In the plane, this gives a single scalar invariant, the affine curvature of the curve.
2.
Such statement does not hold in higher-dimensions since there are higher-dimensional pp-waves of algebraic type II with non-vanishing polynomial scalar invariants.
3.
However, it can also be considered a linear operator acting on bivectors, and as such it has a characteristic polynomial, whose coefficients and roots ( eigenvalues ) are polynomial scalar invariants.
4.
Penrose also pointed out that in a pp-wave spacetime, all the polynomial scalar invariants of the Riemann tensor " vanish identically ", yet the curvature is almost never zero.
5.
The fact that \ mathbf { U } \ cdot \ mathbf { \ partial } is a Lorentz scalar invariant shows that the total derivative with respect to proper time \ frac { d } { d \ tau } is likewise a Lorentz scalar invariant.
6.
The fact that \ mathbf { U } \ cdot \ mathbf { \ partial } is a Lorentz scalar invariant shows that the total derivative with respect to proper time \ frac { d } { d \ tau } is likewise a Lorentz scalar invariant.
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