The Complex Arithmetic Package

  In REDUCE i is the pure imaginary number such that i^2 = -1. Complex quantities can be constructed using i explicitly and the function cnj() will form the complex conjugate by changing the sign of i. This applies to simple expressions such as sums, products, and quotients. Functions can be declared as complex by giving them a conjfn property whose value is the name of a lisp-function that knows how to compute the conjugate of the given function.

A simple variable can be declared complex via the function complex(), and this declaration can be removed with nocomplex(). When cnj() is applied to a complex variable, it returns unevaluated, but the display of the output is of the variable name overtopped with a bar.

Other functions that apply to complex quantities are re() which returns the real part of an expression, im() which return the imaginary part, cmod() which returns the modulus, and rat() which attempts to rationalize a complex quotient. This last function will not be successful unless the product formed by the conjugate of the denominator with the numerator and the denominator do not contain common factors that will cancel out again.

Continuing with the Kerr metric in a null-tetrad, we can compute the tensor metric from the connection and the frame metric with the function tenmetric(). The tensor metric name is the next in the default sequence of names as used by metric(), or the name given as the single argument.

 
#: tenmetrc();

computing g1
  computing z_b
  shift finished.
cofactor finished.
determ finished.
invert finished.
metric finished.

 g1

#: mapfi(g)$

#: mapfi(g_inv)$

 
#: let df (rhob,r) = 1;
#: let df (cnj(rhob),r) = 1;
#: let df (rhob, th) = -i*a*sin(th);
#: let df (cnj(rhob), th) = i*a*sin(th);
#: let df(delta, r) = 2*r-2*m;
#: let df (rho2, r) = 2*r;
#: let df(rho2,th)=-2*a^2*sin(th)*cos(th);

The rotation coeffients are rank-3 objects that are anti-symmetric in their first two indices, and are computed from:

The quantity is used internally by gamma() but is not saved.

Each of the interesting tensors (and scalars) of GR can be computed in the frame with the functions frriemann(), frricci(), frriccisc(), freinstein(), and frweyl(). As with the GR conterparts, each function can take optional argument that is the name to use in place of the default name. The default name is constructed from the tensor metric name appended with an underscore and a trailing segment which is the generic name of the each object and which is itself stored on the name of each function. These functions either compute the required object for the current metric if it does not exist, or they return its name.

 
#: frricci();

 computing eta1_frric
   computing eta1_frri
     computing eta1_gam
       computing z_d
       shift finished.
     gamma finished.
     computing eta1_gam_b
    shift finished.
  frriemann finished.

 eta1_frric

 
#: clear r^2;
#: let rhob = r+i*a*cos(th);
#: let rho2 = r^2 + a^2 * cos(th)^2;
#: let delta = r^2-2*m*r+a^2;
#: mapfi(frric);

 eta1_frric

#: frric[];

 eta1_frric