We saw before that the Riemann tensor is the degree to which second derivatives of a vector field fail to commute. We will now consider the physical implications of that. It has a few different meanings that are (not obviously) equivalent.
Parallel transport
To parallel transport a vector along a path means to carry the vector with you along the path without any change in the vector--that is, with the vector's (covariant) derivative along the path being zero.
It turns out that if you parallel transport a vector along a closed path back to where you started, then it will surprisingly end up different (rotated) from the vector you started with, if there is curvature in the region of the closed path. Likewise if you parallel transport a vector from one point to a second point, the result at the second point may vary with the choice of path. This is most easily seen if you imagine transporting a vector along the surface of the sphere of the earth, say from the equator straight north to the north pole vs first transporting it along the equator for some distance, and then straight north to the north pole.
The change \( \delta v^a \) in vector \( \vec{v} \) after parallel transporting it around a closed parallelogram going first along a small vector \( \vec{T} \), then a small vector \( \vec{S} \), and then completing the parallelogram, is given by the Riemann tensor: \[ \delta v^a = - v^b T^c S^d R^a{}_{b c d} \]
If the curvature is positive (see Gaussian curvature) like on a sphere, \( \vec{v} \) will rotate in the same direction as you go around the closed loop. Negative curvature will rotate \( \vec{v} \) in the opposite direction.
Geodesic deviation
The second meaning of the Riemann tensor is that it also describes geodesic deviation. A geodesic is a curve that is as straight as possible. E.g. a "great circle" on a sphere, or a straight line on a plane. In flat space, two initially parallel geodesics will remain a constant distance between them as they are extended. In curved space, the distance between will grow or shrink.
Given a geodesic with initial tangent \( \vec{V} \), and given a second geodesic starting at a small displacement \( \vec{\xi} \) from the first and initial tangent being the result of \( \vec{V} \) parallel transported along \( \vec{\xi} \), then as we extend the two geodesics forward, the first derivative of the displacement between them will be zero, because we chose them to be initially parallel. But the second derivative (acceleration) of the displacement is given by the Riemann tensor: \[ (\nabla v \nabla v\xi)^a = V^b V^c \xi^d R^a{}_{b c d} \]
(Note this happens to be the same as the change in \( \vec{V} \) when parallel transported around the parallelogram of \( \vec{V} \) and \( \vec{\xi} \).) With positive curvature, the geodesics will converge. With negative curvature, the geodesics will diverge.
Sectional curvature
If \( \vec{V} \) and \( \vec{\xi} \) are orthogonal and unit length, then the above acceleration, dotted with \( \vec{\xi} \) is also negative of the value of the sectional curvature (Gaussian curvature) of the 2-surface made of geodesics through the initial point and tangent to \( \vec{V} \) and \( \vec{\xi} \). That is the flattest 2d surface containing the initial point and tangent to \( \vec{V} \) and \( \vec{\xi} \). Any orthonormal pair of vectors in that same plane will give the same value, because a 2d surface has only one scalar value of curvature at a point.
In addition to telling us how geodesics converge/diverge in this 2-surface, the Gaussian curvature can tell us other things like how much the circumference of a circle divided by its diameter will differ from \( \pi \). And how much its area will differ from \( \pi r^2 \). It will tell us how much the sum of the angles of a triangle formed by geodesics will differ from 180 degrees.
Inability to construct a Cartesian coordinate system.
The above meanings of the Riemann tensor are also equivalent to describing the impossibility of constructing a Cartesian coordinate system in the curved space.
- In Cartesian coordinates we need the basis vectors to be the same everywhere (a basis vector forms a uniform vector field). But the fact that the second covariant derivatives do not commute implies that we cannot construct a vector field whose second derivatives all vanish.
- The failure of a vector to parallel transport around a loop likewise means we cannot construct a uniform vector field for a basis vector. If you transport a basis vector one unit up the x axis and then up the y axis, you will not get the same result if you transport it to the same point by first going up thy y axis and then the x axis. There's no way to consistently define the basis vector at that point.
- In Cartesian coordinates, we need to construct straight and parallel coordinate lines, but geodesic deviation means that initially parallel coordinate lines will inevitably converge or diverge.
We can construct a Cartesian coordinate system if and only if all the values of the Riemann tensor are zero at all points in space.
On the other hand, no matter how curved space is, we can always locally construct an approximately Cartesian coordinate system to first order, within a small enough region of space that appears locally flat. For example, a grid of city streets can approximate a Cartesian coordinate system to first order. But if you try to extend it further across the surface of the globe, you will run into problems. The Riemann tensor shows failure in the second derivatives. We can get the first derivatives to behave. E.g. we can make our coordinate lines initially parallel, with the first derivatave of their separation being zero. The amount of rotation of a parallel-transported vector varies linearly with the area of the closed path, thus it is a second-order property.
