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$$ g=\underbrace{g_{rr}(t,r)\mathrm{d}r^2+2g_{rt}(t,r)\mathrm{d}t\mathrm{d}r+g_{tt}(t,r)\mathrm{d}t^2}_{:=^2g}+\underbrace{h_{AB}(t,r,x^A)\mathrm{d}x^A\mathrm{d}x^B}_{:=h} $$

where ##h## is a Riemannian metric in dimension ##n-1##. Then any causal vector for ##g## is also a causal vector for ##^2g##, and drawing light-cones for ##^2g## gives a good idea of the causal structure of ##(\mathcal{M},g)##.

I really don't understand this proposition. The metric tensor ##^2g## can be represented in a ##2\times 2## matrix, and ##g## corresponds to a ##(n+1) \times (n+1)## matrix. How can I then check ##\bigg(g(x^\mu, x^\nu)<0\bigg) \implies \bigg(\; ^2g(x^\mu, x^\nu)<0\bigg) ## for the same vector ##x=x^\mu \partial_\mu## if the dimension of the corresponding matrices is different?