I was working out the components of the Riemann curvature tensor using the Schwarzschild metric a while back just as an exercise (I’m not a student, and Mathematica is expensive, so I don’t have access to any computing programs that can do it for me, and now that I’m thinking about it, does anyone know of any comparable but less expensive alternatives that I could use to do all the calculating for me?)(adsbygoogle = window.adsbygoogle || []).push({});

Anyways, back to the question at hand: at some point after working all the components out I came to understand that the Schwarzschild metric represents a space that is Ricci-flat— this understanding did not come as a consequence of all my hard work, in fact when I looked back at my work, of course I noticed most of the components that contribute to the Ricci tensor cancel out nicely, except for 1 (well 2 if you count its antisymmetric twin). Of course I thought I made a mistake and rechecked my work, but I still got the same answer. Of course I’m still making a mistake, but I’m failing to see where. So now I’m calling on the aid of the PF, the society so proficient at ferreting out mistakes.

The component in question has a “1” upstairs, and “010” downstairs (I believe I’m following convention when I assign the index “0” for the time direction and “1” for the radial direction). For the sake of brevity, from here on out I will exclude all of the qualifiers like “I calculated...” and “I think...”

*Convention (just for this post): for Christoffel symbols (of the 2nd kind), “a/bc” will mean the Γ with an “a” upstairs and a “bc” downstairs. There shouldn’t be any risk of confusing 1/01 with, say, a fraction in the context of this post.

So, the only non-zero Christoffel symbols this component is affected by are 1/00, 1/11, and 0/01, and we will also need to take the partial derivative of 1/00 with respect to x^1.

Using geometric units, 1/00 = m(r-2m)/r^3 and its derivative is 2m(3m-r)/r^4. 0/01 = m/(r^2-2mr), and 1/11 = -0/01

Our component of R = d(1/00)/dx^1 + (1/00)*(1/11) - (0/01)*(1/00)

Or R = d(1/00)/dx^1 - 2*(1/00)*(0/01)

Which means this component of R = -2m(r-2m)/r^4

Now, I did not find any other non-zero components of R with a “0i0” downstairs for this to cancel with when contracting indices to get the Ricci tensor, and therefore I ended up with the “00” component of the Ricci tensor being non-zero. So where am I going wrong?

p.s. sorry for the notation, I’m typing on an iPad.

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# I Non-zero components of Riemann curvature tensor with Schwarzschild metric

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