I understand energy-momentum tensor with contravariant indices, where

[LoadedAnvils]

I understand energy-momentum tensor with contravariant indices, where

I think I get $T^{αβ}$, but how do I derive the same result for $T_{αβ}$? Why are the contravariant vectors simply changed to covariant ones, and why does it work in Einstein's equation?

 

[Dick]

I think I get $T^{αβ}$, but how do I derive the same result for $T_{αβ}$? Why are the contravariant vectors simply changed to covariant ones, and why does it work in Einstein's equation?

You just lower indices by multiplying both sides of Einstein's equation by the metric tensor, you don't just 'change them'. I'm not sure what you don't understand.