# Schwarzschild Solution in MatLab

1. Jan 6, 2009

### Philosophaie

This is the Schwarzschild Solution for the metric tensor g_ab in MatLab. Could someone take a look at my code. Getting the correct answers for Affinity. Some of the correct answers for Riemann. Why not all? This will yield a zero Ricci Tensor if done correctly. Here is my code:

$$\Gamma^{i}_{jk} = 1/2*g^{il}*(d/dx^{j}*g_{lk}+d/dx^{k}*g_{lj)-d/dx^{l}*g_{jk})$$

$$R^{i}_{jkl} = \Gamma^{r}_{jl}*\Gamma^{i}_{kr}-\Gamma^{r}_{jk}*\Gamma^{i}_{jl}+d/dx^{k}*\Gamma^{i}_{jl}-d/dx^{l}*\Gamma^{i}_{jk}$$

syms r theta m
g_ab = [1/(1-2*m/r) 0 0 0;0 r^2 0 0;0 0 r^2*(sin(h))^2,0;0,0,0,-(1-2*m/r)]
gupab = g_ab^-1
Affinity=sym(zeros(4,4,4))
Riemann=sym(zeros(4,4,4,4))
d(1)=r
d(2)=h
d(3)=p
d(4)=t
for i=1:4
for j=1:4
for k=1:4
for l=1:4
Affinity(i,j,k)=Affinity(i,j,k) + 1/2*gupab(i,l)*(diff(g_ab(l,k),d(j))+diff(g_ab(l,j),d(k))-diff(g_ab(j,k),d(l)))
end
end
end
end
for i=1:4
for j=1:4
for k=1:4
for l=1:4
for r=1:4
Riemann(i,j,k,l)=Riemann(i,j,k,l)+Affinity(r,j,l)*Affinity(i,k,r)-Affinity(r,j,k)*Affinity(i,l,r)+diff(Affinity(i,j,l),d(k))-diff(Affinity(i,j,k),d(l))
end
end
end
end
end

Correct Answers 'not from MatLab' with the last two digit interchangable:

Affinity(1,1,1) = m/r/(2*m-r)
Affinity(1,2,2) = (2*m-r)
Affinity(1,3,3) = (2*m-r)*(sin(theta))^2
Affinity(1,4,4) = m*(2*m-r)/r^3
Affinity(2,2,1) = 1/r
Affinity(2,3,3) = -cos(theta)*sin(theta)
Affinity(3,3,1) = 1/r
Affinity(3,3,2) = cot(theta)
Affinity(4,4,1) = -m/r/(2*m-r)
Rest zero except the interchangable ones(the last two indices).

Riemann(1,2,2,1)=m/r
Riemann(1,3,3,1)=m*(sin(theta))^2/r
Riemann(1,4,4,1)=2*m*(r-2*m)/r^4
Riemann(2,1,2,1)=m/r^2/(2*m-r)
Riemann(2,3,3,2)=-2*m*(sin(theta))^2/r
Riemann(2,4,4,2)=m*(2*m-r)/r^4
Riemann(3,1,3,1)=m/(2*m-r)/r^2
Riemann(3,2,3,2)=2*m/r
Riemann(3,4,4,3)=m*(2*m-r)/r^4
Riemann(4,1,4,1)=-2*m/(2*m-r)/r^2
Riemann(4,2,4,2)=-m/r
Riemann(4,3,4,3)=-m*(sin(theta))^2/r
Rest zero except the interchangable ones (the last two indices).

Why does the Riemann in MatLab not agree with the above result I got from a book?

Last edited: Jan 6, 2009