In the last article I reviewed the problem of having vectors defined in a spacetime where the metric function gives different lengths in different parts of spacetime, and outlined how this could be resolved using connection variables to move vectors around. Although this allows the comparison of two vectors in different parts of spacetime, it creates a new problem.

What happens if we move a vector along two different paths?

Using the covariant derivatives and the connection variables, a vector A can be moved from one place to another along a vector U. And then applying it a second time moves the vector along another vector V to a third location. But what if we move it along V first, and then along U second? It is not necessarily true that the result will be the same final vector in both cases.


In fact in any spacetime that has curvature, the two values will differ. That has almost become the definition of curvature of a surface now.

And so we need to define a new object, called the Riemann tensor. As before, the most common methods of defining it involve calculus, differential geometry, and a growing number of messy indices. However at its root, the Riemann tensor is just a function:

B = R(A,U,V)

in which A is the vector being moved, U and V are the two vectors it will be moved along, and B is a vector representing the difference in the final value of A after moving along the two paths.

In uncurved spacetime, such as is assumed in special relativity or in the Euclidean geometry we all learned in high school, the Riemann tensor is zero. A vector can be moved from one point to another without worrying about the path it takes. As soon as there is mass or energy present, spacetime will become curved and the Riemann tensor becomes important. 

And that will be the subject of the next two entries in this series.