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and say that it "covers" the Rindler wedge in flat space time, which is defined by $$0<x<\infty,-x<t<x$$

I am having difficulty seeing precisely what it means for a metric to "cover" a region. Why doesn't the metric cover all of space-time? What is special about $$x=0$$ or $$t=0$$? Is there a singularity or degeneracy in the metric for some points?