Riemann Curvature Tensor
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Mathematicians define the Riemann curvature tensor through the statement that
in which the Christoffel symbols relate to the metric tensor describing the relevant space as
If we combine those equations and express the Riemann tensor in terms of the metric tensor, we will get a gory algebraic mess. But in most applications of tensor geometry, especially in General Relativity, we use differential displacements and intervals. In those circumstances small patches of a Riemannian manifold (space or spacetime) differ negligibly from local inertial frames, flat manifolds.
To understand that last statement imagine standing on a dry lakebed. On a scale of several miles the lakebed appears flat: if you use a man-sized compass and straightedge, you can draw diagrams on the lakebed and either prove theorems or solve problems in plane geometry with no discernible error. But on a scale of hundreds or thousands of miles Earth痴 curvature necessitates the use of spherical geometry instead of the simpler plane geometry. In like manner we can treat minuscule patches of a curved Riemannian manifold as flat, which means, mathematically, that the first derivatives of the metric tensors describing those patches equal zero, which also necessitates that the corresponding Christoffel symbols equal zero.
In this case we take the second derivatives of the metric tensor as having nonzero values, so we have the first derivatives of the Christoffel symbols as
We then have the Riemann tensor as
In going from the second line to the third line in that equation I took the fact that derivative operators can commute with each other and used it to cancel out the first and fourth terms inside the parentheses.
In order to examine the symmetries of the Riemann tensor a little more clearly we want to lower the contravariant index. To accomplish that task we simply multiply the Riemann tensor by the covariant metric tensor to get
in whichδfe represents the mixed-index Kronecker delta. To examine the symmetries of Rebcd we merely exploit the commutability of the derivative operators and the symmetry of the metric tensor (gab=gba).
Let痴 see what happens when we interchange the first and second indices. We get
In like manner, interchanging the third and fourth indices gives us
and interchanging the first and second indices with the third and fourth indices gives us
Finally, with a little more effort we get
Note that even though we derived Equations 6 through 9 in a local inertial frame, they are nonetheless valid tensor equations and, thus, remain valid in all coordinate systems.
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