The Sato Metric

  This appendix provides an example of a very complicated metric that pushes the limits of the capabilities of REDUCE and REDTEN. A description of the metric may be found in The Sato metric is axially-symmetric and .. The metric coefficients are comprised of rational polynomials This metric provides a prime example of ``intermediate expression swell'', since the combination of the rational coefficients grows explosively, before the final collapse to zero.  

Until the introduction of computers with large amounts of real and virtual memory, this metric could not be shown to be vacuum by REDTEN. The example below causes REDUCE to swell in size to somewhat more than 12 Megabytes, an alternative approach to the problem once yielded a REDUCE of more than 50 Meg. in size.

This example can be found in the demo directory. The output below was generated from REDTEN running in REDUCE 3.4 in CSL on a MIPS M/120.

 
#: coords '(t x y ph)$
#: on time$
#: depend a,x,y$
#: depend b,x,y$
#: depend c,x,y$
#: let x^2 = u + 1$
#: let y^2 = 1 - v$
#: let df(u,x) = 2*x$
#: let df(v,y) = -2*y$
#: z := (m*p/2)*x*y;

       m p x y
 z := ---------
          2

#: rho := (m*p/2)*sqrt(x^2-1)*sqrt(1-y^2);

         sqrt(v) sqrt(u) m p
 rho := ---------------------
                  2

#: om:=2*m*q*c*(1-y^2)/a;

        2 c m q v
 om := -----------
            a

 
#: egam:=a/(p^4*tmp);

            a
 egam := --------
           4
          p  tmp

#: f := a/b;

       a
 f := ---
       b

#: ds := 1/f*(egam*(d(z)^2+d(rho)^2)+rho^2*d(ph)^2)-f*(d(t)-om*d(ph))^2;

             2  2  2  4      2  2           2  2  2  2  2        3
 ds := (d(ph)  b  m  p  tmp u  v  - 16 d(ph)  c  m  p  q  tmp u v

                                2          2         2  2  2
         + 16 d(ph) d(t) a c m p  q tmp u v  - 4 d(t)  a  p  tmp u v

               2    2  2           2    2  2  2       2    2  2  2
         + d(x)  a b  m  u v + d(x)  a b  m  v  + d(y)  a b  m  u

               2    2  2              2
         + d(y)  a b  m  u v)/(4 a b p  tmp u v)

#: metric (ds);

computing g1
cofactor finished.
determ finished.
invert finished.

 g1

Time: 1150 ms

#: off exp$

#: tmp := (x^2-y^2)^4;

               4
 tmp := (u + v)

#: on factor$
#: mapfi (g);

 g1

 
Time: 483 ms

#: mapfi(g_inv);

 g1_inv

Time: 334 ms

#: off factor,exp$
#: christoffel1();

computing g1_c1

 g1_c1

Time: 2133 ms

#: christoffel2();

computing g1_c2

 g1_c2

Time: 1900 ms

#: riemann();

computing g1_R

 g1_R

Time: 11317 ms  plus GC time: 1333 ms

#: ricci();

computing g1_ric

 g1_ric

Time: 7183 ms  plus GC time: 750 ms

 
#: let q^2=1-p^2$
#: A:=(p^2*(x^2-1)^2+q^2*(1-y^2)^2)^2-4*p^2*q^2*(x^2-1)*(1-y^2)*
     (x^2-y^2)^2;

        2  2    2  2    2 2       2             2  2
 A := (p  u  - p  v  + v )  + 4 (p  - 1) (u + v)  p  u v

#: B:=(p^2*x^4+q^2*y^4-1+2*p*x^3-2*p*x)^2+4*q^2*y^2*
     (p*x^3-p*x*y^2+1-y^2)^2;

         2             2          2  2               2
 B := ((p  - 1) (v - 1)  - (u + 1)  p  - 2 p u x + 1)

                            2   2
       + 4 ((u + v) p x + v)  (p  - 1) (v - 1)

#: CC:=p^2*(x^2-1)*((x^2-1)*(1-y^2)-4*x^2*(x^2-y^2))-p^3*x*(x^2-1)*
     (2*(x^4-1)+(x^2+3)*(1-y^2))+q^2*(1+p*x)*(1-y^2)^3;

                                      2
 CC :=  - ((4 (u + v) (u + 1) - u v) p  u

                                    2       3
            + ((u + 4) v + 2 (u + 1)  - 2) p  u x

                2                 3
            + (p  - 1) (p x + 1) v )

#: on peek$
#: mapfi(sub(a=A,b=B,c=CC,ric));

(0 0) is zero.
(0 1) is zero.
(0 2) is zero.
(0 3) is zero.
(1 1) is zero.
(1 2) is zero.
(1 3) is zero.
(2 2) is zero.
(2 3) is zero.
(3 3) is zero.

 g1_ric

Time: 590550 ms  plus GC time: 102667 ms