Einstein Eqs. in Kerr (-Newman) metric

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRic=

with(linalg):

n:=4;

IiIl

NiI=

QyU+SSJRRzYiIiIhIiIiKihJIiVHRiVGJ0klS2VyckdGJUYnSSdtZXRyaWNHRiVGJw==

IiIh

Coordinates Xu=[t, x1, x2 , x3, ...]

Xu:=array(1..n,[t,r,theta,phi]);

PTYiNiM7IiIiIiIlRVxbbCVGJkkidEdGIyIiI0kickdGIyIiJEkmdGhldGFHRiNGJ0kkcGhpR0Yj

dXu:=array(1..n,[dt,dr,dth,dph]);

PTYiNiM7IiIiIiIlRVxbbCVGJkkjZHRHRiMiIiNJI2RyR0YjIiIkSSRkdGhHRiNGJ0kkZHBoR0Yj

Line element of signature (+, -, -, -... ) :

QyQ+SSRyaG9HNiItSSVzcXJ0R0YlNiMsJiokSSJyR0YlIiIjIiIiKiZJImFHRiVGLC1JJGNvc0dGJTYjSSZ0aGV0YUdGJUYsRi1GLQ==

KiQsJiokSSJyRzYiIiIjIiIiKiZJImFHRiZGJy1JJGNvc0c2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJjYjSSZ0aGV0YUdGJkYnRigjRihGJw==

QyQ+SSZkZWx0YUc2IiwqKiZJIk1HRiUiIiJJInJHRiVGKSEiIyokSSJRR0YlIiIjRikqJEkiYUdGJUYuRikqJEYqRi5GKUYp

LCgqJkkiTUc2IiIiIkkickdGJUYmISIjKiRJImFHRiUiIiNGJiokRidGK0Ym

QyQ+SSVsaW5lRzYiLCgqKCwmSSNkdEdGJSIiIiooSSJhR0YlRiotSSRzaW5HRiU2I0kmdGhldGFHRiUiIiNJJGRwaEdGJUYqISIiRjFJJmRlbHRhR0YlRipJJHJob0dGJSEiI0YqKiYsJiomSSNkckdGJUYxRjRGM0YqKiRJJGR0aEdGJUYxRipGKkY1RjFGMyooLCYqJiwmKiRGLEYxRioqJEkickdGJUYxRipGKkYyRipGKiomRixGKkYpRipGM0YxRi1GMUY1RjZGM0Yq

LCgqKCwmSSNkdEc2IiIiIiooSSJhR0YmRictSSRzaW5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0kmdGhldGFHRiYiIiNJJGRwaEdGJkYnISIiRjEsKComSSJNR0YmRidJInJHRiZGJyEiIyokRilGMUYnKiRGN0YxRidGJywmRjpGJyomRilGMS1JJGNvc0dGLEYvRjFGJ0YzRicqJiwmKiZJI2RyR0YmRjFGNEYzRicqJEkkZHRoR0YmRjFGJ0YnRjtGJ0YzKigsJiomLCZGOUYnRjpGJ0YnRjJGJ0YnKiZGKUYnRiVGJ0YzRjFGKkYxRjtGM0Yz

Covariant components of the metric tensor, 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

Gdd:=array(symmetric,1..n,1..n):

for i from 1 to n do

for j from 1 to n do

Der:=(1/2)*diff(line,dXu[i],dXu[j]);Gdd[i,j]:=simplify(Der,symbolic);end do;end do:

print(Gdd);

SSRHZGRHNiI=

Contravariant components of the metric tensor, 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

Guu:=array(symmetric,1..n,1..n):

Guu:=simplify(inverse(Gdd)):

Id:=simplify(multiply(Guu,Gdd));

PTYiNiQ7IiIiIiIlRiVFXFtsMTYkRiYiIiQiIiE2JEYnIiIjRis2JEYtRi1GJjYkRi1GJkYrNiRGJ0YnRiY2JEYnRiZGKzYkRipGJkYrNiRGJ0YqRis2JEYqRipGJjYkRipGLUYrNiRGLUYqRis2JEYmRiZGJjYkRiZGJ0YrNiRGKkYnRis2JEYmRi1GKzYkRi1GJ0Yr

Christoffel symbols 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 and 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

Cddd:=array(1..n,1..n,1..n):

Cddu:=array(1..n,1..n,1..n):

for i from 1 to n do

for j from 1 to n do

for k from 1 to n do

Cddd[i,j,k]:=(1/2)*(diff(Gdd[i,k],Xu[j])+ diff(Gdd[j,k],Xu[i])-diff(Gdd[i,j],Xu[k])); end do; end do; end do:

for i from 1 to n do

for j from 1 to n do

for k from 1 to n do

sym:=add(Guu[k,m]*Cddd[i,j,m],m=1..n);Cddu[i,j,k]:=sym;

end do; end do; end do:

