3         r   4
             e    r  a(r)        e    r        e    r
(%i31) einstein(true);
                              2
                     2       d              2  d         2
(%t31) ein     = (4 r  a(r) (--- (a(r))) - r  (-- (a(r)))
          1, 1                 2               dr
                             dr
                                                 2 c              2 c
                                                 ---              ---
                                 d                r   2            r   2  2
              + 4 c log(e) a(r) (-- (a(r))) - 4 e    r  a(r))/(4 e    r  a (r))
                                 dr

                                                                    2 c
                                                                    ---
                  2  d         2                    d                r   2
                 r  (-- (a(r)))  + 4 c log(e) a(r) (-- (a(r))) - 4 e    r  a(r)
                     dr                             dr
(%t32) ein     = --------------------------------------------------------------
          2, 2                             2 c
                                           ---
                                            r   2  2
                                        4 e    r  a (r)

                              2
                     4       d              4  d         2
(%t33) ein     = (2 r  a(r) (--- (a(r))) - r  (-- (a(r)))
          3, 3                 2               dr
                             dr
               2       d                2    2                     2
 + 4 c log(e) r  a(r) (-- (a(r))) + (8 c  log (e) - 8 c log(e) r) a (r))
                       dr
     2 c
     ---
      r   4  2
/(4 e    r  a (r))

                              2
                     4       d              4  d         2
(%t34) ein     = (2 r  a(r) (--- (a(r))) - r  (-- (a(r)))
          4, 4                 2               dr
                             dr
               2       d                2    2                     2
 + 4 c log(e) r  a(r) (-- (a(r))) + (8 c  log (e) - 8 c log(e) r) a (r))
                       dr
     2 c
     ---
      r   4  2
/(4 e    r  a (r))

(%o34)                               done
(%i35) 

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