123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170171172173174175176177178179180181182183184185186187188189190191192193194195196 |
- % Calculations concerning the special metric of Dieter Egger
- % Small and capital letters are treated as being equivalent
- % Dimension of space-time
- n:=2;
- % turn off extra echoes
- off echo;
- % smaller exponents first
- on revpri;
- % Coordinates
- OPERATOR X$
- X(0):=t$
- X(1):=lambda0$
- % lambda0 depends on t
- DEPEND lambda0,t$
- % Rules
- trig1:={sin(~x)^2=>(1-cos(x)^2)}$
- let trig1$
- % Procedures
- procedure kovab(aa,bb); begin
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO aa(I,J):=DF(bb(I),X(J))+FOR M:=0:n-1 SUM CHRIST(I,J,M)*bb(M)$
- end;
- procedure showMatrix(mm); begin
- MATRIX hh(n,n)$
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO hh(I+1,J+1):=mm(I,J)$
- write hh;
- end;
- procedure showVector(vv); begin
- MATRIX hh(n,1)$
- FOR I:=0:n-1 DO hh(I+1,1):=vv(I)$
- write hh;
- end;
- % Vectors (1-dim arrays start with index 0)
- ARRAY U(n), V(n), LV(n), B(n), LB(n), BG(n)$
- % Arrays (2-dim arrays start with indices (0,0))
- ARRAY G(n,n), GINV(n,n), CHRIST(n,n,n), RIEM(n,n,n,n), RICCI(n,n), EINST(n,n)$
- ARRAY UKV(n,n)$
- % Calculations
- % optionally set maximum radius to 1
- % a0:=1$
- % or leave it open
- a:=a0*sqrt(1-t^2)$
- % Place
- u(0):=a0*asin(t)$
- u(1):=a*lambda0$
- % Metric (cellar indices, covariant, default is zero)
- G(0,0):=a0^2/(1-t^2)$
- G(1,1):=a0^2*(1-t^2)$
- % Inverse Metric (roof indices, contravariant)
- MATRIX MG(n,n), MGINV(n,n)$
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO MG(I+1,J+1):=G(I,J)$
- MGINV:=1/MG$
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO GINV(I,J):=MGINV(I+1,J+1)$
- % show metric
- write "g = ",mg;
- write "ginv = ",mginv;
- write "g*ginv = ",mg*mginv;
- % Christoffel symbols
- for k:=0:n-1 do for l:=0:n-1 do for m:=0:n-1 do CHRIST(k,l,m):=for i:=0:n-1 sum GINV(k,i)/2 * (DF(G(m,i),X(l)) + DF(G(l,i),X(m)) - DF(G(m,l),X(i)));
-
- % curvature tensor
- for m:=0:n-1 do for i:=0:n-1 do for k:=0:n-1 do for p:=0:n-1 do RIEM(m,i,k,p) :=DF(CHRIST(m,i,k),X(p)) - DF(CHRIST(m,i,p),X(k)) + FOR r:=0:n-1 SUM CHRIST(r,i,k)*CHRIST(m,r,p) - CHRIST(r,i,p)*CHRIST(m,r,k)$
-
- % Ricci tensor
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO RICCI(I,J):=FOR M:=0:n-1 SUM RIEM(M,I,J,M)$
- write "ricci = "; showMatrix(ricci);
- % curvature scalar
- R:=FOR I:=0:n-1 SUM FOR J:=0:n-1 SUM GINV(I,J)*RICCI(I,J)$
- write "curvature scalar r = ",r;
- % Einstein tensor
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO EINST(I,J):=RICCI(I,J)-R/2*G(I,J)$
- write "einstein = "; showMatrix(einst);
- % show place
- write "place u = "; showVector(u);
- % covariant derivative of place u
- kovab(ukv,u)$
- write "cov. deriv. of u = "; showMatrix(ukv);
- % classical velocity
- for k:=0:n-1 do v(k):=df(U(k),X(0))$
- write "v = du/dt = "; showVector(v);
- % local velocity with respect to (x0,x1)
- for k:=0:n-1 do LV(k):=V(k)/V(0)$
- write "lv = dx1/dx0 = "; showVector(lv);
- % max. velocity
- Array vmax(n)$
- svmax:=a0/sqrt(1-t^2)$
- for i:=0:n-1 do vmax(i):=svmax$
- svmaxq:=svmax*svmax$
- write "max. velocity = ",svmax;
- % equation of motion
- for k:=0:n-1 do BG(k):=-for m:=0:n-1 sum for n:=0:n-1 sum CHRIST(k,m,n)* vmax(m)*vmax(n)$
- write "equation of motion = "; showVector(bg);
- % local acceleration wrt (x0,x1)
- for k:=0:n-1 do LB(k) :=1/V(0)*df(lv(k),x(0))$
- write "la = dlv/dx0 * 1/v0 = "; showVector(lb);
- %--------------------------------------------------------------
- % write results to file
- OUT "metric2d_results.txt";
- off echo;
- off nat;
- % Metric
- write "metric = ";
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", G(I,J)$
- % Inverse Metric
- WRITE "inverse metric = ";
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", GINV(I,J)$
- % Christoffel symbols
- write "christoffel symbols = ";
- FOR K:=0:n-1 DO FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",K,",",I,",",J,") = ", CHRIST(K,I,J)$
- % curvature tensor
- write "curvature tensor = ";
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO FOR K:=0:n-1 DO FOR L:=0:n-1 DO WRITE "(",I,",",J,",",K,",",L,") = ", RIEM(I,J,K,L)$
-
- % Ricci tensor
- write "ricci tensor = ";
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ", RICCI(I,J)$
- % curvature scalar
- write "curvature scalar = ",R$
- % Einstein tensor
- write "einstein tensor = ";
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",EINST(I,J)$
- % place U
- write "place u = ";
- FOR I:=0:n-1 DO WRITE "(",I,") = ", U(I)$
- % covariant derivative of U
- write "covariant derivative of u = ";
- FOR I:=0:n-1 DO FOR J:=0:n-1 DO WRITE "(",I,",",J,") = ",Ukv(I,J)$
- % velocity V
- write "velocity v = ";
- FOR I:=0:n-1 DO WRITE "(",I,") = ", V(I)$
- % local velocity wrt (x0,x1)
- write "local velocity wrt (x0,x1) = ";
- FOR I:=0:n-1 DO WRITE "(",I,") = ", LV(I)$
- % acceleration
- write "acceleration = ";
- FOR I:=0:n-1 DO WRITE "(",I,") = ", B(I)$
- % local acceleration wrt (x0,x1)
- write "local acceleration wrt (x0,x1) = ";
- FOR I:=0:n-1 DO WRITE "(",I,") = ", LB(I)$
- % equation of motion
- write "equation of motion = ";
- FOR I:=0:n-1 DO WRITE "(",I,") = ", BG(I)$
- % equation of motion
- on factor;
- write "equation of motion = ";
- FOR I:=0:n-1 DO WRITE "(",I,") =", BG(I)$
- off factor;
- SHUT "metric2d_results.txt";
- off revpri;
- on nat;
- END;
|