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riemann

riemann (-f(r)*dt^2 +...

Calculate

riemann

? riemann;

z^2;

Calculate

riemann

? riemann ;

Calculate

riemann-true

riemann(true);

f(r)*dx^2 + dy^2 + dz^2;

Calculate

riemann

? riemann;

z^2;

Calculate

riemann

riemann;

xdx^2 + dy^2;

Calculate

riemann

riemann;

a*dx^2+2*dy^2;

Calculate

riemann-sin

riemann;

a^2/(1-k*x^2)*dx^2 +...

Calculate

riemann

riemann;

-A*dt^2 + dx^2 + dy^2...

Calculate

riemann-sin

riemann;

-dt^2 + a^2/(1-k*x^2)...

Calculate

riemann

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
load(ctensor);
(%o1)          /usr/share/maxima/5.21.1/share/tensor/ctensor.mac
(%i2) (cframe_flag:false,gcd:spmod,ctrgsimp:true,ratfac:true);
(%o2)                                true
(%i3) lg:diag_matrix(1,-r^2,-r^2,-r^2);
                            [ 1   0     0     0   ]
                            [                     ]
                            [       2             ]
                            [ 0  - r    0     0   ]
(%o3)                       [                     ]
                            [             2       ]
                            [ 0   0    - r    0   ]
                            [                     ]
                            [                   2 ]
                            [ 0   0     0    - r  ]
(%i4) ct_coords:[t,r,th,ph];
(%o4)                           [t, r, th, ph]
(%i5) cmetric(false);
(%o5)                                done
(%i6) christof(mcs);
                                             1
(%t6)                           mcs        = -
                                   2, 2, 2   r

                                             1
(%t7)                           mcs        = -
                                   2, 3, 3   r

                                             1
(%t8)                           mcs        = -
                                   2, 4, 4   r

                                              1
(%t9)                          mcs        = - -
                                  3, 3, 2     r

                                              1
(%t10)                         mcs        = - -
                                  4, 4, 2     r

(%o10)                               done
(%i11)  riemann(true);
                                                1
(%t11)                       riem           = - --
                                 2, 3, 2, 3      2
                                                r

                                                1
(%t12)                       riem           = - --
                                 2, 4, 2, 4      2
                                                r

                                               1
(%t13)                        riem           = --
                                  3, 3, 2, 2    2
                                               r

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

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

                                                1
(%t16)                       riem           = - --
                                 4, 4, 3, 3      2
                                                r

(%o16)                               done
(%i17) ricci(true);
                                           2
(%t17)                           ric     = --
                                    2, 2    2
                                           r

(%o17)                               done
(%i18) 
Run Example
load(ctensor);
(%o1)          /usr/share/maxima/5.21.1/share/tensor/ctensor.mac
(%i2) (cframe_flag:false,gcd:spmod,ctrgsimp:true,ratfac:true);
(%o2)                                true
(%i3) lg:diag_matrix(1,-r^2,-r^2,-(r^2));
                            [ 1   0     0     0   ]
                            [                     ]
                            [       2             ]
                            [ 0  - r    0     0   ]
(%o3)                       [                     ]
                            [             2       ]
                            [ 0   0    - r    0   ]
                            [                     ]
                            [                   2 ]
                            [ 0   0     0    - r  ]
(%i4) ct_coords:[t,r,th,ph];
(%o4)                           [t, r, th, ph]
(%i5) cmetric(false);
(%o5)                                done
(%i6) christof(mcs);
                                             1
(%t6)                           mcs        = -
                                   2, 2, 2   r

                                             1
(%t7)                           mcs        = -
                                   2, 3, 3   r

                                             1
(%t8)                           mcs        = -
                                   2, 4, 4   r

                                              1
(%t9)                          mcs        = - -
                                  3, 3, 2     r

                                              1
(%t10)                         mcs        = - -
                                  4, 4, 2     r

(%o10)                               done
(%i11)  riemann(true);
                                                1
(%t11)                       riem           = - --
                                 2, 3, 2, 3      2
                                                r

                                                1
(%t12)                       riem           = - --
                                 2, 4, 2, 4      2
                                                r

                                               1
(%t13)                        riem           = --
                                  3, 3, 2, 2    2
                                               r

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

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

                                                1
(%t16)                       riem           = - --
                                 4, 4, 3, 3      2
                                                r

(%o16)                               done
(%i17) ricci(true);
                                           2
(%t17)                           ric     = --
                                    2, 2    2
                                           r

(%o17)                               done
(%i18) 
[christof,depends,determinant,diag_matrix,diff,exp,gcd,invert,load,matrix,ratfac,riemann,trace] [christof,depends,determinant,diag_matrix,diff,exp,gcd,invert,load,matrix,ratfac,riemann] [christof,depends,diag_matrix,diff,exp,gcd,invert,load,matrix,ratfac,riemann,trace] [christof,depends,diag_matrix,diff,exp,gcd,invert,load,matrix,ratfac,riemann] [christof,depends,diag_matrix,diff,exp,gcd,load,matrix,ratfac,riemann] [christof,depends,diag_matrix,exp,gcd,lambda,load,matrix,ratfac,riemann,scurvature,sin] [christof,depends,diag_matrix,exp,gcd,load,matrix,ratfac,riemann,scurvature] [christof,depends,diag_matrix,exp,gcd,load,matrix,ratfac,riemann] [christof,depends,diag_matrix,gcd,load,matrix,ratfac,riemann,scurvature,sin] [christof,depends,diag_matrix,gcd,load,matrix,ratfac,riemann,scurvature] [christof,depends,exp,gcd,load,matrix,ratfac,riemann,scurvature,sin] [christof,depends,gcd,load,matrix,ratfac,riemann,scurvature,sin] [christof,depends,gcd,load,matrix,ratfac,riemann,scurvature] [christof,diag_matrix,gcd,load,matrix,ratfac,riemann,scurvature,sin] [christof,diag_matrix,gcd,load,matrix,ratfac,riemann,scurvature] [christof,diag_matrix,gcd,load,matrix,ratfac,riemann,sin] [christof,diag_matrix,gcd,load,matrix,ratfac,riemann] [christof,einstein,exp,load,matrix,riemann,scurvature,sin,true] [christof,einstein,exp,load,matrix,riemann,scurvature,true] [christof,einstein,load,matrix,riemann,scurvature,sin,true] [cosh,diag_matrix,gcd,load,matrix,ratfac,riemann,scurvature,sinh] [depends,diag_matrix,einstein,gcd,load,matrix,ratfac,riemann,scurvature,sin] [depends,diag_matrix,gcd,load,matrix,ratfac,riemann,scurvature,sin] [exp,psi,riemann,sin] [exp,riemann] [riemann,sin,true] [riemann,sin] [riemann,true] [riemann]

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