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Ctensor

Introduction to Ctensor

- Component Tensor Manipulation Package. To use the CTENSR package, type TSETUP(); which automatically loads it from within MACSYMA (if it is not already loaded) and then prompts the user to input his coordinate system. The user is first asked to specify the dimension of the manifold. If the dimension is 2, 3 or 4 then the list of coordinates defaults to [X,Y], [X,Y,Z] or [X,Y,Z,T] respectively. These names may be changed by assigning a new list of coordinates to the variable OMEGA (described below) and the user is queried about this. ** Care must be taken to avoid the coordinate names conflicting with other object definitions **. Next, the user enters the metric either directly or from a file by specifying its ordinal position. As an example of a file of common metrics, see TENSOR;METRIC FILE. The metric is stored in the matrix LG. Finally, the metric inverse is computed and stored in the matrix UG. One has the option of carrying out all calculations in a power series. A sample protocol is begun below for the static, spherically symmetric metric (standard coordinates) which will be applied to the problem of deriving Einstein's vacuum equations (which lead to the Schwarzschild solution) as an example. Many of the functions in CTENSR will be displayed for the standard metric as examples.

(C2) TSETUP();

Enter the dimension of the coordinate system: 
4;
Do you wish to change the coordinate names?
N;
Do you want to
1. Enter a new metric?
2. Enter a metric from a file?
3. Approximate a metric with a Taylor series?
Enter 1, 2 or 3 
1;
Is the matrix  1. Diagonal  2. Symmetric  3. Antisymmetric  4. General
Answer 1, 2, 3 or 4
1;
Row 1 Column 1:  A;
Row 2 Column 2:  X^2;
Row 3 Column 3:  X^2*SIN(Y)^2;
Row 4 Column 4:  -D;
Matrix entered.
Enter functional dependencies with the DEPENDS function or 'N' if none 
DEPENDS([A,D],X);
Do you wish to see the metric? 
Y;
                          [ A  0       0        0  ]
                          [                        ]
                          [     2                  ]
                          [ 0  X       0        0  ]
                          [                        ]
                          [         2    2         ]
                          [ 0  0   X  SIN (Y)   0  ]
                          [                        ]
                          [ 0  0       0       - D ]
Do you wish to see the metric inverse? 
N;

Definitions for Ctensor

Function: CHR1 ([i,j,k])

yields the Christoffel symbol of the first kind via the definition

       (g      + g      - g     )/2 .
         ik,j     jk,i     ij,k

To evaluate the Christoffel symbols for a particular metric, the variable METRIC must be assigned a name as in the example under CHR2.

Function: CHR2 ([i,j],[k])

yields the Christoffel symbol of the second kind defined by the relation

                       ks
    CHR2([i,j],[k]) = g    (g      + g      - g     )/2
                             is,j     js,i     ij,s

Function: CHRISTOF (arg)

A function in the CTENSR (Component Tensor Manipulation) package. It computes the Christoffel symbols of both kinds. The arg determines which results are to be immediately displayed. The Christoffel symbols of the first and second kinds are stored in the arrays LCS[i,j,k] and MCS[i,j,k] respectively and defined to be symmetric in the first two indices. If the argument to CHRISTOF is LCS or MCS then the unique non-zero values of LCS[i,j,k] or MCS[i,j,k], respectively, will be displayed. If the argument is ALL then the unique non-zero values of LCS[i,j,k] and MCS[i,j,k] will be displayed. If the argument is FALSE then the display of the elements will not occur. The array elements MCS[i,j,k] are defined in such a manner that the final index is contravariant.

Function: COVDIFF (exp,v1,v2,...)

yields the covariant derivative of exp with respect to the variables vi in terms of the Christoffel symbols of the second kind (CHR2). In order to evaluate these, one should use EV(exp,CHR2).

Function: CURVATURE ([i,j,k],[h])

Indicial Tensor Package) yields the Riemann curvature tensor in terms of the Christoffel symbols of the second kind (CHR2). The following notation is used:

               h            h           h        %1        h
      CURVATURE     = - CHR2      - CHR2     CHR2    + CHR2
               i j k        i k,j       %1 j     i k       i j,k
                              h         %1
                        + CHR2      CHR2
                              %1 k      i j

Variable: DIAGMETRIC

default:[] - An option in the CTENSR (Component Tensor Manipulation) package. If DIAGMETRIC is TRUE special routines compute all geometrical objects (which contain the metric tensor explicitly) by taking into consideration the diagonality of the metric. Reduced run times will, of course, result. Note: this option is set automatically by TSETUP if a diagonal metric is specified.

Variable: DIM

default:[4] - An option in the CTENSR (Component Tensor Manipulation) package. DIM is the dimension of the manifold with the default 4. The command DIM:N; will reset the dimension to any other integral value.

Function: EINSTEIN (dis)

A function in the CTENSR (Component Tensor Manipulation) package. EINSTEIN computes the mixed Einstein tensor after the Christoffel symbols and Ricci tensor have been obtained (with the functions CHRISTOF and RICCICOM). If the argument dis is TRUE, then the non-zero values of the mixed Einstein tensor G[i,j] will be displayed where j is the contravariant index. RATEINSTEIN[TRUE] if TRUE will cause the rational simplification on these components. If RATFAC[FALSE] is TRUE then the components will also be factored.

Function: LRICCICOM (dis)

A function in the CTENSR (Component Tensor Manipulation) package. LRICCICOM computes the covariant (symmetric) components LR[i,j] of the Ricci tensor. If the argument dis is TRUE, then the non-zero components are displayed.

Function: MOTION (dis)

A function in the CTENSR (Component Tensor Manipulation) package. MOTION computes the geodesic equations of motion for a given metric. They are stored in the array EM[i]. If the argument dis is TRUE then these equations are displayed.

Variable: OMEGA

default:[] - An option in the CTENSR (Component Tensor Manipulation) package. OMEGA assigns a list of coordinates to the variable. While normally defined when the function TSETUP is called, one may redefine the coordinates with the assignment OMEGA:[j1,j2,...jn] where the j's are the new coordinate names. A call to OMEGA will return the coordinate name list. Also see DESCRIBE(TSETUP); .

Function: RIEMANN (dis)

A function in the CTENSR (Component Tensor Manipulation) Package. RIEMANN computes the Riemann curvature tensor from the given metric and the corresponding Christoffel symbols. If dis is TRUE, the non-zero components R[i,j,k,l] will be displayed. All the indicated indices are covariant. As with the Einstein tensor, various switches set by the user control the simplification of the components of the Riemann tensor. If RATRIEMAN[TRUE] is TRUE then rational simplification will be done. If RATFAC[FALSE] is TRUE then each of the components will also be factored.

Function: TRANSFORM

- The TRANSFORM command in the CTENSR package has been renamed to TTRANSFORM.

Function: TSETUP ()

A function in the CTENSR (Component Tensor Manipulation) package which automatically loads the CTENSR package from within MACSYMA (if it is not already loaded) and then prompts the user to make use of it. Do DESCRIBE(CTENSR); for more details.

Function: TTRANSFORM (matrix)

A function in the CTENSR (Component Tensor Manipulation) package which will perform a coordinate transformation upon an arbitrary square symmetric matrix. The user must input the functions which define the transformation. (Formerly called TRANSFORM.)

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