Let $x_1, x_2$ and $y_1, y_2$ be 4 nonzero future pointing null vectors in 4-dimensional Minkowski spacetime. Define $$ B(x_1, x_2; y_1, y_2) = 2(x_1, y_1)(x_2, y_2) + 2(x_1, y_2)(x_2, y_1) - (x_1, x_2)(y_1, y_2),$$ where $(-, -)$ denotes the Minkowski inner product.
I can show, in a somewhat convoluted way, though I am sure there is a simpler way to do things, that $B$ is nonnegative (for $x_i$s and $y_j$s as above). Can one characterize when $B$ vanishes?
I actually have one such function $B$ for each positive integer $m$, depending on $2m$ nonzero future pointing null vectors in 4-dimensional spacetime. I don't currently have the final expressions for such functions $B$ if $m > 2$, but I know that $B$ is symmetric in the $x_i$s, while keeping the $y_j$s fixed, and it is also symmetric in the $y_j$s, while keeping the $x_i$s fixed. If $m = 1$, then $B$ is just the Minkowski inner product. Another remark is that the vanishing of $B$ corresponds to a pair of polynomials of degree (at most) $m$ being $m$-apolar. This is essentially where my definition comes from.