Cd:=array(1..n):

for i from 1 to n do

sym:=add(Cddu[i,l,l],l=1..n); Cd[i]:=sym; end do:

Riemann tensor 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

Rdddd:=array(1..n,1..n,1..n,1..n);

PTYiNiY7IiIiIiIlRiVGJUYlRVxbbCE=

Ric_dd:=array(1..n,1..n);

PTYiNiQ7IiIiIiIlRiVFXFtsIQ==

for i from 1 to n do

for k from 1 to n do

for l from 1 to n do

for m from 1 to n do

coco:=1/2*simplify(diff(Gdd[i,m],Xu[k],Xu[l])+diff(Gdd[k,l],Xu[i],Xu[m])-diff(Gdd[i,l],Xu[k],Xu[m])-diff(Gdd[k,m],Xu[i],Xu[l]))+add(Cddd[k,l,p]*Cddu[i,m,p]-Cddd[k,m,p]*Cddu[i,l,p],p=1..n);Rdddd[i,k,l,m]:=simplify(coco,symbolic);end do;end do;end do;end do;

for j from 1 to n do

for k from 1 to n do

Ric_dd[j,k]:=simplify(add(add(Guu[p,q]*Rdddd[j,p,k,q],q=1..n),p=1..n));end do;end do:

Ricci tensor 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 and 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

Rdd:=Ric_dd:

Rud:=simplify(multiply(Guu,Rdd)):

Rsc:=simplify(add(Rud[s,s],s=1..n));

IiIh

for k from 1 to n do

for i from k to n do rezu:=Rud[k,i]; print(Ricci_ud[k,i],rezu); end do; end do:

NiQmSSlSaWNjaV91ZEc2IjYkIiIiRiciIiE=

NiQmSSlSaWNjaV91ZEc2IjYkIiIiIiIjIiIh

NiQmSSlSaWNjaV91ZEc2IjYkIiIiIiIkIiIh

NiQmSSlSaWNjaV91ZEc2IjYkIiIiIiIlIiIh

NiQmSSlSaWNjaV91ZEc2IjYkIiIjRiciIiE=

NiQmSSlSaWNjaV91ZEc2IjYkIiIjIiIkIiIh

NiQmSSlSaWNjaV91ZEc2IjYkIiIjIiIlIiIh

NiQmSSlSaWNjaV91ZEc2IjYkIiIkRiciIiE=

NiQmSSlSaWNjaV91ZEc2IjYkIiIkIiIlIiIh

NiQmSSlSaWNjaV91ZEc2IjYkIiIlRiciIiE=

Einstein tensor LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEkRXVkRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi9GMg==

Eud:=array(1..n,1..n):

for i from 1 to n do

for j from 1 to n do

Eud[i,j]:=simplify(Rud[i,j]-(1/2)*Rsc*Id[i,j]); end do;end do:

for k from 1 to n do

for i from k to n do rez:=Eud[k,i]; print(Ein_ud[k,i],rez); end do; end do:

NiQmSSdFaW5fdWRHNiI2JCIiIkYnIiIh

NiQmSSdFaW5fdWRHNiI2JCIiIiIiIyIiIQ==

NiQmSSdFaW5fdWRHNiI2JCIiIiIiJCIiIQ==

NiQmSSdFaW5fdWRHNiI2JCIiIiIiJSIiIQ==

NiQmSSdFaW5fdWRHNiI2JCIiI0YnIiIh

NiQmSSdFaW5fdWRHNiI2JCIiIyIiJCIiIQ==

NiQmSSdFaW5fdWRHNiI2JCIiIyIiJSIiIQ==

NiQmSSdFaW5fdWRHNiI2JCIiJEYnIiIh

NiQmSSdFaW5fdWRHNiI2JCIiJCIiJSIiIQ==

NiQmSSdFaW5fdWRHNiI2JCIiJUYnIiIh

Gravitational sources (density sigma and pressure p)

QyQ+SSZzaWdtYUc2IiomLUkpc2ltcGxpZnlHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiU2IyZJJEV1ZEdGJTYkIiIiRjBGMC1JIi9HRik2IywkKiZJI1BpR0YqRjBJIkdHRiVGMCIiKUYwRjA=

IiIh

QyQ+SSJwRzYiLCQqJiZJJEV1ZEdGJTYkIiIjRisiIiItSSIvRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJComSSNQaUdGMEYsSSJHR0YlRiwiIilGLCEiIkYs

IiIh

Riemann tensor

NiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=