The Frame Package

  The Frame package implements a standard tetrad formalism in which one sets up a basis of four contravariant vectors that form a ``connection'' between the tensor and frame indices (following Chandra, ?? ref):

The index enclosed in parenthesis distinguishes tetrad indices from normal tensor indices. As well, we may define the covariant vectors

and the matrix inverse of such that:

and we may also define a constant symmetric matrix so that

and

This matrix functions as a metric for the tetrad indices, and can be used to raise and lower indices. This ``metric'' is often assumed to be diagonal and Minkowskian so that the vectors form an orthonormal basis, but other forms can also be assumed, as shown below.

An important relation between the tensor metric and the connections exists:

The tetrad components of any tensor can be found via

and the tensor components of a tetrad can be found via the inverse relation:

In REDTEN the tetrad metric is created via the function frmetric(), which by default creates a metric of the form

and with a non-nil second argument creates a Minkowskian metric. For example,

 
#: frmetric()

computing eta1
cofactor finished.
determ finished.
invert finished.

  eta1

 
#: frmetric(nil, t)
computing eta2
invert finished.

  eta2

To use the frame package the user must determine in advance the structure of the particular tetrad for the chosen metric. These are entered into the system and, along with the frame metric, the tensor metric can be constructed from them with the function tenmetric() (see below). In the current implementation the user must enter components for the and declare this as a connection via the setcon() command. Again using the Kerr metric as an example, we have

 
#: coords '(t r th ph);

#: frmetric();

computing eta1
cofactor finished.
determ finished.
invert finished.

  eta1

#: mkobj ('(l n m), '(-1));

  l
   a

  n
   a

  m
   a

#: ias (l);
 l[0] = 1;
 l[1] = -rho2/delta;
 l[2] = 0;
 l[3] = -a*sin(th)^2;

 l

#: ias (n);
 n[0] = delta;
 n[1] = rho2;
 n[2] = 0;
 n[3] = -a*delta*sin(th)^2;

 n

#: ias(m);
 m[0] = i*a*sin(th);
 m[1] = 0;
 m[2] = -rho2;
 m[3] = -i*(r^2+a^2)*sin(th);

 m

#: complex (rhob)$
#: let r^2=sin(th)^2*a^2-a^2+rho2$
#: let rhob*cnj(rhob)=rho2$

#: mkobj('z,'(2 -1),'(),'(),'connection);

  a
 z
    b
#: z[0,i]==l[i];

  0
 z     = l
    i     i
#: z[1,i]==n[i];

  1
 z     = n
    i     i

#: z[2,i]==m[i];

  3
 z     = m
    i     i

#: z[3,i]==cnj(m[i]);

  3
 z     = cnj (m )
    i          i

#: setcon (z);

 z

At this point it is convenient to discuss the complex arithmetic package in REDTEN. Other complex packages may exist and could be used in place of the one supplied with REDTEN, but they must use the function cnj() to carry out conjugation, since this name is coded into the REDTEN source.

• The Complex Arithmetic Package
• Converting basis
• Covariant differentiation