South Pointing Chariots in Modern Science

Consider a nice even plane and lay an arrow (the Robin Hood kind of thing) flat down onto it. Now if you move the arrow around, it is said to be parallel transported, if

   a) it remains flat on the plane all the time and
   b) point and tail where moved the same distance.

Simple concept. Mathematically spoken, the arrow (vector) has to stay in the tangent space of the plane (rule a)  and has to stay as parallel as possible to its previous location (rule b).

Next step is to wrap the plane aound a sphere. It gets impossible to lay it down flat now, but if it was very short and/or the sphere was very big that doesn't really matter. Again you can start moving around the arrow in the plane (called manifold
generally). Big treat !

But things start getting interesting as soon as you track the way of the arrow's tail (called path). Consider the longer arrows in the figure nearby. Starting at point 1, it is moved forward following its tip) to point 3, then sideways to point 5 and backwards to point 7. Note that the two rules given above are followed without exception. Nevertheless the arrow at point 7 is no longer parallel to the arrow at point 1 !

This phenomenon - called Berry's Phase - is found in more and more corners of modern science. For example Einstein's universe isn't flat any longer, but "deformed" by gravity. Consequently arrows (think of waves of polarized light) get twisted around in spite of being parallel transported. Too abstract ? Try Foucault's pendulum: The plane of oscillation is parallel transported, nonetheless it turns 360° each day. Something more lively ? Try amoebae; these tiny creatures move around by parallel transporting their intestines on parallel paths...

Back to M. Santander. What our chariot does, is parallel transporting the emperor's arm. So by moving the vehicle over non-flat manifolds, many experiments can be performed - without boring math beforehand. Some results:

• If a vector is parallel transported along two different paths from point A to point B, the final vectors may not be parallel.
• If a vector is parallel transported in a closed loop, it may end up turned by an angle.
• This angle depends on the area enclosed by the loop. The ratio between area and angle is defined as Gaussian curvature.
• There may be paths on a manifold which do not produce an angle (e.g. around the earth along the equator). These are called geodesic.
• The deviation of the pointer relative to the chariot can be measured. The ratio between deviation and path length is called the geodesic curvature.