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== tensor fields in general relativity ==tensor fields on a manifold are maps which attach a tensor to each point of the manifold.
==tensor fields in general relativity==tensor fields on a manifold are maps which attach a tensor to each point of the manifold.

noticing that formula_82 can be pulled out of the equation,:formula_83here is our equation, and now we have to give it a name,:formula_84you'll notice that the left side of the equation has 3 indices and the right side has 4, so we will have to sum over a pair of indices,:formula_85finally the riemann curvature tensor is written as;:formula_86you can contract indices to make the tensor covariant simply by multiplying by the metric, which will be useful when working with einstein's field equations,:formula_87and by further decomposition,:formula_88this tensor is called the ricci tensor which can also be derived by setting formula_89 and formula_90 in the riemann tensor to the same indice and summing over them.
noticing that formula_82 can be pulled out of the equation,:formula_83here is our equation, and now we have to give it a name,:formula_84you'll notice that the left side of the equation has 3 indices and the right side has 4, so we will have to sum over a pair of indices,:formula_85finally the riemann curvature tensor is written as;:formula_86you can contract indices to make the tensor covariant simply by multiplying by the metric, which will be useful when working with einstein's field equations,:formula_87and by further decomposition,:formula_88this tensor is called the ricci tensor which can also be derived by setting formula_89 and formula_90 in the riemann tensor to the same indice and summing over them.