Meaning of cross terms in line element

  • #1

Main Question or Discussion Point

In a problem from Hartle's Gravity, we are asked to express the line element in non-Cartesian coordinates u, v which are defined with respect to x, y. I have no problem getting the new expression for the line element, but then we are asked if the new coordinate curves intersect at right angles, and the solution says they do because there are no cross terms, du * dv. What is the logic here?

I've taken vector calculus, and try as I might I cannot seem to figure out why the absence of such terms indicates the curves intersect at right angles. I think I might be a little confused since I am interpreting du and dv as scalar values.
 

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  • #2
Orodruin
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The line element is given by ##ds^2 = g_{ij} dx^i dx^j##. Hence, if there are cross terms, then there are off-diagonal entries in the metric. Since the coordinate lines have the holonomic basis vectors ##\partial_i## as their tangent vectors, it follows that the inner product between the tangent vectors of two coordinate lines is given by
$$
g(\partial_i,\partial_j) = g_{ij}
$$
by definition. Therefore, if ##g_{ij} = 0## for ##i \neq j##, then the coordinate lines are orthogonal to each other since their tangent vectors are orthogonal.
 

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