Introduction to Ctensor Ctensor

37.2: 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.)

Introduction to Ctensor

Ctensor