Introduction to the mathematics of general relativity: Difference between revisions

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{{main|Riemann curvature tensor}}

The [[Riemann curvature tensor]] {{math|''R^ρ_σμν''}} tells us, mathematically, how much curvature there is in any given region of space. In flat space this tensor is zero.

Contracting the tensor produces 2 more mathematical objects:

# The [[Riemann curvature tensor]]: {{math|''R^ρ_σμν''}}, which gives the most information on the curvature of a space and is derived from derivatives of the [[metric tensor]]. In flat space this tensor is zero.

# The [[Ricci tensor]]: {{math|''R_σν''}}, comes from the need in Einstein's theory for a curvature tensor with only 2 indices. It is obtained by averaging certain portions of the Riemann curvature tensor.

# The [[scalar curvature]]: {{math|''R''}}, the simplest measure of curvature, assigns a single scalar value to each point in a space. It is obtained by averaging the Ricci tensor.