Re: When would the metric tensor have offdiagonal elements--what would it mean

Metrics usually only have diagonal elements because they're *much* easier to work with that way. Furthermore, *any* constant-spacetime metric can be diagonalized by a transforming the coordinates into a frame where the metric is diagonal.

To understand the *meaning* of a metric, I'll direct you to Greg Egan's excellent Foundations article. Off-diagonal terms appear when things change, such as when a reference frame is accelerated or a gravitational field is turned on. You can avoid this (and thereby keep the metric diagonal) by changing coordinates, but if you want to maintain the same reference frame for the whole problem you may end up working with a non-diagonal metric. The only time I've seen this done was in a careful analysis of the old Twin Paradox.